This page intentionally left blank Process Control Process Control emphasizes the importance of computers in this modern age of teaching and practicing process control. An introductory textbook, it covers the most essential aspects of process control suitable for a one-semester course. The text covers classical techniques, but also includes discussion of state-space modeling and control, a modern control topic lacking in most chemical process control introductory texts. MATLAB R , a popular engineering software package, is used as a powerful yet approachable computational tool. Text examples demonstrate how root locus, Bode plots, and time-domain simulations can be integrated to tackle a control problem. Classical control and state-space designs are compared. Despite the reliance on MATLAB, theory and analysis of process control are well presented, creating a well-rounded pedagogical text. Each chapter concludes with problem sets, to which hints or solutions are provided. A Web site provides excellent support in the way of MATLAB outputs of text examples and MATLAB sessions, references, and supplementary notes. A succinct and readable text, this book will be useful for students studying process control, as well as for professionals undertaking industrial short courses or looking for a brief reference. Pao C. Chau is Professor of Chemical Engineering at the University of California, San Diego. He also works as a consultant to the biotechnology industry on problems dealing with bioreactor design and control and molecular modeling. CAMBRIDGE SERIES IN CHEMICAL ENGINEERING Series Editor: Arvind Varma, University of Notre Dame Editorial Board: Alexis T. Bell, University of California, Berkeley John Bridgwater, University of Cambridge Robert A. Brown, MIT L. Gary Leal, University of California, Santa Barbara Massimo Morbidelli, ETH, Zurich Stanley I. Sandler, University of Delaware Michael L. Shuler, Cornell University Arthur W. Westerberg, Carnegie Mellon University Books in the Series: E. L. Cussler, Diffusion: Mass Transfer in Fluid Systems, second edition Liang-Shih Fan and Chao Zhu, Principles of GasвЂ“Solid Flows Hasan Orbey and Stanley I. Sandler, Modeling VaporвЂ“Liquid Equilibria: Cubic Equations of State and Their Mixing Rules T. Michael Duncan and Jeffrey A. Reimer, Chemical Engineering Design and Analysis: An Introduction John C. Slattery, Advanced Transport Phenomena A. Varma, M. Morbidelli, H. Wu, Parametric Sensitivity in Chemical Systems M. Morbidelli, A. Gavriilidis, and A. Varma, Catalyst Design: Optimal Distribution of Catalyst in Pellets, Reactors, and Membranes E. L. Cussler and G. D. Moggridge, Chemical Product Design Pao C. Chau, Process Control: A First Course with MATLAB Process Control A First Course with MATLAB Pao C. Chau University of California, San Diego пќЈпќЎпќпќўпќІпќ©пќ¤пќ§пќҐ пќµпќ®пќ©пќ¶пќҐпќІпќіпќ©пќґпќ№ пќ°пќІпќҐпќіпќі Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, SГЈo Paulo Cambridge University Press The Edinburgh Building, Cambridge пќЈпќўпњІ пњІпќІпќµ, United Kingdom Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521807609 В© Cambridge University Press 2002 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2002 пќ©пќіпќўпќ®-пњ±пњі пќ©пќіпќўпќ®-пњ±пњ° пњ№пњ·пњё-пњ°-пњµпњ±пњ±-пњ°пњ·пњ¶пњґпњґ-пњІ eBook (EBL) пњ°-пњµпњ±пњ±-пњ°пњ·пњ¶пњґпњґ-пњґ eBook (EBL) пќ©пќіпќўпќ®-пњ±пњі пќ©пќіпќўпќ®-пњ±пњ° пњ№пњ·пњё-пњ°-пњµпњІпњ±-пњёпњ°пњ·пњ¶пњ°-пњ№ hardback пњ°-пњµпњІпњ±-пњёпњ°пњ·пњ¶пњ°-пњі hardback пќ©пќіпќўпќ®-пњ±пњі пќ©пќіпќўпќ®-пњ±пњ° пњ№пњ·пњё-пњ°-пњµпњІпњ±-пњ°пњ°пњІпњµпњµ-пњґ paperback пњ°-пњµпњІпњ±-пњ°пњ°пњІпњµпњµ-пњ№ paperback Cambridge University Press has no responsibility for the persistence or accuracy of пќµпќІпќ¬s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Preface page xi 1 Introduction 2 Mathematical Preliminaries 2.1. A Simple Differential Equation Model 2.2. Laplace Transform 2.3. Laplace Transforms Common to Control Problems 2.4. Initial- and Final-Value Theorems 2.5. Partial-Fraction Expansion 2.6. Transfer Function, Pole, and Zero 2.7. Summary of Pole Characteristics 2.8. Two Transient Model Examples 2.9. Linearization of Nonlinear Equations 2.10. Block-Diagram Reduction Review Problems 6 7 8 12 15 16 21 24 26 33 37 40 3 Dynamic Response 3.1. First-Order Differential Equation Models 3.2. Second-Order Differential Equation Models 3.3. Processes with Dead Time 3.4. Higher-Order Processes and Approximations 3.5. Effect of Zeros in Time Response Review Problems 44 45 48 52 53 58 61 4 State-Space Representation 4.1. State-Space Models 4.2. Relation of State-Space Models to Transfer Function Models 4.3. Properties of State-Space Models Review Problems 64 64 71 78 81 5 Analysis of Single-Loop Control Systems 5.1. PID controllers 5.2. Closed-Loop Transfer Functions 83 83 90 1 vii Contents 5.3. 5.4. Closed-Loop System Response Selection and Action of Controllers Review Problems 95 102 105 6 Design and Tuning of Single-Loop Control Systems 6.1. Tuning Controllers with Empirical Relations 6.2. Direct Synthesis and Internal Model Control Review Problems 108 108 115 127 7 Stability of Closed-Loop Systems 7.1. Deп¬Ѓnition of Stability 7.2. The RouthвЂ“Hurwitz Criterion 7.3. Direct-Substitution Analysis 7.4. Root-Locus Analysis 7.5. Root-Locus Design 7.6. Final Remark on Root-Locus Plots Review Problems 129 129 130 135 137 143 145 146 8 Frequency-Response Analysis 8.1. Magnitude and Phase Lag 8.2. Graphical Analysis Tools 8.3. Stability Analysis 8.4. Controller Design Review Problems 147 147 152 161 168 176 9 Design of State-Space Systems 9.1. Controllability and Observability 9.2. Pole-Placement Design 9.3. State Estimation Design Review Problems 178 178 182 188 194 Multiloop Systems 10.1. Cascade Control 10.2. Feedforward Control 10.3. FeedforwardвЂ“Feedback Control 10.4. Ratio Control 10.5. Time-Delay Compensation вЂ“ The Smith Predictor 10.6. Multiple-Input Multiple-Output Control 10.7. Decoupling of Interacting Systems Review Problems 198 198 203 206 207 209 210 217 221 MATLAB Tutorial Sessions Session 1. Important Basic Functions Session 2. Partial-Fraction and Transfer Functions Session 3. Time-Response Simulation Session 4. State-Space Functions Session 5. Feedback Simulation Functions Session 6. Root-Locus Functions Session 7. Frequency-Response Functions 226 226 234 238 243 249 254 260 10 viii Contents Homework Problems Part I. Basic Problems Part II. Intermediate Problems Part III. Extensive Integrated Problems 267 267 275 290 References 305 Index 307 ix Preface This is an introductory text written from the perspective of a student. The major concern is not of how much material is covered, but rather, how the most important and basic concepts that one should grasp in a п¬Ѓrst course are presented. If your instructor is using some other text that you are struggling to understand, I hope that I can help you too. The material here is the result of a process of elimination. The writing and the examples are succinct and selfexplanatory, and the style is purposely unorthodox and conversational. To a great extent, the style, content, and the extensive use of footnotes are molded heavily by questions raised in class. I left out very few derivation steps. If they are left out, the missing steps are provided as hints in the Review Problems at the back of each chapter. I strive to eliminate those вЂњeasily obtainedвЂќ results that bafп¬‚e many of us. Most of you should be able to read the material on your own. You just need basic knowledge in differential equations, and it helps if you have taken a course on writing material balances. With the exception of Chapters 4, 9, and 10, which should be skipped in a quarter-long course, it also helps if you proceed chapter by chapter. The presentation of material is not intended for someone to just jump right in the middle of the text. A very strong emphasis is placed on developing analytical skills. To keep pace with the modern computer era, a coherent and integrated approach is taken to using a computational tool. I believe in active learning. When you read the chapters, it is very important that you have MATLAB with its Control Toolbox to experiment and test the examples п¬Ѓrsthand. Notes to Instructors There are probably more introductory texts on control than on any other engineering disciplines. It is arguable whether we need another control text. As we move into the era of hundred-dollar textbooks, I believe we can lighten the economic burden and, with the Internet, assemble a new generation of modularized texts that soften the printing burden by off-loading selected material to the Web. Still, a key resolve is to scale back on the scope of a text to the most crucial basics. How much students can or be enticed to learn is inversely proportional to the number of pages that they have to read вЂ“ akin to diminished magnitude and increased lag in frequency response. Therefore, as textbooks become thicker over the years in attempts to reach out to students and are excellent resources from the perspective of xi Preface instructors, these texts are by no means more effective pedagogical tools. This project was started as a set of review notes when I found that students were having trouble identifying the key concepts in these expansive texts. I also found that these texts in many circumstances deter students from active learning and experimenting on their own. At this point, the contents are scaled down to п¬Ѓt a one-semester course. On a quarter system, Chapters 4, 9, and 10 can be omitted. With the exception of Chapters 4 and 9, on state-space models, the organization has вЂњevolvedвЂќ to become very classical. The syllabus is chosen such that students can get to tuning proportionalвЂ“integralвЂ“differential controllers before they lose interest. Furthermore, discrete-time analysis has been discarded. If there is to be one introductory course in the undergraduate curriculum, it is very important to provide an exposure to state-space models as a bridge to a graduate-level course. Chapter 10, on mutliloop systems, is a collection of topics that are usually handled by several chapters in a formal text. This chapter is written such that only the most crucial concepts are illustrated and that it could be incorporated comfortably into a one-semester curriculum. For schools with the luxury of two control courses in the curriculum, this chapter should provide a nice introductory transition. Because the material is so restricted, I emphasize that this is a вЂњп¬ЃrstcourseвЂќ textbook, lest a student mistakenly ignore the immense expanse of the control п¬Ѓeld. I also have omitted appendices and extensive references. As a modularized tool, I use the Web Support to provide references, support material, and detailed MATLAB plots and results. Homework problems are also handled differently. At the end of each chapter are short, mostly derivation-type problems that are called Review Problems. Hints or solutions are provided for these exercises. To enhance the skill of problem solving, the extreme approach is taken, more so than that of Stephanopoulos (1984), of collecting major homework problems at the back and not at the end of each chapter. My aim is to emphasize the need to understand and integrate knowledge, a virtue that is endearing to ABET, the engineering accreditation body in the United States. These problems do not even specify the associated chapter as many of them involve different techniques. A student has to determine the appropriate route of attack. An instructor may п¬Ѓnd it aggravating to assign individual parts of a problem, but when all the parts are solved, I hope the exercise will provide a better perspective on how different ideas are integrated. To be an effective teaching tool, this text is intended for experienced instructors who may have a wealth of their own examples and material, but writing an introductory text is of no interest to them. The concise coverage conveniently provides a vehicle with which they can take a basic, minimalist set of chapters and add supplementary material that they deem appropriate. Even without supplementary material, however, this text contains the most crucial material, and there should not be a need for an additional expensive, formal text. Although the intended teaching style relies heavily on the use of MATLAB, the presentation is very different from texts that prepare elaborate M-п¬Ѓles and even menu-driven interfaces. One of the reasons why MATLAB is such a great tool is that it does not have a steep learning curve. Students can quickly experiment on their own. Spoon-feeding with our misguided intention would only destroy the incentive to explore and learn on oneвЂ™s own. To counter this pitfall, strong emphasis is placed on what students can accomplish easily with only a few MATLAB statements. MATLAB is introduced as walk-through tutorials that encourage students to enter commands on their own. As a strong advocate of active learning, I do not duplicate MATLAB results. Students again are encouraged to execute the commands themselves. In case help is needed, the Web Support, however, has the complete xii Preface set of MATLAB results and plots. This organization provides a more coherent discourse on how one can make use of different features of MATLAB, not to mention save signiп¬Ѓcant printing costs. Finally, the tutorials can easily be revised to keep up with the continual upgrade of MATLAB. At this writing, the tutorials are based on MATLAB Version 6.1 and the object-oriented functions in the Control System Toolbox Version 5.1. Simulink Version 4.1 is also utilized, but its scope is limited to simulating more complex control systems. As a п¬Ѓrst-course text, the development of models is limited to stirred tanks, stirred-tank heaters, and a few other examples that are used extensively and repeatedly throughout the chapters. My philosophy is one step back in time. The focus is the theory and the building of a foundation that may help to solve other problems. The design is also formulated to be able to launch into the topic of tuning controllers before students may lose interest. The coverage of Laplace transforms is not entirely a concession to remedial mathematics. The examples are tuned to illustrate immediately how pole positions may relate to time-domain response. Furthermore, students tend to be confused by the many different design methods. As much as I could, especially in the controller design chapters, I used the same examples throughout. The goal is to help a student understand how the same problem can be solved by different techniques. I have given up the pretense that we can cover controller design and still have time to do all the plots manually. I rely on MATLAB to construct the plots. For example, I take a unique approach to root-locus plots. I do not ignore it as some texts do, but I also do not go into the hand-sketching details. The same can be said of frequency-response analysis. On the whole, I use root-locus and Bode plots as computational and pedagogical tools in ways that can help students to understand the choice of different controller designs. Exercises that may help such thinking are in the MATLAB tutorials and homework problems. Finally, I have to thank Costas Pozikidris and Florence Padgett for encouragement and support on this project, Raymond de Callafon for revising the chapters on state-space models, and Allan Cruz for proofreading. Last but not least, Henry Lim combed through the manuscript and made numerous insightful comments. His wisdom is sprinkled throughout the text. Web Support (MATLAB outputs of text examples and MATLAB sessions, references, supplementary notes, and solution manual) is available at http://us.cambridge.org/ titles/0521002559.html. xiii 1 Introduction ontrol systems are tightly intertwined in our daily lives so much so that we take them for granted. They may be as low tech and unglamorous as our п¬‚ush toilet. Or they may be as high tech as electronic fuel injection in our cars. In fact, there is more than a handful of computer control systems in a typical car that we now drive. In everything from the engine to transmission, shock absorber, brakes, pollutant emission, temperature, and so forth, there is an embedded microprocessor controller keeping an eye out for us. The more gadgetry, the more tiny controllers pulling the trick behind our backs.1 At the lower end of consumer electronic devices, we can bet on п¬Ѓnding at least one embedded microcontroller. In the processing industry, controllers play a crucial role in keeping our plants running вЂ“ virtually everything from simply п¬Ѓlling up a storage tank to complex separation processes and chemical reactors. As an illustration, letвЂ™s take a look at a bioreactor (Fig. 1.1). To п¬Ѓnd out if the bioreactor is operating properly, we monitor variables such as temperature, pH, dissolved oxygen, liquid level, feed п¬‚ow rate, and the rotation speed of the impeller. In some operations, we may also measure the biomass and the concentration of a speciп¬Ѓc chemical component in the liquid or the composition of the gas efп¬‚uent. In addition, we may need to monitor the foam head and make sure it does not become too high. We most likely need to monitor the steam п¬‚ow and pressure during the sterilization cycles. We should note that the schematic diagram is far from complete. By the time we have added enough details to implement all the controls, we may not recognize the bioreactor. These features are not pointed out to scare anyone; on the other hand, this is what makes control such a stimulating and challenging п¬Ѓeld. For each quantity that we want to maintain at some value, we need to ensure that the bioreactor is operating at the desired conditions. LetвЂ™s use the pH as an example. In control calculations, we commonly use a block diagram to represent the problem (Fig. 1.2). We will learn how to use mathematics to describe each of the blocks. For now, the focus is on some common terminology. To consider pH as a controlled variable, we use a pH electrode to measure its value and, with a transmitter, send the signal to a controller, which can be a little black box or a computer. The controller takes in the pH value and compares it with the desired pH, what C 1 In the 1999 Mercedes-Benz S-class sedan, there are approximately 40 вЂњelectronic control unitsвЂќ that control up to 170 different variables. 1 Introduction Measurements: pH, temperature liquid level, off gas analysis, etc. Control Algorithm Performance specifications Off gas Impeller Air sparge Acid Base Antifoam Medium Feed Cooling water Product Figure 1.1. Schematic diagram of instrumentation associated with a fermentor. The steam sterilization system and all sensors and transmitters are omitted for clarity. The thick solid lines represent process streams. The thin solid lines represent information п¬‚ow. is called the set point or the reference. If the values are not the same, there is an error, and the controller makes proper adjustments by manipulating the acid or the base pump вЂ“ the actuator.2 The adjustment is based on calculations made with a control algorithm, also called the control law. The error is calculated at the summing point, where we take the desired pH minus the measured pH. Because of how we calculate the error, this is a negative-feedback mechanism. This simple pH control example is what we call a single-input single-output (SISO) system; the single input is the set point and the output is the pH value.3 This simple feedback Desired + pH Error в€’ pH Control Algorithm Acid/base Pump Mixed Vessel Controller Function Actuator Process Measured pH pH Transducer pH electrode with transmitter Figure 1.2. Block-diagram representation of a single-input singleoutput negative-feedback system. Labels within the boxes are general. Labels outside the boxes apply to the simpliп¬Ѓed pH control discussion. 2 3 2 In real life, bioreactors actually use onвЂ“off control for pH. We will learn how to identify input and output variables, how to distinguish among manipulated variables, disturbances, measured variables, and so forth. Do not worry about remembering all the terms here; they will be introduced properly in subsequent chapters. Introduction mechanism is also what we call a closed loop. This single-loop system ignores the fact that the dynamics of the bioreactor involves complex interactions among different variables. If we want to take a more comprehensive view, we need to design a multiple-input multipleoutput (MIMO), or multivariable, system. When we invoke the term system, we are referring to the process4 (the bioreactor here), the controller, and all other instrumentation, such as sensors, transmitters, and actuators (like valves and pumps) that enable us to control the pH. When we change a speciп¬Ѓc operating condition, meaning the set point, we would like, for example, the pH of the bioreactor to follow our command. This is what we call servocontrol. The pH value of the bioreactor is subjected to external disturbances (also called load changes), and the task of suppressing or rejecting the effects of disturbances is called regulatory control. Implementation of a controller may lead to instability, and the issue of system stability is a major concern. The control system also has to be robust such that it is not overly sensitive to changes in process parameters. What are some of the issues when we design a control system? In the п¬Ѓrst place, we need to identify the role of various variables. We need to determine what we need to control, what we need to manipulate, what the sources of disturbances are, and so forth. We then need to state our design objective and speciп¬Ѓcations. It may make a difference whether we focus on the servo or on the regulator problem, and we certainly want to make clear, quantitatively, the desired response of the system. To achieve these goals, we have to select the proper control strategy and controller. To implement the strategy, we also need to select the proper sensors, transmitters, and actuators. After all is done, we have to know how to tune the controller. Sounds like we are working with a musical instrument, but thatвЂ™s the jargon. The design procedures depend heavily on the dynamic model of the process to be controlled. In more advanced model-based control systems, the action taken by the controller actually depends on the model. Under circumstances for which we do not have a precise model, we perform our analysis with approximate models. This is the basis of a п¬Ѓeld called system identiп¬Ѓcation and parameter estimation. Physical insight that we may acquire in the act of model building is invaluable in problem solving. Although we laud the virtue of dynamic modeling, we will not duplicate the introduction of basic conservation equations. It is important to recognize that all of the processes that we want to control, e.g., bioreactor, distillation column, п¬‚ow rate in a pipe, drug delivery system, etc., are what we have learned in other engineering classes. The so-called model equations are conservation equations in heat, mass, and momentum. We need force balance in mechanical devices, and, in electrical engineering, we consider circuit analysis. The difference between what we now use in control and what we are more accustomed to is that control problems are transient in nature. Accordingly, we include the time-derivative (also called accumulation) term in our balance (model) equations. What are some of the mathematical tools that we use? In classical control, our analysis is based on linear ordinary differential equations with constant coefп¬Ѓcients вЂ“ what is called linear time invariant (LTI). Our models are also called lumped-parameter models, meaning that variations in space or location are not considered. Time is the only independent variable. Otherwise, we would need partial differential equations in what is called distributed-parameter models. To handle our linear differential equations, we rely heavily 4 In most of the control world, a process is referred to as a plant. Here вЂњprocessвЂќ is used because, in the process industry, a plant carries the connotation of the entire manufacturing or processing facility. 3 Introduction Table 1.1. Examples used in different chapters Example Page no. Example 4.7 Example 4.7A Example 4.7B Example 4.7C Example 4.8 Example 4.8A 72 73 186 187 75 181 Example 5.7 Example 5.7A Example 5.7B Example 5.7C Example 5.7D 101 112 123 123 172 Example 7.2 Example 7.2A Example 7.2B Example 7.2C Example 7.2D Example 7.3 Example 7.3A Example 7.3B Example 7.4 Example 7.4A Example 7.5 Example 7.5A Example 7.5B 133 135 139 170 171 133 135 140 136 172 137 143 188 on Laplace transform, and we invariably rearrange the resulting algebraic equation into the so-called transfer functions. These algebraic relations are presented graphically as block diagrams (as in Fig. 1.2). However, we rarely go as far as solving for the time-domain solutions. Much of our analysis is based on our understanding of the roots of the characteristic polynomial of the differential equation вЂ“ what we call the poles. At this point, a little secret should be disclosed. Just from the terminology, it may be inferred that control analysis involves quite a bit of mathematics, especially when we go over stability and frequency-response methods. That is one reason why these topics are not immediately introduced. Nonetheless, we have to accept the prospect of working with mathematics. It would be a lie to say that one can be good in process control without sound mathematical skills. Starting in Chap. 6, a select set of examples is repeated in some subsections and chapters. To reinforce the thinking that different techniques can be used to solve the same problem, these examples retain the same numeric labeling. These examples, which do not follow conventional numbering, are listed in Table 1.1 to help you п¬Ѓnd them. It may be useful to point out a few topics that go beyond a п¬Ѓrst course in control. With certain processes, we cannot take data continuously, but rather in certain selected slow intervals (e.g., titration in freshmen chemistry). These are called sampled-data systems. With 4 Introduction computers, the analysis evolves into a new area of its own вЂ“ discrete-time or digital control systems. Here, differential equations and Laplace transform do not work anymore. The mathematical techniques to handle discrete-time systems are difference equations and z transforms. Furthermore, there are multivariable and state-space controls, which we will encounter in a brief introduction. Beyond the introductory level are optimal control, nonlinear control, adaptive control, stochastic control, and fuzzy-logic control. Do not lose the perspective that control is an immense п¬Ѓeld. Classical control appears insigniп¬Ѓcant, but we have to start somewhere, and onward we crawl. 5 2 Mathematical Preliminaries lassical process control builds on linear ordinary differential equations (ODEs) and the technique of the Laplace transform. This is a topic that we no doubt have come across in an introductory course on differential equations вЂ“ like two years ago? Yes, we easily have forgotten the details. Therefore an attempt is made here to refresh the material necessary to solve control problems; other details and steps will be skipped. We can always refer back to our old textbook if we want to answer long-forgotten but not urgent questions. C What Are We Up to? r The properties of Laplace transform and the transforms of some common functions. We need them to construct a table for doing an inverse transform. r Because we are doing an inverse transform by means of a look-up table, we need to r r r r 6 break down any given transfer functions into smaller parts that match what the table has вЂ“ what are called partial fractions. The time-domain function is the sum of the inverse transform of the individual terms, making use of the fact that Laplace transform is a linear operator. The time-response characteristics of a model can be inferred from the poles, i.e., the roots of the characteristic polynomial. This observation is independent of the input function and singularly the most important point that we must master before moving onto control analysis. After a Laplace transform, a differential equation of deviation variables can be thought of as an inputвЂ“output model with transfer functions. The causal relationship of changes can be represented by block diagrams. In addition to transfer functions, we make extensive use of steady-state gain and time constants in our analysis. Laplace transform is applicable to only linear systems. Hence we have to linearize nonlinear equations before we can go on. The procedure of linearization is based on a п¬Ѓrst-order Taylor series expansion. 2.1. A Simple Differential Equation Model 2.1. A Simple Differential Equation Model First an impetus is provided for solving differential equations in an approach unique to control analysis. The mass balance of a well-mixed tank can be written (see Review Problems) as dC П„ = Cin в€’ C, with C(0) = C0 , dt where C is the concentration of a component, Cin is the inlet concentration, C0 is the initial concentration, and П„ is the space time. In classical control problems, we invariably rearrange the equation as П„ dC + C = Cin dt (2.1) and further redeп¬Ѓne variables C = C в€’ C0 and Cin = Cin в€’ C0 .1 We designate C and Cin as deviation variables вЂ“ they denote how a quantity deviates from the original value at t = 0.2 Because C0 is a constant, we can rewrite Eq. (2.1) as dC (2.2) + C = Cin , with C (0) = 0. dt Note that the equation now has a zero initial condition. For reference, the solution to Eq. (2.2) is3 П„ C (t) = 1 П„ t 0 Cin (z)eв€’(tв€’z)/П„ dz. (2.3) If Cin is zero, we have the trivial solution C = 0. It is obvious from Eq. (2.2) immediately. For a more interesting situation in which C is nonzero or for C to deviate from the initial C0 , Cin must be nonzero, or in other words, Cin is different from C0 . In the terminology of differential equations, the right-hand side (RHS) Cin is called the forcing function. In control, it is called the input. Not only is Cin nonzero, it is, under most circumstances, a function of time as well, Cin = Cin (t). In addition, the time dependence of the solution, meaning the exponential function, arises from the left-hand side (LHS) of Eq. (2.2), the linear differential operator. In fact, we may recall that the LHS of Eq. (2.2) gives rise to the so-called characteristic equation (or characteristic polynomial). Do not worry if you have forgotten the signiп¬Ѓcance of the characteristic equation. We will come back to this issue again and again. This example is used just as a prologue. Typically in a class on differential equations, we learn to transform a linear ordinary equation into 1 2 3 At steady state, 0 = Cins в€’ C s , and if Cins = C0 , we can also deп¬Ѓne Cin = Cin в€’ Cins . We will come back to this when we learn to linearize equations. We will see that we should choose C0 = C s. Deviation variables are analogous to perturbation variables used in chemical kinetics or in п¬‚uid mechanics (linear hydrodynamic stability). We can consider a deviation variable as a measure of how far it is from steady state. When you come across the term convolution integral later in Eq. (4.10) and wonder how it may come about, take a look at the form of Eq. (2.3) again and think about it. If you wonder where Eq. (2.3) comes from, review your old ODE text on integrating factors. We skip this detail as we will not be using the time-domain solution in Eq. (2.3). 7 Mathematical Preliminaries an algebraic equation in the Laplace domain, solve for the transformed dependent variable, and п¬Ѓnally get back the time-domain solution with an inverse transformation. In classical control theory, we make extensive use of a Laplace transform to analyze the dynamics of a system. The key point (and at this moment the trick) is that we will try to predict the time response without doing the inverse transformation. Later, we will see that the answer lies in the roots of the characteristic equation. This is the basis of classical control analyses. Hence, in going through Laplace transform again, it is not so much that we need a remedial course. Our old differential equation textbook would do п¬Ѓne. The key task here is to pitch this mathematical technique in light that may help us to apply it to control problems. 2.2. Laplace Transform Let us п¬Ѓrst state a few important points about the application of Laplace transform in solving differential equations (Fig. 2.1). After we have formulated a model in terms of a linear or a linearized differential equation, dy/dt = f (y), we can solve for y(t). Alternatively, we can transform the equation into an algebraic problem as represented by the function G(s) in the Laplace domain and solve for Y (s). The time-domain solution y(t) can be obtained with an inverse transform, but we rarely do so in control analysis. What we argue (of course it is true) is that the Laplace-domain function Y (s) must contain the same information as y(t). Likewise, the function G(s) must contain the same dynamic information as the original differential equation. We will see that the function G(s) can be вЂњclean lookingвЂќ if the differential equation has zero initial conditions. That is one of the reasons why we always pitch a control problem in terms of deviation variables.4 We can now introduce the deп¬Ѓnition. The Laplace transform of a function f (t) is deп¬Ѓned as в€ћ L[ f (t)] = f (t)eв€’st dt, (2.4) 0 where s is the transform variable.5 To complete our deп¬Ѓnition, we have the inverse transform, f (t) = Lв€’1 [F(s)] = 1 2ПЂ j Оі + jв€ћ F(s)est ds, (2.5) Оі в€’ jв€ћ where Оі is chosen such that the inп¬Ѓnite integral can converge.6 Do not be intimidated by f(t) y(t) L F(s) Input/forcing function (disturbances, manipulated variables) Y(s) G(s) dy/dt = f(t) Output (controlled variable) Input Output Figure 2.1. Relationship between time domain and Laplace domain. 4 5 6 8 But! What we measure in an experiment is the вЂњrealвЂќ variable. We have to be careful when we solve a problem that provides real data. There are many acceptable notations for a Laplace transform. Here we use a capital letter, and, if confusion may arise, we further add (s) explicitly to the notation. If you insist on knowing the details, they can be found on the Web Support. 2.2. Laplace Transform Eq. (2.5). In a control class, we never use the inverse transform deп¬Ѓnition. Our approach is quite simple. We construct a table of the Laplace transform of some common functions, and we use it to do the inverse transform by means of a look-up table. An important property of the Laplace transform is that it is a linear operator, and the contribution of individual terms can simply be added together (superimposed): L[a f 1 (t) + b f 2 (t)] = aL[ f 1 (t)] + bL[ f 2 (t)] = a F1 (s) + bF2 (s). (2.6) Note: The linear property is one very important reason why we can do partial fractions and inverse transforms by means of a look-up table. This is also how we analyze more complex, but linearized, systems. Even though a text may not state this property explicitly, we rely heavily on it in classical control. We now review the Laplace transforms of some common functions вЂ“ mainly the ones that we come across frequently in control problems. We do not need to know all possibilities. We can consult a handbook or a mathematics textbook if the need arises. (A summary of the important transforms is in Table 2.1.) Generally, it helps a great deal if you can do the following common ones without having to use a look-up table. The same applies to simple algebra, such as partial fractions, and calculus, such as linearizing a function. (1) A constant: f (t) = a, F(s) = (a/s). (2.7) The derivation is в€ћ L[a] = a 0 a eв€’st dt = в€’ eв€’st s в€ћ =a 0+ 0 1 s = a . s (2) An exponential function (Fig. 2.2): f (t) = eв€’at , L[eв€’at ] = a with a > 0, F(s) = [1/(s + a)], в€ћ в€’1 в€’(a+s)t e (s + a) eв€’at eв€’st dt = 0 в€ћ 0 = (2.8) 1 . (s + a) (3) A ramp function (Fig. 2.2): f (t) = at for t в‰Ґ 0 and a = constant, в€ћ L[at] = a 0 a = s Exponential decay 1 t eв€’st dt = a в€’t eв€’st s в€ћ 0 в€ћ 0 F(s) = (a/s 2 ), в€ћ + 0 (2.9) 1 в€’st e dt s a eв€’st dt = 2 . s Linear ramp slope a Figure 2.2. Illustration of exponential and ramp functions. 9 Mathematical Preliminaries Table 2.1. Summary of a handful of common Laplace transforms Function F (s) f (t) The very basic functions a/s a/s 2 1/(s + a) П‰/(s 2 + П‰2 ) s/(s 2 + П‰2 ) П‰/[(s + a)2 + П‰2 ] (s + a)/[(s + a)2 + П‰2 ] a or au(t) at eв€’at sin П‰t cos П‰t eв€’at sin П‰t eв€’at cos П‰t d2 f dt 2 t f (t) dt 0 s 2 F(s) в€’ s f (0) в€’ f (0) F(s) s eв€’st0 F(s) A Transfer functions in time-constant form Transfer functions in pole-zero form f (t в€’ t0 ) AОґ(t) 1/(П„ s + 1) 1 (П„ s + 1)n 1/[s(П„ s + 1)] 1/[(П„1 s + 1)(П„2 s + 1)] (1/П„ )eв€’t/П„ 1 t nв€’1 eв€’t/П„ П„ n (n в€’ 1)! 1 в€’ eв€’t/П„ eв€’t/П„1 в€’ eв€’t/П„2 /П„1 в€’ П„2 1 s(П„1 s + 1)(П„2 s + 1) (П„3 s + 1) (П„1 s + 1)(П„2 s + 1) (П„3 s + 1) s(П„1 s + 1)(П„2 s + 1) 1+ 1/(s + a) 1/[(s + a)2 ] 1 (s + a)n 1/[s(s + a)] 1/[(s + a)(s + b)] s/[(s + a)2 ] s/[(s + a)(s + b)] eв€’at t eв€’at 1 t nв€’1 eв€’at (n в€’ 1)! (1/a) (1 в€’ eв€’at ) [1/(b в€’ a)](eв€’at в€’ eв€’bt ) (1 в€’at) eв€’at [1/(b в€’ a)] (beв€’bt в€’ aeв€’at ) 1 s(s + a)(s + b) 1 1 1+ (beв€’at в€’ aeв€’bt ) ab aв€’b П„1 eв€’t/П„1 в€’ П„2 eв€’t/П„2 П„2 в€’ П„ 1 1 П„1 в€’ П„3 в€’t/П„1 1 П„2 в€’ П„3 в€’t/П„2 e + e П„1 П„1 в€’ П„2 П„2 П„2 в€’ П„ 1 П„3 в€’ П„1 в€’t/П„1 П„3 в€’ П„2 в€’t/П„2 1+ e + e П„1 в€’ П„2 П„2 в€’ П„ 1 Note: We may п¬Ѓnd many more Laplace transforms in handbooks or texts, but here we stay with the most basic ones. The more complex ones may actually be a distraction to our objective, which is to understand pole positions. (4) Sinusoidal functions 10 f (t) = sin П‰t, F(s) = [П‰/(s 2 + П‰2 )], (2.10) f (t) = cos П‰t, F(s) = [s/(s 2 + П‰2 )]. (2.11) 2.2. Laplace Transform We use the fact that sin П‰t = (1/2 j)(e jП‰t в€’ eв€’ jП‰t ) and the result with an exponential function to derive L[sin П‰t] = = в€ћ 1 2j (e jП‰t в€’ eв€’ jП‰t )eв€’st dt 0 в€ћ 1 2j в€ћ eв€’(sв€’ jП‰)t dt в€’ 0 0 1 1 в€’ s в€’ jП‰ s + jП‰ 1 = 2j eв€’(s+ jП‰)t dt = s2 П‰ . + П‰2 The Laplace transform of cos П‰t is left as an exercise in the Review Problems. If you need a review on complex variables, the Web Support has a brief summary. (5) Sinusoidal function with exponential decay: f (t) = eв€’at sin П‰t, F(s) = П‰ . (s + a)2 + П‰2 (2.12) Making use of previous results with the exponential and sine functions, we can pretty much do this one by inspection. First, we put the two exponential terms together inside the integral: в€ћ sin П‰t eв€’(s+a)t dt = 0 = в€ћ 1 2j 0 1 2j в€ћ eв€’(s+aв€’ jП‰)t dt в€’ eв€’(s+a+ jП‰)t dt 0 1 1 в€’ . (s + a) в€’ jП‰ (s + a) + jП‰ The similarity to the result of sin П‰t should be apparent now, if it was not the case with the LHS. (6) First-order derivative, d f /dt: L df dt = s F(s) в€’ f (0); (2.13) second-order derivative: L d2 f dt 2 = s 2 F(s) в€’ s f (0) в€’ f (0). (2.14) We have to use integration by parts here: L df dt в€ћ = 0 d f в€’st e dt = f (t)eв€’st dt в€ћ в€ћ +s f (t)eв€’st dt 0 0 = в€’ f (0) + s F(s), L d2 f dt 2 в€ћ = 0 df =в€’ dt d dt df dt eв€’st dt = d f в€’st e dt в€ћ 0 в€ћ +s 0 d f в€’st e dt dt + s[s F(s) в€’ f (0)]. 0 We can extend these results to п¬Ѓnd the Laplace transform of higher-order derivatives. The key is that, if we use deviation variables in the problem formulation, all the 11 Mathematical Preliminaries initial-value terms will drop out in Eqs. (2.13) and (2.14). This is how we can get these clean-looking transfer functions in Section 2.6. (7) An integral: t L f (t) dt = 0 F(s) . s (2.15) We also need integration by parts here: в€ћ t 0 2.3. 0 в€ћ t 1 f (t) dt eв€’st dt = в€’ eв€’st s f (t) dt 0 + 0 1 s в€ћ f (t)eв€’st dt = 0 F(s) . s Laplace Transforms Common to Control Problems We now derive the Laplace transform of functions common in control analysis. (1) Step function: f (t) = Au(t), F(s) = (A/s). (2.16) We п¬Ѓrst deп¬Ѓne the unit-step function (also called the Heaviside function in mathematics) and its Laplace transform7 : u(t) = 1, t > 0 ; 0, t < 0 1 . s L[u(t)] = U (s) = (2.17) The Laplace transform of the unit-step function (Fig. 2.3) is derived as follows: L[u(t)] = lim+ в†’0 в€ћ Оµ u(t)eв€’st dt = в€ћ 0+ eв€’st dt = в€’1 в€’st e s в€ћ = 0 1 . s f(t) Unit step t 1 t=0 0 t to t' = t в€’ t o 0 Rectangular pulse Impulse function A t=0 Time-delay function f(t вЂ“ t o ) Area = 1 T t=0 Figure 2.3. Unit-step, time-delay, rectangular, and impulse functions. 7 12 Strictly speaking, the step function is discontinuous at t = 0, but many engineering texts ignore it and simply write u(t) = 1 for t в‰Ґ 0. 2.3. Laplace Transforms Common to Control Problems With the result for the unit step, we can see the results of the Laplace transform of any step function f (t) = Au(t): A, t > 0 ; 0, t < 0 f (t) = Au(t) = L[Au(t)] = A . s The Laplace transform of a step function is essentially the same as that of a constant in Eq. (2.7). When we do the inverse transform of A/s, which function we choose depends on the context of the problem. Generally, a constant is appropriate under most circumstances. (2) Dead-time function (Fig. 2.3): L[ f (t в€’ t0 )] = eв€’st0 F(s). f (t в€’ t0 ), (2.18) The dead-time function is also called the time-delay, transport-lag, translated, or time-shift function (Fig. 2.3). It is deп¬Ѓned such that an original function f (t) is вЂњshiftedвЂќ in time t0 , and no matter what f (t) is, its value is set to zero for t < t0 . This time-delay function can be written as 0, t в€’ t0 < 0 = f (t в€’ t0 )u(t в€’ t0 ). f (t в€’ t0 ), t в€’ t0 > 0 f (t в€’ t0 ) = The second form on the far right of the preceding equation is a more concise way of saying that the time-delay function f (t в€’ t0 ) is deп¬Ѓned such that it is zero for t < t0 . We can now derive the Laplace transform, в€ћ L[ f (t в€’ t0 )] = в€ћ f (t в€’ t0 )u(t в€’ t0 )eв€’st dt = 0 f (t в€’ t0 )eв€’st dt, t0 and, п¬Ѓnally, в€ћ в€ћ f (t в€’ t0 ) eв€’st dt = eв€’st0 t0 f (t в€’ t0 )eв€’s(tв€’t0 ) d(t в€’ t0 ) t0 в€ћ = eв€’st0 f (t )eв€’st dt = eв€’st0 F(s), 0 where the п¬Ѓnal integration step uses the time-shifted axis t = t в€’ t0 . (3) Rectangular-pulse function (Fig. 2.3): пЈ± пЈІ 0, t < 0 A f (t) = A, 0 < t < T = A[u(t) в€’ u(t в€’ T )]; L[ f (t)] = (1 в€’ eв€’sT ). пЈі s 0, t > T (2.19) We can generate the rectangular pulse by subtracting a step function with dead time T from a step function. We can derive the Laplace transform by using the formal deп¬Ѓnition в€ћ L[ f (t)] = 0 f (t)eв€’st dt = A T 0+ eв€’st dt = A в€’1 в€’st e s T = 0 A (1 в€’ eв€’sT ), s 13 Mathematical Preliminaries or, better yet, by making use of the results of a step function and a dead-time function: A A L[ f (t)] = L[A u(t) в€’ A u(t в€’ T )] = в€’ eв€’sT . s s (4) Unit-rectangular-pulse function: пЈ± 0, t <0 пЈґ пЈІ 1 1 f (t) = 1/T, 0 < t < T = [u(t) в€’ u(t в€’ T )]; L[ f (t)] = (1 в€’ eв€’sT ). пЈґ sT T пЈі 0, t>T (2.20) This is a prelude to the important impulse function. We can deп¬Ѓne a rectangular pulse such that the area is unity. The Laplace transform follows that of a rectangular pulse function: 1 1 1 L[ f (t)] = L u(t) в€’ u(t в€’ T ) = (1 в€’ esT ). T T Ts (5) Impulse function (Fig. 2.3): L[Оґ(t)] = 1, L[AОґ(t)] = A. (2.21) The (unit) impulse function is called the Dirac function (or simply the delta function) in mathematics.8 If we suddenly dump a bucket of water into a bigger tank, the impulse function is how we describe the action mathematically. We can consider the impulse function as the unit-rectangular function in Eqs. (2.20) as T shrinks to zero while the height 1/T goes to inп¬Ѓnity: 1 Оґ(t) = lim [u(t) в€’ u(t в€’ T )]. T в†’0 T The area of this вЂњsqueezed rectangleвЂќ nevertheless remains at unity: lim T в†’0 T 1 T = 1, or, in other words, в€ћ в€’в€ћ Оґ(t) dt = 1. The impulse function is rarely deп¬Ѓned in the conventional sense, but rather by its important property in an integral: в€ћ в€’в€ћ f (t)Оґ(t) dt = f (0), в€ћ в€’в€ћ f (t)Оґ(t в€’ t0 ) dt = f (t0 ). (2.22) We easily obtain the Laplace transform of the impulse function by taking the limit of the unit-rectangular-function transform of Eqs. (2.20) with the use of lвЂ™HЛ†opitalвЂ™s rule: 1 в€’ eв€’sT seв€’sT = lim = 1. T в†’0 T в†’0 Ts s From this result, it is obvious that L[AОґ(t)] = A. L[Оґ(t)] = lim 8 14 In mathematics, the unit-rectangular function is deп¬Ѓned with a height of 1/2T and a width of 2T from в€’T to T . We simply begin at t = 0 in control problems. Furthermore, the impulse function is the time derivative of the unit-step function. 2.4. Initial- and Final-Value Theorems 2.4. Initial- and Final-Value Theorems Two theorems are now presented that can be used to п¬Ѓnd the values of the time-domain function at two extremes, t = 0 and t = в€ћ, without having to do the inverse transform. In control, we use the п¬Ѓnal-value theorem quite often. The initial-value theorem is less useful. As we have seen from our п¬Ѓrst example in Section 2.1, the problems that we solve are deп¬Ѓned to have exclusively zero initial conditions. Initial-Value Theorem: lim [s F(s)] = lim f (t). sв†’в€ћ (2.23) tв†’0 Final-Value Theorem: lim [s F(s)] = lim f (t). (2.24) tв†’в€ћ sв†’0 The п¬Ѓnal-value theorem is valid provided that a п¬Ѓnal-value exists. The proofs of these theorems are straightforward. We will do the one for the п¬Ѓnal-value theorem. The proof of the initial-value theorem is in the Review Problems. Consider the deп¬Ѓnition of the Laplace transform of a derivative. If we take the limit as s approaches zero, we п¬Ѓnd в€ћ lim sв†’0 0 d f (t) в€’st e dt = lim [s F(s) в€’ f (0)]. sв†’0 dt If the inп¬Ѓnite integral exists,9 we can interchange the limit and the integration on the LHS to give в€ћ d f (t) в€’st e dt = sв†’0 dt в€ћ lim 0 d f (t) = f (в€ћ) в€’ f (0). 0 Now if we equate the RHSs of the previous two steps, we have f (в€ћ) в€’ f (0) = lim [s F(s) в€’ f (0)]. sв†’0 We arrive at the п¬Ѓnal-value theorem after we cancel the f (0) terms on both sides. Example 2.1: Consider the Laplace transform F(s) = {[6(s в€’ 2)(s + 2)]/[s(s + 1) Г— (s + 3)(s + 4)]}. What is f (t = в€ћ)? lim s sв†’0 6(в€’2)(2) 6(s в€’ 2)(s + 2) = = в€’2. s(s + 1)(s + 3)(s + 4) (3)(4) Example 2.2: Consider the Laplace transform F(s) = [1/(s в€’ 2)]. What is f (t = в€ћ)? Here, f (t) = e2t . There is no upper bound for this function, which is in violation of the existence of a п¬Ѓnal value. The п¬Ѓnal-value theorem does not apply. If we insist on applying the theorem, we will get a value of zero, which is meaningless. Example 2.3: Consider the Laplace transform F(s) = {[6(s 2 в€’ 4)]/[(s 3 + s 2 в€’ 4s в€’ 4)]}. What is f (t = в€ћ)? Yes, another trick question. If we apply the п¬Ѓnal-value theorem without 9 This is a key assumption and explains why Examples 2.2 and 2.3 do not work. When a function has no bound вЂ“ what we call unstable вЂ“ the assumption is invalid. 15 Mathematical Preliminaries thinking, we would get a value of 0, but this is meaningless. With Matlab R , we can use roots([1 1 -4 -4]) to п¬Ѓnd that the polynomial in the denominator has roots в€’1, в€’2, and +2. This implies that f (t) contains the term e2t , which increases without bound. As we move on, we will learn to associate the time-exponential terms to the roots of the polynomial in the denominator. From these examples, we can gather that, to have a meaningful, i.e., п¬Ѓnite, bounded value, the roots of the polynomial in the denominator must have negative real parts. This is the basis of stability, which will formally be deп¬Ѓned in Chap. 7. 2.5. Partial-Fraction Expansion Because we rely on a look-up table to do a reverse Laplace transform, we need the skill to reduce a complex function down to simpler parts that match our table. In theory, we should be able to вЂњbreak upвЂќ a ratio of two polynomials in s into simpler partial fractions. If the polynomial in the denominator, p(s), is of an order higher than the numerator, q(s), we can derive10 О±1 О±2 О±i О±n q(s) = + + В·В·В· + + В·В·В· + , (2.25) F(s) = p(s) (s + a1 ) (s + a2 ) (s + ai ) (s + an ) where the order of p(s) is n and the ai are the negative values of the roots of the equation p(s) = 0. We then perform the inverse transform term by term: f (t) = Lв€’1 [F(s)] = Lв€’1 + В· В· В· + Lв€’1 О±1 О±2 + Lв€’1 (s + a1 ) (s + a2 ) О±i О±n + В· В· В· + Lв€’1 . (s + ai ) (s + an ) (2.26) This approach works because of the linear property of the Laplace transform. The next question is how to п¬Ѓnd the partial fractions in Eq. (2.25). One of the techniques is the so-called Heaviside expansion, a fairly straightforward algebraic method. Three important cases are illustrated with respect to the roots of the polynomial in the denominator: (1) distinct real roots, (2) complex-conjugate roots, and (3) multiple (or repeated) roots. In a given problem, we can have a combination of any of the above. Yes, we need to know how to do them all. 2.5.1. Case 1: p(s) Has Distinct, Real Roots Example 2.4: Find f (t) of the Laplace transform F(s) = [(6s 2 в€’ 12)/(s 3 + s 2 в€’ 4s в€’ 4)]. From Example 2.3, the polynomial in the denominator has roots в€’1, в€’2, and +2, values 10 16 If the order of q(s) is higher, we п¬Ѓrst need to carry out вЂњlong divisionвЂќ until we are left with a partial-fraction вЂњresidue.вЂќ Thus the coefп¬Ѓcients О±i are also called residues. We then expand this partial fraction. We would encounter such a situation in only a mathematical problem. The models of real physical processes lead to problems with a higher-order denominator. 2.5. Partial-Fraction Expansion that will be referred to as poles in Section 2.6. We should be able to write F(s) as 6s 2 в€’ 12 О±1 О±2 О±3 = + + . (s + 1)(s + 2)(s в€’ 2) (s + 1) (s + 2) (s в€’ 2) The Heaviside expansion takes the following idea. If we multiply both sides by (s + 1), we obtain О±3 О±2 6s 2 в€’ 12 = О±1 + (s + 1) + (s + 1), (s + 2)(s в€’ 2) (s + 2) (s в€’ 2) which should be satisп¬Ѓed by any value of s. Now if we choose s = в€’1, we should obtain О±1 = 6s 2 в€’ 12 (s + 2)(sв€’ 2) = 2. s=в€’1 Similarly, we can multiply the original fraction by (s + 2) and (s в€’ 2) to п¬Ѓnd О±2 = О±3 = 6s 2 в€’ 12 (s + 1)(s в€’ 2) s=в€’2 6s 2 в€’ 12 (s + 1)(s + 2) s=2 = 3, = 1, respectively. Hence, F(s) = 3 1 2 + + , (s + 1) (s + 2) (s в€’ 2) and the look-up table would give us f (t) = 2eв€’t + 3eв€’2t + e2t . When you use MATLAB to solve this problem, be careful when you interpret the results. The computer is useless unless we know what we are doing. We provide only the necessary statements.11 For this example, all we need is [a,b,k]=residue([6 0 -12],[1 1 -4 -4]) Example 2.5: Find f (t) of the Laplace transform F(s) = [(6s)/(s 3 + s 2 в€’ 4s в€’ 4)]. Again, the expansion should take the form 6s О±1 О±2 О±3 = + + . (s + 1)(s + 2)(s в€’ 2) (s + 1) (s + 2) (s в€’ 2) One more time, for each term, we multiply the denominators on the RHS and set the resulting equation to its root to obtain О±1 = О±3 = 11 6s (s + 2)(s в€’ 2) s=в€’1 6s (s + 1)(s + 2) s=2 = 2, О±2 = 6s (s + 1)(s в€’ 2) = в€’3, s=в€’2 = 1. From here on, it is important that you go over the MATLAB sessions. Explanation of residue() is in Session 2. Although we do not print the computer results, they can be found on the Web Support. 17 Mathematical Preliminaries The time-domain function is f (t) = 2eв€’t в€’ 3eв€’2t + e2t . Note that f (t) has the identical functional dependence in time as in Example 2.4. Only the coefп¬Ѓcients (residues) are different. The MATLAB statement for this example is [a,b,k]=residue([6 0],[1 1 -4 -4]) Example 2.6: Find f (t) of the Laplace transform F(s) = {6/[(s + 1)(s + 2)(s + 3)]}. This time, we should п¬Ѓnd О±1 = О±3 = 6 (s + 2)(s + 3) s=в€’1 6 (s + 1)(s + 2) s=в€’3 = 3, О±2 = 6 (s + 1)(s + 3) = в€’6, s=в€’2 = 3. The time-domain function is f (t) = 3eв€’t в€’ 6eв€’2t + 3eв€’3t . The eв€’2t and eв€’3t terms will decay faster than the eв€’t term. We consider the eв€’t term, or the pole at s = в€’1, as more dominant. We can conп¬Ѓrm the result with the following MATLAB statements: p=poly([-1 -2 -3]); [a,b,k]=residue(6,p) Note: (1) The time dependence of the time-domain solution is derived entirely from the roots of the polynomial in the denominator (what we will refer to in Section 2.6 as the poles). The polynomial in the numerator affects only the coefп¬Ѓcients О±i . This is one reason why we make qualitative assessment of the dynamic response characteristics entirely based on the poles of the characteristic polynomial. (2) Poles that are closer to the origin of the complex plane will have corresponding exponential functions that decay more slowly in time. We consider these poles more dominant. (3) We can generalize the Heaviside expansion into the fancy form for the coefп¬Ѓcients О±i = (s + ai ) q(s) p(s) , s=в€’ai but we should always remember the simple algebra that we have gone through in the preceding examples. 2.5.2. Case 2: p(s) Has Complex Roots Example 2.7: Find f (t) of the Laplace transform F(s) = [(s + 5)/(s 2 + 4s + 13)]. We п¬Ѓrst take the painful route just so we better understand the results from MATLAB. If we have 18 2.5. Partial-Fraction Expansion to do the chore by hand, we much prefer the completing the perfect-square method in Example 2.8. Even without MATLAB, we can easily п¬Ѓnd that the roots of the polynomial s 2 + 4s + 13 are в€’2 В± 3 j and F(s) can be written as the sum of s+5 О± О±в€— s+5 = = + . s 2 + 4s + 13 [s в€’ (в€’2 + 3 j)][s в€’ (в€’2 в€’ 3 j)] s в€’ (в€’2 + 3 j) s в€’ (в€’2 в€’ 3 j) We can apply the same idea as in Example 2.4 to п¬Ѓnd О±= s+5 [s в€’ (в€’2 в€’ 3 j)] = s=в€’2+3 j ( j + 1) (в€’2 + 3 j) + 5 1 = = (1 в€’ j), (в€’2 + 3 j) + 2 + 3 j 2j 2 and its complex conjugate12 is О±в€— = 1 (1 + j). 2 The inverse transform is hence 1 1 (1 в€’ j)e(в€’2+3 j)t + (1 + j)e(в€’2в€’3 j)t 2 2 1 в€’2t = e [(1 в€’ j)e j3t + (1 + j)eв€’ j3t ]. 2 We can apply EulerвЂ™s identity to the result: f (t) = 1 в€’2t e [(1 в€’ j)(cos 3t + j sin 3t) + (1 + j)(cos 3t в€’ j sin 3t)] 2 1 = eв€’2t [2(cos 3t + sin 3t)], 2 which we further rewrite as в€љ f (t) = 2eв€’2t sin(3t + П†), f (t) = where П† = tanв€’1 (1) = ПЂ/4 or 45в—¦ . The MATLAB statement for this example is simply [a,b,k]=residue([1 5],[1 4 13]) Note: (1) Again, the time dependence of f (t) is affected by only the roots of p(s). For the general complex-conjugate roots в€’aВ± bj, the time-domain function involves eв€’at and (cos bt + sin bt). The polynomial in the numerator affects only the constant coefп¬Ѓcients. (2) We seldom use the form (cos bt + sin bt). Instead, we use the phase-lag form as in the п¬Ѓnal step of Example 2.7. Example 2.8: Repeat Example 2.7 with a look-up table. In practice, we seldom do the partial-fraction expansion of a pair of complex roots. Instead, we rearrange the polynomial 12 If you need a review of complex-variable deп¬Ѓnitions, see the Web Support. Many steps in Example 2.7 require these deп¬Ѓnitions. 19 Mathematical Preliminaries p(s) by noting that we can complete the squares: s 2 + 4s + 13 = (s + 2)2 + 9 = (s + 2)2 + 32 . We then write F(s) as F(s) = s2 s+5 3 (s + 2) + . = 2 2 + 4s + 13 (s + 2) + 3 (s + 2)2 + 32 With a Laplace-transform table, we п¬Ѓnd f (t) = eв€’2t cos 3t + eв€’2t sin 3t, which is the answer with very little work. Compared with how messy the partial fraction was in Example 2.7, this example also suggests that we want to leave terms with conjugatecomplex roots as one second-order term. 2.5.3. Case 3: p(s) Has Repeated Roots Example 2.9: Find f (t) of the Laplace transform F(s) = {2/[(s + 1)3 (s + 2)]}. The polynomial p(s) has the roots в€’1 repeated three times and в€’2. To keep the numerator of each partial fraction a simple constant, we have to expand to О±1 О±2 2 О±3 О±4 = + . + + (s + 1)3 (s + 2) (s + 1) (s + 1)2 (s + 1)3 (s + 2) To п¬Ѓnd О±3 and О±4 is routine: О±3 = 2 (s + 2) = 2, s=в€’1 О±4 = 2 (s + 1)3 = в€’2. s=в€’2 The problem is with п¬Ѓnding О±1 and О±2 . We see that, say, if we multiply the equation with (s + 1) to п¬Ѓnd О±1 , we cannot select s = в€’1. What we can try is to multiply the expansion with (s + 1)3 , О±4 (s + 1)3 2 = О±1 (s + 1)2 + О±2 (s + 1) + О±3 + , (s + 2) (s + 2) and then differentiate this equation with respect to s: в€’2 = 2О±1 (s + 1) + О±2 + 0 + [О±4 terms with (s + 1)]. (s + 2)2 Now we can substitute s = в€’1, which provides О±2 = в€’2. We can be lazy with the last О±4 term because we know its derivative will contain (s + 1) terms and they will drop out as soon as we set s = в€’1. To п¬Ѓnd О±1 , we differentiate the equation one more time to obtain 4 = 2 О±1 + 0 + 0 + [О±4 terms with (s + 1)], (s + 2)3 20 2.6. Transfer Function, Pole, and Zero which of course will yield О±1 = 2 if we select s = в€’1. Hence, we have 2 в€’2 2 в€’2 2 = + , + + 3 2 3 (s + 1) (s + 2) (s + 1) (s + 1) (s + 1) (s + 2) and the inverse transform by means of the look-up table is f (t) = 2 1в€’t + t2 2 eв€’t в€’ eв€’2t . We can also arrive at the same result by expanding the entire algebraic expression, but that actually takes more work (!), and we leave this exercise to the Review Problems. The MATLAB command for this example is p=poly([-1 -1 -1 -2]); [a,b,k]=residue(2,p) Note: In general, the inverse transform of repeated roots takes the form Lв€’1 О±1 О±n О±2 + В·В·В· + + 2 (s + a) (s + a) (s + a)n = О±1 + О±2 t + О±3 2 О±n t + В·В·В· + t nв€’1 eв€’at . 2! (n в€’ 1)! The exponential function is still based on the root s = в€’a, but the actual time dependence will decay slower because of the (О±2 t + В· В· В· +) terms. 2.6. Transfer Function, Pole, and Zero Now that we can do Laplace transform, let us return to our п¬Ѓrst example. The Laplace transform of Eq. (2.2) with its zero initial condition is (П„ s + 1)C (s) = Cin (s), which we rewrite as C (s) 1 = = G(s). Cin (s) П„s + 1 (2.27) We deп¬Ѓne the RHS as G(s), our ubiquitous transfer function. It relates an input to the output of a model. Recall that we use deviation variables. The input is the change in the inlet concentration, Cin (t). The output, or response, is the resulting change in the tank concentration, C (t). Example 2.10: What is the time domain response C (t) in Eq. (2.27) if the change in inlet concentration is (1) a unit-step function and (2) an impulse function? (1) With a unit-step input, Cin (t) = u(t) and Cin (s) = 1/s. Substitution for Cin (s) in Eq. (2.27) leads to C (s) = 1 1 1 в€’П„ = + . П„s + 1 s s П„s + 1 21 Mathematical Preliminaries After an inverse transform by means of a look-up table, we have C (t) = 1 в€’ eв€’t/П„ . The change in tank concentration eventually will be identical to the unit-step change in inlet concentration. (2) With an impulse input, Cin (s) = 1, and substitution for Cin (s) in Eq. (2.27) leads to simply 1 , C (s) = П„s + 1 and the time-domain solution is C (t) = П„1 eв€’t/П„ . The effect of the impulse eventually will decay away. Finally, you may want to keep in mind that the results of this example can also be obtained by means of the general time-domain solution in Eq. (2.3). The key of this example is to note that, irrespective of the input, the time-domain solution contains the time-dependent function eв€’t/П„ , which is associated with the root of the polynomial in the denominator of the transfer function. The inherent dynamic properties of a model are embedded in the characteristic polynomial of the differential equation. More speciп¬Ѓcally, the dynamics is related to the roots of the characteristic polynomial. In Eq. (2.27), the characteristic equation is П„ s + 1 = 0, and its root is в€’1/П„ . In a general sense, that is, without specifying what Cin is and without actually solving for C (t), we can infer that C (t) must contain a term with eв€’t/П„ . We refer to the root в€’1/П„ as the pole of the transfer function G(s). We can now state the deп¬Ѓnitions more generally. For an ODE13 an y (n) + anв€’1 y (nв€’1) + В· В· В· + a1 y (1) + a0 y = bm x (m) + bmв€’1 x (mв€’1) + В· В· В· + b1 x (1) + b0 x, (2.28) with n > m and zero initial conditions y (nв€’1) = В· В· В· = y = 0 at t = 0, the corresponding Laplace transform is bm s m + bmв€’1 s mв€’1 + В· В· В· + b1 s + b0 Y (s) Q(s) = . = G(s) = n nв€’1 X (s) an s + anв€’1 s + В· В· В· + a1 s + a0 P(s) (2.29) Generally we can write the transfer function as the ratio of two polynomials in s.14 When we talk about the mathematical properties, the polynomials are denoted as Q(s) and P(s), but the same polynomials are denoted as Y (s) and X (s) when the focus is on control problems or transfer functions. The orders of the polynomials are such that n в‰Ґ m for physically realistic processes.15 We know that G(s) contains information on the dynamic behavior of a model as represented by the differential equation. We also know that the denominator of G(s) is the 13 Yes, we try to be general and use an nth-order equation. If you have trouble with the development in this section, think of a second-order equation in all the steps: a2 y (2) + a1 y (1) + a0 y = b1 x (1) + b0 x. 14 15 22 All the features about poles and zeros can be obtained from this simpler equation. The exception is when we have dead time. We will come back to this term in Chap. 3. For real physical processes, the orders of polynomials are such that n в‰Ґ m. A simple explanation is to look at a so-called leadвЂ“lag element when n = m and y (1) + y = x (1) + x. The LHS, which is the dynamic model, must have enough complexity to reп¬‚ect the change of the forcing on the RHS. Thus if the forcing includes a rate of change, the model must have the same capability too. 2.6. Transfer Function, Pole, and Zero characteristic polynomial of the differential equation. The roots of the characteristic equation, P(s) = 0: p1 , p2 , . . . , pn , are the poles of G(s). When the poles are real and negative, we also use the time-constant notation: 1 1 1 p1 = в€’ , p2 = в€’ , . . . , pn = в€’ . П„1 П„2 П„n The poles reveal qualitatively the dynamic behavior of the model differential equation. The term вЂњroots of the characteristic equationвЂќ is used interchangeably with вЂњpoles of the transfer function.вЂќ For the general transfer function in Eq. (2.29), the roots of the polynomial Q(s), i.e., of Q(s) = 0, are referred to as the zeros. They are denoted by z 1 , z 2 , . . . , z m , or, in timeconstant notation, by z1 = в€’ 1 , П„a z2 = в€’ 1 ,..., П„b zm = в€’ 1 . П„m We can factor Eq. (2.29) into the so-called pole-zero form: G(s) = Q(s) = P(s) bm an (s в€’ z 1 )(s в€’ z 2 ) В· В· В· (s в€’ z m ) . (s в€’ p1 )(s в€’ p2 ) В· В· В· (s в€’ pn ) (2.30) If all the roots of the two polynomials are real, we can factor the polynomials such that the transfer function is in the time-constant form: G(s) = Q(s) = P(s) b0 a0 (П„a s + 1)(П„b s + 1) В· В· В· (П„m s + 1) . (П„1 s + 1)(П„2 s + 1) В· В· В· (П„n s + 1) (2.31) Equations (2.30) and (2.31) will be a lot less intimidating when we come back to using the examples in Section 2.8. These forms are the mainstays of classical control analysis. Another important quantity is the steady-state gain.16 With reference to general differential equation model (2.28) and its Laplace transform in Eq. (2.29), the steady-state gain is deп¬Ѓned as the п¬Ѓnal change in y(t) relative to a unit change in the input x(t). Thus an easy derivation of the steady-state gain is to take a unit-step input in x(t), or X (s) = 1/s, and п¬Ѓnd the п¬Ѓnal value in y(t): y(в€ћ) = lim [s G(s) X (s)] = lim s G(s) sв†’0 sв†’0 b0 1 = . s a0 (2.32) The steady-state gain is the ratio of the two constant coefп¬Ѓcients. Take note that the steadystate gain value is based on the transfer function only. From Eqs. (2.31) and (2.32), one easy way to вЂњspotвЂќ the steady-state gain is to look at a transfer function in the time-constant form. Note: (1) When we talk about the poles of G(s) in Eq. (2.29), the discussion is regardless of the input x(t). Of course, the actual response y(t) also depends on x(t) or X (s). (2) Recall from the examples of partial-fraction expansion that the polynomial Q(s) in the numerator, or the zeros, affects only the coefп¬Ѓcients of the solution y(t), but not 16 This quantity is also called the static gain or dc gain by electrical engineers. When we talk about the model of a process, we also use the term process gain quite often to distinguish it from a system gain. 23 Mathematical Preliminaries the time-dependent functions. That is why, for qualitative discussions, we focus on only the poles. (3) For the time-domain function to be made up of only exponential terms that decay in time, all the poles of a transfer function must have negative real parts. This is the key to stability analysis, which will be covered in Chap. 7. 2.7. Summary of Pole Characteristics We now put one and one together. The key is that we can вЂњreadвЂќ the poles вЂ“ telling what the form of the time-domain function is. We should have a pretty good idea from our exercises in partial fractions. Here, the results are provided one more time in general notation. Suppose we have taken a characteristic polynomial, found its roots, and completed the partial-fraction expansion; this is what we expect in the time domain for each of the terms: (A) Real distinct poles: Terms of the form ci /(s в€’ pi ), where the pole pi is a real number, have the time-domain function ci e pi t . Most often, we have a negative real pole such that pi = в€’ai and the time-domain function is ci eв€’ai t . (B) Real poles, repeated m times: Terms of the form ci,2 ci,1 ci,m + , + В·В·В· + 2 (s в€’ pi ) (s в€’ pi ) (s в€’ pi )m with the root pi repeated m times, have the time-domain function ci,1 + ci,2 t + ci,3 2 ci,m t + В·В·В· + t mв€’1 e pi t . 2! (m в€’ 1)! When the pole pi is negative, the decay in time of the entire response will be slower (with respect to only one single pole) because of the terms involving time in the bracket. This is the reason why we say that the response of models with repeated roots (e.g., tanks-in-series in Section 3.4) tends to be slower or вЂњsluggish.вЂќ (C) Complex-conjugate poles: Terms of the form [ci /(s в€’ pi )] + [ciв€— /(s в€’ piв€— )], where pi = О± + jОІ and piв€— = О± в€’ jОІ are the complex poles, have the time-domain funcв€— tion ci e pi t + ciв€— e pi t , which is a form we seldom use. Instead, we rearrange them to give the form (some constant) eО±t sin(ОІt + П†), where П† is the phase lag. It is cumbersome to write the partial fraction with complex numbers. With complex-conjugate poles, we commonly combine the two п¬Ѓrst-order terms into a second-order term. With notation that is introduced formally in Chap. 3, we can write the second-order term as П„ 2s2 as + b , + 2О¶ П„ s + 1 where the coefп¬Ѓcient О¶ is called the damping ratio. To have complex roots in the denominator, we need 0 < О¶ < 1. The complex poles pi and piв€— are now written as pi , piв€— = в€’ 24 О¶ В±j П„ 1 в€’ О¶2 , with 0 < О¶ < 1, П„ 2.7. Summary of Pole Characteristics and the time-domain function is usually rearranged to give the form (some constant)e(в€’О¶ t)/П„ sin 1 в€’ О¶2 t +П† , П„ where, again, П† is the phase lag. (D) Poles on the imaginary axis: If the real part of a complex pole is zero, then p = В±П‰j. We have a purely sinusoidal behavior with frequency П‰. If the pole is zero, it is at the origin and corresponds to the integrator 1/s. In the time domain, we would have a constant, or a step function. (E) If a pole has a negative real part, it is in the left-hand plane (LHP). Conversely, if a pole has a positive real part, it is in the right-hand plane (RHP) and the time-domain solution is deп¬Ѓnitely unstable. Note: Not all poles are born equal! The poles closer to the origin are dominant. It is important to understand and be able to identify dominant poles if they exist. This is a skill that is used later in what is called model reduction. This is a point that we п¬Ѓrst observed in Example 2.6. Consider two terms such that 0 < a1 < a2 (Fig. 2.4): c1 c2 c1 c2 + + В·В·В·+ = + + В·В·В· + (s в€’ p1 ) (s в€’ p2 ) (s + a1 ) (s + a2 ) c2 /a2 c1 /a1 + + В·В·В·. = (П„1 s + 1) (П„2 s + 1) Y (s) = Their corresponding terms in the time domain are y(t) = c1 eв€’a1 t + c2 eв€’a2 t + В· В· В· + = c1 eв€’t/П„1 + c2 eв€’t/П„2 + В· В· В· . As time progresses, the term associated with П„2 (or a2 ) will decay away faster. We consider the term with the larger time constant П„1 as the dominant pole.17 Finally, for a complex pole, we can relate the damping ratio (О¶ < 1) to the angle that the pole makes with the real axis (Fig. 2.5). Taking the absolute values of the dimensions of the triangle, we can п¬Ѓnd Оё = tanв€’1 1 в€’ О¶2 , О¶ (2.33) Im Large a Small П„ Small a Large П„ в€’a 2 в€’a 1 Re Exponential term e decays quickly в€’a 2 t в€’a t Exponential term e 1 decays slowly Figure 2.4. Poles with small and large time constants. 17 Our discussion is valid only if П„1 is вЂњsufп¬ЃcientlyвЂќ larger than П„2 . We could establish a criterion, but at the introductory level, we shall keep this as a qualitative observation. 25 Mathematical Preliminaries p j в€љ1 в€’ О¶2 П„ Оё в€’ О¶/П„ Figure 2.5. Complex pole angular position on the complex plane. and, more simply, Оё = cosв€’1 О¶. (2.34) Equation (2.34) is used in the root-locus method in Chap. 7 when we design controllers. 2.8. Two Transient Model Examples We now use two examples to review how deviation variables relate to the actual ones, and we can go all the way to п¬Ѓnd the solutions. 2.8.1. A Transient-Response Example We routinely test the mixing of continuous-п¬‚ow stirred-tanks (Fig. 2.6) by dumping some kind of inert tracer, say a dye, into the tank and see how they get вЂњmixed up.вЂќ In more dreamy moments, you can try the same thing with cream in your coffee. However, you are a highly paid engineer, and you must take a more technical approach. The solution is simple. We can add the tracer in a well-deп¬Ѓned вЂњmanner,вЂќ monitor the efп¬‚uent, and analyze how the concentration changes with time. In chemical reaction engineering, you will see that the whole business is elaborated into the study of residence time distributions. In this example, we have a stirred-tank with a volume V1 of 4 m3 being operated with an inlet п¬‚ow rate Q of 0.02 m3 /s and that contains an inert species at a concentration Cin of 1 gmol/m3 . To test the mixing behavior, we purposely turn the knob that doses in the tracer and we jack up its concentration to 6 gmol/m3 (without increasing the total п¬‚ow rate) for a duration of 10 s. The effect is a rectangular pulse input (Fig. 2.7). What is the pulse response in the efп¬‚uent? If we do not have the patience of 10 s and dump all the extra tracer in at one shot, what is the impulse response? Q, c in Q, C 1 V1 Figure 2.6. A constantvolume continuous-п¬‚ow well-mixed vessel. 26 2.8. Two Transient Model Examples C in 6 C in 1 s C in 5 1 0 10 time [s] C s in 0 0 10 Figure 2.7. A rectangular pulse in real and deviation variables. The model equation is a continuous-п¬‚ow stirred-tank without any chemical reaction: dC1 = Q(Cin в€’ C1 ). dt In terms of space time П„1 , it is written as V1 dC1 = Cin в€’ C1 , dt where V1 4 = = 200 s. П„1 = Q 0.02 П„1 (2.35) The initial condition is C(0) = C1s , where C1s is the value of the steady-state solution. The inlet concentration is a function of time, Cin = Cin (t), and will become our input. The analytical results are presented here, and the simulations are done with MATLAB in the Review Problems. At steady state, Eq. (2.35) is18 0 = Cins в€’ C1s . (2.36) As suggested in Section 2.1, we deп¬Ѓne the deviation variables C1 = C1 в€’ C1s , Cin = Cin в€’ Cins , and combining Eqs. (2.35) and (2.36) would give us П„1 dC1 = Cin в€’ C1 dt with the zero initial condition C (0) = 0. We further rewrite the equation as П„1 dC1 + C1 = Cin dt (2.37) to emphasize that Cin is the input (or forcing function). The Laplace transform of Eq. (2.37) is C1 (s) 1 = , Cin (s) П„1 s + 1 18 (2.38) At steady state, it is obvious from Eq. (2.35) that the solution must be identical to the inlet concentration. You may п¬Ѓnd it redundant that we add a superscript s to Cins . The action is taken to highlight the particular value of Cin (t) that is needed to maintain the steady state and to make the deп¬Ѓnitions of deviation variables a bit clearer. 27 Mathematical Preliminaries where the RHS is the transfer function. Here, it relates changes in the inlet concentration to changes in the tank concentration. This is a convenient form with which we can address different input cases. Now we have to п¬Ѓt the pieces together for this problem. Before the experiment, that is, at steady state, we must have Cins = C1s = 1. (2.39) Hence the rectangular pulse is really a perturbation in the inlet concentration: пЈ± пЈІ 0, t < 0 Cin = 5, 0 < t < 10 пЈі 0, t > 10. This input can be written succinctly as Cin = 5 [u(t) в€’ u(t в€’ 10)], which then can be applied to Eq. (2.37). Alternatively, we apply the Laplace transform of this input, Cin (s) = 5 (1 в€’ eв€’10s ), s and substitute it into Eq. (2.38) to arrive at C1 (s) = 1 5 (1 в€’ eв€’10s ). (П„1 s + 1) s (2.40) The inverse transform of Eq. (2.40) gives us the time-domain solution for C1 (t): C1 (t) = 5 1 в€’ eв€’t/П„1 в€’ 5 1 в€’ eв€’(tв€’10)/П„1 u(t в€’ 10). The most important time dependence of eв€’t/П„1 arises only from the pole of the transfer function in Eq. (2.38). Again, we can вЂњspell outвЂќ the function if we want to: For t < 10, C1 (t) = 5 1 в€’ eв€’t/П„1 , and for t > 10, C1 (t) = 5 1 в€’ eв€’t/П„1 в€’ 5 1 в€’ eв€’(tв€’10)/П„1 = 5 eв€’(tв€’10)/П„1 в€’ eв€’t/П„1 . In terms of the actual variable, we have, for t < 10, C1 (t) = C1s + C1 = 1 + 5 1 в€’ eв€’t/П„1 , and for t > 10, C1 (t) = 1 + 5 eв€’(tв€’10)/П„1 в€’ eв€’t/П„1 . We now want to use an impulse input of equivalent вЂњstrength,вЂќ i.e., the same amount of inert tracer added. The amount of additional tracer in the rectangular pulse is 5(gmol/m3 ) 0.02(m3 /s)10(s) = 1 gmol, 28 2.8. Two Transient Model Examples which should also be the amount of tracer in the impulse input. Let the impulse input be Cin = MОґ(t). Note that Оґ(t) has the unit of inverse time and M has a funny and physically meaningless unit, and we calculate the magnitude of the input by matching the quantities в€ћ 1 (gmol) = m3 s 0.02 0 M = 50 gmol s m3 M Оґ(t) 1 dt(s) = 0.02M s or gmol s . m3 Thus, Cin (t) = 50Оґ(t), Cin (s) = 50, and, for an impulse input, Eq. (2.38) is simply C1 (s) = 50 . (П„1 s + 1) (2.41) After inverse transform, the solution is C1 (t) = 50 в€’t/П„1 e , П„1 and, in real variables, C1 (t) = 1 + 50 в€’t/П„1 e . П„1 We can do a mass balance based on the outlet в€ћ Q 0 C1 (t) dt = 0.02 50 П„1 в€ћ eв€’t/П„1 dt = 1 (gmol). 0 Hence mass is conserved and the mathematics is correct. We now raise a second question. If the outlet of the vessel is fed to a second tank with a volume V2 of 3 m3 (Fig. 2.8), what is the time response at the exit of the second tank? With the second tank, the mass balance is П„2 V2 dC2 = (C1 в€’ C2 ), where П„2 = , dt Q П„2 dC2 + C2 = C1 , dt or (2.42) Q, c in c1 c2 V1 V2 Figure 2.8. Two well-mixed vessels in series. 29 Mathematical Preliminaries where C1 and C2 are the concentrations in tanks one and two, respectively. The equation analogous to Eq. (2.37) is dC2 + C2 = C1 , dt and its Laplace transform is П„2 C2 (s) = 1 C (s). П„2 s + 1 1 (2.43) (2.44) With the rectangular pulse to the п¬Ѓrst vessel, we use the response in Eq. (2.40) and substitute into Eq. (2.44) to give C2 (s) = 5(1 в€’ eв€’10s ) . s(П„1 s + 1)(П„2 s + 1) With the impulse input, we use the impulse response in Eq. (2.41) instead, and Eq. (2.44) becomes 50 C2 (s) = , (П„1 s + 1)(П„2 s + 1) from which C2 (t) can be obtained by means of a proper look-up table. The numerical values 4 3 = 200 s and П„2 = = 150 s П„1 = 0.02 00.2 can be used. We will skip the inverse transform. It is not always instructive to continue with an algebraic mess. To sketch the time response, we will do so with MATLAB in the Review Problems. 2.8.2. A Stirred-Tank Heater Temperature control in a stirred-tank heater is a common example (Fig. 2.9). We will come across it many times in later chapters. For now, the basic model equation is presented, and we use it as a review of transfer functions. The heat balance, in standard heat transfer notation, is dT = ПЃC p Q(Ti в€’ T ) + U A(TH в€’ T ), (2.45) dt where U is the overall heat transfer coefп¬Ѓcient, A is the heat transfer area, ПЃ is the п¬‚uid density, C p is the heat capacity, and V is the volume of the vessel. The inlet temperature Ti = Ti (t) and steam-coil temperature TH = TH (t) are functions of time and are presumably given. The initial condition is T (0) = T s , the steady-state temperature. ПЃC p V Q, Tin Heating fluid, TH V Q, T Figure 2.9. A continuous-п¬‚ow stirred-tank heater. 30 2.8. Two Transient Model Examples Before we go on, it should be emphasized that what we subsequently п¬Ѓnd are nothing but different algebraic manipulations of the same heat balance. First, we rearrange Eq. (2.45) to give V Q dT UA = (Ti в€’ T ) + (TH в€’ T ). dt ПЃC p Q The second step is to deп¬Ѓne П„= V , Q Оє= UA , ПЃC p Q which leads to dT П„ + (1 + Оє)T = Ti + Оє TH . dt At steady state, (2.46) (1 + Оє)T s = Tis + Оє THs . (2.47) We now deп¬Ѓne the deviation variables: T = T в€’ T s, Ti = Ti в€’ Tis , TH = TH в€’ THs , dT d(T в€’ T s ) dT = = . dt dt dt Subtracting Eq. (2.47) from transient equation (2.46) would give dT + (1 + Оє)(T в€’ T s ) = Ti в€’ Tis + Оє TH в€’ THs dt or, in deviation variables, П„ П„ dT + (1 + Оє)T = Ti + Оє TH . dt (2.48) The initial condition is T (0) = 0. Equation (2.48) is identical in form to Eq. (2.46). This is typical of linear equations. Once you understand the steps, you can jump from Eq. (2.46) to (2.48), skipping over the formality. From here on, we omit the prime ( ) where it would not cause confusion, as it goes without saying that we work with deviation variables. We now further rewrite the same equation as dT + aT = K i Ti + K H TH , dt where 1 Оє (1 + Оє) , Ki = , K H = . a= П„ П„ П„ A Laplace transform gives us sT (s) + aT (s) = K i Ti (s) + K H TH (s). (2.48a) (2.49) Hence, Eq. (2.48a) becomes T (s) = Ki s+a Ti (s) + KH s+a TH (s) = G d (s)Ti (s) + G p (s)TH (s), (2.49a) 31 Mathematical Preliminaries where Ki KH , G p (s) = . s+a s+a Of course, G d (s) and G p (s) are the transfer functions, and they are in pole-zero form. Once again (!), we are working with deviation variables. The interpretation is that changes in the inlet temperature and the steam temperature lead to changes in the tank temperature. The effects of the inputs are additive and mediated by the two transfer functions. Are there more manipulated forms of the same old heat balance? You bet. In fact, we very often rearrange Eq. (2.48), without the primes, as G d (s) = П„p dT + T = K d Ti + K p TH , dt (2.48b) where19 П„p = 1 П„ , = a (1 + Оє) Kd = Ki 1 , = a (1 + Оє) Kp = KH Оє . = a (1 + Оє) After Laplace transform, the model equation is T (s) = G d (s)Ti (s) + G p (s)TH (s), (2.49b) which is identical to Eq. (2.49a) except that the transfer functions are in the time-constant form: G d (s) = Kd , П„ps + 1 G p (s) = Kp . П„ps + 1 In these rearrangements, П„ p is the process time constant and K d and K p are the steady-state gains.20 The denominators of the transfer functions are identical; they both are from the LHS of the differential equation вЂ“ the characteristic polynomial that governs the inherent dynamic characteristic of the process. Let us try one simple example. If we keep the inlet temperature constant at our desired steady state, the statements in deviation variables (without the prime) are Ti (t) = 0, Ti (s) = 0. Now we want to know what happens if the steam temperature increases by 10 в—¦ C. The changes in the deviation variables are TH = Mu(t), TH (s) = (M/s), where M = 10 в—¦ C. We can write Kp M T (s) = . П„ps + 1 s 19 20 32 (2.50) If the heater is well designed, Оє ( = UA/ПЃC p Q) should be much larger than 1. The steady-state gain K p approaches unity, which means that changing the steam temperature is an effective means of changing the tank temperature. In contrast, K d is very small, and the tank temperature is insensitive to changes in the inlet temperature. At п¬Ѓrst reading, you may п¬Ѓnd the notation confusing вЂ“ and in some ways this was done on purpose. This is as bad as it gets once you understand the different rearrangements. So go through each step slowly. K i and K H in Eq. (2.49a) are referred to as gains, but not the steady-state gains. The process time constant is also called a п¬Ѓrst-order lag or linear lag. 2.9. Linearization of Nonlinear Equations 1 T 0.8 MKp 0.632 0.6 0.4 0.2 П„ p = 1.5 0 0 2 4 6 t 8 10 Figure 2.10. Illustration of a п¬Ѓrst-order response [Eq. (2.51)] normalized by MK p . The curve is plotted with П„ p = 1.5 (arbitrary time unit). At t = П„ p , the normalized response is 63.2%. After partial fraction expansion, T (s) = MK p П„p 1 в€’ . s П„ps + 1 An inverse transform by means of a look-up table gives our time-domain solution for the deviation in T 21 : T (t) = MK p 1 в€’ eв€’t/П„ p . (2.51) Keep a mental imprint of the shape of this п¬Ѓrst-order step response as shown in Fig. 2.10. As time progresses, the exponential term decays away, and the temperature approaches the new value MK p . Also illustrated in the п¬Ѓgure is the much-used property that, at t = П„ p , the normalized response is 63.2%. After this exercise, letвЂ™s hope that we have a better appreciation of the different forms of a transfer function. With one, it is easier to identify the pole positions. With the other, it is easier to extract the steady-state gain and time constants. It is very important for us to learn how to interpret qualitatively the dynamic response from the pole positions and to make a physical interpretation with the help of quantities like steady-state gains and time constants. 2.9. Linearization of Nonlinear Equations Because a Laplace transform can be applied to only a linear differential equation, we must вЂњп¬ЃxвЂќ a nonlinear equation. The goal of control is to keep a process running at a speciп¬Ѓed condition (the steady state). For the most part, if we do a good job, the system should be only slightly perturbed from the steady state such that the dynamics of returning to the steady state is a п¬Ѓrst-order decay, i.e., a linear process. This is the cornerstone of classical control theory. What we do is a freshmen calculus exercise in п¬Ѓrst-order Taylor series expansion about the steady state and reformulate the problem in terms of deviation variables. This is illustrated 21 Note that if we had chosen also TH = 0, T (t) = 0 for all t, i.e., nothing happens. Recall once again from Section 2.1 that this is a result of how we deп¬Ѓne a problem by using deviation variables. 33 Mathematical Preliminaries with one simple example. Consider the differential equation that models the liquid level h in a tank with cross-sectional area A: A dh = Q in (t) в€’ ОІh 1/2 . dt (2.52) The initial condition is h(0) = h s , the steady-state value. The inlet п¬‚ow rate Q in is a function of time. The outlet is modeled as a nonlinear function of the liquid level. Both the tank cross-section A and the coefп¬Ѓcient ОІ are constants. We next expand the nonlinear term about the steady-state value h s (also our initial condition by choice) to provide22 A dh 1 в€’1/2 (h в€’ h s ) . = Q in в€’ ОІ h 1/2 s + hs dt 2 (2.53) At steady state, we can write Eq. (2.52) as 0 = Q sin в€’ ОІh 1/2 s , (2.54) where h s is the steady-state solution and Q sin is the particular value of Q in to maintain steady state. If we subtract the steady-state equation from the linearized differential equation, we have A dh 1 в€’1/2 (h в€’ h s ) . = Q in в€’ Q sin в€’ ОІ h dt 2 s (2.55) We now deп¬Ѓne the deviation variables: h = h в€’ hs , Q in = Q in в€’ Q sin . Substituting them into the linearized equation and moving the h term to the left should give A dh + dt ОІ в€’1/2 h = Q in (t), h 2 s (2.56) with the zero initial condition h (0) = 0. It is important to note that the initial condition in Eq. (2.52) has to be h s , the original steady-state level. Otherwise, we will not obtain a zero initial condition in Eq. (2.56). On the other hand, because of the zero initial condition, the forcing function Q in must be п¬Ѓnite to have a nontrivial solution. Repeating our mantra for the umpteenth time, we п¬Ѓnd that the LHS of Eq. (2.56) gives rise to the characteristic polynomial and describes the inherent dynamics. The actual response is subject to the inherent dynamics and the input that we impose on the RHS. 22 We casually ignore the possibility of a more accurate second-order expansion. That is because the higher-order terms are nonlinear, and we need a linear approximation. Needless to say that, with a п¬Ѓrst-order expansion, it is acceptable only if h is sufп¬Ѓciently close to h s . In case you forgot, the п¬Ѓrst-order Taylor series expansion can be written as f (x1 , x2 ) в‰€ f (x1s , x2s ) + в€‚ f /в€‚ x1 |x1s ,x2s (x1 в€’ x1s ) + в€‚ f /в€‚ x2 |x1s ,x2s (x2 в€’ x2s ). 34 2.9. Linearization of Nonlinear Equations Note: r Always do the linearization before you introduce the deviation variables. r As soon as we п¬Ѓnish the п¬Ѓrst-order Taylor series expansion, the equation is linearized. All steps that follow are to clean up the algebra with the understanding that terms of the steady-state equation should cancel out and to change the equation to deviation variables with zero initial condition. A more general description is now provided. Consider an ODE, dy = f (y; u), with y(0) = ys , dt (2.57) where u = u(t) contains other parameters that may vary with time. If f (y; u) is nonlinear, we approximate with TaylorвЂ™s expansion: в€‚f dy в‰€ f (ys ; us ) + dt в€‚y ys ,us (y в€’ ys ) + в€‡ T f (ys ; us )(u в€’ us ), (2.58) where в€‡ f (ys ; us ) is a column vector of the partial derivatives of the function with respect to elements in u, в€‚ f /в€‚u i , and evaluated at ys and us . At steady state, Eq. (2.57) is 0 = f (ys ; us ), (2.59) where ys is the steady-state solution and us are the values of parameters needed to maintain the steady state. We again deп¬Ѓne the deviation variables, y = y в€’ ys , u = u в€’ us , and subtract Eq. (2.59) from Eq. (2.58) to obtain the linearized equation with zero initial condition: в€‚f dy + в€’ dt в€‚y ys ,us y = [в€‡ T f (ys ; us )]u , (2.60) where we have put quantities being evaluated at steady-state conditions in brackets. When we solve a particular problem, they are just constant coefп¬Ѓcients after the substitution of numerical values. Example 2.11: Linearize the differential equation for the concentration in a mixed vessel; [V (dC/dt)] = Q in (t)Cin (t) в€’ Q in (t)C, where the п¬‚ow rate and the inlet concentration are functions of time. A п¬Ѓrst-term Taylor expansion of the RHS leads to the approximation V dC в‰€ [Q in,s Cin,s + Cin,s (Q in в€’ Q in,s ) + Q in,s (Cin в€’ Cin,s )] dt в€’ [Q in,s Cs + Cs (Q in в€’ Q in,s ) + Q in,s (C в€’ Cs )], and the steady-state equation, without canceling the п¬‚ow variable, is 0 = Q in,s Cin,s в€’ Q in,s Cs . 35 Mathematical Preliminaries We subtract the two equations and at the same time introduce deviation variables for the dependent variable C and all the parametric variables to obtain V dC в‰€ [Cin,s Q in + Q in,s Cin ] в€’ [Cs Q in + Q in,s C ], dt and, after moving the C term to the LHS, we obtain V dC + [Q in,s ]C = [Cin,s в€’ Cs ]Q in + [Q in,s ]Cin . dt The п¬Ѓnal result can be interpreted as stating how changes in the п¬‚ow rate and the inlet concentration lead to changes in the tank concentration, as modeled by the dynamics on the LHS. Again, we put the constant coefп¬Ѓcients evaluated at steady-state conditions in brackets. We can arrive at this result quickly if we understand Eq. (2.60) and apply it carefully. The п¬Ѓnal step also has zero initial condition C (0) = 0, and we can take the Laplace transform to obtain the transfer functions if they are requested. As a habit, we can deп¬Ѓne П„ = V /Q in,s and the transfer functions will be in the time-constant form. Example 2.12: Linearize the differential equation (dy/dt) = в€’x y в€’ ОІy 2 в€’ Оі yв€’1 , where x = x(t). Each nonlinear term can be approximated as x y в‰€ xs ys + ys (x в€’ xs ) + xs (y в€’ ys ) = xs ys + ys x + xs y , y 2 в‰€ ys2 + 2ys (y в€’ ys ) = ys2 + 2ys y , Оі yв€’1 в‰€ Оі ys в€’1 + (ln Оі )Оі ys в€’1 y . With the steady-state equation, 0 = xs ys + ОІys + Оі ys в€’1 , and the usual algebraic work, we arrive at (dy /dt) + [xs + 2ОІys + (ln Оі )Оі ys в€’1 ]y = в€’ys x . Example 2.13: What is the linearized form of the reaction rate term r A = в€’k(T )C A = в€’k0 eв€’E/RT C A , where both temperature T and concentration C A are functions of time? r A в‰€ в€’ k0 eв€’E/RTs C A,s + k0 eв€’E/RTs (C A в€’ C A,s ) + E RTs2 k0 eв€’E/RTs C A,s (T в€’ Ts ) . In terms of deviation variables, the linearized approximation is r A в‰€ r A,s 1 + E 1 C + T , C A,s A RTs2 where r A,s = в€’k0 eв€’E/RTs C A,s . Note: Although our analyses use deviation variables and not the real variables, examples and homework problems can keep bouncing back and forth. The reason is that, when we do 36 2.10. Block-Diagram Reduction (a) GL + в€’ Gc Ga Gp + (b) + в€’ G + Gm G CL Figure 2.11. (a) Example of a feedback-system block diagram, (b) typical reduced block diagrams. an experiment, we measure the actual variable, not the deviation variable. You may п¬Ѓnd this really confusing. All we can do is to be extra careful when we solve a problem. 2.10. Block-Diagram Reduction The use of block diagrams to illustrate a cause-and-effect relationship is prevalent in control. We use operational blocks to represent transfer functions and lines for unidirectional information transmission. It is a nice way to visualize the interrelationships of various components. They will be crucial in helping us identify manipulated and controlled variables and input(s) and output(s) of a system. Many control systems are complicated-looking networks of blocks. The simplest control system looks like Fig. 2.11(a). The problem is that many theories in control are based on a simple closed-loop or single-block structure [Fig. 2.11(b)]. Hence we must learn how to read a block diagram and reduce it to the simplest possible form. We will learn in later chapters how the diagram is related to an actual physical system. First, we do some simple algebraic manipulation and, better yet, do it graphically. It is important to remember that all (graphical) block-diagram reduction is a result of formal algebraic manipulation of transfer functions. When all imagination fails, always refer back to the actual algebraic equations.23 Of all manipulations, the most important one is the reduction of a feedback loop. Here is the so-called block-diagram reduction and corresponding algebra. For a negative-feedback system (Fig. 2.12), we have E = R в€’ H Y, (2.61) Y = G E. (2.62) Using Eq. (2.61) to substitute for E in Eq. (2.62) leads to Y = G[R в€’ H Y ], which can be rearranged to give, for a negative-feedback loop,24 G Y = . R 1 + GH 23 24 (2.63) See the Web Support for a comment on MasonвЂ™s gain formula. Similarly, we can write for the case of positive feedback that E = R + H Y and Y = G(R + H Y ), and we have instead (Y/R) = {[G(s)]/[1 в€’ G(s)H (s)]}. 37 Mathematical Preliminaries (a) R + Y E G в€’ H (b) R Y G 1 + GH Figure 2.12. Simple negativefeedback loop and its reduced single closed-loop transfer function form. The RHS of Eq. (2.63) is what will be referred to as the closed-loop transfer function in later chapters. Note: The important observation is that when we вЂњcloseвЂќ a negative-feedback loop, the numerator consists of the product of all the transfer functions along the forward path. The denominator is 1 plus the product of all the transfer functions in the entire feedback loop (i.e., both forward and feedback paths). The denominator is also the characteristic polynomial of the closed-loop system. If we have positive feedback, the sign in the denominator is minus. Here, we try several examples and take the conservative route of writing out the relevant algebraic relations.25 Example 2.14: Derive the closed-loop transfer functions C/R and C/L for the system shown in Fig. E2.14. We identify two locations after the summing points with lower case e and a to help us.26 We can write at the summing point below H , a = в€’C + K R, and substitute this relation into the equation for the summing point above H to give e = R + H a = R + H (K R в€’ C). L R e + + K Gp + C + H a + в€’ Figure E2.14. 25 26 38 A few more simple examples are in the Web Support. How do we decide the proper locations? We do not know for sure, but what should help is after a summing point where information has changed. We may also use the location before a branch-off point, helping us to trace where the information is routed. 2.10. Block-Diagram Reduction We substitute the relation for e into the equation about G p to obtain C = L + G p e = L + G p (R + H K R в€’ H C). The п¬Ѓnal result is a rearrangement to get C out explicitly: 1 1 + GpH C= G p (1 + H K ) 1 + GpH L+ R. Example 2.15: Derive the closed-loop transfer function C/R for the system with three overlapping negative-feedback loops in Fig. E2.15(a). The key to this problem is to proceed in steps and вЂњuntieвЂќ the overlapping loops п¬Ѓrst. We identify various locations with lowercase a, b, d, f , and g to help us. We п¬Ѓrst move the branch-off point over to the RHS of G 4 [Fig. E2.15(b)]. We may note that we can write a = H1 G 3 d = H1 (G 3 G 4 ) d, G4 that is, to maintain the same information at the location a, we must divide the branch-off information at C by G 4 . Similarly, we note that at the position g in Fig. E2.15(a), g = G 1 (R в€’ f ) в€’ bH1 = G 1 R в€’ f в€’ bH1 , G1 that is, if we move the break-in point from g out to the left of G 1 , we need to divide the information by G 1 before breaking in. The block diagram after both the branch-off and the break-in points are moved are shown as Steps 1 and 2 in Fig. E2.15(b). (We could have drawn it such that the loops are п¬‚ush with one another at R.) a (a) R + в€’ G1 в€’ + g G2 H1 f + b d в€’ G3 G4 C H2 (b) Steps 1 and 2 a в€’ + R в€’ + G1 g G2 H1 G 1G 4 f + d в€’ G3 G4 C H2 (c) Step 3 в€’ R + H1 G1G4 G 1G 2 G 3G 4 1 + G 1G 2 1 + H 2 G 3G 4 C Figure E2.15. 39 Mathematical Preliminaries U K (a) X3 + в€’ + + 1 s X2 X1 1 s 2О¶П‰ П‰2 U + в€’ 1 s + 2О¶П‰ 1 s X1 (b) П‰2 Figure E2.16. Once the loops are no longer overlapping, the block diagram is easy to handle. We п¬Ѓrst close the two small loops, shown as Step 3 in Fig. E2.15(c). The п¬Ѓnal result is to close the big loop. The resulting closed-loop transfer function is G1G2G3G4 C = . R (1 + G 1 G 2 )(1 + H2 G 3 G 4 ) + H1 G 2 G 3 Example 2.16: Derive the closed-loop transfer function X 1 /U for the block diagram in Fig. E2.16(a). We will see this one again in Chap. 4 on state-space models. With the integrator 1/s, X 2 is the Laplace transform of the time derivative of x1 (t), and X 3 is the second-order derivative of x1 (t). We can write the algebraic equations about the summing point (the comparator) for X 3 and the two equations about the two integrators 1/s. We should arrive at the result after eliminating X 2 and X 3 . However, we can also obtain the result quickly by recognizing the two feedback loops. We п¬Ѓrst вЂњcloseвЂќ the inner loop to arrive at Fig. E2.16(b). With that, we can see the answer. We вЂњcloseвЂќ this loop to obtain X1 1 = 2 . U s + 2О¶ П‰s + П‰2 Review Problems (1) Derive Eq. (2.1). (2) (a) Check that when the RHS is zero, the solution to Eq. (2.2) is zero. (b) Derive Eq. (2.3) by using the method of integrating factor. (c) Derive the solution c(t) in Eq. (2.3) with an impulse input, Cin = Оґ(t), and a unit-step input, Cin = u(t). Show that they are identical when we use the Laplace transform technique, as in Example 2.10. (3) Prove the linear property of the Laplace transform in Eq. (2.6). (4) Derive the Laplace transforms of (a) 1/(П„ s + 1), (b) cos П‰t, (c) eв€’at cos П‰t. (5) Prove the initial-value theorem. (6) Show that the inverse transform of F(s) = [6/(s 3 + s 2 в€’ 4s в€’ 4)] is f (t) = в€’2eв€’t + 32 eв€’2t + 12 e2t . 40 2.10. Block-Diagram Reduction (7) Double check О± в€— in the complex root of Example 2.7 with the Heaviside expansion. (8) Find the inverse Laplace transform of the general expression Y (s) = {[c/(s в€’ p)] + [cв€— /(s в€’ p в€— ]}, where c = a в€’ bj and p = О± + П‰j. Rearrange the result to a sine function with time lag. (9) With respect to the repeated root of Example 2.9, show that we could have written 2 = О±1 (s + 1)2 (s + 2) + О±2 (s + 1)(s + 2) + О±3 (s + 2) + О±4 (s + 1)3 , and that after expanding and collecting terms of the same power in s, we can form the matrix equation пЈ® пЈ№пЈ® пЈ№ пЈ® пЈ№ 0 1 0 0 1 О±1 пЈЇ 4 1 0 3 пЈє пЈЇ О±2 пЈє пЈЇ 0 пЈє пЈЇ пЈєпЈЇ пЈє пЈЇ пЈє пЈ° 5 3 1 3 пЈ» пЈ° О±3 пЈ» = пЈ° 0 пЈ» 2 2 2 2 1 О±4 (10) (11) (12) (13) (14) (15) from which we can solve for the coefп¬Ѓcients. Yes, this is a chore not worth doing even with MATLAB. The route that we take in the example is far quicker. For a general transfer function G(s) = Y (s)/ X (s), how can we п¬Ѓnd the steady-state gain? Do the partial fractions of {(s + 1)/[s 2 (s 2 + 4s в€’ 5)]}. Find the transfer functions as suggested in Example 2.11. Derive Eqs. (3.33) and (3.34). Do the numerical simulation for Subsection 2.8.1. Regarding Eqs. (2.50) and (2.51) in Subsection 2.8.2: (a) What is T (t) as t в†’ в€ћ? What is the actual temperature that we measure? (b) What are the effects of K p on the п¬Ѓnal tank temperature? What is the signiп¬Ѓcance if K p approaches unity? (c) What is the time constant for the process? Hints: (1) Equation (2.1) is straight from material balance. With Q denoting the volumetric п¬‚ow rate and V the volume, the balance equation of some species A as denoted by C with reference to Fig. 2.6 is V dC = QCin в€’ QC. dt Physically, each term in the equation represents {Accumulation} = {Flow in} в€’ {Flow out}. Equation (2.1) is immediately obvious with the deп¬Ѓnition of space time П„ = V /Q. (2) (c) With the impulse input, C (t) = П„1 eв€’t/П„ , and with the unit-step input, C (t) = 1 в€’ eв€’t/П„ . (3) This is just a simple matter of substituting the deп¬Ѓnition of Laplace transform. (4) Answers are in the Laplace transform summary table. The key to (a) is to rearrange the function as (1/П„ )/(s + 1/П„ ), and the result should then be immediately obvious 41 Mathematical Preliminaries with Eq. (2.9). The derivations of (b) and (c) are very similar to the case involving sin П‰t. (5) The starting point is again the Laplace transform of a derivative, but this time we take the limit as s в†’ в€ћ and the entire LHS with the integral becomes zero. (6) This follows Examples 2.4 and 2.5. (7) О±в€— = s+5 [s в€’ (в€’2 + 3 j)] = s=в€’2в€’3 j (1 в€’ j) 1 (в€’2 в€’ 3 j) + 5 = = (1 + j). (в€’2 в€’ 3 j) + 2 в€’ 3 j в€’2 j 2 (8) We should have в€— y(t) = c e pt + cв€— e p t = (a в€’ bj)e(О±+П‰j)t + (a + bj)e(О±в€’П‰j)t . We now do the expansion: y(t) = eО±t [(a в€’ bj)(cos П‰t + j sin П‰t) + (a + bj)(cos П‰t в€’ j sin П‰t)]. After cancellation, y(t) = 2eО±t (a cos П‰t + b sin П‰t), which of course is y(t) = 2eО±t A sin(П‰t + П†), (10) (11) (12) (13) (14) where A = (a 2 + b2 )1/2 , and П† = tanв€’1 (a/b) Use X = 1/s and the п¬Ѓnal-value theorem. This is what we did in Eq. (2.32). Note that y(в€ћ) is not the same as g(в€ћ). Use MATLAB to check your algebra. With the time constant deп¬Ѓned as П„ = V /Q in,s , the steady-state gain for the transfer function for the inlet п¬‚ow rate is rate is (Cin,s в€’ Cs )/Q in,s , and it is 1 for the inlet concentration transfer function. To п¬Ѓnd tan Оё is easy. To п¬Ѓnd cos Оё, we need to show that the distance from the pole p to the origin (the hypotenuse) is 1/П„ . These are the statements that you can use: tau1=200; G1=tf(1,[tau1 1]); pulselength=10; delt=5; t=0:delt:1000; u=zeros(size(t)); % Transfer function of the first vessel % % % % % Generate a vector to represent the rectangular pulse (We are jumping ahead. These steps are explained in MATLAB Session 3.) u(1:pulselength/delt+1)=5; lsim(G1,u,t); % The response of the rectangular pulse hold y=50*impulse(G1,t); % Add on the impulse response plot(t,y) 42 2.10. Block-Diagram Reduction tau2=150; G2=tf(1,[tau2 1]); G=G1*G2; lsim(G,u,t) y=50*impulse(G,t); plot(t,y) % Generate the transfer function for % both vessels % Add the responses (15) (a) As t в†’ в€ћ, the deviation in T (t) should be 10K p в—¦ C. We have to know the original steady-state temperature in order to calculate the actual temperature. (b) Because K p < 1, the п¬Ѓnal change in temperature will not be 10 в—¦ C. However, if K p = 1, the п¬Ѓnal change will be 10 в—¦ C. We can make this statement without doing the inverse transform. (c) The time constant of the stirred-tank heater is П„ p , not the space time. 43 3 Dynamic Response e now derive the time-domain solutions of п¬Ѓrst- and second-order differential equations. It is not that we want to do the inverse transform, but comparing the time-domain solution with its Laplace transform helps our learning process. What we hope to establish is a better feel between pole positions and dynamic characteristics. We also want to see how different parameters affect the time-domain solution. The results are useful in control analysis and in measuring model parameters. At the end of the chapter, dead time, the reduced-order model, and the effect of zeros are discussed. W What Are We Up to? r Even as we speak of a time-domain analysis, we invariably still work with a r r r r Laplace transform. Time domain and Laplace domain are inseparable in classical control. In establishing the relationship between time domain and Laplace domain, we use only п¬Ѓrst- and second-order differential equations. That is because we are working strictly with linearized systems. As we have seen in partial-fraction expansion, any function can be вЂњbroken upвЂќ into п¬Ѓrst-order terms. Terms of complex roots can be combined together to form a second-order term. Repeated roots (of multicapacity processes) lead to a sluggish response. Tanks-in-series is a good example in this respect. With higher-order models, we can construct approximate reduced-order models based on the identiп¬Ѓcation of dominant poles. This approach is used in empirical controller tuning relations in Chap. 6. The dead-time transfer function has to be handled differently in classical control, and we use the PadВґe approximation for this purpose. A brief review is in order: Recall that the Laplace transform is a linear operator. The effects of individual inputs can be superimposed to form the output. In other words, an observed output change can be attributed to the individual effects of the inputs. From the stirred-tank heater example in Subsection 2.8.2 we found T (s) = G d (s)Ti (s) + G p (s)TH (s). 44 3.1. First-Order Differential Equation Models We can analyze the change in tank temperature as a result of individual changes in either inlet or steam temperatures without doing the inverse transform. The compromise is that we do not have the time-domain analytical solution, T (t), and cannot as easily think of time. We can put the example in more general terms. LetвЂ™s consider an nth-order differential equation and two forcing functions, x1 (t) and x2 (t), dn y dnв€’1 y dy + a + В· В· В· + a1 + a0 y = b1 x1 (t) + b2 x2 (t), nв€’1 n nв€’1 dt dt dt where y is the output deviation variable. We also have the zero initial conditions, an (3.1) y(0) = y (0) = y (0) = В· В· В· = y (nв€’1) (0) = 0. (3.2) The Laplace transform of Eq. (3.1) leads to Y (s) = G 1 (s)X 1 (s) + G 2 (s)X 2 (s), (3.3) where b1 , an s n + anв€’1 s nв€’1 + В· В· В· + a1 s + a0 b2 G 2 (s) = an s n + anв€’1 s nв€’1 + В· В· В· + a1 s + a0 G 1 (s) = (3.4) are the two transfer functions for the two inputs X 1 (s) and X 2 (s), respectively. Take note (again!) that the characteristic polynomials in the denominators of both transfer functions are identical. The roots of the characteristic polynomial (the poles) are independent of the inputs. It is obvious, because they come from the same differential equation (same process or system). The poles tell us what the time-domain solution, y(t), generally would вЂњlookвЂќ like. A п¬Ѓnal reminder: No matter how high the order of n may be in Eqs. (3.4), we can always use partial fractions to break up the transfer functions into п¬Ѓrst- and second-order terms. 3.1. First-Order Differential Equation Models This section is a review of the properties of a п¬Ѓrst-order differential equation model. Our Chap. 2 examples of mixed vessels, stirred-tank heater, and homework problems of isothermal stirred-tank chemical reactors all fall into this category. Furthermore, the differential equation may represent either a process or a control system. What we cover here applies to any problem or situation as long as it can be described by a linear п¬Ѓrst-order differential equation. We usually try to identify features that are characteristic of a model. Using the examples in Section 2.8 as a guide, we can write a п¬Ѓrst-order model by using deviation variables with one input and with constant coefп¬Ѓcients a1 , a0 , and b in general notation as1 dy + a0 y = bx(t) a1 (3.5) dt with a1 = 0 and y(0) = 0. The model, as in Eq. (2.2), is rearranged as dy + y = K x(t), (3.6) П„ dt where П„ is the time constant and K is the steady-state gain. 1 Whether the notation is y or y is immaterial. The key is to п¬Ѓnd the initial condition of the problem statement. If the initial condition is zero, the notation must refer to a deviation variable. 45 Dynamic Response In the event that we are modeling a process, we would use a subscript p (П„ = П„ p , K = K p ). Similarly, the parameters would be the system time constant and system steady-state gain when we analyze a control system. To avoid confusion, we may use a different subscript for a system. The Laplace transform of Eq. (3.6) is K Y (s) = G(s) = , X (s) П„s + 1 (3.7) where G(s) denotes the transfer function. There is one real pole at в€’1/П„ . (What does it imply in terms of the time-domain function? If you are stuck, refer back to Example 2.10.) 3.1.1. Step Response of a First-Order Model Consider a step input, x(t) = Mu(t), and X (s) = M/s; the output is Y (s) = 1 П„ K M = MK в€’ , (П„ s + 1) s s (П„ s + 1) (3.8) and the inverse transform gives the solution y(t) = M K 1 в€’ eв€’t/П„ . (3.9) We п¬Ѓrst saw a plot of this function in Fig. 2.10. The output y(t) starts at zero and increases exponentially to a new steady-state M K . A process with this property is called self-regulating. The larger the time constant, the slower the response [Fig. 3.1(a)]. We can check the result with the п¬Ѓnal-value theorem: lim [sY (s)] = lim s sв†’0 sв†’0 MK = M K. s(П„ s + 1) The new steady state is not changed by a magnitude of M, but is scaled by the gain K [Fig. 3.1(b)]. Consequently we can consider the steady-state gain as the ratio of the observed change in output in response to a unit change in an input, y/M. In fact, this is how we measure K . The larger the steady-state gain, the more sensitive the output to changes in the input. As noted in Fig. 2.10, at t = П„ , y(t) = 0.632 M K . This is a result that we often use to estimate the time constant from experimental data. y 1.2 y (a) MK 1 M 2 (b) K=2 1.5 0.8 0.6 increasing П„ 0.4 K = 0.5 0.5 0.2 0 0 K=1 1 2 4 6 t 8 10 0 0 2 4 6 t 8 10 Figure 3.1. Properties of a п¬Ѓrst-order transfer function in time domain. (a) y/M K : effect of changing the time constant, plotted with П„ = 0.25, 0.5, 1, and 2 (arbitrary time unit); (b) y/M: effect of changing the steady-state gain; all curves have П„ = 1.5. 46 3.1. First-Order Differential Equation Models 3.1.2. Impulse Response of a First-Order Model Consider an impulse input, x(t) = MОґ(t), and X (s) = M; the output is now Y (s) = MK . (П„ s + 1) (3.10) The time-domain solution, as in Example 2.10, is y(t) = M K в€’t/П„ , e П„ (3.11) which implies that the output rises instantaneously to some value at t = 0 and then decays exponentially to zero. 3.1.3. Integrating Process When the coefп¬Ѓcient a0 = 0 in differential equation (3.5), we have dy = dt b a1 x(t), (3.12) Y (s) KI = G(s) = , X (s) s (3.13) where K I = (b/a1 ). Here, the pole of the transfer function G(s) is at the origin, s = 0. The solution of Eq. (3.12), which we could have written immediately without any transform, is t y(t) = K I x(t) dt. (3.14) 0 This is called an integrating (also capacitive or non-self-regulating) process. We can associate the name with charging a capacitor or п¬Ѓlling up a tank. We can show that, with a step input, the output is a ramp function. When we have an impulse input, the output will not return to the original steady-state value, but will accumulate whatever we have added. (Both items are exercises in the Review Problems.) Example 3.1: Show that a storage tank with pumps at its inlet and outlet (Fig. E3.1) is an integrating process. At constant density, the mass balance of a continuous-п¬‚ow mixed tank is simply A dh = qin в€’ q, dt with h(0) = h s , where A is the cross section and h is the liquid level. The inlet and the outlet п¬‚ow rates, qin and q, respectively, as dictated by the pumps, are functions of time but not of the liquid q in q Figure E3.1. 47 Dynamic Response level. At steady state, the п¬‚ow rates must be the same. Thus we can deп¬Ѓne deviation variables h = h в€’ h s , qin = qin в€’ qs , and q = q в€’ qs , and the mass balance becomes dh = qin в€’ q , with h (0) = 0. dt The general solution is A h (t) = 1 A t 0 (qin в€’ q ) dt, where the change in the liquid level is a simple time integral on the change in п¬‚ow rates. In terms of a Laplace transform, the differential equation leads to H (s) = 1 Q in (s) в€’ Q(s) . A s The transfer function has the distinct feature that a pole is at the origin. Because a step input in either qin or q would lead to a ramp response in h , there is no steady-state gain at all. To better observe how the tank works like a capacitor, assume that the inlet п¬‚ow rate is constant and that we have a pump only at the outlet. The transfer function is now just H (s) = в€’Q(s) . As If for some reason the outlet pump slows down, the liquid level in the tank will back up until it overп¬‚ows. Similarly, if the outlet pump speeds up, the tank will be drained. The tank level will not reach a new steady state with respect to a step change in the input. 3.2. Second-Order Differential Equation Models We have not encountered examples with a second-order equation, especially one that exhibits oscillatory behavior. One reason is that processing equipment tends to be self-regulating. An oscillatory behavior is most often the result of implementing a controller, and we shall see that in the control chapters. For now, this section provides several important deп¬Ѓnitions. A model that leads to a second-order differential equation, a2 d2 y dy + a1 + a0 y = b x(t), a2 = 0, 2 dt dt y(0) = y (0) = 0, (3.15) is usually rearranged to take the forms П„2 dy d2 y + 2О¶ П„ + y = K x(t), or dt 2 dt dy d2 y + 2О¶ П‰n + П‰n2 y = K П‰n2 x(t), dt 2 dt (3.16) where П„2 = a2 , a0 2О¶ П„ = a1 b , K = , a0 a0 П‰n = 1 . П„ The corresponding Laplace transforms are G(s) = 48 Y (s) K = 2 2 X (s) П„ s + 2О¶ П„ s + 1 or X (s) K П‰n2 = 2 , X (s) s + 2О¶ П‰n s + П‰n2 (3.17) 3.2. Second-Order Differential Equation Models where П„ is the natural period of oscillation, П‰n is the natural (undamped) frequency, О¶ is the damping ratio (also called the damping coefп¬Ѓcient or factor), and K is the steady-state gain. The characteristic polynomial is p(s) = П„ 2 s 2 + 2О¶ П„ s + 1 = (s в€’ p1 )(s в€’ p2 ), (3.18) which provides the poles О¶ 4О¶ 2 П„ 2 в€’ 4П„ 2 О¶2 в€’ 1 = в€’ В± . (3.19) 2П„ 2 П„ П„ A stable process (or system) requires О¶ > 0 because we need П„ > 0 to be physically meaningful. In addition, the two pole positions, and thus time response, take on four possibilities, depending on the value of О¶ : p1,2 = в€’2О¶ П„ В± (1) О¶ > 1: Two distinct real poles. This case is called overdamped. Here, we can factor the polynomial in terms of two time constants, П„1 and П„2 , G(s) = П„ 2s2 K K = , + 2О¶ П„ s + 1 (П„1 s + 1)(П„2 s + 1) (3.20) such that the two real poles are at в€’1/П„1 and в€’1/П„2 .2 (2) О¶ = 1: Two repeating poles at в€’1/П„ . This case is termed critically damped. The natural period П„ may be considered the вЂњtime constantвЂќ in the sense that it is associated with the exponential function. In actual fact, the time response is not strictly exponential, as we saw in Example 2.9 and conп¬Ѓrmed in the time-domain solution in Eq. (3.22). (3) 0 < О¶ < 1: Two complex-conjugate poles. This situation is considered underdamped. We also write О¶ 2 в€’ 1 = j 1 в€’ О¶ 2 . It is very important to note that П„ is not the time constant here. The real part of the pole in Eq. (3.19) is в€’О¶ /П„ , and this is the value that determines the exponential decay, as in Eq. (3.23). In this sense, the time constant is П„/О¶ . (4) О¶ = 0: Two purely imaginary conjugate poles with frequency П‰n = 1/П„ . This is equivalent to an oscillation with no damping and explains why П‰n is referred to as the natural frequency. 3.2.1. Step-Response Time-Domain Solutions Consider a step input, x(t) = Mu(t), with X (s) = M/s, and the different cases with respect to the value of О¶ . We can derive the output response y(t) for the different cases. We rarely 2 Here, we can п¬Ѓnd that П„ 2 = П„1 П„2 , 2О¶ П„ = (П„1 + П„2 ) or П„= в€љ П„1 П„2 , П„1 + П„2 О¶ = в€љ . 2 П„1 П„2 In this case of having real poles, we can also relate П„ П„ П„1 = , П„2 = . О¶ в€’ О¶2 в€’ 1 О¶ + О¶2 в€’ 1 49 Dynamic Response use these results. They are provided for reference. In the case of the underdamped solution, it is used to derive the characteristic features in the next subsection. (1) О¶ > 1, overdamped: The response is sluggish compared with that of critically damped or underdamped cases: О¶2 в€’ 1 t+ П„ y(t) = M K 1 в€’ eв€’О¶ t/П„ cosh О¶ О¶2 в€’ 1 sinh О¶2 в€’ 1 t П„ . (3.21) This form is unnecessarily complicated. When we have an overdamped response, we typically use the simple exponential form with the exp(в€’t/П„1 ) and exp(в€’t/П„2 ) terms. (YouвЂ™ll get to try this in the Review Problems.) (2) О¶ = 1, critically damped: The response is the вЂњfastestвЂќ without oscillatory behavior: t П„ y(t) = M K 1 в€’ 1 + eв€’t/П„ . (3.22) (3) 0 в‰¤ О¶ < 1, underdamped: The response is fast initially, then overshoots and decays to steady state with oscillations. The oscillations are more pronounced and persist longer with smaller О¶ : 1 в€’ О¶2 t П„ y(t) = M K 1 в€’ eв€’О¶ t/П„ cos + О¶ 1 в€’ О¶2 sin 1 в€’ О¶2 t П„ . (3.23) This equation can be rearranged as eв€’О¶ t/П„ y(t) = M K 1 в€’ where П† = tan в€’1 в€љ 1 в€’ О¶2 1в€’О¶ 2 О¶ 1 в€’ О¶2 t +П† П„ sin , (3.23a) . 3.2.2. Time-Domain Features of Underdamped Step Response From the solution of the underdamped step response (0 < О¶ < 1), we can derive the following characteristics (Fig. 3.2). They are useful in two respects: (1) п¬Ѓtting experimental data in the measurements of natural period and damping factor, and (2) making control system design speciп¬Ѓcations with respect to the dynamic response. (1) Overshoot (OS): OS = A B = exp в€’ПЂ О¶ 1 в€’ О¶2 , (3.24) where A and B are shown in Fig. 3.2. We compute only the п¬Ѓrst or maximum overshoot in the response. The overshoot increases as О¶ becomes smaller. The OS becomes zero as О¶ approaches 1. 50 3.2. Second-Order Differential Equation Models y T MK C A 1 B 0 tr Tp Ts Time, t Figure 3.2. Key features in an underdamped response. See text for equations. The time to reach the peak value is ПЂП„ ПЂ = . Tp = 1 в€’ О¶2 П‰n 1 в€’ О¶ 2 (3.25) This peak time is less as О¶ becomes smaller and is meaningless when О¶ = 1. We can also derive the rise time вЂ“ time for y(t) to cross or hit the п¬Ѓnal value for the п¬Ѓrst time вЂ“ as П„ (ПЂ в€’ cosв€’1 О¶ ). (3.26) tr = 1 в€’ О¶2 (2) Frequency (or period of oscillation) T: П‰= 1 в€’ О¶2 П„ or T = 2ПЂ П„ 1 в€’ О¶2 as П‰ = 2ПЂ . T (3.27) Note that T = 2T p and the unit of the frequency is radian per time unit. (3) Settling time: The real part of a complex pole in Eq. (3.19) is в€’О¶ /П„ , meaning that the exponential function forcing the oscillation to decay to zero is eв€’О¶ t/П„ , as in Eq. (3.23). If we draw an analogy to a п¬Ѓrst-order transfer function, the time constant of an underdamped second-order function is П„/О¶ . Thus, to settle within В±5% of the п¬Ѓnal value, we can choose the settling time as3 Ts = 3 П„ 3 , = О¶ О¶ П‰n (3.28) and if we choose to settle within В±2% of the п¬Ѓnal value, we can use Ts = 4П„/О¶ . (4) Decay ratio (DR): DR = C A = exp в€’2ПЂ О¶ 1в€’ О¶2 = OS2 . (3.29) The DR is the square of the OS, and both quantities are functions of О¶ only. The deп¬Ѓnitions of C and A are shown in Fig. 3.2. 3 Refer to Review Problem (1) to see why we may pick factors of 3 or 4. 51 Dynamic Response 3.3. Processes with Dead Time Many physiochemical processes involve a time delay between the input and the output. This delay may be due to the time required for a slow chemical sensor to respond or for a п¬‚uid to travel down a pipe. A time delay is also called dead time or transport lag. In controller design, the output will not contain the most current information, and systems with dead time can be difп¬Ѓcult to control. From Eq. (2.18) in Section 2.3, the Laplace transform of a time delay is an exponential function. For example, п¬Ѓrst- and second-order models with dead time will appear as Y (s) K eв€’td s = , X (s) П„s + 1 Y (s) K eв€’td s = 2 2 . X (s) П„ s + 2О¶ П„ s + 1 Many classical control techniques are developed to work only with polynomials in s, and we need some way to tackle the exponential function. To handle the time delay, we do not simply expand the exponential function as a Taylor series. We use the so-called PadВґe approximation, which puts the function as a ratio of two polynomials. The simplest is the п¬Ѓrst-order (1/1) PadВґe approximation: eв€’td s в‰€ 1 в€’ t2d s . 1 + t2d s (3.30) This is a form that serves many purposes. The term in the denominator introduces a negative pole in the LHP and thus introduces probable dynamic effects to the characteristic polynomial of a problem. The numerator introduces a positive zero in the RHP, which is needed to make a problem to become unstable. (This point will become clear when we cover Chap. 7.) Finally, the approximation is more accurate than a п¬Ѓrst-order Taylor series expansion.4 There are higher-order approximations. For example, the second-order (2/2) PadВґe approximation is eв€’td s в‰€ td2 s 2 в€’ 6td s + 12 . td2 s 2 + 6td s + 12 (3.31) Again, this form introduces poles in the LHP and at least one zero is in the RHP. At this point, the important task is to observe the properties of the PadВґe approximation in numerical simulations. Example 3.2: Use the п¬Ѓrst-order PadВґe approximation to plot the unit-step response of the п¬Ѓrst order with a dead-time function: eв€’3s Y = . X 10s + 1 4 52 We will skip the algebraic details. The simple idea is that we can do long division of a function of the form in approximation (3.30) and match the terms to a Taylor expansion of the exponential function. If we do, weвЂ™ll п¬Ѓnd that the (1/1) PadВґe approximation is equivalent to a third-order Taylor series. 3.4. Higher-Order Processes and Approximations 1 0.8 0.6 y 0.4 0.2 0 0 10 20 30 t 40 50 Figure E3.2. Making use of approximation (3.30), we can construct a plot with the approximation (в€’1.5s + 1) Y . = X (10s + 1)(1.5s + 1) The result is the thin curve in Fig. E3.2. Note how it dips below zero near t = 0. This behavior has to do with the п¬Ѓrst-order PadВґe approximation, and we can improve the result with a second-order PadВґe approximation. We will try that in the Review Problems. Here the observation is that when compared with the original transfer function (the thick solid curve), the approximation is acceptable at larger times. How did we generate the solid curve? We computed the result for the п¬Ѓrst-order function and then shifted the curve down three time units (td = 3). The MATLAB statements are td=3; P1=tf([-td/2 1],[td/2 1]); % First-order PadГ© approximation t=0:0.5:50; taup=10; G1=tf(1,[taup 1]); y1=step(G1*P1,t); % y1 is first order with PadГ© approximation of % dead time y2=step(G1,t); t2=t+td; % Shift the time axis for the actual time-delay % function plot(t,y1,'--',t2,y2); We now move onto a few so-called higher-order or complex processes. We should remind ourselves that all linearized higher-order systems can be broken down into simple п¬Ѓrst- and second-order units. Other so-called complex processes, like two interacting tanks, are just a math problem in coupled differential equations; these problems are still linear. The following sections serve to underscore these points. 3.4. Higher-Order Processes and Approximations Staged processes in chemical engineering or compartmental models in bioengineering give rise to higher-order models. The higher-order models are due to a cascade of п¬Ѓrst-order elements. Numerical calculation is used to illustrate that the resulting response becomes 53 Dynamic Response more sluggish, thus conп¬Ѓrming our analysis in Example 2.9. We shall also see how the response may be approximated by a lower-order function with dead time. An example of two interacting vessels is given last. 3.4.1. Simple Tanks-in-Series Consider a series of well-mixed vessels (or compartments) in which the volumetric п¬‚ow rate and the respective volumes are constant (Fig. 3.3). If we write the mass balances of the п¬Ѓrst two vessels as in Subection 2.8.1, we have5 dc1 = c0 в€’ c1 , dt dc2 = c1 в€’ c2 , П„2 dt П„1 (3.32) (3.33) where П„1 = V1 /q0 and П„2 = V2 /q0 are the space times of each vessel. Again following Subsection 2.8.1, we п¬Ѓnd that the Laplace transforms of the mass balance in deviation variables would be C1 1 1 C2 = = , . C0 П„1 s + 1 C1 П„2 s + 1 (3.34) The effect of changes in c0 (t) on the efп¬‚uent of the second vessel is evaluated as C2 C1 1 C2 1 . = = C0 C1 C0 (П„2 s + 1) (П„1 s + 1) (3.35) Obviously, we can generalize to a series of n tanks as in 1 Cn = . C0 (П„1 s + 1) В· В· В· (П„nв€’1 s + 1)(П„n s + 1) (3.36) In this example, the steady-state gain is unity, which is intuitively obvious. If we change the color of the inlet with a food dye, all the mixed tanks will have the same color eventually. qo c o c1 c2 V1 V2 c nвЂ“1 qn cn Vn Figure 3.3. Illustration of compartments or tanks-in-series. 5 54 Many texts illustrate with a model on the change of inlet п¬‚ow rate. In such a case, we usually need to assume that the outlet п¬‚ow rate of each vessel is proportional to the liquid level or the hydrostatic head. The steady-state gains will not be unity. 3.4. Higher-Order Processes and Approximations 1.2 y 1 n=1 0.8 0.6 n=5 0.4 0.2 0 0 10 20 t 30 40 Figure E3.3. In addition, the more tanks we have in a series, the longer we have to wait until the nth tank вЂњseesвЂќ the changes that we have made in the п¬Ѓrst one. The more tanks in the series, the more sluggish the response of the overall process. Processes that are products of п¬Ѓrst-order functions are also called multicapacity processes. Finally, if all the tanks have the same space time, П„1 = П„2 = В· В· В· = П„ , Eq. (3.36) becomes 1 Cn = . C0 (П„ s + 1)n (3.37) This particular case is not common in reality, but is a useful textbook illustration. Example 3.3: Making use of Eq. (3.37), show how the unit-step response Cn (t) becomes more sluggish as n increases from 1 to 5. The exercise is almost trivial with MATLAB. To generate Fig. E3.3, the statements are tau=3; G=tf(1,[tau 1]); step(G); hold step(G*G); step(G*G*G); step(G*G*G*G); step(G*G*G*G*G); % Just an arbitrary time constant % First-order function unit-step response % % % % Second-order response Third-order response Fourth-order response Fifth-order response It is clear that, as n increases, the response, as commented in Example 2.9, becomes slower. If we ignore the вЂњdataвЂќ at small times, it appears that the curves might be approximated with п¬Ѓrst-order with dead-time functions. We shall do this exercise in the Review Problems. 3.4.2. Approximation with Lower-Order Functions with Dead Time Following the lead in Example 3.3, we now make use of the result in Example 2.6 and the comments about dominant poles in Section 2.7 to see how we may approximate a transfer function. LetвЂ™s say we have a high-order transfer function that has been factored into partial fractions. If there is a large enough difference in the time constants of individual terms, we may try to throw away the small time-scale terms and retain the ones with dominant poles (large time constants). This is our reduced-order model approximation. From Fig. E3.3, we also need to add a time delay in this approximation. The extreme of this idea is to use a п¬Ѓrst-order 55 Dynamic Response with dead-time function. It obviously cannot do an adequate job in many circumstances. Nevertheless, this simple approximation is all we use when we learn to design controllers with empirical tuning relations. A second-order function with dead time generally provides a better estimate, and this is how we may make a quick approximation. Suppose we have an nth-order process that is broken down into n п¬Ѓrst-order processes in series with time constants П„1 , П„2 , . . . , П„n . If we can identify, say, two dominant time constants (or poles) П„1 and П„2 , we can approximate the process as G(s) в‰€ K eв€’td s , (П„1 s + 1)(П„2 s + 1) (3.38) where n td в‰€ П„i . i=1,2 The summation to estimate the dead time is over all the other time constants (i = 3, 4, etc.). This idea can be extended to the approximation of a п¬Ѓrst-order with dead-time function. Example 3.4: Find the simplest lower-order approximation of the following transfer function 3 Y = . X (0.1s + 1)(0.5s + 1)(s + 1)(3s + 1) In this example, the dominant pole is at в€’1/3, corresponding to the largest time constant at 3 (time unit). Accordingly, we may approximate the full-order function as Y 3eв€’1.6s , = X (3s + 1) where 1.6 is the sum of dead times 0.1, 0.5, and 1. With X representing a unit-step input, the response of the full-order function (solid curve) and that of the п¬Ѓrst-order with deadtime approximation (dotted curve) are shown in Fig. E3.4. The plotting of the dead-time function is further approximated by the PadВґe approximation. Even so, the approximation is reasonable when time is large enough. The pole at в€’1/3 can indeed be considered as dominant. 3 2.5 y 2 1.5 1 0.5 0 -0.5 0 Figure E3.4. 56 6 t 12 18 3.4. Higher-Order Processes and Approximations The MATLAB statements are p2=conv([1 1],[3 1]); p4=conv(conv(p2,[0.5 1]) , [0.1 1]); G4=tf(3,p4); % The original full-order function t=0:0.2:18; y4=step(G4,t); % Unit-step response td=1+0.1+0.5; % Approximate dead time P1=tf([-td/2 1],[td/2 1]); G1=tf(3,[3 1]); y1=step(P1*G1,t); plot(t,y1,t,y4) If we follow approximation (3.38), we should approximate the original function with a second-order function with time constants 1 and 3, and dead time 0.6. We will п¬Ѓnd it to be a much better п¬Ѓt; we will do this in the Review Problems. 3.4.3. Interacting Tanks-in-Series To complete the discussion, the balance equations are included for the case in which two differential equations may be coupled. The common example is two tanks connected such that there is only a valve between them (Fig. 3.4). Thus the п¬‚ow between the two tanks depends on the difference in the hydrostatic heads. With constant density, we can write the mass balance of the п¬Ѓrst tank as A1 dh 1 = q0 в€’ dt h1 в€’ h2 . R1 (3.39) Similarly, for the second vessel we have A2 dh 2 = dt h1 в€’ h2 R1 в€’ h2 . R2 (3.40) Here we model the п¬‚ow through the valves with resistances R1 and R2 , both of which are constants. We rearrange the balance equations a bit, and because both equations are linear, we can quickly rewrite them in deviation variables (without the primes) as П„1 dh 1 = в€’h 1 + h 2 + R1 q0 , dt П„2 R2 R2 dh 2 = h1 в€’ 1 + dt R1 R1 h 1 (0) = 0, h2, h 2 (0) = 0, (3.41) (3.42) qo h1 h2 R1 R2 Figure 3.4. Illustration of two tanks interacting in their liquid levels. 57 Dynamic Response where we have deп¬Ѓned П„1 = A1 R1 and П„2 = A2 R2 . The Laplace transforms of these equations are (П„1 s + 1)H1 в€’ H2 = R1 Q 0 , в€’R2 R2 H1 + П„2 s + 1 + R1 R1 (3.43) H2 = 0. (3.44) We have written the equations in a form that lets us apply CramerвЂ™s rule. The result is H1 = R1 П„2 s + (R1 + R2 ) Q0, p(s) H2 = R2 Q0, p(s) (3.45) where the characteristic polynomial is p(s) = (П„1 s + 1)(П„2 s + 1 + R2 /R1 ) в€’ R2 /R1 . (3.46) We do not need to carry the algebra further. The points to be made are clear. First, even the п¬Ѓrst vessel has a second-order transfer function; it arises from the interaction with the second tank. Second, if we expand Eq. (3.46), we should see that the interaction introduces an extra term in the characteristic polynomial, but the poles should remain real and negative.6 That is, the tank responses remain overdamped. Finally, we may be afraid (!) that the algebra might become hopelessly tangled with more complex models. Indeed, we would prefer to use state-space representation based on Eqs. (3.41) and (3.42). After Chaps. 4 and 9, you can try this problem in Homework Problem II (39). 3.5. Effect of Zeros in Time Response The inherent dynamics is governed by the poles, but the zeros can impart п¬Ѓner вЂњп¬ЃngerprintвЂќ features by modifying the coefп¬Ѓcients of each term in the time-domain solution. That was the point that was attempted to be made with the examples in Section 2.5. Two common illustrations on the effects of zeros are the leadвЂ“lag element and the sum of two functions in parallel. 3.5.1. LeadвЂ“Lag Element The so-called leadвЂ“lag element is a semiproper function with a п¬Ѓrst-order lead divided by a п¬Ѓrst-order lag: П„z s + 1 Y = , (3.47) X П„s + 1 where П„z and П„ are two time constants. We have to wait until the chapters on controllers to see that this function is the basis of a derivative controller and not till the frequency 6 To see this, you need to go one more step to get p(s) = П„1 П„2 s 2 + (П„1 + П„2 + П„1 R2 /R1 )s + 1 and compare the roots of this polynomial with the case with no interaction: p(s) = (П„1 + 1)(П„2 + 1). Note how we have an extra term when the tanks interact. 58 3.5. Effect of Zeros in Time Response 2 4 1.5 y 1 0.5 0 в€’4 -0.5 -1 -1.5 -2 0 2 4 6 8 Time (s) 10 Figure 3.5. Time response of a leadвЂ“ lag element with П„ = 2 s. The curves from top to bottom are plotted with П„z = 4, 3, 2, 1, в€’1, в€’2, and в€’4 s. response chapter to appreciate the terms lead and lag. For now, we take a quick look at its time response. For the case with a unit-step input such that X = 1/s, we have, after partial-fraction expansion, 1 П„z в€’ П„ + , s П„s + 1 and the inverse transform by means of a look-up table yields Y = (3.48) П„z в€’t/П„ e . (3.49) П„ There are several things that we want to take note of. First, the exponential function is dependent on only П„ , or in other words, the pole at в€’1/П„ . Second, with Eq. (3.49), the actual time response depends on whether П„ < П„z , П„ > П„z , or П„z < 0 (Fig. 3.5). Third, when П„ = П„z , the time response is just a horizontal line at y = 1, corresponding to the input x = u(t). This is also obvious from Eq. (3.47), which becomes just Y = X . When a zero equals a pole, we have what is called a pole-zero cancellation. Finally, the value of y is nonzero at time zero. We may wonder how that could be the case when we use differential equation models that have zero initial conditions. The answer has to do with the need for the response to match the rate of change term in the input. We will get a better picture in Chap. 4 when we cover state-space models. y(t) = 1 в€’ 1 в€’ 3.5.2. Transfer Functions in Parallel There are circumstances in which a complex process may involve two competing (i.e., opposing) dynamic effects that have different time constants. One example is the increase in inlet temperature to a tubular catalytic reactor with exothermic kinetics. The initial effect is that the exit temperature will momentarily decrease as increased conversion near the entrance region depletes reactants at the distal, exit end. Given time, however, higher reaction rates lead to higher exit temperatures. To model this highly complex and nonlinear dynamics properly, we need the heat and the mass balances. In classical control, however, we would replace them with a linearized 59 Dynamic Response model that is the sum of two functions in parallel: Y K1 K2 = + . X П„1 s + 1 П„2 s + 1 (3.50) We can combine the two terms to give the second-order function K (П„z s + 1) Y = , X (П„1 s + 1)(П„2 s + 1) (3.51) where K = K1 + K2, П„z = K 1 П„2 + K 2 П„1 . K1 + K2 For circumstances in which the two functions represent opposing effects, one of them has a negative steady-state gain. In the following illustration, we choose to have K 2 < 0. From Eq. (3.51), the time response y(t) should be strictly overdamped. However, this is not necessarily the case if the zero is positive (or П„z < 0). We can show with algebra how various ranges of K i and П„i may lead to different zeros (в€’1/П„z ) and time responses. However, we will not do that. (We will use MATLAB to take a closer look in the Review Problems, though.) The key, once again, is to appreciate the principle of superposition with linear models. Thus we should get a rough idea of the time response simply based on the form in Eq. (3.50). The numerical calculation is illustrated in Fig. 3.6. The input is a unit step, X = 1/s, and the two steady-state gains are K 1 = 3 and K 2 = в€’1 such that |K 1 | > |K 2 |. We consider the three cases in which П„1 is roughly the same as П„2 [Fig. 3.6(a)], П„1 П„2 [Fig. 3.6(b)], and (c) П„1 П„2 [Fig. 3.6(c)]. We should see that the overall response is overdamped in the case of Fig. 3.6(a), but in the case of Fig. 3.6(b) we can have an overshoot and in the case of Fig. 3.6(c) an initial inverse response. Note that all three cases have the same overall steady gain of K = 2.7 3 3 (a) (b) 2.5 2.5 3 2 2 2 1.5 1.5 1.5 y 1 y 1 y 1 0.5 0.5 0.5 0 0 0 -0.5 -0.5 -0.5 -1 0 2 4 t 6 8 10 -1 0 (c) 2.5 2 4 t 6 8 10 -1 0 2 4 t 6 8 10 Figure 3.6. Time-response calculations with different time constants. In all cases, K 1 = 3, K 2 = в€’1, and the individual terms in Eq. (3.50) are indicated by the dashed curves. Their superimposed responses y are the solid curves. (a) П„1 = П„2 = 2; (b) П„1 = 0.5, П„2 = 2; (c) П„1 = 2, П„2 = 0.5. 7 60 When you repeat this exercise with MATLAB in the Review Problems, check that П„z is negative in the case of Fig. 3.6(c). More commonly, we say that this is the case with a positive zero. After we have learned frequency response, we will see that this is an example of what is referred to as a nonminimum phase. 3.5. Effect of Zeros in Time Response Review Problems (1) With respect to the step response of the п¬Ѓrst-order model in Eq. (3.9), make a table for y(t)/M K p , with t/П„ p = 1, 2, 3, 4, and 5. It is helpful to remember the results for t/П„ p = 1, 3, and 5. (2) It is important to understand that the time constant П„ p of a process, say, a stirredtank, is not the same as the space time П„ . Review this point with the stirred-tank heater example in Chap. 2. Further, derive the time constant of a continuous-п¬‚ow stirred-tank reactor (CSTR) with a п¬Ѓrst-order chemical reaction. (3) Write the time-response solutions to the integrating process in Eq. (3.14) when the input is (1) a unit step and (2) an impulse. How are they different from the solutions to a self-regulating process? (4) Derive the time-constant relationships given in footnote 2. (5) With respect to the overdamped solution of second-order equation (3.21), derive the step response y(t) in terms of the more familiar exp(в€’t/П„1 ) and exp(в€’t/П„2 ). This is much easier than Eq. (3.21) and more useful too! (6) Show that when О¶ = 0 (natural period of oscillation, no damping), the process (or system) oscillates with a constant amplitude at the natural frequency П‰n . (The poles are at В±П‰n and the period is 2ПЂ П„ .) (7) Use MATLAB to make plots of OS and DR as functions of the damping ratio. (8) What is the expected time response when the real part of the pole is zero in a second-order function? The pole can be just zero or have purely imaginary parts. (9) Plot the unit-step response by using just п¬Ѓrst- and second-order PadВґe approximations (3.30) and (3.31). Try also the step response of a п¬Ѓrst-order function with dead time as in Example 3.2. Note that, although the approximation to the exponential function itself is not that good, the approximation to the entire transfer function is not as bad as long as td П„ . How do you plot the exact solution in MATLAB? (10) Use MATLAB to observe the effect of higher-order multicapacity models as in Example 3.3. Try to п¬Ѓt the п¬Ѓfth-order case with a п¬Ѓrst-order with dead-time function. (11) With respect to Example 3.4, try also a second-order with dead-time approximation. (12) We do not have a rigorous criterion to determine when a pole is absolutely dominant. Plot the exponential decay with different time constants to get an idea when the terms associated with smaller time constants can be omitted. (13) With MATLAB, try to do a unit-step response of a leadвЂ“lag element as in Eq. (3.49). (14) Repeat the time-response simulation of inverse response in Section 3.5.2. Calculate the value of zero in each case. Hints: (1) y(t)/M K p at t/П„ p = 1, 2, 3, 4, and 5 are 0.63, 0.86, 0.95, 0.98, and 0.99. (2) The mass balance of a CSTR with a п¬Ѓrst-order chemical reaction is very similar to the problem in Subsection 2.8.1. We just need to add the chemical reaction term. The balance written for reactant A will appear as V dC A = q(C0 в€’ C A ) в€’ V kC A , dt where C A is the molar concentration of A, V is the reactor volume, q is the volumetric п¬‚ow rate, C0 is the inlet concentration of A, and k is the п¬Ѓrst-order reaction rate 61 Dynamic Response constant. If we deп¬Ѓne space time П„ = V /q, the equation can be rewritten as П„ (3) (5) (6) (7) (8) (9) dC A + (1 + kП„ )C A = C 0 . dt This is a linear equation if k and П„ are constants. Now if we follow the logic in Subsection 2.8.2, we should п¬Ѓnd that the time constant of a CSTR with a п¬Ѓrst-order reaction is П„/(1 + kП„ ). Part of the answer is already in Example 3.1. This is really an algebraic exercise in partial fractions. The answer hides in Table 2.1. This is obvious from Eq. (3.17) or Eq. (3.19). Plot Eq. (3.29) with 0 < О¶ < 1. You can write a small M-п¬Ѓle to do the plotting too. See Review Problem (6). Follow Example 3.2. For the п¬Ѓrst order approximation, we can try, for example, td=3; % Use an M-file to rerun with different % values P1=tf([-td/2 1],[td/2 1]); step(P1); % Note how the response starts from % negative values t=0:0.1:50; taup=10; G1=tf(1,[taup 1]); y1=step(G1*P1,t); % y1 is first order with PadГ© approx of % dead time y2=step(G1,t); % y2 has no time delay t2=t+td; plot(t,y1, t2,y2,'-.'); % Note how the PadГ© approx has a dip at the beginning (10) The MATLAB statements to do the unit-step response are already in Example 3.4. You may repeat the computation with a different time constant. The statements to attempt п¬Ѓtting the п¬Ѓve tanks-in-series responses are tau=3; G=tf(1,[tau 1]); [y5,t]=step(G*G*G*G*G); G1=tf(1,[12 1]); y1=step(G1,t); t1=t+3; plot(t,y5, t1,y1) % The fifth-order calculation % Using a time shift to do the % first-order with dead-time plot The choice of the time constant and dead time is meant as an illustration. The п¬Ѓt will not be particularly good in this example because there is no one single dominant pole in the п¬Ѓfth-order function with a pole repeated п¬Ѓve times. A п¬Ѓrst-order with dead-time function will never provide a perfect п¬Ѓt. (11) Both п¬Ѓrst- and second-order approximation statements are here: q=3; p2=conv([1 1],[3 1]); 62 % Second-order reduced model 3.5. Effect of Zeros in Time Response p4=conv(conv(p2,[0.5 1]),[0.1 1]); roots(p4) % check G2=tf(q,p2); G4=tf(q,p4); step(G4) hold td=0.1+0.5; P1=tf([-td/2 1],[td/2 1]); step(P1*G2); % Not bad! td=1+0.1+0.5; G1=tf(q,[3 1]); % First-order approximation step(P1*G1); % is not that good in this case hold off (12) Below are the MATLAB statements that we may use for a visual comparison of exponential decay with different time constants. In rough engineering calculations, a pole may already exhibit acceptable dominant behavior if other time constants are 1/3 or less: tau=1; t=0:0.1:5; f=exp(-t/tau); plot(t,f) hold % Now add curves with smaller time constants frac=0.3; f=exp(-t/(frac*tau)); plot(t,f) frac=0.2; f=exp(-t/(frac*tau)); plot(t,f) frac=0.1; f=exp(-t/(frac*tau)); plot(t,f) (13) Try to vary the zero as in tz=3; % Try to vary tz, zero is -1/tz G=tf([tz 1],[5 1]); step(G); (14) Try to vary the values of П„1 and П„2 . To display the value of zero is trivial: k1=3; k2=-1; tau1=2; tau2=0.5; k=k1+k2; tz=(k1*tau2+k2*tau1)/k; G=tf(k*[tz 1], conv([tau1 1],[tau2 1])); step(G); 63 4 State-Space Representation he limitation of transfer function representation becomes obvious as we tackle more complex problems. For complex systems with multiple inputs and outputs, transfer function matrices can become very clumsy. In the so-called modern control, the method of choice is state-space or state variables in the time domain вЂ“ essentially a matrix representation of the model equations. The formulation allows us to make use of theories in linear algebra and differential equations. It is always a mistake to tackle modern control without a п¬Ѓrm background in these mathematical topics. For this reason, we will not overreach by doing both the mathematical background and the control together. Without a formal mathematical framework, the explanation is made by means of examples as much as possible. The actual state-space control has to be delayed until after we tackle classical transfer function feedback systems. T What Are We Up to? r Learning how to write the state-space representation of a model. r Understanding the how a state-space representation is related to the transfer function representation. 4.1. State-Space Models Just as we are feeling comfortable with transfer functions, we now switch gears totally. Nevertheless, we are still working with linearized differential equation models in this chapter. Whether we have a high-order differential equation or multiple equations, we can always rearrange them into a set of п¬Ѓrst-order differential equations. Bold statements indeed! We will see that when we go over the examples. With state-space models, a set of differential equations is put in standard matrix form, 64 xЛ™ = Ax + Bu, (4.1) y = Cx + Du, (4.2) 4.1. State-Space Models where x is the state-variable vector, u is the input, and y is the output. The time derivative is denoted by the overdot. In addition, A is the process (plant) matrix, B is the input matrix, C is the output matrix, and D is the feedthrough matrix. Very few processes (and systems) have an input that has a direct inп¬‚uence on the output. Hence D is usually zero. When we discuss single-input single-output (SISO) models, scalar variables should be used for the input, output, and feedthrough: u, y, and D. For convenience, we keep the notation for B and C, but keep in mind that, in this case, B is a column vector and C is a row vector. If x is of the order of n, then A is n Г— n, B is n Г— 1, and C is 1 Г— n.1 The idea behind the use of Eqs. (4.1) and (4.2) is that we can make use of linear system theories, and we can analyze complex systems much more effectively. There is no unique way to deп¬Ѓne the state variables. What will be shown is just one of many possibilities. Example 4.1: Derive the state-space representation of a second-order differential equation in a form similar to that of Eq. (3.16): d2 y dy + П‰n2 y = K u(t). + 2О¶ П‰n (E4.1) 2 dt dt We can do blindfolded the transfer function of this equation with zero initial conditions: G p (s) = K Y (s) = 2 . U (s) s + 2О¶ П‰n s + П‰n2 (E4.2) Now letвЂ™s do something different. First, we rewrite the differential equation as dy d2 y в€’ П‰n2 y + K u(t), = в€’2О¶ П‰n dt 2 dt and deп¬Ѓne the state variables,2 dx1 , (E4.3) dt which allow us to redeп¬Ѓne the second-order equation as a set of two coupled п¬Ѓrst-order equations. The п¬Ѓrst differential equation is the deп¬Ѓnition of the state variable x2 in Eq. (E4.3); the second is based on the differential equation x1 = y, x2 = dx2 = в€’2О¶ П‰n x2 в€’ П‰n2 x1 + K u(t). dt We now put the result in a matrix equation: xЛ™ 1 xЛ™ 2 = 0 1 в€’П‰n2 в€’2О¶ П‰n x1 0 + u(t). K x2 (E4.4) (E4.5) We further write y = [1 0] 1 2 x1 x2 (E4.6) If you are working with only SISO problems, it would be more appropriate to replace the notation B with b and C with cT , and write d for D. This exercise is identical to how higher-order equations are handled in numerical analysis and would come as no surprise if you have taken a course on numerical methods. 65 State-Space Representation u(t) x 2(t) U K 1 s + в€’ + + x 2(t) X2 x 1 (t) = y(t) 1 s X1 = Y 2О¶П‰ П‰2 Figure E4.2. as a statement that x1 is our output variable. Compare the results with Eqs. (4.1) and (4.2), and we see that, in this case, A= 0 1 , в€’П‰n2 в€’2О¶ П‰n B= 0 , C = [1 0], K D = 0. To п¬Ѓnd the eigenvalues of A, we solve its characteristic equation: |О»I в€’ A| = О»(О» + 2О¶ П‰n ) + П‰n2 = 0. (E4.7) We can use the MATLAB function tf2ss() to convert the transfer function in Eq. (E4.2) to state-space form: z=0.5; wn=1.5; % Pick two sample numbers for О¶ and П‰n p=[1 2*z*wn wn*wn]; [a,b,c,d]=tf2ss(wn*wn,p) However, you will п¬Ѓnd that the MATLAB result is not identical to Eq. (E4.5). It has to do with the fact that there is no unique representation of a state-space model. To avoid unnecessary confusion, the differences with MATLAB are explained in MATLAB Session 4. One important observation that we should make immediately: The characteristic polynomial of the matrix A [Eq. (E4.7)] is identical to that of the transfer function [Eq. (E4.2)]. Needless to say, the eigenvalues of A are the poles of the transfer function. It is a reassuring thought that different mathematical techniques provide the same information. It should come as no surprise if we remember our linear algebra. Example 4.2: Draw the block diagram of the state-space representation of the second-order differential equation in Example 4.1. The result is in Fig. E4.2. It is quite easy to understand if we note that the transfer function of an integrator is 1/s. Thus the second-order derivative is located before the two integrations are made. The information at the summing point also adds up to the terms of the second-order differential equation. The resulting transfer function is identical to Eq. (E4.2). The reduction to a closed-loop transfer function was done in Example 2.16. Example 4.3: LetвЂ™s try another model with a slightly more complex input. Derive the statespace representation of the differential equation d2 y du dy +y= + 3u, + 0.4 2 dt dt dt 66 y(0) = dy/dt(0) = 0, u(0) = 0, 4.1. State-Space Models which has the transfer function Y s+3 = 2 . U s + 0.4s + 1 The method that we will follow is more for illustration than for its generality. LetвЂ™s introduce a variable X 1 between Y and U : Y X1 Y = = U U X1 1 (s + 3). s 2 + 0.4s + 1 The п¬Ѓrst part, X 1 /U , is a simple problem itself: X1 = U s2 1 + 0.4s + 1 is the Laplace transform of d2 x 1 dx1 + x1 = u. + 0.4 dt 2 dt With Example 4.1 as the hint, we deп¬Ѓne the state variables x1 = x1 (i.e., the same), and x2 = dx1 /dt. Using steps similar to those of Example 4.1, we п¬Ѓnd that the result, as equivalent to Eq. (4.1), is xЛ™ 1 xЛ™ 2 = 0 1 в€’1 в€’0.4 x1 0 + u. 1 x2 (E4.8) As for the second part Y / X 1 = (s + 3), it is the Laplace transform of y = (dx1 /dt) + 3x1 . We can use the state variables deп¬Ѓned above to rewrite as y = x2 + 3x1 , (E4.9) or, in matrix form, y = [3 1] x1 , x2 which is the form of Eq. (4.2). With MATLAB, the statements for this example are q=[1 3]; p=[1 0.4 1]; roots(p) [a,b,c,d]=tf2ss(q,p) eig(a) Comments at the end of Example 4.1 also apply here. The result should be correct, and we should п¬Ѓnd that both the roots of the characteristic polynomial p and the eigenvalues of the matrix a are в€’0.2 В± 0.98 j. We can also check by going backward: [q2,p2]=ss2tf(a,b,c,d,1) and the original transfer function is recovered in q2 and p2. 67 State-Space Representation u x + 1 s в€’ x в€’1 + y + 3 Figure E4.4. Example 4.4: Derive the state-space representation of the leadвЂ“lag transfer function: s+2 Y = . U s+3 We follow the hint in Example 4.3 and write the transfer function as XY 1 Y = = s + 2. U U X s+3 From X/U = 1/(s + 3), we have s X = в€’3X + U , or in time domain, dx = в€’3x + u dt and from Y/ X = s + 2, we have Y = s X + 2X and substitution for sX leads to (E4.10) Y = (в€’3X + U ) + 2X = в€’X + U. The corresponding time-domain equation is y = в€’x + u. (E4.11) Thus all the coefп¬Ѓcients in Eqs. (4.1) and (4.2) are scalar, with A = в€’3, B = 1, C = в€’1, and D = 1. Furthermore, (E4.10) and (E4.11) can be represented by the block diagram in Fig. E4.4. We may note that the coefп¬Ѓcient D is not zero, meaning that, with a leadвЂ“lag element, an input can have an instantaneous effect on the output. Thus, although the state variable x has zero initial condition, it is not necessarily so with the output y. This analysis explains the mystery with the inverse transform of this transfer function in Eq. (3.49). The MATLAB statement for this example is [a,b,c,d]=tf2ss([1 2], [1 3]) The next two examples illustrate how state-space models can handle a multiple-input multiple output (MIMO) problem. A simple example shows how to translate information in a block diagram into a state-space model. Some texts rely on signal-п¬‚ow graphs, but we do not need them with simple systems. Moreover, we can handle complex problems easily with MATLAB. Go over MATLAB Session 4 before reading Example 4.7A. Example 4.5: Derive the state space representation of two continuous-п¬‚ow stirred-tank reactors in series (CSTR-in-series). Chemical reaction is п¬Ѓrst order in both reactors. The reactor volumes are п¬Ѓxed, but the volumetric п¬‚ow rate and inlet concentration are functions of time. This example is used to illustrate how state-space representation can handle complex models. First, we use the solution to Review Problem 2 in Chap. 3 and write the mass 68 4.1. State-Space Models balances of reactant A in chemical reactors 1 and 2: dc1 = q(c0 в€’ c1 ) в€’ V1 k1 c1 , dt dc2 = q(c1 в€’ c2 ) в€’ V2 k2 c2 . V2 dt (E4.12) V1 (E4.13) Because q and c0 are input functions, the linearized equations in deviation variables and with zero initial conditions are (with all primes omitted in the notation) dc1 = qs c0 + (c0s в€’ c1s )q в€’ (qs + V1 k1 )c1 dt dc2 V2 = qs c1 + (c1s в€’ c2s )q в€’ (qs + V2 k2 )c2 . dt V1 (E4.14) (E4.15) The missing steps are similar to the steps that we did Example 2.11. Dividing the equations by the respective reactor volumes and deп¬Ѓning space times П„1 = V1 /qs and П„2 = V2 /qs , we obtain 1 dc1 = c0 + dt П„1 c0s в€’ c1s qв€’ V1 1 + k 1 c1 , П„1 (E4.16) 1 dc2 = c1 + dt П„2 c1s в€’ c2s qв€’ V2 1 + k 2 c2 . П„2 (E4.17) Up to this point, the exercise is identical to what we learned in Chap. 2. In fact, we can now take the Laplace transform of these two equations to derive the transfer functions. In state-space models, however, we would put the two linearized equations in matrix form. Analogous to Eq. (4.1), we now have пЈ№ пЈ® пЈ® пЈ№ 1 1 c0s в€’ c1s в€’ 0 + k1 пЈє пЈЇ пЈЇ П„1 пЈє c0 П„1 d c1 V1 пЈє c1 пЈЇ пЈє =пЈЇ +пЈЇ . (E4.18) пЈє пЈ° c1s в€’ c2s пЈ» q пЈ» c2 пЈ° dt c2 1 1 0 в€’ + k2 V2 П„2 П„2 The output y in Eq. (4.2) can be deп¬Ѓned as y1 y2 = 1 0 0 1 c1 c2 = c1 c2 (E4.19) if we are to use two outputs. In SISO problems, we likely would measure and control only c2 , and hence we would deп¬Ѓne instead y = [0 1] c1 c2 (E4.20) with c2 as the only output variable. Example 4.6: Derive the transfer function Y/U and the corresponding state-space model of the block diagram in Fig. E4.6. 69 State-Space Representation U X2 + 2 s+1 в€’ X3 X1= Y 1 s 1 s + 10 Figure E4.6. From block-diagram reduction in Chap. 2, we can easily spot that Y = U 1+ 2 s(s+1) 2 s(s+1)(s+10) , which is reduced to 2(s + 10) Y = 3 . U s + 11s 2 + 10s + 2 (E4.21) This transfer function has closed-loop poles at в€’0.29, в€’0.69, and в€’10.02. (Of course, we computed them by using MATLAB.) To derive the state-space representation, one visual approach is to identify locations in the block diagram where we can assign state variables and write out the individual transfer functions. In this example, we have chosen to use (Fig. E4.6) X2 X3 1 2 1 X1 , , = , = = X2 s U в€’ X3 s + 1 X1 s + 10 Y = X 1; the п¬Ѓnal equation is the output equation. We can now rearrange each of the three transfer functions from which we write their time-domain equivalent: s X 1 = X 2, s X 2 = в€’X 2 в€’ 2X 3 + 2U, s X 3 = в€’10X 3 + X 1 , dx1 = x2 ; dt dx2 = в€’x2 в€’ 2x3 + 2u; dt dx3 = x1 в€’ 10x3 . dt The rest is trivial. We rewrite the differential equations in matrix form as пЈ№пЈ® пЈ№ пЈ® пЈ№ пЈ® пЈ№ пЈ® x1 x 0 1 0 0 d пЈ° 1пЈ» пЈ° x2 = 0 в€’1 в€’2 пЈ» пЈ° x2 пЈ» + пЈ° 2 пЈ»u, dt 1 0 в€’10 0 x3 x3 пЈ® пЈ№ x1 y = [1 0 0] пЈ° x2 пЈ». x3 (E4.22a) (E4.22b) (E4.22c) (E4.23) (E4.24) We can check with MATLAB that the model matrix A has eigenvalues в€’0.29, в€’0.69, and в€’10.02. They are identical to the closed-loop poles. Given a block diagram, MATLAB can put the state-space model together for us easily. To do that, we need to learn some closed-loop MATLAB functions; we will leave this illustration to MATLAB Session 5. 70 4.2. Relation of State-Space Models to Transfer Function Models An important reminder: Eq. (E4.23) has zero initial conditions x(0) = 0. This is a direct consequence of deriving the state-space representation from transfer functions. Otherwise, Eq. (4.1) is not subjected to this restriction. 4.2. Relation of State-Space Models to Transfer Function Models From Example 4.6, we may see why the primary mathematical tools in modern control are based on linear system theories and time-domain analysis. Part of the confusion in learning these more advanced techniques is that the umbilical cord to Laplace transform is not entirely severed, and we need to appreciate the link between the two approaches. On the bright side, if we can convert a state-space model to transfer function form, we can still make use of classical control techniques. A couple of examples in Chap. 9 will illustrate how classical and state-space techniques can work together. We can take the Laplace transform of matrix equation (4.1) to give sX(s) = AX(s) + BU (s), (4.3) where the capital X does not indicate that it is a matrix, but rather is used in keeping with our notation of Laplace variables. From Eq. (4.3), we can extract X explicitly as X(s) = (sI в€’ A)в€’1 BU (s) = О¦(s)BU (s), (4.4) where О¦(s) = (sI в€’ A)в€’1 (4.5) is the resolvent matrix. More commonly, we refer to the state-transition matrix (also called the fundamental matrix), which is its inverse transform: О¦(t) = Lв€’1 [(sI в€’ A)в€’1 ]. (4.6) We use the same notation, О¦, for the time function and its Laplace transform, and we add the t or s dependence only when it is not clear in which domain the notation is used. Setting D = 0 and X(s) as derived in Eq. (4.4), we п¬Ѓnd that the output Y (s) = CX(s) becomes Y (s) = CО¦(s)BU (s). (4.7) In this case, in which U and Y are scalar quantities, CО¦B must also be scalar.3 In fact, if we make an association between Eq. (4.7) and what we have learned in Chap. 2, CО¦B is our ubiquitous transfer function. We can rewrite Eq. (4.7) as Y (s) = G p (s)U (s), (4.7a) where G p (s) = CО¦(s)B. (4.8) Hence we can view the transfer function as how the Laplace transform of the state-transition matrix О¦ mediates the input B and the output C matrices. We may wonder how this output 3 From Eq. (4.5), we see that О¦ is an n Г— n matrix. Because B is n Г— 1 and C is 1 Г— n, CО¦B must be 1 Г— 1. 71 State-Space Representation equation is tied to the matrix A. With linear algebra, we can rewrite the deп¬Ѓnition of О¦ in Eq. (4.5) as О¦(s) = (sI в€’ A)в€’1 = adj(sI в€’ A) . det(sI в€’ A) (4.5a) Substitution of this form in Eq. (4.8) provides a more informative view of the transfer function: G p (s) = C[adj(sI в€’ A)]B . det(sI в€’ A) (4.8a) The characteristic polynomial clearly is det(sI в€’ A) = 0. (4.9) This is the result that we have arrived at, albeit less formally, in Example 4.1. Again, the poles of G p are identical to the eigenvalues of the model matrix A. Example 4.7: The results in this section are illustrated with a numerical version of Example 4.5. Consider again two CSTR-in-series, with V1 = 1 m3 , V2 = 2 m3 , k1 = 1 minв€’1 , k2 = 2 minв€’1 , and, initially at steady state, П„1 = 0.25 min, П„2 = 0.5 min, and inlet concentration c0s = 1 kmol/m3 . Derive the transfer functions and the state-transition matrix in which both c0 and q are input functions. With the steady-state form of Eqs. (E4.12) and (E4.13), we can calculate c1s = c0s 1 = = 0.8, 1 + k1 П„1 1 + 0.25 c2s = c1s 0.8 = = 0.4. 1 + k2 П„2 1 + 2(0.5) In addition, we п¬Ѓnd that 1/П„1 = 4 minв€’1 , 1/П„2 = 2 minв€’1 , (1/П„1 + k1 ) = 5 minв€’1 , (1/П„2 + k2 ) = 4 minв€’1 , (c0s в€’ c1s )/V1 = 0.2 kmol/m6 , and (c1s в€’ c2s )/V2 = 0.2 kmol/m6 . We substitute these numerical values into Eqs. (E4.16) and (E4.17) and take the Laplace transform of these equations to obtain (for more general algebraic result, we should take the transform п¬Ѓrst) C1 (s) = 4 0.2 C0 (s) + Q(s), s+5 s+5 C2 (s) = 2 0.2 C1 (s) + Q(s). s+4 s+4 (E4.25) Further substitution for C1 (s) with Eq. (E4.25) into C2 (s) gives C2 (s) = 8 0.2(s + 7) C0 (s) + Q(s) (s + 4)(s + 5) (s + 4)(s + 5) (E4.26) Equations (E4.25) and (E4.26) provide the transfer functions relating changes in п¬‚ow rate Q and inlet concentration C0 to changes in the two tank concentrations. With the state-space model, the substitution of numerical values into Eq. (E4.18) leads to the dynamic equation d dt 72 c1 c2 = в€’5 0 2 в€’4 c1 4 0.2 + 0 0.2 c2 c0 . q (E4.27) 4.2. Relation of State-Space Models to Transfer Function Models With the model matrix A, we can derive (sI в€’ A) = s+5 0 , в€’2 s + 4 О¦(s) = (sI в€’ A)в€’1 = 1 s+4 0 . 2 s + 5 (s + 5)(s + 4) (E4.28) We consider Eq. (E4.19) in which both concentrations c1 and c2 are outputs of the model. The transfer function in Eq. (4.7) is now a matrix: G p (s) = CО¦(s)B = 1 s+4 0 2 s + 5 (s + 5)(s + 4) 4 0.2 , 0 0.2 (E4.29) where C is omitted as it is just the identity matrix [Eq. (E4.19)].4 With input u(s) = [C0 (s) Q(s)]T , we can write output equation (4.6) as 1 4(s + 4) 0.2(s + 4) C1 (s) = CО¦(s)Bu(s) = 8 0.2(s + 7) C2 (s) (s + 5)(s + 4) C0 (s) . Q(s) (E4.30) This is how MATLAB returns the results. We can, of course, clean up the algebra to give пЈ№ пЈ® 4 0.2 пЈє C0 (s) пЈЇ (s + 5) (s + 5) C1 (s) пЈє =пЈЇ , (4.30a) пЈ° C2 (s) 8 0.2(s + 7) пЈ» Q(s) (s + 5)(s + 4) (s + 5)(s + 4) which is identical to what we have obtained earlier in Eqs. (E4.25) and (E4.26). The case of only one output, as in Eq. (E4.20), is easy, and that is covered in Example 4.7A. To wrap things up, we can take the inverse transform of Eq. (E4.30a) to get the timedomain solutions: c1 c2 = 0.2eв€’5t 4eв€’5t 8(eв€’4t в€’ eв€’5t ) 0.2(3eв€’4t в€’ 2eв€’5t ) c0 . q (E4.31) Example 4.7A: Repeat Example 4.7 using MATLAB. If you understand the equations in Example 4.7, you are ready to tackle the same problem with MATLAB. t1=0.25; t2=0.5; k1=1; k2=2; V1=1; V2=2; cos=1; % Define the variables % Calculate the steady-state values. MATLAB should return c1s=cos/(1+k1*t1); % 0.8 c2s=c1s/(1+k2*t2); % 0.4 4 Be careful with the notation. Uppercase italic C with subscripts 1 and 2 denotes concentration in the Laplace domain. The boldface uppercase C is the output matrix. 73 State-Space Representation a11=-(1/t1+k1); a12=0; a21=1/t2; a22=-(1/t2+k2); b11=1/t1; b12=(cos-c1s)/V1; b21=0; b22=(c1s-c2s)/V2; a=[a11 a12; a21 a22]; b=[b11 b12; b21 b22]; eig(a) c=[1 0; 0 1]; % Coefficients of A and B using (E4.18) % -5 % -4 % 0.2 % 0.2 % % % % Finally build A and B in (E4.27) [-5 0; 2 4] [4 0.2; 0 0.2] Check that they are -4, -5 % Define C such that both C1 and C2 are % outputs d=[0 0; 0 0]; With all the coefп¬Ѓcient matrices deп¬Ѓned, we can do the conversion to transfer function. The function ss2tf() works with only one designated input variable. Thus for the п¬Ѓrst input variable C0 we use [q1,p]=ss2tf(a,b,c,d,1) % MATLAB returns, for input no. 1 % q1=[0 4 16; 0 0 8] % p =[1 9 20] = (s+4)(s+5) The returned vector p is obviously the characteristic polynomial. The matrix q1 is really the п¬Ѓrst column of the transfer function matrix in Eq. (E4.30), denoting the two terms describing the effects of changes in C0 on C1 and C2 . Similarly, the second column of the transfer function matrix in Eq. (E4.30) is associated with changes in the second input Q and can be obtained with [q2,p]=ss2tf(a,b,c,d,2) % q2=[0 .2 .8; 0 .2 1.4] % The first term is 0.2(s+4) because % MATLAB retains p=(s+4)(s+5) If C2 is the only output variable, we deп¬Ѓne C according to output equation (E4.20). Matrices A and B remain the same. The respective effects of changes of C0 and Q on C2 can be obtained with c=[0 1]; d=[0 0]; % C2 is the only output [q21,p]=ss2tf(a,b,c,d,1) % Co as input; q21=[0 0 8] [q22,p]=ss2tf(a,b,c,d,2) % Q as input; q22=[0 .2 1.4] We should п¬Ѓnd that the result is the same as the second row of Eq. (E4.30), denoting the two terms describing the effects of changes in C0 and Q on C2 . 74 4.2. Relation of State-Space Models to Transfer Function Models Similarly, if C1 is the only output variable, we use instead c=[1 0]; d=[0 0]; % C1 is the only output [q11,p]=ss2tf(a,b,c,d,1) % q11=[0 4 16] [q12,p]=ss2tf(a,b,c,d,2) % q12=[0 .2 .8] and the result is the п¬Ѓrst row of (E4.30). Example 4.8: Develop a fermentor model that consists of two mass balances, one for the cell mass (or yeast), C1 , and the other for glucose (or substrate), C2 . We have to forget about the alcohol for now. The cell mass balance [Eq. (E4.32)] has two terms on the RHS. The п¬Ѓrst one describes cell growth by use of the speciп¬Ѓc growth rate Вµ = Вµ(C2 ). The second term accounts for the loss of cells that is due to outlet п¬‚ow Q, which in turn is buried inside the notation D, the dilution rate: dC1 = ВµC1 в€’ DC 1 . dt (E4.32) The speciп¬Ѓc growth rate and dilution rate are deп¬Ѓned as Вµ = Вµ(C2 ) = Вµm C2 , K m + C2 D= Q . V The glucose balance has three terms on the RHS. The п¬Ѓrst accounts for the consumption by the cells. The last two terms account for the п¬‚ow of glucose into and out of the fermentor: dC2 ВµC1 =в€’ + D (C20 в€’ C2 ) dt Y (E4.33) The maximum speciп¬Ѓc growth rate Вµm , Monod constant K m , and cell yield coefп¬Ѓcient Y are constants. In Eq. (E4.33), C20 is the inlet glucose concentration. The dilution rate D is dependent on the volumetric п¬‚ow rate Q and the volume V and really is the reciprocal of the space time of the fermentor. In this problem, the fermentor volume V is п¬Ѓxed, and we vary the п¬‚ow rate Q. Hence it is more logical to use D (and easier to think) as it is proportional to Q. Our question is to formulate this model under two circumstances: (1) when we vary only the dilution rate, and (2) when we vary both the dilution rate and the amount of glucose input. We also derive the transfer function model in the second case. In both cases, C1 and C2 are the two outputs. To solve this problem, we obviously have to linearize the equations. In vector form, the nonlinear model is dx = f (x, D), dt (E4.34) where x = [C1 C2 ]T and пЈ№ [Вµ(C2 ) в€’ D]C1 пЈ». = пЈ° Вµ(C2 ) C1 + D(C20 в€’ C2 ) в€’ Y пЈ® f (x, D) = f 1 (x, D) f 2 (x, D) (E4.35) 75 State-Space Representation We п¬Ѓrst take the inlet glucose, C20 , to be a constant (i.e., no disturbance) and vary only the dilution rate D. From the steady-state form of Eqs. (E4.32) and (E4.33), we can derive (without special notation for the steady-state values) ВµC1 , C1 = Y (C20 в€’ C2 ). (E4.36) Y Now we linearize the two equations about the steady state. We expect to arrive at (with the primes dropped from all the deviation variables and partial derivatives evaluated at the steady state) пЈ№ пЈ® пЈ® пЈ№ в€‚ f1 в€‚ f1 в€‚ f1 пЈє пЈЇ пЈЇ в€‚D пЈє d C1 пЈЇ в€‚C в€‚C2 пЈє C1 пЈє (E4.37) =пЈЇ 1 +пЈЇ пЈє пЈ° в€‚ f 2 пЈ» D. пЈ° в€‚ f 2 в€‚ f 2 пЈ» C2 dt C2 в€‚ D ss в€‚C1 в€‚C2 ss D(C20 в€’ C2 ) = Using Eq. (E4.35) to evaluate the partial derivatives, we should п¬Ѓnd пЈ№ пЈ№ пЈ® пЈ® 0 C1 Вµ в€’C1 d C1 C 1 пЈ» =пЈ° Вµ + пЈ° C1 пЈ» D = Ax + BD, C1 C2 dt C2 в€’ Вµ в€’Вµ в€’ Y Y Y ss ss where Вµ is the derivative with respect to the substrate C2 : Вµ = dВµ Km = Вµm . dC2 (K m + C2 )2 (E4.38) (E4.39) All the coefп¬Ѓcients in A and B are evaluated at steady-state conditions. From Eq. (E4.32), D = Вµ at steady state. Hence the coefп¬Ѓcient a11 in A is zero. To complete the state-space model, we п¬Ѓnd that the output equation is C1 C2 = 1 0 0 1 C1 . C2 (E4.40) In this case, the output matrix C is just an identity matrix. Now, we will see what happens with two inputs. In practice, we most likely would make a highly concentrated glucose stock and dose it into a main feed stream that contains the other ingredients. What we manipulate is the dosage rate. Consider that the glucose feedstock has a п¬Ѓxed concentration C2 f and an adjustable feed rate q f , and the other nutrients are being fed at a rate of q0 . The effective glucose feed concentration is C20 = q f C2 f q f C2 f , = q f + q0 Q (E4.41) where Q = q f + q0 is the total inlet п¬‚ow rate and the dilution rate is q f + q0 Q = . V V The general fermentor model equation as equivalent to Eq. (E4.34) is D= (E4.42) dx = f (x, u), dt (E4.43) where the state-space remains x = [C1 C2 ]T , but the input is the vector u = [D0 D f ]T . 76 4.2. Relation of State-Space Models to Transfer Function Models Here, D0 = q0 /V and D f = q f /V are the respective dilution rates associated with the two inlet streams. That is, we vary the main nutrient feed and glucose dosage п¬‚ow rates to manipulate this system. The function f is пЈ® f 1 (x, u) f 2 (x, u) f (x, u) = =пЈ° Вµ(C2 )C1 в€’ (D0 + D f )C1 в€’ пЈ№ пЈ». Вµ(C2 ) C1 + D f C2 f в€’ (D0 + D f )C2 Y (E4.44) At steady state, Вµ = (D0 + D f ) = D, C1 = в€— Y (C20 (E4.45) в€’ C2 ), (E4.46) where в€— = C20 D f C2 f . D0 + D f The equations linearized about the steady state [with the primes dropped from the deviation variables as in Eq. (E4.38)] are d dt пЈ® пЈ№ 0 C1 Вµ в€’C1 C1 пЈ» C1 + в€’C1 =пЈ° Вµ C1 C2 C в€’C2 (C2 f в€’ C2 ) 2 в€’ в€’ Вµ в€’Вµ Y Y ss ss D0 = Ax + Bu. Df (E4.47) The output equation remains the same as in Eq. (E4.40). Laplace transform of the model equations and rearrangement lead us to C1 C2 = G 11 G 12 G 21 G 22 D0 , Df (E4.48) where the four open-loop plant transfer functions are G 11 = G 12 = G 21 = G 22 = в€’ C1 Y ss s в€’ C1 C1 Вµ Y + Вµ + C1 Вµ C2 ss p(s) в€’ C1 Y ss s + C1 Вµ (C2 f в€’ C2 ) в€’ C1 p(s) в€’(C2 )ss s + C1 Вµ Y ss p(s) (C2 f в€’ C2 )ss s + p(s) , C1 Вµ Y ss C1 Вµ Y , +Вµ (E4.49) ss , (E4.50) (E4.51) , (E4.52) 77 State-Space Representation and the characteristic polynomial is p(s) = s 2 + C1 Вµ +Вµ Y s+ ss C1 Вµ Вµ Y . (E4.53) ss Until we can substitute numerical values and turn the problem over to a computer, we have to admit that the state-space form of Eq. (E4.47) is much cleaner to work with. This completes our вЂњfeel-goodвЂќ examples. It may not be too obvious, but the hint is that linear system theory can help us analyze complex problems. We should recognize that statespace representation can do everything in classical control and more, and we should feel at ease with the language of state-space representation. 4.3. Properties of State-Space Models This section contains brief remarks on some transformations and the state-transition matrix. The scope is limited to materials that we may draw on from introductory linear algebra. 4.3.1. Time-Domain Solution We can п¬Ѓnd the solution to Eq. (4.1), which is simply a set of п¬Ѓrst-order differential equations. As analogous to how Eq. (2.3) was obtained, we now use the matrix exponential function as the integration factor, and the result is (see the hints in the Review Problems) t x(t) = eAt x(0) + eв€’A(tв€’П„ ) Bu(П„ ) dП„, (4.10) 0 where the п¬Ѓrst term on the RHS evaluates the effect of the initial condition and the second term is the so-called convolution integral that computes the effect of the input u(t). The point is that state-space representation is general and is not restricted to problems with zero initial conditions. When Eq. (4.1) is homogeneous (i.e., Bu = 0), the solution is simply x(t) = eAt x(0). (4.11) We can also solve the equation by using a Laplace transform. Starting again from Eq. (4.1), we can п¬Ѓnd (see Review Problems) t x(t) = О¦(t)x(0) + О¦(t в€’ П„ )Bu(П„ ) dП„, (4.12) 0 where О¦(t) is the state-transition matrix deп¬Ѓned in Eq. (4.6). Comparing Eq. (4.10) with Eq. (4.12), we can see that О¦(t) = eAt . (4.13) We have shown how the state-transition matrix can be derived in a relatively simple problem in Example 4.7. For complex problems, there are numerical techniques that we can use to compute О¦(t), or even the Laplace transform О¦(s), but, of course, we shall skip them. 78 4.3. Properties of State-Space Models One idea (not that we will really do this) is to apply the Taylor series expansion on the exponential function of A and evaluate the state-transition matrix with О¦(t) = eAt = I + At + 1 2 2 1 A t + A3 t 3 + В· В· В· . 2! 3! (4.14) We can derive, instead of an inп¬Ѓnite series, a closed-form expression for the exponential function. For an n Г— n matrix A, we have eAt = О±0 (t)I + О±1 (t)A + О±2 (t)A2 + В· В· В· + О±nв€’1 (t)Anв€’1 . (4.15) The challenge is to п¬Ѓnd those coefп¬Ѓcients О±i (t), which we shall skip.5 4.3.2. Controllable Canonical Form Although there is no unique state-space representation of a system, there are вЂњstandardвЂќ ones that control techniques make use of. Given any state equations (and if some conditions are met), it is possible to convert them to these standard forms. A couple of important canonical forms are covered in this subsection. A tool that we should be familiar with from introductory linear algebra is the similarity transform, which allows us to transform a matrix into another one but retains the same eigenvalues. If a state x and another xВЇ are related by a so-called similarity transform, the state-space representations constructed with x and xВЇ are considered to be equivalent.6 For the nth-order differential equation7 y (n) + anв€’1 y (nв€’1) + В· В· В· + a1 y (1) + a0 y = u(t), (4.16) we deп¬Ѓne x1 = y, x2 = y (1) , x3 = y (2) , . . . , and xn = y (nв€’1) . (4.17) The original equation in Eq.(4.16) can now be reformulated as a set of п¬Ѓrst-order differential equations: xЛ™ 1 = x2 , xЛ™ 2 = x3 , .. . xЛ™ nв€’1 = xn , 5 6 7 (4.18) We need only the general form of Eq. (4.15) later in Chap. 9. There are other properties of the stateвЂ“ transition matrix that have been skipped, but the text has been structured such that they are not needed here. That includes transforming a given system to the controllable canonical form. We can say that statespace representations are unique up to a similarity transform. As for transfer functions, we can say that they are unique up to scaling in the coefп¬Ѓcients in the numerator and the denominator. However, the derivation of canonical transforms requires material from Chap. 9 and is not crucial for the discussion here. These details are provided on the Web Support. Be careful when you read the MATLAB manual; it inverts the index of coefп¬Ѓcients as in y (n) + a1 y (nв€’1) + В· В· В· + anв€’1 y (1) + an y. Furthermore, here we use a simple RHS in the ODE. You п¬Ѓnd more general, and thus messier, derivations in other texts. 79 State-Space Representation and п¬Ѓnally xЛ™ n = в€’a0 x1 в€’ a1 x2 в€’ В· В· В· в€’ anв€’1 xn + u(t). This set of equations, of course, can be put in matrix form as in Eq. (4.1): пЈ® 0 1 0 пЈЇ 0 0 1 пЈЇ . пЈЇ . xЛ™ = пЈЇ . пЈ° 0 0 0 в€’a0 в€’a1 в€’a2 В·В·В· В·В·В· 0 0 .. . В·В·В· 1 В· В· В· в€’anв€’1 пЈ® пЈ№ 0 пЈє пЈЇ0пЈє пЈє пЈЇ пЈє пЈє x + пЈЇ ... пЈє u = Ax + Bu. пЈє пЈЇ пЈє пЈ» пЈ°0пЈ» пЈ№ (4.19) 1 The output equation equivalent to Eq. (4.2) is y = [1 0 0 В· В· В· 0]x = Cx. (4.20) The system of equations in Eqs. (4.19) and (4.20) is called the controllable canonical form (also phase-variable canonical form). As the name implies, this form is useful in doing controllability analysis and in doing pole-placement system design вЂ“ topics that are covered in Chap. 9. With all the zeros along the leading diagonal, we can п¬Ѓnd relatively easily that the characteristic equation of A, |sI в€’ A| = 0, is s n + anв€’1 s nв€’1 + В· В· В· + a1 s + a0 = 0, (4.21) which is immediately obvious from Eq. (4.16) itself. We may note that the coefп¬Ѓcients of the characteristic polynomial are contained in matrix A in Eq. (4.19). Matrices with this property are called the companion form. When we use MATLAB, its canon() function returns a companion matrix that is the transpose of A in Eq. (4.19); this form is called the observable canonical form. We shall see that in MATLAB Session 4. 4.3.3. Diagonal Canonical Form Here, we want to transform a system matrix A into a diagonal matrix О› that is made up of the eigenvalues of A. In other words, all the differential equations are decoupled after the transformation. For a given system of equations in Eq. (4.1), in which A has distinct eigenvalues, we should п¬Ѓnd a transformation with a matrix P, xВЇ = Pв€’1 x or x = PВЇx, (4.22) such that ВЇ xЛ™ВЇ = О›x + Bu, (4.23) ВЇ = Pв€’1 B, and О› = Pв€’1 AP is a diagonal matrix made up of the eigenvalues of where now B A. The transformation matrix (also called the modal matrix) P is made up of the eigenvectors of A. In control, Eq. (4.23) is called the diagonal canonical form. If A has repeated eigenvalues (multiple roots of the characteristic polynomial), the result, again from introductory linear algebra, is the Jordan canonical form. Brieп¬‚y, the transformation matrix P now needs a set of generalized eigenvectors, and the transformed matrix 80 4.3. Properties of State-Space Models J = Pв€’1 AP is made of Jordan blocks for each of the repeated eigenvalues. For example, if matrix A has three repeated eigenvalues О»1 , the transformed matrix should appear as ... .. 11 ...... ....................... ... ... ... 22 . where J11 0 , J J 0 J= (4.24) пЈ® пЈ№ О»1 1 0 = пЈ° 0 О»1 1 пЈ» 0 0 О»1 and J22 is a diagonal matrix made up of eigenvalues О»4 , . . . , О»n . Because in Chap. 9 such cases are not used, the details are left to a second course in modern control. Example 4.9: For a model with the following transfer function, Y /U = 1 , (s + 3)(s + 2)(s + 1) п¬Ѓnd the diagonal and observable canonical forms with MATLAB. The statements to use are G=zpk([],[-1 -2 -3],1); S=ss(G); canon(S) canon(S,'companion') % S is the state--space system % Default is the diagonal form % This is the observable companion There is no messy algebra. We can be spoiled! Further details are in MATLAB Session 4. Review Problems (1) Fill in the gaps in the derivation of Eqs. (E4.25) and (E4.26) in Example 4.7. (2) Write the dimensions of all the matrixes in Eq. (4.6) for the general case of MIMO models. Take x to be n Г— 1, y to be m Г— 1, and u to be k Г— 1. (3) Derive Eq. (4.9) using Eq. (4.3) as the starting point. (4) For the SISO system shown in Fig. R4.4, derive the state-space representation. Show that the characteristic equation of the model matrix is identical to the closed-loop characteristic polynomial as derived from the transfer functions. (5) Derive Eq. (4.10). (6) Derive Eq. (4.12). (7) Derive Eq. (4.23). u + в€’ K x2 3 s+2 x1= y 1 s+5 Figure R4.4. 81 State-Space Representation Hints: (2) A is (n Г— n), B (n Г— k), C (m Г— n), О¦ (n Г— n), and CО¦B (m Г— k). (4) Multiply the K to the transfer function to give a gain of 3K . Then the rest is easier than Example 4.6. (5) We multiply Eq. (4.1) by exp(в€’At) to give eв€’At [Л™x в€’ Ax] = eв€’At Bu, which is d в€’At (e x) = eв€’At Bu. dt Integration with the initial condition gives t eв€’At x(t) в€’ x(0) = eв€’AП„ Bu(П„ ) dП„, 0 which is one step away from Eq. (4.10). (6) The Laplace transform of Eq. (4.1) with nonzero initial conditions is sX в€’ x(0) = AX + BU or X = (sI в€’ A)в€’1 x(0) + (sI в€’ A)в€’1 BU. Substituting in the deп¬Ѓnition О¦(s) = (sI в€’ A)в€’1 , we have X = О¦(s)x(0) + О¦(s)BU. The time-domain solution vector is the inverse transform x(t) = Lв€’1 [О¦(s)]x(0) + Lв€’1 [О¦(s)BU ], and if we invoke the deп¬Ѓnition of a convolution integral (from calculus), we have Eq. (4.12). (7) We п¬Ѓrst substitute x = PВЇx into Eq. (4.1) to give P d xВЇ = APВЇx + Bu. dt Then we multiply the equation by the inverse, Pв€’1 , d xВЇ = Pв€’1 APВЇx + Pв€’1 Bu, dt which is Eq. (4.23). 82 5 Analysis of Single-Loop Control Systems e now п¬Ѓnally launch into the material on controllers. State-space representation is more abstract, and it helps to understand controllers in the classical sense п¬Ѓrst. We will come back to state-space controller design later. The introduction stays with the basics. Our primary focus is to learn how to design and tune a classical proportionalвЂ“integraвЂ“derivative (PID) controller. Before that, we п¬Ѓrst need to know how to set up a problem and derive the closed-loop characteristic equation. W What Are We Up to? r Introducing the basic PID control schemes r Deriving the closed-loop transfer function of a system and understanding its properties 5.1. PID controllers We use a simple liquid-level controller to illustrate the concept of a classic feedback control system.1 In this example (Fig. 5.1), we monitor the liquid level in a vessel and use the information to adjust the opening of an efп¬‚uent valve to keep the liquid level at some userspeciп¬Ѓed value (the set point or reference). In this case, the liquid level is both the measured variable and the controlled variable вЂ“ they are the same in a SISO system. In this respect, the controlled variable is also the output variable of the SISO system. A system refers to the process that we need to control plus the controller and accompanying accessories, such as sensors and actuators.2 LetвЂ™s say we want to keep the liquid level at the set point h s , but a sudden surge in the inlet п¬‚ow rate qi (the disturbance or load) increases h such that there is a deviation h = h в€’ h s > 0. The deviation can be rectiп¬Ѓed if we open up the valve (or we can think in 1 2 In Fig. 5.1, we use the actual variables because they are what we measure. Regardless of the notations in a schematic diagram, the block diagram in Fig. 5.2 is based on deviation variables and their Laplace transform. Recall footnote 4 in Chap. 1: A process is referred to as a plant in control engineering. 83 Analysis of Single-Loop Control Systems qi h s h LC LT p q R Figure 5.1. Schematic diagram of a liquidlevel control system. terms of lowering the п¬‚ow resistance R). Here we assume that the level controller will send out an appropriate signal to the valve to accomplish the task. It is logical to think that the signal from the controller, p(t), should be a function of the deviation. However, because we are to implement negative feedback, we base our decision on the error, which is deп¬Ѓned as e(t) = h s (t) в€’ h(t), which is the negative of the deviation (Fig. 5.2). The actual controller output is p(t) = p s + f [e(t)] = p s + f [h s в€’ h(t)], (5.1) where f is some function of e(t) and p s is the actuating signal at steady state when the deviation is h = 0; it is commonly referred to as the controller bias signal. Our task is to determine plausible choices of the controller functions вЂ“ what are also called control laws. The classical controller functions are explained in the following subsections, but if you need a more physical feel of how a control system is put together now, you may jump ahead and read Section 5.2 п¬Ѓrst. 5.1.1. Proportional Control The simplest idea is that the compensation signal (actual controller output) is proportional to the error e(t): p(t) = p s + K c e(t) = p s + K c [h s в€’ h(t)], (5.2) where K c is the proportional gain of the controller. It is obvious that the value of K c determines the controller вЂњsensitivityвЂќ вЂ“ how much compensation to enact for a given change in error. + s H (s) E(s) вЂ“ P(s) Gc H(s) Figure 5.2. Information around the summing point in a negativefeedback system. 84 5.1. PID controllers For all commercial devices, the proportional gain is a positive quantity. Because we use negative feedback (see Fig. 5.2), the controller output moves in the reverse direction of the controlled variable.3 In the liquid-level control example, if the inlet п¬‚ow is disturbed such that h rises above h s , then e < 0, and that leads to p < p s , i.e., the controller output is decreased. In this case, we of course will have to select or purchase a valve such that a lowered signal means opening the valve (decreasing п¬‚ow resistance). Mathematically, this valve has a negative steady-state gain (в€’K v ).4 Now what if our valve has a positive gain? (An increased signal means opening the valve.) In this case, we need a negative proportional gain. Commercial devices provide such a вЂњswitchвЂќ on the controller box to invert the signal. Mathematically, we have changed the sign of the compensation term to p = p s в€’ K c e.5 By the deп¬Ѓnition of a control problem, there should be no error at t = 0, i.e., es = 0, and the deviation variable of the error is simply the error itself: e (t) = e(t) в€’ es = e(t). Hence Eq. (5.2) is a relation between the deviation variables of the error and the controller output: p(t) в€’ p s = K c [e(t) в€’ es ], or p (t) = K c e (t), and the transfer function of a proportional controller is simply G c (s) = P(s) = Kc. E(s) (5.3) Generally, the proportional gain is dimensionless [i.e., p(t) and e(t) have the same units]. Many controller manufacturers use the percent proportional band (PB), which is deп¬Ѓned as 100 PB = . (5.4) Kc 3 4 5 You may come across the terms reverse and direct acting; they are extremely confusing. Some authors consider the action between the controller output and the controlled variable, and thus a negativefeedback loop with positive K c is considered reverse acting. However, most commercial vendors consider the action between the error (controller input) and the controller output, and now a controller with a positive K c is direct acting, exactly the opposite terminology. We will avoid using these confusing terms. The important point is to select the proper signs for all the steady-state gains, and we will get back to this issue in Section 5.4. Take note that, from the mass balance of the tank, the process gain associated with the outlet п¬‚ow rate is also negative. A simple-minded check is that in a negative-feedback system there can only be one net negative sign вЂ“ at the feedback summing point. If one unit in the system has a negative steady-state gain, we know something else must have a negative steady-state gain too. More confusion may have been introduced here than in texts that ignore this tiny detail. To reduce confusion, we keep K c a positive number. For problems in which the proportional gain is negative, we use the notation в€’K c . We can imagine that the minus sign is likened to having п¬‚ipped the action switch on the controller. 85 Analysis of Single-Loop Control Systems A high proportional gain is equivalent to a narrow PB and a low gain is wide PB. We can interpret a PB as the range over which the error must change to drive the controller output over its full range.6 Before doing any formal analysis, we state a few qualitative features of each type of controller. This is one advantage of classical control. We can make a fairly easy physical interpretation of the control algorithm. The analyses in Section 5.3 will conп¬Ѓrm these qualitative observations. General Qualitative Features of Proportional Control r We expect that a proportional controller will improve or accelerate the response of a process. The larger K c is, the faster and more sensitive is change in compensation with respect to a given error. However, if K c is too large, we expect the control compensation to overreact, leading to oscillatory response. In the worst case, the system may become unstable. r There are physical limits to a control mechanism. A controller (like an ampliп¬Ѓer) can deliver only so much voltage or current; a valve can deliver only so much п¬‚uid when fully opened. At these limits, the control system is saturated.7 r We expect a system with only a proportional controller to have a steady-state error (or an offset). A formal analysis is introduced in Section 5.3. This is one simplistic way to see why. LetвЂ™s say we change the system to a new set point. The proportional controller output, p = p s + K c e, is required to shift away from the previous bias p s and move the system to a new steady state. For p to be different from p s , the error must have a п¬Ѓnite nonzero value.8 r To tackle a problem, consider a simple proportional controller п¬Ѓrst. This may be all we need (lucky break!) if the offset is small enough (for us to bear with) and the response is adequately fast. Even if this is not the case, the analysis should help us plan the next step. 5.1.2. ProportionalвЂ“Integral Control To eliminate offset, we can introduce integral action into the controller. In other words, we use a compensation that is related to the history of the error: p (t) = 1 П„I t e (t) dt, 0 1 P(s) = , E(s) s П„I where П„ I is the integral time constant (reset time, or minutes per repeat9 ). Commercial devices may also use 1/П„ I , which is called the reset rate (repeats per minute). 6 7 8 9 86 In some commercial devices, the proportional gain is deп¬Ѓned as the ratio of the percentage of controller output to the percentage of controlled variable change [%/%]. In terms of the control system block diagram that we will go through in the next subsection, we just have to add вЂњgainsвЂќ to do the unit conversion. Typical ranges of device outputs are 0вЂ“10 V, 0вЂ“1 V, 4вЂ“20 mA, and 3вЂ“15 psi. The exception is when a process contains integrating action, i.e., 1/s in the transfer functions вЂ“ a point that is illustrated in Example 5.5. Roughly, the reset time is the time that it takes the controller to repeat the proportional action. This is easy to see if we take the error to be a constant in the integral. 5.1. PID controllers The integral action is such that we accumulate the error from t = 0 to the present. Thus the integral is not necessarily zero even if the current error is zero. Moreover, the value of the integral will not decrease unless the integrand e (t) changes its sign. As a result, integral action forces the system to overcompensate and leads to oscillatory behavior, i.e., the closedloop system will exhibit an underdamped response. If there is too much integral action, the system may become unstable. In practice, integral action is never used by itself. The norm is a proportionalвЂ“integral (PI) controller. The time-domain equation and the transfer function are p (t) = K c e (t) + 1 П„I t e (t) dt , G c (s) = K c 1 + 0 1 . П„I s (5.5) If the error cannot be eliminated within a reasonable period, the integral term can become so large that the controller is saturated вЂ“ a situation referred to as integral or reset windup. This may happen during start-up or large set-point changes. It may also happen if the proportional gain is too small. Many industrial controllers have вЂњantiwindupвЂќ that temporarily halts the integral action whenever the controller output becomes saturated.10 On the plus side, the integration of the error allows us to detect and eliminate very small errors. To obtain a simple explanation of why integral control can eliminate offsets, refer back to the intuitive explanation of offset with only a proportional controller. If we desire e = 0 at steady state and to shift controller output p away from the previous bias p s , we must have a nonzero term. Here, it is provided by the integral in the п¬Ѓrst equation of Eqs. (5.5). That is, as time progresses, the integral term takes on a п¬Ѓnal nonzero value, thus permitting the steady-state error to stay at zero. General Qualitative Features of PI control r PI control can eliminate offset. We must use a PI controller in our design if the offset is unacceptably large. r The elimination of the offset is usually at the expense of a more underdamped system response. The oscillatory response may have a short rise time, but is penalized by excessive overshoot or exceedingly long settling time.11 r Because of the inherent underdamped behavior, we must be careful with the choice of the proportional gain. In fact, we usually lower the proportional gain (or detune the controller) when we add integral control. 10 Another strategy is to implement the PI algorithm in the so-called reset-feedback conп¬Ѓguration. The basis of internal reset feedback is to rearrange and implement the PI transfer function as 1+ 11 1 П„I s + 1 1 1 = = = . П„I s П„I s П„ I s/(П„ I s + 1) 1 в€’ 1/(П„ I s + 1) Now, the internal state of the controller, whether it be electronics or a computer algorithm for integration, will have an upper limit. External reset feedback, on the other hand, makes use of measurements of the manipulated variable. You may п¬Ѓnd such implementation details in more applied control books. Some texts use the term вЂњsluggishвЂќ here without further qualiп¬Ѓcation. The sluggishness in this case refers to the long settling time, not the initial response. 87 Analysis of Single-Loop Control Systems 5.1.3. ProportionalвЂ“Derivative Control We certainly want to respond very differently if the temperature of a chemical reactor is changing at a rate of 100 в—¦ C/s as opposed to 1 в—¦ C/s. In a way, we want to вЂњprojectвЂќ the error and make corrections accordingly. In contrast, proportional and integral controls are based on the present and the past. Derivative controller action is based on how fast the error is changing with time (rate action control). We can write p (t) = П„ D P(s) = П„ D s, E(s) de , dt where П„ D is the derivative time constant (sometimes just rate time). Here, the controller output is zero as long as the error stays constant, that is, even if the error is not zero. Because of the proportionality to the rate of change, the controller response is very sensitive to noise. If there is a sudden change in error, especially when we are just changing the set point, the controller response can be unreasonably large вЂ“ leading to what is called a derivative kick. Derivative action is never used by itself. The simplest implementation is a proportionalвЂ“ derivative (PD) controller. The time-domain equation and the transfer function of an вЂњidealвЂќ PD controller are p (t) = K c e (t) + П„ D de , dt G c (s) = K c [1 + П„ D s]. (5.6) In practice, we cannot build a pneumatic device or a passive circuit that provides ideal derivative action. Commercial (real!) PD controllers are designed on the basis of a leadвЂ“lag element: G c (s) = K c П„D s + 1 , О±П„ D s + 1 (5.7) where О± is a small number, typically 0.05 в‰¤ О± в‰¤ 0.2. In effect, we are adding a very large real pole to the derivative transfer function. Later, after learning root-locus and frequency-response analysis, we can make more rational explanations, including why the function is called a leadвЂ“lag element. We will see that this is a nice strategy that is preferable to using the ideal PD controller. To reduce derivative kick (the sudden jolt in response to set-point changes), the derivative action can be based on the rate of change of the measured (controlled) variable instead of the rate of change of the error. One possible implementation of this idea is in Fig. 5.3. This PI action R + E вЂ“ P 1 Kc (1 + П„ s ) I П„D s + 1 Y О±П„ D s + 1 Derivative action Figure 5.3. Implementation of derivative control on the measured variable. 88 5.1. PID controllers way, the derivative control action ignores changes in the reference and just tries to keep the measured variable constant.12 General Qualitative Features of Derivative Control r PD control is not useful for systems with large dead time or noisy signals. r The sign of the rate of change in the error could be opposite that of the proportional or the integral terms. Thus adding derivative action to PI control may counteract the overcompensation of the integrating action. PD control may improve system response while reducing oscillations and overshoot. (Formal analysis in Chap. 7 will show that the problem is more complex than is implied by this simple statement.) r If simple proportional control works п¬Ѓne (in the sense of acceptable offset), we may try PD control. Similarly, we may try PID on top of PI control. The additional stabilizing action allows us to use a larger proportional gain and obtain a faster system response. 5.1.4. ProportionalвЂ“IntegralвЂ“Derivative Control Finally, we can put all the components together to make a PID (or three-mode) controller. The time-domain equation and the transfer function of an вЂњidealвЂќ PID controller are p (t) = K c e (t) + G c (s) = K c 1 + 1 П„I t e (t) dt + П„ D 0 de , dt 1 П„I П„D s2 + П„I s + 1 + П„D s = Kc . П„I s П„I s (5.8a) (5.8b) We also п¬Ѓnd it rather usual that the proportional gain is multiplied into the bracket to give the integral and the derivative gains: G c (s) = K c + KI + K D s, s (5.8c) where K I = K c /П„ I and K D = K c П„ D . With a higher-order polynomial in the numerator, the ideal PID controller is not considered physically realizable. We nevertheless use this ideal controller in analyses because of the cleaner algebra, and more importantly because we can gain valuable insight with it. We can say the same with the use of the ideal PD controller too. In real life, different manufacturers implement the вЂњrealвЂќ PID controller slightly differently.13 One possibility is to modify the derivative action as G c (s) = K c 1 + 12 13 1 (О± + 1)П„ D s + 1 П„D s 1 + = Kc + П„I s О±П„ D s + 1 О±П„ D s + 1 П„I s (5.9a) For review after Chap. 7 on root locus: With the strategy in Fig. 5.3, the closed-loop characteristic polynomial and thus the poles remain the same, but not the zeros. You may also wonder how to write the function G c (s), but it is much easier and more logical just to treat the PI action and derivation action as two function blocks when analyzing a problem. Not only that, but most implementations are based on some form of the digitized control law. An illustration of the positional digital algorithm along with concepts such as bumpless transfer, externalrate feedback, and bias tracking is in the LabView liquid-level simulation module on the Web Support. 89 Analysis of Single-Loop Control Systems Another implementation of the actual PID control is to introduce the derivative control in series with PI control: G c (s) = K c П„I s + 1 П„I s П„D s + 1 . О±П„ D s + 1 (5.9b) This conп¬Ѓguration is also referred to as interacting PID, series PID, or rate-before-reset. To eliminate derivative kick, the derivative leadвЂ“lag element is implemented on the measured (controlled) variable in the feedback loop. 5.2. Closed-Loop Transfer Functions We п¬Ѓrst establish the closed-loop transfer functions of a fairly general SISO system. After that, we walk through the diagram block by block to gather the thoughts that we must have in synthesizing and designing a control system. An important detail is the units of the physical properties. 5.2.1. Closed-Loop Transfer Functions and Characteristic Polynomials Consider the stirred-tank heater again, this time in a closed loop (Fig. 5.4). The tank temperature can be affected by variables such as the inlet and the jacket temperatures and the inlet п¬‚ow rate. Back in Chap. 2, we derived the transfer functions for the inlet and the jacket temperatures. In Laplace transform, the change in temperature is given in Eq. (2.49b) as T (s) = G L (s)Ti (s) + G p (s)TH (s). (5.10) This is our process model. Be careful with the context when using the word вЂњinput.вЂќ The inlet and the jacket temperatures are the inputs to the process, but they are not necessarily the inputs to the system. One of them will become the manipulated variable of the system. In a SISO system, we manipulate only one variable, so we must make a decision. Because our goal is to control the tank temperature, it would be much more sensible to manipulate the steam temperature TH instead of the inlet temperature. We can arrive at this decision with physical intuition, or we can base it on the fact that, from Chap. 2, the steam temperature has a higher process gain. Hence with respect to the control system, we choose TH as the manipulated variable (M), which is governed by the actuator function G a and the controller signal P. The tank temperature T is the system output (also the controlled variable C). L (Ti) Process (Plant) Controller R (Tsp ) Km GL + E в€’ P Gc Ga Cm M (TH) Gm Gp + + C Figure 5.4. Block diagram of a simple SISO closed-loop system. 90 C (T) 5.2. Closed-Loop Transfer Functions The system input is the set point Tsp (or reference R) вЂ“ our desired steady-state tank temperature.14 There are other system inputs that can affect our closed-loop response, and we consider them load (or disturbance) variables. In this case, the load variable is the inlet temperature, Ti . Now you may understand why we denote the two transfer functions as G p and G L . The important point is that input means different things for the process and for the closed-loop system. For the rest of the control loop, G c is obviously the controller transfer function. The measuring device (or transducer) function is G m . Although it is not shown in the block diagram, the steady-state gain of G m is K m . The key is that the summing point can compare only quantities with the same units. Hence we need to introduce K m on the reference signal, which should have the same units as C. The use of K m , in a way, performs unit conversion between what we вЂњdial inвЂќ and what the controller actually uses in comparative tests.15 The next order of business is to derive the closed-loop transfer functions. For better readability, we write the Laplace transforms without the s dependence explicitly. Around the summing point, we observe that E = K m R в€’ G m C. From the process we have C = G p (G a G c E) + G L L . Substituting for E, we п¬Ѓnd that the equation becomes C = G p G a G c (K m R в€’ G m C) + G L L . This step can be rearranged to give C= Km Gc Ga G p 1 + Gm Gc Ga G p R+ GL 1 + Gm GcGa G p L = G sp R + G load L , (5.11) which provides us with the closed-loop transfer functions G sp and G load . Based on Eq. (5.11), the inputs to the system are the reference R and the load variable L; the controlled variable is the system output. The п¬Ѓrst transfer function G sp accounts for the effect of a set-point change and is also called the command tracking function. The second function G load accounts for the effect of changes in disturbance. The important point is that the dynamics and the stability of the system are governed by the closed-loop characteristic polynomial: 1 + G m G c G a G p = 0, (5.12) which is the same whether we are looking at set-point or disturbance changes. As an abbreviation, many authors write G OL = G m G c G a G p and refer to it as the open-loop transfer 14 15 There is no standard notation. We could have used Y in place of C for system output or replaced G a with G v for valve (G f is also used), G L and L with G d and D for disturbance, and G m with G T for transducer. Here P was selected to denote controller output, more or less for pneumatic. Many texts, especially those in electrical engineering, ignore the need for K m , and the п¬Ѓnal result is slightly different. They do not have to check the units because all they deal with are electrical signals. 91 Analysis of Single-Loop Control Systems function as if the loop were disconnected.16 We may also refer to G c G a G p as the forwardloop transfer function. Our analyses of SISO systems seldom take into account simultaneous changes in set point and load.17 We denote the two distinct possibilities as two different problems: (1) Servo problems: Consider changes in set point with no disturbance (L = 0); C = G sp R. Ideally (meaning unlikely to be encountered in reality), we would like to achieve perfect tracking of set-point changes: C = R. Reminder: We are working with deviation variables. (2) Regulator problems: Consider changes in disturbance with a п¬Ѓxed set point (R = 0); C = G load L. The goal is to reject disturbances, i.e., keep the system output at its desired value in spite of load changes. Ideally, we would like to have C = 0, i.e., perfect disturbance rejection. 5.2.2. How Do We Choose the Controlled and the Manipulated Variables? In homework problems, by and large, the variables are stated. Things will be different when we are on the job. Here are some simple ideas on how we may make the decision. Choice of controlled variables: r r r r those that are dictated by the problem (for instance, temperature of a refrigerator) those that are not self-regulating those that may exceed equipment or process constraints those that may interact with other variables (for example, reactor temperature may affect product yield). Choice of manipulated variables: r r r r those that have a direct effect on the process, especially the output variable those that have a large steady-state gain (good sensitivity) those that have no dead time those that have no interaction with other control loops. After we have chosen the controlled and the manipulated variables, the remaining ones are taken as load variables in a SISO system. 5.2.3. Synthesis of a Single-Loop Feedback System We now walk through the stirred-tank heater system once again. This time, we will take a closer look at the transfer functions and the units (Fig. 5.5). 16 17 92 Can an open loop be still a loop? You may wonder, what is an open loop? Often, we loosely refer to elements or properties of part of a system as open loop, as opposed to a complete closed-loop system. You will see more of this language in Chap. 7. In real life, we expect probable simultaneous reference and disturbance inputs. As far as analysis goes, the mathematics is much simpler if we consider one case at a time. In addition, either case shares the same closed-loop characteristic polynomial. Hence they should also share the same stability and dynamic response characteristics. Later when we talk about integral error criteria in controller design, there are minor differences, but not sufп¬Ѓcient to justify analyzing a problem with simultaneous reference and load inputs. 5.2. Closed-Loop Transfer Functions Ti (В°C) Stirred-tank heater GL Tsp (В°C) TH (В°C) + Km (mV) в€’ Gc Ga Gp (mV) (mV) Gm + T (В°C) + T (В°C) Figure 5.5. Block diagram of a simple SISO closed-loop system with physical units. Process Model The п¬Ѓrst item on the agenda is вЂњprocess identiп¬Ѓcation.вЂќ We either derive the transfer functions of the process based on scientiп¬Ѓc or engineering principles, or we simply do a step-input experiment and п¬Ѓt the data to a model. Either way, we need to decide what the controlled variable is. We then need to decide which should be the manipulated variable. All remaining variables are delegated to become disturbances. With the stirred-tank heater, we know quite well by now that we want to manipulate the heating-coil temperature to control the tank temperature. The process function G p is deп¬Ѓned based on this decision. In this simple illustration, the inlet temperature is the only disturbance, and the load function is deп¬Ѓned accordingly. From Subsection 2.8.2 and Eq. (2.49b), we have the п¬Ѓrst-order process model: T = G L Ti + G p TH = KL Ti + П„ps + 1 Kp TH . П„ps + 1 (5.13) From Subsection 2.8.2, we know that the steady-state gain and the time constant are dependent on the values of п¬‚ow rate, liquid density, heat capacity, heat transfer coefп¬Ѓcient, and so on. For the sake of illustration, the heat transfer analysis is skipped. LetвЂ™s presume that we have done our homework, substituted in numerical values, and have found K p = 0.85 в—¦ C/в—¦ C and П„ p = 20 min. Signal Transmitter Once we know what to control, we need to п¬Ѓnd a way to measure the quantity. If the transducer (sensor and transmitter packaged together) is placed far downstream or is too well insulated and the response is slow, the measurement transfer function may appear as Tm K m eв€’td s = Gm = , T П„m s + 1 (5.14) where K m is the measurement gain of the transducer, П„m is the time constant of the device, and td accounts for transport lag. In the worst case, the sensor may be nonlinear, meaning that the measurement gain would change with the range of operating conditions. With temperature, we can use a thermocouple, which typically has a resolution of the order of 0.05 mV/в—¦ C. [We could always use a resistance temperature detector (RTD) for better resolution and response time.] That is too small a change in output for most 12-bit analog-digital converters, so we must have an ampliп¬Ѓer to boost the signal. This is something we do in a laboratory, but commercially, we should п¬Ѓnd off-the-shelf transducers with the 93 Analysis of Single-Loop Control Systems sensor and ampliп¬Ѓer packaged together. Many of them have a scaled output of, for example, 0вЂ“1 V or 4вЂ“20 mA. For the sake of illustration, letвЂ™s presume that the temperature transmitter has a built-in ampliп¬Ѓer that allows us to have a measurement gain of K m = 5 mV/в—¦ C. LetвЂ™s also presume that there is no transport lag and that the thermocouple response is rapid. The measurement transfer function in this case is simply G m = K m = 5 mV/в—¦ C. This so-called measurement gain is really the slope of a calibration curve вЂ“ an idea that we are familiar with. We do a least-squares п¬Ѓt if this curve is linear and п¬Ѓnd the tangent at the operating point if the curve is nonlinear. Controller The ampliп¬Ѓed signal from the transmitter is sent to the controller, which can be a computer or a little black box. Not much can be said about the controller function now, except that it is likely a PID controller or a software application with a similar interface. A reminder is that a controller has a front panel with physical units such as degrees celsius. (Some also have relative scales of 0вЂ“100%.) Therefore, when we dial a change in the set point, the controller needs to convert the change into electrical signals. ThatвЂ™s why K m is part of the controller in the block diagram (Fig. 5.5). Actuator Control Valve Last but not least, designing a proper actuator can create the most headaches. We have to п¬Ѓnd an actuator that can drive the range of the manipulated variable. We also want the device to have a faster response than the process. After that, we have to п¬Ѓnd a way to interface the controller to the actuator. A lot of work is masked by the innocent-looking notation G a . For the stirred-tank heater example, several comments should be made here. We need to consider safety. If the system fails, we want to make sure that no more heat is added to the tank. Thus we want a fail-close valve вЂ“ meaning that the valve requires energy (or a positive signal change) to open it. In other words, the valve gain is positive. We can check the thinking as follows: If the tank temperature drops below the set point, the error increases. With a positive proportional gain, the controller output will increase, hence opening up the valve. If the process plant has a power outage, the valve closes and shuts off the steam. But how can the valve shut itself off without power? This leads to the second comment. One may argue for emergency power or a springloaded valve, but to reduce п¬Ѓre hazards, the nominal industrial practice is to use pneumatic (compressed-air-driven) valves that are regulated by a signal of 3вЂ“15 psi. The electrical signal to and from the controller is commonly 4вЂ“20 mA. A current signal is less susceptible to noise than a voltage signal is over longer transmission distances. Hence, in a more applied setting, we expect to п¬Ѓnd a current-to-pressure transducer (I/P) situated between the controller output and the valve actuator. Finally, we have been sloppy in associating the п¬‚ow rate of steam with the heating-coil temperature. The proper analysis that includes a heat balance of the heating medium is in the Review Problems. To sidestep the actual calculations, we have to make a few more 94 5.3. Closed-Loop System Response assumptions for the valve gain to illustrate what we need to do in reality: (1) Assume that we have the proper ampliп¬Ѓer or transducer to interface the controller output with the valve, i.e., converting electrical information into п¬‚ow rate. (2) We use a valve with linear characteristics such that the п¬‚ow rate varies linearly with the opening.18 (3) The change in steam п¬‚ow rate can be вЂњtranslatedвЂќ to changes in heating-coil temperature. When the steady-state gains of all three assumptions are lumped together, we may arrive at a valve gain K v with units of degrees celsius per millivolt. For this illustration, letвЂ™s say the valve gain is 0.6 в—¦ C/mV and the time constant is 0.2 min. The actuator controller function would appear as Gv = 0.6 в—¦ C/mV Kv = . П„v s + 1 0.2 s + 1 The closed-loop characteristic equation of the stirred-tank heater system is hence 1 + GcGv G p Gm = 1 + Gc (0.6)(0.85)(5) = 0. (0.2s + 1)(20s + 1) We will not write the entire closed-loop function C/R or, in this case, T /Tsp . The main reason is that our design and analysis will be based on only the characteristic equation. The closed-loop function is handy to do only time-domain simulation, which we can easily compute by using MATLAB. That being said, we need to analyze the closed-loop transfer function for several simple cases so we have a better theoretical understanding. 5.3. Closed-Loop System Response In this section, we will derive the closed-loop transfer functions for several examples. The scope is limited by how much sense we can make out of the algebra. Nevertheless, the steps that we go through are necessary to learn how to set up problems properly. The analysis also helps us to better understand why a system may have a faster response, why a system may become underdamped, and when there is an offset. When the algebra is clean enough, we can also make observations as to how controller settings may affect the closed-loop system response. The results generally reafп¬Ѓrm the qualitative statements that we have made concerning the characteristics of different controllers. The actual work is rather cookbook-like: (1) With a given problem statement, draw the control loop and derive the closed-loop transfer functions. (2) Pick either the servo or the regulator problem. Reminder: the characteristic polynomial is the same in either case. 18 In reality, the valve characteristic curve is likely nonlinear and we need to look up the technical speciп¬Ѓcation in the manufacturerвЂ™s catalog. After that, the valve gain can be calculated from the slope of the characteristic curve at the operating point. See Homework Problem I.33 and the Web Support. 95 Analysis of Single-Loop Control Systems R + E в€’ C Gc Gp Figure 5.6. Simple unity feedback system. (3) With the given choices of G c (P, PI, PD, or PID), G p , G a , and G m , plug their transfer functions into the closed-loop equation. The characteristic polynomial should fall out nicely. (4) Rearrange the expressions such that we can redeп¬Ѓne the parameters as time constants and steady-state gains for the closed-loop system. All analyses follow the same general outline. What we must accept is that there are no handy-dandy formulas to plug and chug. We must be competent in deriving the closed-loop transfer function, steady-state gain, and other relevant quantities for each speciп¬Ѓc problem. In our examples, we will take G m = G a = 1 and use a servo system with L = 0 to highlight the basic ideas. The algebra tends to be more tractable in this simpliп¬Ѓed unity feedback system with only G c and G p (Fig. 5.6), and the closed-loop transfer function is GcG p C , = R 1 + GcG p (5.15) which has the closed-loop characteristic equation 1 + G c G p = 0. Example 5.1: Derive the closed-loop transfer function of a system with proportional control and a п¬Ѓrst-order process. What is the value of the controlled variable at steady state after a unit-step change in set point? In this case, we consider G c = K c and G p = [K p /(П„ p s + 1)], and substitution into Eq. (5.15) leads to19 Kc K p C . = R П„ ps + 1 + Kc K p 19 (E5.1) You may wonder how transfer functions are related to differential equations. This is a simple illustration. We use y to denote the controlled variable. The п¬Ѓrst-order process function G p arises from Eq. (3.6): П„p dy + y = K p x. dt In the unity feedback loop with G c = K c , we have x = K c (r в€’ y). Substitution for x in the ODE leads to dy П„p + y = K c K p (r в€’ y) dt or dy П„p + (1 + K c K p )y = K c K p r. dt It is obvious that Eq. (E5.1) is the Laplace transform of this equation. This same idea can be applied to all other systems, but, of course, nobody does that. We all work within the Laplace transform domain. 96 5.3. Closed-Loop System Response We now divide both the numerator and denominator with (1 + K c K p ) to obtain K c K p /(1 + K c K p ) K C = = , R [П„ p /(1 + K c K p )]s + 1 П„s + 1 (E5.2) where K = Kc K p , 1 + Kc K p П„= П„p 1 + Kc K p are the closed-loop steady-state gain and time constant. Recall Eq. (5.11); the closed-loop characteristic equation is the denominator of the closedloop transfer function, and the probable locations of the closed-loop pole are given by s = в€’(1 + K c K p )/П„ p . There are two key observations. First, K < 1, meaning that the controlled variable will change in magnitude less than a given change in set point, the source of offset. The second is that П„ < П„ p , meaning that the system has a faster response than the open-loop process. The system time constant becomes smaller as we increase the proportional gain. This is consistent with the position of the closed-loop pole, which should вЂњmove awayвЂќ from the origin as K c increases. We now take a formal look at the steady-state error (offset). LetвЂ™s consider a more general step change in set point, R = M/s. The eventual change in the controlled variable, by means of the п¬Ѓnal-value theorem, is K M = MK. П„s + 1 s The offset is the relative error between the set point and the controlled variable at steady state, i.e., (r в€’ cв€ћ )/r : c (в€ћ) = lim s sв†’0 ess = Kc K p M в€’ MK 1 = . =1в€’K =1в€’ M 1 + Kc K p 1 + Kc K p (E5.3) We can reduce the offset if we increase the proportional gain. LetвЂ™s take another look at the algebra for evaluating the steady-state error. The error that we have derived in the example is really the difference between the change in controlled variable and the change in set point in the block diagram (Fig. 5.6). Thus we can write E = Rв€’C = R 1в€’ GcG p 1 + GcG p =R 1 . 1 + GcG p Now if we have a unit-step change R = 1/s, the steady-state error by means of the п¬Ѓnal-value theorem is (recall that e = e ) ess = lim s sв†’0 1 1 1 1 = = , 1 + GcG p s 1 + lim G c G p 1 + K err (5.16) sв†’0 where K err = lim G c G p . We call K err the position error constant.20 For the error to sв†’0 20 In many control texts, we also п¬Ѓnd the derivation of the velocity error constant (by using R = s в€’2 ) and acceleration error constant (by using R = s в€’3 ) and a subscript p is used on what we call K err here. 97 Analysis of Single-Loop Control Systems approach zero, K err must approach inп¬Ѓnity. In Example 5.1, the error constant and the steady-state error are K err = lim G c G p = sв†’0 Kc K p 1 = K c K p , and again ess = . П„ps + 1 1 + Kc K p (5.17) Example 5.2: Derive the closed-loop transfer function of a system with proportional control and a second-order overdamped process. If the second-order process has time constants 2 and 4 min and process gain 1.0 (units), what proportional gain would provide us with a system with damping ratio of 0.7? In this case, we consider G c = K c , and G p = {K p /[(П„1 s + 1)(П„2 s + 1)]}, and substitution into Eq. (5.15) leads to Kc K p C = = R (П„1 s + 1)(П„2 s + 1) + K c K p K c K p /(1 + K c K p ) П„1 П„2 П„1 +П„2 s 2 + 1+K s 1+K c K p cKp +1 . (E5.4) The key is to recognize that the system may exhibit underdamped behavior even though the open-loop process is overdamped. The closed-loop characteristic polynomial can have either real or complex roots, depending on our choice of K c . (This is much easier to see when we work with a root locus in Chap. 7.) For now, we rewrite the closed-loop function as C K , = 2 2 R П„ s + 2О¶ П„ s + 1 (E5.4a) where the closed-loop steady-state gain is K = [(K c K p )/(1 + K c K p )], and the system natural time period and damping ratio are П„= П„1 П„2 , 1 + Kc K p О¶ = 1 2 (П„1 + П„2 ) . П„1 П„2 (1 + K c K p ) (E5.5) If we substitute О¶ = 0.7, K p = 1, П„1 = 2, and П„2 = 4 into the second expression, we should п¬Ѓnd the proportional gain K c to be 1.29. Last, we should see immediately that the system steady-state gain in this case is the same as that in Example 5.1, meaning that this second-order system will have the same steady-state error. In terms of controller design, we can take an entirely analytical approach when the system is simple enough. Of course, such circumstances are not common in real life. Furthermore, we often have to compromise between conп¬‚icting criteria. For example, we cannot require a system to have both a very fast rise time and a very short settling time. If we want to provide a smooth response to a set-point change without excessive overshoot, we cannot also expect a fast and snappy initial response. As engineers, it is our job to decide. In terms of design speciп¬Ѓcation, it is not uncommon to use the decay ratio (DR) as the design criterion. Repeating Eq. (3.29), we know that the DR [or the overshoot (OS)] is a function of the damping ratio: DR = (OS)2 = exp 98 в€’2ПЂ О¶ 1 в€’ О¶2 . (5.18) 5.3. Closed-Loop System Response From this equation we can derive О¶2 = (ln DR)2 . 4ПЂ 2 + (ln DR)2 (5.19) If we have a second-order system, we can derive an analytical relation for the controller. If we have a proportional controller with a second-order process, as in Example 5.2, the solution is unique. However, if we have, for example, a PI controller (two parameters) and a п¬Ѓrst-order process, there are no unique answers as we only have one design equation. We must specify one more design constraint in order to have a well-posed problem. Example 5.3: Derive the closed-loop transfer function of a system with PI control and a п¬Ѓrst-order process. What is the offset in this system? We substitute G c = K c [(П„ I s + 1)/П„ I s] and G p = [K p /(П„ p s + 1)] into Eq. (5.15), and we п¬Ѓnd that the closed-loop servo transfer function is K c K p (П„ I s + 1) C = = R П„ I s(П„ p s + 1) + K c K p (П„ I s + 1) (П„ I s + 1) П„I П„ p Kc K p s2 + П„ I (1+K c K p ) s Kc K p +1 . (E5.6) There are two noteworthy items. First, the closed-loop system is now second order. The integral action adds another order. Second, the system steady-state gain is unity and will not have an offset. This is a general property of using PI control. [If this is not immediately obvious, try taking R = 1/s and apply the п¬Ѓnal-value theorem. We should п¬Ѓnd the eventual change in the controlled variable to be c (в€ћ) = 1.] With the expectation that the second-order system may exhibit underdamped behavior, we rewrite the closed-loop function as C (П„ I s + 1) , = 2 2 R П„ s + 2О¶ П„ s + 1 (E5.6a) where the natural time period and DR of the system are П„= П„I П„ p , Kc K p О¶ = 1 (1 + K c K p ) 2 П„I . Kc K pП„ p (E5.7) Although we have the analytical results, it is not obvious how choices of the integral time constant and the proportional gain may affect the closed-loop poles or the system DR. (We 1.) Again, the may get a partial picture if we consider circumstances under which K c K p analysis is deferred until we cover root locus; we should п¬Ѓnd that to be a wonderful tool in assessing how controller design may affect system response. Example 5.4: Derive the closed-loop transfer function of a system with PD control and a п¬Ѓrst-order process. Closed-loop transfer function (5.15) with G c = K c (1 + П„ D s) and G p = [K p /(П„ p s + 1)] is K c K p (П„ D s + 1) K c K p (П„ D s + 1) C = = . R (П„ p s + 1) + K c K p (П„ D s + 1) (П„ p + K c K p П„ D )s + 1 + K c K p (E5.8) 99 Analysis of Single-Loop Control Systems The closed-loop system remains п¬Ѓrst order and the function is that of a leadвЂ“lag element. We can rewrite the closed-loop transfer function as K (П„ D s + 1) C = , R П„s + 1 (E5.8a) where the system steady-state gain and time constant are K = Kc K p , 1 + Kc K p П„= П„ p + Kc K p П„D . 1 + Kc K p The system steady-state gain is the same as that with proportional control in Example 5.1. We, of course, expect the same offset with PD control too. The system time constant depends on various parameters. Again, this analysis is deferred until we discuss root locus. Example 5.5: Derive the closed-loop transfer function of a system with proportional control and an integrating process. What is the offset in this system? LetвЂ™s consider G c = K c and G p = 1/As; substitution into Eq. (5.15) leads to Kc C 1 = . = R As + K c (A/K c )s + 1 (E5.9) We can see quickly that the system has unity gain and there should be no offset. The point is that integral action can be introduced by the process and we do not need PI control under such circumstances. We come across processes with integral action in the control of rotating bodies and liquid levels in tanks connected to pumps (e.g., Example 3.1). Example 5.6: Provide illustrative closed-loop time-response simulations. Most texts have schematic plots to illustrate the general properties of a feedback system. This is something that we can do ourselves by using MATLAB. Simulate the observations that we have made in previous examples. Use a unity feedback system. We consider Example 5.3 again; letвЂ™s pick П„ p to be 5 min and K p be 0.8 (unit). Instead of using the equation that we derived in Example 5.3, we can use the following statements in MATLAB to generate a simulation for the case of a unit-step change in the set point. This approach is much faster than using Simulink. kc=1; taui=10; % The two tuning parameters to % be varied % The following statements are best saved in an M-file Gc=tf(kc*[taui 1],[taui 0]); % The PI controller function Gp=tf(0.8,[5 1]); % The process function Gcl=feedback(Gc*Gp,1) % Unity closed-loop function % GcGp/(1 + GcGp) step(Gcl); 100 % Personalize your own plotting % and put a hold for additional % curves 5.3. Closed-Loop System Response In these statements, we have used feedback() to generate the closed-loop function C/R. The unity feedback loop is indicated by the вЂњ1вЂќ in the function argument. Try п¬Ѓrst with K c = 1 and П„ I with values of 10, 1, and 0.1. Next, select П„ I = 0.1 and repeat with K c = 0.1, 1, 5, and 10. In both cases, the results should follow the qualitative trends that we anticipate. If we repeat the calculation with a larger integral time П„ I = 1 and use K c = 0.1, 1, 5, 10, and 50, we may п¬Ѓnd the results to be rather unexpected. However, we do not have enough theory to explain them now. Keep the results in mind; it is hoped that this will be motivation to explore the later chapters. We could also modify the M-п¬Ѓle by changing the PI controller to a PD or a PID controller to observe the effects of changing the derivative time constant. (Help is in MATLAB Session 5.) We will understand the features of these dynamic simulations better in later chapters. For now, the simulations should give us a qualitative feel of the characteristics of a PID controller and (we hope) also the feeling that we need a better way to select controller settings. Example 5.7: We have to design a servo controller for a mixing system (see Fig. E5.7A). A blue dye for making denim is injected into a stream of water. The injected dye is blended into the pipe п¬‚ow with the aid of in situ static mixers. A photodetector downstream is used to monitor the dye concentration. The analog output of the detector is transmitted to a controller, which in turn sends a signal to the dye-injection regulating valve. In designing this mixing system, we have to be careful with the location of the photodetector. It has to be downstream enough to ensure a good reading of the mixed stream. However, if the photodetector is too far downstream, the larger transport lag can destabilize the system. The water п¬‚ow rate in the pipe is 2 L/s and the pipe has a cross-sectional area of 5 cm2 . The regulating valve is especially designed so that the dye dosage, in milliliters per second, varies linearly with the valve position. The regulating valve is thus п¬Ѓrst order with a time constant of 0.2 s and a steady-state gain of 0.6 mL sв€’1 mVв€’1 . The mixing process itself can also be modeled as п¬Ѓrst order with a steady-state gain of 0.8 ppm s mLв€’1 (where ppm indicates parts per million). A previous experiment indicated that a step change in the regulating valve resulted in a response in a dye concentration that is 99.3% complete in 20 s. The magic photodetector is extremely fast, and the response is linear over a large concentration range. The manufacturer provided the calibration as v = 0.3 + 2.6 (dye), where the voltage output is in millivolts and the concentration of the dye is in parts per million. This is a problem that we have to revisit many times in later chapters. For now, letвЂ™s draw the block diagram of the dye control system and provide the necessary transfer functions. We identify units in the diagram and any possible disturbances to the system. In addition, CT CC Dye injection Mixers Q L Figure E5.7A. 101 Analysis of Single-Loop Control Systems Q L/s GL R Km + mV mV Gc mL/s Ga Gp вЂ“ ppm mV + + C ppm Gm Figure E5.7B. we do not know where to put the photodetector at this point. LetвЂ™s just presume that the photodetector is placed 290 cm downstream. The block diagram is shown in Fig. E5.7B, where the dye concentration is denoted by C and the set point by R. The п¬‚ow rate is one probable source of disturbance. Based on the information given, the transfer functions are Gp = Kp Kv , Ga = , G m = K m eв€’td s , П„ps + 1 П„v s + 1 and we do not know the steady-state gain of the load function G L . The values of various parameters are K p = 0.8 ppm s mLв€’1 , П„ p в‰€ 20/5 = 4 s, K a = 0.6 mL sв€’1 mVв€’1 , П„a = 0.2 s, and K m = 2.6 mV/ppm. The average п¬‚uid velocity is 2000/5 = 400 cm/s. The transport lag is hence td = 290/400 = 0.725 s. We presumably will use a PID transfer function for the controller G c ; we will continue with this problem in Example 5.7A. 5.4. Selection and Action of Controllers A few words need to be added on the action of controllers. The point to make is that we have to do a bit of physical reasoning when we design a real system. We also have to factor in safety and determine what the controller and actuator may do if there is a system failure. A standard textbook system has a controller with a positive proportional gain. All the other blocks such as the process and actuator have positive steady-state gains as well. However, this is not always the case. Here liquid-level control is used to illustrate the idea. Keep Fig. 5.7 in mind in the discussion below. Say we want to control the liquid level in a tank by manipulating the inlet п¬‚ow rate (Fig. 5.8). If the liquid level drops below the set point, the controller will increase its output signal to open up the inlet valve and increase liquid п¬‚ow. The changes in controlled variable and controller output are in opposite directions. This is a consequence of how the error is deп¬Ѓned in a negative-feedback system. R + E в€’ Kc Kv Kp C Figure 5.7. Simple system used in the discussion of controller actions. Depending on the situation, K c , K v , and K p can be either positive or negative. 102 5.4. Selection and Action of Controllers qi R p h LC LT h s q Figure 5.8. Manipulating the liquid level with an inlet valve. In this particular case, we use an air-to-open valve, meaning that we need to increase the signal to open up the valve. That is, the valve has a positive steady-state gain (+K v ). A pneumatic air-to-open valve also means that energy is required for keeping it open. Under a system failure in which power is lost, the valve closes and prevents п¬‚ooding the tank. We refer to the valve here as a fail-close valve, which is the preferred safety design in Fig. 5.8. The opposite case is to use an air-to-close valve, which has a negative steady-state gain (в€’K v ); an increase in signal will close the valve.21 Hence this is a fail-open valve, which, for safety reasons, is not a wise choice here. Nevertheless, if we insist on installing this air-to-close valve, we will need a controller with a negative gain (в€’K c ). Now, if the liquid level drops, the controller output signal will also decrease, opening up the air-to-close valve. LetвЂ™s take a look at the logic when we control the liquid level by manipulating the outlet valve (Fig. 5.9). In this case the process gain K p associated with the outlet п¬‚ow is negative. If the liquid level drops below the set point, we now want to reduce the outlet п¬‚ow rate by closing up the valve. Again, there are two possible cases. If we use a controller with positive gain (+K c ), the controller output increases as the liquid level drops. We can reduce the п¬‚ow only if we use an air-to-close valve (в€’K v ). In case of power outage, the valve will stay open. This fail-open valve can drain the entire tank, an event that we may not want to happen. On the other hand, we can choose an air-to-open valve (+K v ) at the outlet location. Now the only way to reduce the п¬‚ow rate as the liquid level drops is to вЂњswitchвЂќ to a controller qi h s h LT LC p q R Figure 5.9. Manipulating the liquid level with an outlet valve. The process gain is negative in this case. 21 This point can easily get lost in the long explanation: An air-to-open valve has a positive gain and is failed close. An air-to-close valve has a negative gain (в€’K v ) and is failed open. 103 Analysis of Single-Loop Control Systems with a negative gain (в€’K c ). With в€’K c as the proportional gain, a drop in liquid level will lead to a decrease in controller output. In this case, we have a fail-close valve, which is desirable if we do not want perturbations to propagate downstream. There is no question that the terminology is confusing. Do not let it confuse you. The best strategy is to вЂњwalkвЂќ through the sign (action) of every single steady-state gain of the block diagram, including the process and the sensor, and determine the probable and logical signs (actions) of the controller and the actuator. As a consistency check, we should note that within the feedback loop there should only be one net negative sign. There is no getting away from doing some thinking of your own. 5.4.1. A Few Comments on the Choice of Controllers In process engineering, the most common types of controlled variables are liquid level, п¬‚ow rate, temperature, pressure, and sometimes concentration. Here are some very general ideas. To fully appreciate these tidbits, we also need to know something about the actual hardware вЂ“ actuators or control elements вЂ“ that we п¬Ѓnd in handbooks or equipment manuals. Flow Control PI controllers are the most common. They eliminate offsets and have acceptable speeds of response in most industrial settings. We usually pick a low to intermediate gain (wide proportional band, PB в‰€ 150) to reduce the effect of noisy signals (from п¬‚ow turbulence; also why we do not use D control). We also use a low reset time (в‰€ 0.1 min/repeat; i.e., relatively large I action) to get fast set-point tracking. Level Control We usually need to keep the liquid level within a certain range only around the desired set point. Speed is not a great concern. Recall that, depending on how we implement the outlet pump, we can have a process with integrating action itself. Also, if the outlet п¬‚ow rate is used as the manipulated variable, the controller setting must be conservative to avoid sudden surges in the exit п¬‚ow rate. Thus a simple P controller is usually adequate, presuming that we have no complications such as boiling or vaporization. Make sure you check whether the valve that you are using (or buying) is air-to-open or air-to-close. Pressure Control The control of pressure depends on the situation and cannot be generalized. We can see this from a couple of examples: For the vapor pressure in a п¬‚ash drum (and thus also vapor п¬‚ow rate), we need a fast and tight response loop. We need at least a PI controller as in п¬‚ow control. For the top of a distillation column, we usually control the pressure indirectly by means of the condensation of vapor in the condenser, which in turn is controlled by the amount of cooling water. Heat transfer through a heat exchanger has very slow dynamics. Thus we cannot use PI control. We either use P control or, when response time is important, PID. Temperature Control Heat transfer lags can be signiп¬Ѓcant, and the nature of the problem can be quite different in various processes. If there is a sensor lag, it is mostly due to heat transfer between the sensor and the п¬‚uid medium. (Thermocouples, depending on how we make them, can have very 104 5.4. Selection and Action of Controllers L G*L R + Km вЂ“ Gc + Ga + C Gp Gm Figure R5.1. fast response times.) The overall response is sluggish, and PI control will make it more so. It is unlikely we can live with any offsets. PID control is the appropriate choice. Concentration Control The strategy depends on the situation and how we measure the concentration. If we can rely on pH or absorbance (UV, visible, or IR spectrometer), the sensor response time can be reasonably fast, and we can make our decision based on the actual process dynamics. Most likely we would be thinking along the lines of PI or PID controllers. If we can use only gas chromatography (GC) or other slow analytical methods to measure concentration, we must consider discrete data-sampling control. Indeed, prevalent time delay makes chemical process control unique and, in a sense, more difп¬Ѓcult than many mechanical or electrical systems. In terms of the situation, if we use a PI controller on a slow multicapacity process, the resulting system response will be even more sluggish. We should use PID control to increase the speed of the closed-loop response (being able to use a higher proportional gain) while maintaining stability and robustness. This comment applies to other cases such as temperature control as well. Review Problems (1) An alternative way of drawing a feedback system block diagram is shown in Fig. R5.1. How is G в€—L related to G L as in Fig. 5.4? (2) Try to obtain the closed-loop transfer functions in Eq. (5.11) by means of observation, i.e., without using the algebraic steps in the text. (3) Consider the liquid п¬‚ow-rate controller in Fig. R5.3. We want to keep the п¬‚ow rate q constant no matter how the upstream pressure п¬‚uctuates. Consider what happens if the upstream п¬‚ow Q drops below the steady-state value. How would you choose the regulating valve when you have (a) a positive and (b) a negative proportional gain? (4) What is the OS and DR if we pick О¶ = 0.707? If the DR is 1/4, what is the damping ratio? FC Q FT P q Figure R5.3. 105 Analysis of Single-Loop Control Systems (5) Refer back to Example 5.1. What is the offset if we consider the regulating problem (R = 0, L = 1/s)? (6) When we developed the model for the stirred-tank heater, we ignored the dynamics of the heating coil. Provide a slightly more realistic model that takes into consideration the п¬‚ow rate of condensing steam. (7) Do the time-domain simulations with MATLAB in Example 5.6. Try also with a PD or a PID controller. Hints: (1) G в€—L = G L /G p . (2) The G sp function is obvious. To obtain the load function G load , set R = 0, and try to visualize the block diagram such that it is unity in the forward path and all the functions are in the feedback loop. (4) OS = 4.32% and DR = 1.87 Г— 10в€’3 . When DR = 0.25, О¶ = 0.215. (5) Now, C= GL L, 1 + GcG p R = 0, and thus E = Rв€’C =0в€’ GL L. 1 + GcG p With L = 1/s and the п¬Ѓnal-value theorem, GL . sв†’0 1 + G c G p e(в€ћ) = в€’ lim Substitution of the п¬Ѓrst-order functions and a proportional controller gives e(в€ћ) = в€’ KL , 1 + Kc K p which becomes smaller if we increase K c . (6) How we model the stirred-tank heater is subject to the actual situation. At a slightly more realistic level, we may assume that heat is provided by condensing steam and that the coil metal is at the same temperature as the condensing steam. The heat balance and the Laplace transform of the tank remains identical to those given in Chap. 2: ПЃC p V dT = ПЃC p Q(Ti в€’ T ) + U A(TH в€’ T ), dt T = Kd Ti + П„ps + 1 Kp TH . П„ps + 1 We also need a heat balance for the heating coil, which can be written as dTH = m s О» в€’ U A(TH в€’ T ), dt where TH is the temperature and M H and C H are the mass and the heat capacity, respectively, of the heating coil. The steam mass п¬‚ow rate is m s , and О» is the heat of MH C H 106 5.4. Selection and Action of Controllers condensation. We should obtain the Laplace transform of the form TH = 1 T+ П„H s + 1 Ks П„H s + 1 Ms . You should be able to п¬Ѓll in the gaps and п¬Ѓnish the rest of the work in deriving the transfer functions. In this case, you may want to use the steam mass п¬‚ow rate as the manipulated variable. The transfer function relating its effect on T will be second order, and the characteristic polynomial does not have the clean form in our simpler examples. (7) The basic statements are provided already in the example. For more details, see the Web Support for the MATLAB statements and plots. 107 6 Design and Tuning of Single-Loop Control Systems e will go through a whole bundle of tuning methods. We only need to вЂњpickвЂќ three numbers for a PID controller, but this is one of the most confusing parts of learning control. Different tuning techniques give similar but not identical results. There are no вЂњbestвЂќ or вЂњabsolutely correctвЂќ answers. The methods all have pros and cons and, working together, they complement each other. We need to make proper selection and sound judgment вЂ“ very true to the act (and art) of design. W What Are We Up to? r Tuning a controller with empirical relations r Tuning a controller with internal model control relations 6.1. Tuning Controllers with Empirical Relations LetвЂ™s presume that we have selected the valves and the transducers and even installed a controller. We now need to determine the controller settings вЂ“ a practice that is called tuning a controller. Trial-and-error tuning can be extremely time consuming (and dumb!), to the extent that it may not be done. A large distillation column can take hours to reach steady state. A chemical reactor may not reach steady state at all if you have reactor вЂњrunaway.вЂќ Some systems are unstable at high and low feedback gains; they are stable only in some intermediate range. These are reasons why we have to go through all the theories to learn how to design and tune a controller with well-educated (or so we hope) guesses. Empirical tuning roughly involves doing either an open-loop or a closed-loop experiment and п¬Ѓtting the response to a model. The controller gains are calculated on the basis of this п¬Ѓtted function and some empirical relations. When we use empirical tuning relations, we cannot dictate system dynamic response speciп¬Ѓcations. The controller settings are seldom optimal and most often require п¬Ѓeld tuning after installation to meet more precise dynamic response speciп¬Ѓcations. Empirical tuning may not be appealing from a theoretical viewpoint, but it gives us a quick-and-dirty starting point. Two remarks before we begin. 108 6.1. Tuning Controllers with Empirical Relations r Most empirical tuning relations that we use here are based on open-loop data п¬Ѓtted to a п¬Ѓrst-order with dead-time transfer function. This feature is unique to process engineering, in which most units are self-regulating. The dead time is either an approximation of multistage processes or a result of transport lag in the measurement. With large uncertainties and the need for п¬Ѓeld tuning, models more elaborate than the п¬Ѓrst-order with dead-time function are usually not warranted with empirical tuning. r Some empirical tuning relations, such as that of Cohen and Coon, are developed to achieve a one-quarter DR response in handling disturbances. When we apply the settings of these relations to a servo problem, it tends to be very oscillatory and is not what is considered as slightly underdamped.1 The controller design depends on the speciп¬Ѓc problem at hand. We certainly need to know how to tune a controller after using empirical tuning relations to select the initial settings.2 6.1.1. Controller Settings Based on a Process-Reaction Curve To make use of empirical tuning relations, one approach is to obtain the so-called process-reaction curve. We disable the controller and introduce a step change to the actuator. We then measure the open-loop step response. This practice can simply be called an open-loop step test. Although we disconnect the controller in the schematic diagram (Fig. 6.1), we usually only need to turn the controller to the вЂњmanualвЂќ mode in reality. As shown in the block diagram, what we measure is a lumped response, representing the dynamics of the blocks G a , G p , and G m . We denote the lumped function as G PRC , the process-reaction-curve function: G PRC = Cm = Ga G p Gm . P (6.1) From the perspective of doing the experiment, we need the actuator to effect a change in the manipulated variable and the sensor to measure the response. L=0 Process Step input P = R Km Rm + вЂ“ M s Gc Measure C m GL Ga Gm Gp + C + C Figure 6.1. Block-diagram illustration of an open-loop step test. 1 2 If we assume that an oscillatory system response can be п¬Ѓtted to a second-order underdamped function. With Eq. (3.29), we can calculate that with a DR of 0.25, the damping ratio О¶ is 0.215, and the maximum percent OS is 50%, which is not insigniп¬Ѓcant. (These values came from Revew Problem (4) back in Chap. 5.) By and large, a quarter DR response is acceptable for disturbances but not desirable for set-point changes. Theoretically, we can pick any DR of our liking. Recall from Section 2.7 that the position of the closed-loop pole lies on a line governed by Оё = cosв€’1 О¶ . In the next chapter, we will locate the pole position on a root-locus plot based on a given damping ratio. 109 Design and Tuning of Single-Loop Control Systems K y 0 td П„ t Figure 6.2. Illustration of п¬Ѓtting Eq. (6.2) (solid curve) to open-loop step-test data representative of self-regulating and multicapacity processes (dotted curve). The time-constant estimation shown here is based on the initial slope and a visual estimation of dead time. The ZieglerвЂ“Nichols tuning relation (see Table 6.2) also uses the slope through the inп¬‚ection point of the data (not shown). Alternative estimation methods are provided on the Web Support. The measurement of G PRC is how we may design a system if we know little about our process and are incapable of constructing a model (what an excuse!). Even if we know what the functions G a and G m should be, we do not need them because the controller empirical tuning relations were developed for the lumped function G PRC . On the other hand, if we know precisely what the functions G a , G p , and G m are, we may use them to derive G PRC as a reduced-order approximation of the product of G a G p G m . The real-time data (the process-reaction curve) in most processing unit operations take the form of a sigmoidal curve, which is п¬Ѓtted to a п¬Ѓrst-order with dead-time function (Fig. 6.2): G PRC = K eв€’td s Cm в‰€ . P П„s + 1 (6.2) One reason why this approximation works is that process unit operations are generally openloop stable, and many are multicapacity in nature. Reminder: An underdamped response of the system is due to the controller, which is taken out in the open-loop step test. Using the п¬Ѓrst-order with dead-time function, we can go ahead and determine the controller settings with empirical tuning relations. The most common ones are the ZieglerвЂ“ Nichols relations. In process unit operation applications, we can also use the CohenвЂ“Coon or the CianconeвЂ“Marlin relations. These relations are listed in Table 6.1. 6.1.2. Minimum Error-Integral Criteria The open-loop test response п¬Ѓtted to a п¬Ѓrst-order with dead-time function G PRC can be applied to other tuning relations. One such possibility is a set of relations derived from the minimization of error integrals. Here, only the basic idea behind the use of error integrals is provided. To derive the tuning equations, we would use the theoretical time-domain closed-loop system response as opposed to a single quantity, such as the DR. The time-domain solution 110 6.1. Tuning Controllers with Empirical Relations is dependent on the type of controller and the nature of the input (set-point or disturbance changes) and, in our case, a вЂњprocessвЂќ function that is п¬Ѓrst order with dead-time. We can also calculate the error вЂ“ the difference between the set point and the controlled variable. We then п¬Ѓnd controller settings that may minimize the error over time (the error integral), using for instance, Lagrange multipliers as in introductory calculus. Of course, we are not doing the work; the actual analysis is better left for a course in optimal control. There are different ways to deп¬Ѓne the error function to be minimized. A few possibilities are as follows: (1) Integral of the square error (ISE): в€ћ ISE = [e (t)]2 dt. (6.3) 0 The ISE magniп¬Ѓes large errors вЂ“ squaring a small number (< 1) makes it even smaller. Thus minimization of this integral should help to suppress large initial errors. The resulting controller setting tends to have a high proportional gain and the system is very underdamped. (2) Integral of the absolute error (IAE): в€ћ IAE = |e (t)| dt. (6.4) 0 The IAE simply integrates the absolute value and puts equal weight to large and small errors. (3) Integral of the time-weighted absolute error (ITAE): в€ћ ITAE = t|e (t)| dt. (6.5) 0 The time-weighting function puts a heavy penalty on errors that persist for long periods of time. This weighting function also helps to derive controller settings that allow for low settling times. Before we move on, a few comments and reminders: r As far as we are concerned, using the error-integral criteria is just another empirical method. We are simply using the results of minimization obtained by other people, not to mention that the п¬Ѓrst-order with dead-time function is from an open-loop test. r The controller setting is different depending on which error integral we minimize. Set-point and disturbance inputs have different differential equations, and because the optimization calculation depends on the time-domain solution, the result will depend on the type of input. The closed-loop poles are the same, but the zeros, which affect the time-independent coefп¬Ѓcients, are not. r The time integral is from t = 0 to t = в€ћ, and we can minimize it only if it is bounded. In other words, we cannot minimize the integral if there is a steady-state error. Only PI and PID controllers are applicable to this design method.3 r Theoretically, we can minimize the integral by using other criteria. On the whole, the controller settings based on minimizing the ITAE provide the most conservative 3 If you come across a proportional controller here, it is possible only if the derivation has ignored the steady-state error or shifted the reference such that the so-called offset is zero. 111 Design and Tuning of Single-Loop Control Systems controller design and are highly recommended. This is the only set of tuning relations included in Table 6.1. 6.1.3. ZieglerвЂ“Nichols Ultimate-Cycle Method This empirical method is based on closed-loop testing (also called on-line tuning) of processes that are inherently stable, but in which the system may become unstable. We use only proportional control in the experiment. If it is not possible to disable the integral and derivative control modes, we set the integral time to its maximum value and the derivative time to its minimum. The proportional gain is slowly increased until the system begins to exhibit sustained oscillations with a given small-step set-point or load change. The proportional gain and the period of oscillation at this point are the ultimate gain K cu and the ultimate period Tu . These two quantities are used in a set of empirical tuning relations developed by Ziegler and Nichols вЂ“ again listed in Table 6.1. Two more comments: r A preview: We can derive the ultimate gain and the ultimate period (or frequency) with stability analyses. In Chap. 7, we use the substitution s = jП‰ in the closed-loop characteristic equation. In Chap. 8, we make use of what is called the Nyquist stability criterion and Bode plots. r One may question the meaning of вЂњsustained oscillations.вЂќ We may gather that the ultimate gain and ultimate period are associated with marginal stability вЂ“ the instance when the system is just about to become unstable. Of course, we never want to push that far in real life. With large uncertainties involved in empirical tuning and п¬Ѓeld tuning, it is not necessary to have accurate measurements of K cu and Tu . When we do an experiment, we just increase the proportional gain until we achieve a fairly underdamped response. Example 5.7A: What would be the PID controller settings for the dye-mixing problem in Example 5.7? Based on what we have obtained in Example 5.7, if we did an open-loop experiment as suggested in Eq. (6.1), our step response would п¬Ѓt well to the function: G PRC = G a G p G m = (0.8)(0.6)(2.6)eв€’0.725s . (4s + 1)(0.2s + 1) However, to use the empirical tuning relations, we need to п¬Ѓt the data to a п¬Ѓrst-order function with dead-time. Thus at this stage, we probably would have obtained the following approximation: G PRC в‰€ 1.25eв€’0.9s . 4s + 1 Here we assume that the data п¬Ѓtting allows us to recover the time constant of the dominant pole reasonably well, and the dead-time is roughly 0.9s. We are not adding exactly 0.2 to 0.725 as a way to emphasize that, in reality, we would be doing data п¬Ѓtting and the result will vary. How good an approximation is depends very much on the relative differences in the time constants. (Try with MATLAB simulation to see how good the approximation is. For the numbers chosen in this example, it is easy to obtain a good approximation.) 112 6.1. Tuning Controllers with Empirical Relations Now, with Table 6.1,4 we can calculate the following PID controller settings: CohenвЂ“Coon ZieglerвЂ“Nichols ITAE (set point) CianconeвЂ“Marlin (set point) Kc П„I П„D 4.9 4.3 2.8 1.2 2.0 1.8 3.1 4.4 0.31 0.45 0.31 0.07 All tuning relations provide different results. Generally, the CohenвЂ“Coon relation has the largest proportional gain and the dynamic response tends to be the most underdamped. The CianconeвЂ“Marlin relation provides the most conservative setting, and it uses a very small derivative time constant and a relatively large integral time constant. In a way, their correlation reп¬‚ects a common industrial preference for PI controllers. WeвЂ™ll see how they compare in time response simulations when we come back to this problem in Example 5.7C. A point to be made is that empirical tuning is a very imprecise science. There is no reason to worry about the third signiп¬Ѓcant п¬Ѓgure in your tuning parameters. The calculation only serves to provide us with an initial setting with which we begin to do п¬Ѓeld or computational tuning. Although the calculations in the preceding example may appear as simple plug-and-chug, we should take a closer look at the tuning relations. The CohenвЂ“Coon equations for the proportional gain taken from Table 6.1 are П„ 1 , + td 3 P: Kc K = PI: K c K = 0.9 PID: K c K = П„ 1 , + td 12 4П„ 1 . + 3 td 4 (6.6) (6.7a) (6.8a) The choice of the proportional gain is affected by two quantities: the product K c K and the ratio of dead time to time constant td /П„ . It may not be obvious why the product K c K is important now, but we shall see in Section 6.2 how it arises from direct synthesis and appreciate in Chap. 8 how it helps determine system stability. Under circumstances in which the dead time is relatively small, only the п¬Ѓrst term on the RHS is important in the three tuning equations. When dead time becomes larger (or П„/td smaller), we need to decrease the proportional gain, and this is how the tuning relations are constructed. When we add integral control, we need to decrease K c . Indeed, in Eq. (6.7a), the П„/td term is decreased by 10% and the constant term is reduced to 1/12. With the implementation of PID control, we can afford to have a larger K c . This is reп¬‚ected in Eq. (6.8a). We can make similar observations with the ZieglerвЂ“Nichols relations in Table 6.1. Furthermore, we may also see in Table 6.1 that if the dead-time increases, we should raise the integral time constant. 4 Really calculated with our M-п¬Ѓle recipe.m, which can be found on our Web Support. 113 Design and Tuning of Single-Loop Control Systems Table 6.1. Table of tuning relations A. Tuning relations based on open-loop testing and response п¬Ѓtted to a п¬Ѓrst-order with dead-time function: G PRC = K eв€’td s П„s + 1 Controller Cohen вЂ“ Coon P Kc K = PI K c K = 0.9 П„ 1 + td 3 Kc K = (6.6) П„ 1 + td 12 П„ td (6.9) П„ td (6.7a) K c K = 0.9 30 + 3(td /П„ ) 9 + 20(td /П„ ) (6.7b) П„ I = 3.3td 4П„ 1 + 3 td 4 (6.8a) K c K = 1.2 П„ I = td 32 + 6(td /П„ ) 13 + 8(td /П„ ) (6.8b) П„ I = 2td (6.11b) П„ D = td 4 11 + 2(td /П„ ) (6.8c) П„ D = 0.5td (6.11c) П„ I = td Kc K = PID Ziegler вЂ“ Nichols (6.10a) (6.10b) П„ td (6.11a) Minimum ITAE criterion For load change: Kc = a1 K П„ td b1 , П„I = П„ a2 td П„ b2 , П„ D = a3 П„ td П„ b3 . (6.12) Controller a1 b1 a2 b2 a3 b3 PI PID 0.859 1.357 0.977 0.947 0.674 0.842 0.680 0.738 вЂ” 0.381 вЂ” 0.995 For set-point change: Kc = 114 a1 П„ K td b1 , П„I = П„ , a2 в€’ b2 (td /П„ ) П„ D = a3 П„ td П„ b3 . (6.13) Controller a1 b1 a2 b2 a3 b3 PI PID 0.586 0.965 0.916 0.855 1.03 0.796 0.165 0.147 вЂ” 0.308 вЂ” 0.929 6.2. Direct Synthesis and Internal Model Control B. Tuning relations based on closed-loop testing and the ZieglerвЂ“Nichols ultimate-gain (cycle) method with given ultimate proportional gain K cu and ultimate period Tu . ZieglerвЂ“Nichols ultimate-gain method Controller P PI K c = 0.5K cu K c = 0.455K cu П„ I = 0.833Tu PID Quarter decay K c = 0.6K cu П„ I = 0.5Tu П„ D = 0.125 Tu (6.14) (6.15a) (6.15b) Just a bit of overshoot (6.16a) (6.16b) (6.16c) K c = 0.33K cu П„ I = 0.5Tu П„ D = 0.333Tu No overshoot (6.17a) (6.17b) (6.17c) K c = 0.2K cu П„ I = 0.5Tu П„ D = 0.333Tu (6.18a) (6.18b) (6.18c) Note: All formulas in Table 6.1, and the PID settings in Table 6.2 later, are implemented in the M-п¬Ѓle recipe.m, available from our Web Support. The Ciancone and Marlin tuning relations are graphical, and we have omitted them from the tables. The correlation plots, explanations, and the interpolation calculations are provided by our M-п¬Ѓle ciancone.m, which is also used by recipe.m. 6.2. Direct Synthesis and Internal Model Control We now apply a different philosophy to controller design. Up until now, we have had a preconceived idea of what a controller should be, and we tune it until we have the desired system response. On the other hand, we can be more proactive: We deп¬Ѓne what our desired closed-loop response should be and design the controller accordingly. The resulting controller is not necessarily a PID controller. This is acceptable with computer-based controllers because we are not restricted to off-the-shelf hardware. In this chapter, however, our objective is more restricted. We purposely choose simple cases and make simplifying assumptions such that the results are PID controllers. We will see how the method helps us select controller gains based on process parameters (i.e., the process model). The method provides us with a more rational controller design than do the empirical tuning relations. Because the result depends on the process model, this method is what we consider a model-based design. 6.2.1. Direct Synthesis We consider a servo problem (i.e., L = 0) and set G m = G a = 1. The closed-loop function is the familiar GcG p C = , R 1 + GcG p (6.19) which we now rearrange as Gc = 1 C/R . G p 1 в€’ C/R (6.20) 115 Design and Tuning of Single-Loop Control Systems The implication is that if we deп¬Ѓne our desired system response C/R, we can derive the appropriate controller function for a speciп¬Ѓc process function G p . A couple of quick observations: First, G c is the reciprocal of G p . The poles of G p are related to the zeros of G c and vice versa вЂ“ this is the basis of the so-called pole-zero cancellation.5 Second, the choice of C/R is not entirely arbitrary; it must satisfy the closedloop characteristic equation: 1 + GcG p = 1 + C/R 1 в€’ C/R = 0. (6.21) From Eq. (6.20), it is immediately clear that we cannot have an ideal servo response where C/R = 1, which would require an inп¬Ѓnite controller gain. Now Eq. (6.21) indicates that C/R cannot be some constant either. To satisfy Eq. (6.21), the closed-loop response C/R must be some function of s, meaning that the system cannot respond instantaneously and must have some п¬Ѓnite response time. LetвЂ™s select a more realistic system response, say, a simple п¬Ѓrst-order function with unity steady-state gain: C 1 = , R П„c s + 1 (6.22) where П„c is the system time constant, a design parameter that we specify. The unity gain means that we should eliminate offset in the system. Substitution of Eq. (6.22) into Eq. (6.20) leads to the controller function: Gc = 1 1 . G p П„c s (6.23) The closed-loop characteristic equation, corresponding to Eq. (6.21), is 1+ 1 = 0, П„c s (6.24) which really is 1 + П„c s = 0, as dictated by Eq. (6.22). The closed-loop pole is at s = в€’1/П„c . This result is true no matter what G p is вЂ“ as long as we can physically build or program the controller on a computer. Because the system time constant П„c is our design parameter, it appears that direct synthesis magically allows us to select whatever response time we want. Of course this cannot be the case in reality. There are physical limitations, such as saturation. Example 6.1: Derive the controller function for a system with a п¬Ѓrst-order process and a system response dictated by Eq. (6.22). The process transfer function is G p = [K p /(П„ p s + 1)], and the controller function according to Eq. (6.23) is Gc = 5 116 П„p (П„ p s + 1) 1 1 = , 1+ Kp П„c s K p П„c П„ps (E6.1) The controller function will take on a positive pole if the process function has a positive zero. It is not desirable to have an inherently unstable element in our control loop. This is an issue that internal model control will address. 6.2. Direct Synthesis and Internal Model Control Table 6.2. Summary of PID controller settings based on internal model control or direct synthesis Process model Controller Kc П„I П„D Kp П„ps + 1 PI П„p K p П„c П„p вЂ” Kp (П„1 s + 1)(П„2 s + 1) PID П„1 + П„2 П„1 П„2 П„1 + П„2 П„1 П„2 П„1 вЂ” PID with П„1 > П„2 PI (underdamped) П„1 + П„2 K p П„c П„1 K p П„c П„1 4K p О¶ 2 П„2 Kp П„ 2 s 2 + 2О¶ П„ s + 1 PID 2О¶ П„ K p П„c 2О¶ П„ П„ 2О¶ Kp s(П„ p s + 1) PD 1 K p П„c вЂ” П„p K p eв€’td s П„ps + 1 PI П„p K p (П„c + td ) П„p вЂ” PID 1 2П„ p /td + 1 K p 2П„c /td + 1 П„ p + td /2 П„p 2П„ p /td + 1 P 1 K p П„c вЂ” вЂ” Kp s which is obviously a PI controller with K c = П„ p /K p П„c and П„ I = П„ p . Note that the proportional gain is inversely proportional to the process gain. Speciп¬Ѓcation of a small system time constant П„c also leads to a large proportional gain. Reminder: The controller settings K c and П„ I are governed by the process parameters, and the system response, which we choose. The one and only tuning parameter is the system response time constant П„c . Example 6.2: Derive the controller function for a system with a second-order overdamped process and system response as dictated by Eq. (6.22). The process transfer function is G p = {K p /[(П„1 s + 1)(П„2 s + 1)]}, and the controller function according to Eq. (6.23) is Gc = (П„1 s + 1)(П„2 s + 1) 1 . Kp П„c s We may see that this is a PID controller. Nevertheless, there are two ways to manipulate the function. One is to expand the terms in the numerator and factor out (П„1 + П„2 ) to obtain Gc = 1 (П„1 + П„2 ) 1 1+ + K p П„c (П„1 + П„2 ) s П„1 П„2 s . П„ 1 + П„2 (E6.2) The proportional gain, integral time, and derivative time constants are provided by the respective terms in the transfer function. If you have trouble spotting them, they are summarized in Table 6.2. 117 Design and Tuning of Single-Loop Control Systems The second approach is to consider the controller function as a series PID such that we write Gc = П„1 1 1+ (П„2 s + 1), K p П„c П„1 s (E6.3) with П„1 > П„2 . We can modify the derivative term to be the вЂњrealвЂќ derivative action as written in Eqs. (5.9a) and (5.9b). From our experience that the derivative time constant should be smaller than the integral time constant, we should pick the larger time constant as the integral time constant. Thus we select П„1 to be the integral time constant and П„2 to be the derivative time constant. In the limit П„1 П„2 , both arrangements [Eqs. (E6.2) and (E6.3)] of the controller function are the same. When dead-time is inherent in a process, it is difп¬Ѓcult to avoid dead-time in the system. Thus we deп¬Ѓne the system response as C eОёs = , R П„c s + 1 (6.25) where Оё is the dead-time in the system. The controller function, by means of Eq. (6.20), is hence Gc = 1 Gp eв€’Оё s (П„c s + 1) в€’ eв€’Оё s в‰€ 1 eв€’Оёs . G p (П„c + Оё)s (6.26) To arrive at the last term, we use a simple Taylor expansion (eв€’Оёs в‰€ 1 в€’ Оёs) of the exponential term. This is purposely done to simplify the algebra, as shown in Example 6.3. [We could have used the PadВґe approximation in Eq. (6.26), but the result will not be the simple PI controller.] Example 6.3: Derive the controller function for a system with a п¬Ѓrst-order process with dead-time and system response as dictated by Eq. (6.25). The process transfer function is G p = [(K p eв€’td s )/(П„ p s + 1)]. To apply Eq. (6.26), we make an assumption about the dead time, that Оё = td . The result is a PI controller: Gc = П„p 1 1+ . K p (П„c + Оё) П„ps (E6.4) Even though this result is based on what we say is a process function, we could apply Eq. (E6.4) as if the derivation were for the п¬Ѓrst-order with dead-time function G PRC obtained from an open-loop step test. This is a question that invariably arises: What is a reasonable choice of the system time constant П„c ? Various sources in the literature have different recommendations. For example, one guideline suggests that we need to ensure that П„c > 1.7Оё for a PI controller and П„c > 0.25Оё for a PID controller. A reasonably conservative choice has been programmed into the M-п¬Ѓle recipe.m available from the Web Support. The important reminder is that we 118 6.2. Direct Synthesis and Internal Model Control should have a habit of checking the П„c setting with time-response simulation and tuning analysis. In contrast to Eq. (6.22), we can dictate a second-order underdamped system response: 1 C = 2 2 , R П„ s + 2О¶ П„ s + 1 (6.27) where П„ and О¶ are the system natural period and damping ratio yet to be determined. Substitution of Eq. (6.27) into Eq. (6.20) leads to Gc = 1 1 , G p П„ 2 s 2 + 2О¶ П„ s (6.28) which is a slightly more complicated form than that of Eq. (6.23). Again, with simpliп¬Ѓed cases, we can arrive at PID-type controllers. Example 6.4: Derive the controller function for a system with a second-order overdamped process but an underdamped system response as dictated by Eq. (6.27). The process transfer function is Gp = Kp , (П„1 s + 1)(П„2 s + 1) and the controller function according to Eq. (6.28) is Gc = (П„1 s + 1)(П„2 s + 1) . K p П„ s(П„ s + 2О¶ ) We now deп¬Ѓne П„ f = П„/2О¶ , and G c becomes Gc = (П„1 s + 1)(П„2 s + 1) . 2О¶ K p П„ s(П„ f s + 1) Suppose that П„2 is associated with the slower pole (П„2 > П„1 ); we now require П„ f = П„2 such that the pole and the zero cancel each other. The result is a PI controller: Gc = 1 (П„1 s + 1) . 2О¶ K p П„ s With our deп¬Ѓnition of П„ f and the requirement that П„ f = П„2 , we can write П„ = 2О¶ П„2 , and the п¬Ѓnal form of the controller is Gc = 1 П„1 1+ 4K p О¶ 2 П„2 П„1 s = Kc 1 + 1 . П„I s (E6.5) The integral time constant is П„ I = П„1 , and the group multiplying the term in parentheses is the proportional gain K c . In this problem, the system damping ratio О¶ is the only tuning parameter. 6.2.2. Pole-Zero Cancellation We used the term pole-zero cancellation at the beginning of this section. A few more words should be said as we can better appreciate the idea behind direct synthesis. Pole-zero cancellation is also referred to as cancellation compensation or dominant-pole design. Of course, 119 Design and Tuning of Single-Loop Control Systems it is unlikely to have perfect pole-zero cancellation in real life, and this discussion is aimed more toward helping our theoretical understanding. The idea is that we may cancel the (undesirable open-loop) poles of our process and replace them with a desirable closed-loop pole. Recall in Eq. (6.20) that G c is sort of the reciprocal of G p . The zeros of G c are by choice the poles of G p . The product of G c G p cancels everything out вЂ“ hence the term pole-zero cancellation. To be redundant, we can rewrite the general design equation as GcG p = C/R . 1 в€’ C/R (6.20a) That is, no matter what G p is, we deп¬Ѓne G c such that their product is dictated entirely by a function (the RHS) in terms of our desired system response (C/R). For the speciп¬Ѓc closed-loop response as dictated by Eq. (6.22), we can also rewrite Eq. (6.23) as GcG p = 1 . П„c s (6.23a) Because the system characteristic equation is 1 + G c G p = 0, our closed-loop poles are dependent on only our design parameter П„c . A closed-loop system designed on the basis of pole-zero cancellation has drastically different behavior than a system without such cancellation. LetвЂ™s try to illustrate with a PI controller and a п¬Ѓrst-order process function, and the simpliп¬Ѓcation that G m = G a = 1. The closed-loop characteristic equation is 1 + Gc G p = 1 + Kc Kp П„I s + 1 = 0. П„I s П„ps + 1 (6.29) Under normal circumstances, we would pick a П„ I that we deem appropriate. Now if we pick П„ I to be identical to П„ p , the zero of the controller function cancels the pole of the process function. We are left with only one open-loop pole at the origin. Equation (6.29), when П„ I = П„ p , is reduced to 1+ Kc K p Kc K p . = 0, or s = в€’ П„ps П„p There is now only one real and negative closed-loop pole (presuming that K c > 0). This situation is exactly what direct synthesis leads us to. Recall from Example 6.1 that, based on the chosen C/R in Eq. (6.22), the PI controller function is Gc = Kc П„I s + 1 П„I s = П„ps + 1 П„p , K p П„c П„ps where П„ I = П„ p and K c = П„ p /K p П„c . Substitution of K c one step back into the characteristic equation shows that the closed-loop pole is indeed at s = в€’1/П„c . The product G c G p is also consistent with Eq. (6.23a) and П„c . 6.2.3. Internal Model Control A more elegant approach than direct synthesis is internal model control (IMC). The premise of IMC is that, in reality, we have only an approximation of the actual process. Even if we 120 6.2. Direct Synthesis and Internal Model Control L (a) GL R + E G*c вЂ“ P C Gp ~ Gp ~ CвЂ“C ~ C вЂ“ + (b) R + E вЂ“ P Gc C Gp Figure 6.3. A system with (a) IMC compared with (b) a conventional system (b). have the correct model, we may not have accurate measurements of the process parameters. Thus the imperfect model should be factored as part of the controller design. In the block diagram that implements IMC [Fig. 6.3(a)], our conventional controller G c consists of the (theoretical) model controller G в€—c and the approximate function GЛњ p . Again, our objective is limited. We use the analysis in very restrictive and simpliп¬Ѓed cases to arrive at results in Example 6.5 to help us tune PID controllers as in Fig. 6.3(b). We п¬Ѓrst need to derive the closed-loop functions for the system. Based on the block diagram, the error is Лњ E = R в€’ (C в€’ C) and the model controller output is Лњ P = G в€—c E = G в€—c (R в€’ C + C). If we substitute CЛњ = GЛњ p P, we have P = G в€—c (R в€’ C + GЛњ p P), (6.30) from which we can rearrange to obtain P= G в€—c (R в€’ C). 1 в€’ G в€—c GЛњ p (6.28a) The gist of this step is to show the relationship between the conventional controller function G c and the other functions: Gc = G в€—c . 1 в€’ G в€—c GЛњ p (6.31) This is an equation that we will use to retrieve the corresponding PID controller gains. For now, we substitute Eq. (6.28a) in an equation around the process: C = GL L + G p P = GL L + G p G в€—c (R в€’ C). 1 в€’ G в€—c GЛњ p 121 Design and Tuning of Single-Loop Control Systems From this step, we derive the closed-loop equation: C= G p G в€—c (1 в€’ G в€—c GЛњ p )G L L + R. 1 + G в€—c (G p в€’ GЛњ p ) 1 + G в€—c (G p в€’ GЛњ p ) (6.32) The terms in the brackets are the two closed-loop transfer functions. As always, they have the same denominator вЂ“ the closed-loop characteristic polynomial. There is still one piece of unп¬Ѓnished business. We do not know how to choose G в€—c yet. Before we make this decision, we may recall that, in direct synthesis, the poles of G c are вЂњinheritedвЂќ from the zeros of G p . If G p has positive zeros, it will lead to a G c function with positive poles. To avoid that, we вЂњsplitвЂќ the approximate function as a product of two parts: GЛњ p = GЛњ p+ GЛњ pв€’ , (6.33) with GЛњ p+ containing all the positive zeros, if present. The controller will be designed on the basis of GЛњ pв€’ only. We now deп¬Ѓne the model controller function in a way similar to direct synthesis6 : G в€—c = 1 1 Лњ G pв€’ П„c s + 1 r , (6.34) where r = 1, 2, etc. Like direct synthesis, П„c is the closed-loop time constant and our only tuning parameter. The п¬Ѓrst-order function raised to an integer power of r is used to ensure that the controller is physically realizable.7 Again, we would violate this intention in our simple example just so that we can obtain results that resemble an ideal PID controller. Example 6.5: Repeat the derivation of a controller function for a system with a п¬Ѓrst-order process with dead time using an IMC. Say we model our process (read: п¬Ѓtting the open-loop step-test data) as a п¬Ѓrst-order function with time delay; expecting experimental errors or uncertainties, we п¬Ѓnd that our measured or approximate model function GЛњ p is K p eв€’td s GЛњ p = . П„ps + 1 We use the п¬Ѓrst-order PadВґe approximation for the dead-time and isolate the positive zero term as in Eq. (6.33): GЛњ p в‰€ 6 7 122 Kp (П„ p s + 1) t2d s + 1 в€’ td s + 1 = GЛњ pв€’ GЛњ p+ , 2 (E6.6) If the model is perfect, G p = GЛњ p and Eq. (6.32) becomes simply C = G p G в€—c R if we also set L = 0. We choose C/R to be a п¬Ѓrst-order response with unity gain, and we arrive at a choice of G в€—c very similar to the deп¬Ѓnition in Eq. (6.34). The literature refers the term as a п¬Ѓrst-order п¬Ѓlter. It makes sense only if you recall your linear circuit analysis or if you wait until the chapter on frequency-response analysis. 6.2. Direct Synthesis and Internal Model Control where GЛњ p+ = в€’ td s+1 . 2 If we choose r = 1, Eq. (6.34) gives G в€—c = (П„ p s + 1) t2d s + 1 1 . Kp (П„c s + 1) (E6.7) Substitution of Eq. (E6.7) and approximation (E6.6) into Eq. (6.31), after some algebraic work, will lead to the tuning parameters of an ideal PID controller: П„ p П„p 1 2 td + 1 td ; П„I = П„ p + ; П„D = П„p Kc = . П„c K p 2 td + 1 2 2 td + 1 (E6.8) Example 5.7B: What would be the PID controller settings for the dye-mixing problem if we use IMC-based tuning relations? With the same п¬Ѓrst-order with dead-time approximation as that in Example 5.7A, and the choice of П„c being two thirds the value of dead time, the IMC relations in Eqs. (E6.8) provide the following PID settings (as computed with our M-п¬Ѓle recipe.m): IMC Kc П„I П„D 3.4 4.5 0.4 Compare this result by using other tuning relations in Example 5.7A. The IMC proportional gain falls in between the CohenвЂ“Coon and ITAE settings, but the integral time constant is relatively high. With less integrating action, we expect this IMC tuning to be less oscillatory. Indeed, we shall see that when we do Example 5.7C (or you can cheat and read the plotting result from the Web Support). Example 5.7C: How do the different controller settings affect the system time response in the dye-mixing problem? We can use the following MATLAB statements to do time-response simulations (explanations are in MATLAB Session 5). Better yet, save them in an M-п¬Ѓle. The plotting can be handled differently to suit your personal taste. (Of course, you can use Simulink instead.) alfa=0.1; % Real PID Gc=tf(kc*[taui*taud (taui+taud) 1],[alfa*taui*taud taui 0]); td=0.725; Gm=tf([-td/2 1],[td/2 1]); % PadГ© approximation for dead time Km=2.6; % Move Km into the forward path Gp=tf(0.8,[4 1]); Ga=tf(0.6,[0.2 1]); Gcl=feedback(Km*Gc*Ga*Gp,Gm); % The closed-loop function step(Gcl) % Plotting... 123 124 ZieglerвЂ“Nichols continuous cycling (empirical tuning with closed-loop test) Increase proportional gain of only a proportional controller until system sustains oscillation. Measure ultimate gain and ultimate period. Apply empirical design relations. Apply design relations derived from minimization of an error integral of the theoretical time-domain response. вЂў Time integral performance criteria (ISE, IAE, ITAE) вЂў CohenвЂ“Coon вЂў ZieglerвЂ“Nichols вЂў CiaconeвЂ“Marlin Measure open-loop step response, the so-called process-reaction curve. Fit data to п¬Ѓrst-order with dead-time function. Apply empirical design relations. Derive closed-loop damping ratio from a second-order system characteristic polynomial. Relate the damping ratio to the proportional gain of the system. What to do? Empirical tuning with open-loop step test Transient response criteria вЂў Analytical derivation Method Table 6.3. Summary of methods to select controller gains Proportional gain, integral, and derivative time constants of PID controllers Proportional gain, integral, and derivative time constants to PI and PID controllers Proportional gain, integral, and derivative time constants to PI and PID controllers Usually the proportional gain What is evaluated? вЂў Experimental analog of the s = jП‰ substitution calculation вЂў Not necessarily feasible with chemical systems in practice вЂў Tuning relations allow for choices from one-quarter DR to little oscillations. вЂў CohenвЂ“Coon was designed to handle disturbances by preventing a large initial deviation from the set point. The one-quarter DR response is generally too underdamped for set point changes. вЂў Different settings for load and set-point changes вЂў Different settings for different deп¬Ѓnitions of the error integral вЂў The minimum ITAE criterion provides the least oscillatory response вЂў Limited to second-order systems. No unique answer other than a P controller вЂў Theoretically can use other transient response criteria. вЂў 1/4 DR provides a 50% OS. Comments 125 (Model-based) direct synthesis вЂў Root-locus вЂў Direct substitution Stability analysis methods вЂў RouthвЂ“Hurwitz criterion For a given system, synthesize the controller function according to a speciп¬Ѓed closed-loop response. The system time constant, П„c , is the only tuning parameter. Substitute s = jП‰ into characteristic polynomial and solve for closed-loop poles on the Im axis. The Im and the Re parts of the equation allow the ultimate gain and ultimate frequency to be solved. With each chosen value of proportional gain, plot the closed-loop poles. Generate the loci with either hand-sketching or computer. Apply the Routh test on the closed-loop characteristic polynomial to п¬Ѓnd if there are closed-loop poles on the RHP. Proportional gain, integral, and derivative time constants where appropriate The loci of closed-loop poles reveal the effect of controller gain on the probable closed-loop dynamic response. Together with speciп¬Ѓcations of damping ratio and time constant, the loci can be a basis of selecting proportional gain. Ultimate gain and ultimate period (Pu = 2ПЂ/П‰u ) that can be used in the ZieglerвЂ“Nichols continuouscycling relations Establish limits on the controller gain. (cont.) вЂў The design is not necessarily PID, but where the structure of a PID controller results, this method provides insight into the selection of the controller mode (PI, PD, PID) and settings. вЂў Theory is based on pole-zero cancellation, which is rare in practice. вЂў Especially useful with system that has no dead time вЂў Rarely used in the п¬Ѓnal controller design because of difп¬Ѓculty in handling dead time вЂў Method is instructive and great pedagogical tool вЂў Usually applies to relatively simple systems with the focus on the proportional gain вЂў Need be careful on interpretation when the lower limit on proportional gain is negative вЂў Result on ultimate gain is consistent with the Routh array analysis вЂў Limited to relatively simple systems 126 вЂў Maximum closed-loop log modulus вЂў Nichols chart вЂў Nyquist plot вЂў Bode plot Frequency-domain methods вЂў IMC Method Table 6.3. (cont.) Nyquist plot is a frequency parametric plot of the magnitude and the argument of the open-loop transfer function in polar coordinates. Bode plot is magnitude vs. frequency and phase angle vs. frequency plotted individually. Nichols chart is a frequency parametric plot of open-loop function magnitude vs. phase angle. The closed-loop magnitude and phase angle are overlaid as contours. A plot of the magnitude vs. frequency of the closed-loop transfer function Extension of direct synthesis. Controller design includes an internal approximation process function. What to do? The peak amplitude ratio for a chosen proportional gain With gain or phase margin, calculate proportional gain. Can also estimate the peak amplitude ratio, and assess the degree of oscillation. Calculate proportional gain needed to satisfy the gain or phase margin. For a п¬Ѓrst-order function with dead time, the proportional gain, integral, and derivative time constants of an ideal PID controller What is evaluated? вЂў Nichols chart is usually constructed for unity feedback loops only. вЂў Can handle dead time easily and rigorously вЂў The Nyquist criterion allows the use of open-loop functions in Nyquist or Bode plots to analyze the closed-loop problem вЂў The stability criteria have no use for simple п¬Ѓrst- and second-order systems with no positive open-loop zeros. вЂў These plots address the stability problem but need other methods to reveal the probable dynamic response. Comments 6.2. Direct Synthesis and Internal Model Control We reset the three controller parameters each time we execute the M-п¬Ѓle. For example, to use the CohenвЂ“Coon results, we would take from Example 5.7A kc=4.9; taui=2; taud=0.31; MATLAB calculation details and plots can be found on the Web Support. You should observe that CohenвЂ“Coon and ZieglerвЂ“Nichols tuning relations lead to roughly 74% and 64% OS, respectively, which are more signiп¬Ѓcant than what we expect with a quarter DR criterion. The ITAE, with в€ј14% OS, is more conservative. CianconeвЂ“Marlin tuning relations are ultraconservative; the system is slow and overdamped. With the IMC tuning setting in Example 5.7B, the resulting time-response plot is (very nicely) slightly underdamped, even though the derivation in Example 6.4 is predicated on a system response without oscillations. Part of the reason lies in the approximation of the dead-time function and part of the reason is due to how the system time constant was chosen. Generally, it is important to double check IMC settings with simulations. At this point, you may be sufп¬Ѓciently confused with respect to all the different controller tuning methods. Use Table 6.3 as a guide to review and compare different techniques from this chapter and also from Chaps. 7 and 8. Review Problems (1) Repeat Example 6.1 when we have Gp = Kp . s(П„ p s + 1) What is the offset of the system? (2) What are the limitations to an IMC? (Especially with respect to the choice of П„c ?) (3) What control action increases the order of the system? (4) Refer back to Example 6.4. If we have a third-order process, Gp = Kp , (П„1 s + 1)(П„2 s + 1)(П„3 s + 1) what is the controller function if we follow the same strategy as that in the example? (5) Complete the time-response simulations in Example 5.7C using settings in Example 5.7A. (6) (Optional) How would you implement the PID algorithm in a computer program? Hints: (1) The result is an ideal PD controller with the choice of П„ D = П„ p . See that you can obtain the same result with an IMC too. Here, take the process function as the approximate model; it has no parts that we need to consider as having positive zeros. There is no offset; the integrating action is provided by G p . (2) Too small a value of П„c means too large a K c and therefore saturation. System response is subject to imperfect pole-zero cancellation. (3) Integration is 1/s. 127 Design and Tuning of Single-Loop Control Systems (4) The intermediate step is Gc = (П„1 s + 1)(П„2 s + 1)(П„3 s + 1) , 2О¶ K p П„ s(П„ f s + 1) where П„ f = П„/2О¶ , and now we require that П„ f = П„3 , presuming it is the largest time constant. The п¬Ѓnal result, after some of the ideas also taken from Example 6.2, is an ideal PID controller with the form П„1 П„2 (П„1 + П„2 ) 1 1 Gc = + s . 1+ 4K p О¶ 2 П„3 П„1 + П„2 s П„1 + П„2 The necessary choices of K c , П„ I , and П„ D are obvious. Again, О¶ is the only tuning parameter. (5) See the Web Support for the simulations. (6) Use п¬Ѓnite difference. The ideal PID in Eq. (5.8a) can be discretized as pn = p s + K c en + t П„I n ek + k=1 П„D (en в€’ enв€’1 ) , t where pn is the controller output at the nth sampling period, p s is the controller bias, t is the sampling period, and en is the error. This is referred to as the position-form algorithm. The alternative approach is to compute the change in the controller output based on the difference between two samplings: pn = pn в€’ pnв€’1 = K c (en в€’ enв€’1 ) + t П„I en + П„D (en в€’ 2enв€’1 + enв€’2 ) . t This is the velocity-form algorithm, which is considered to be more attractive than the position form. The summation of error is not computed explicitly and thus the velocity form is not as susceptible to reset windup. 128 7 Stability of Closed-Loop Systems hen we design a closed-loop system, the speciп¬Ѓcations may dictate features in dynamic response. However, we cannot do that unless the system is stable. Thus the foremost concern in a control system design is to keep the system stable, which in itself can be used as a design tool. W What Are We Up to? Analyzing stability of a closed-loop system with three techniques: r RouthвЂ“Hurwitz criterion for the stability region r Substitution of s = jП‰ to п¬Ѓnd roots at marginal stability r Root-locus plots of the closed-loop poles 7.1. Deп¬Ѓnition of Stability Our objective is simple. We want to make sure that the controller settings will not lead to an unstable system. Consider the closed-loop system response that we derived in Section 5.2: C= Km Gc Ga G p R+ 1 + Gm Gc Ga G p GL L 1 + Gm Gc Ga G p (7.1) with the characteristic equation 1 + G m G c G a G p = 0. (7.2) The closed-loop system is stable if all the roots of the characteristic polynomial have negative real parts. Or we can say that all the poles of the closed-loop transfer function lie in the lefthand plane (LHP). When we make this statement, the stability of the system is deп¬Ѓned entirely on the inherent dynamics of the system and not on the input functions. In other words, the results apply to both servo and regulating problems. We also see another common deп¬Ѓnition вЂ“ bounded-input bounded-output (BIBO) stability: A system is BIBO stable if the output response is bounded for any bounded input. 129 Stability of Closed-Loop Systems One illustration of this deп¬Ѓnition is to consider a hypothetical situation with a closed-loop pole at the origin. In such a case, we know that if we apply an impulse input or a rectangular pulse input, the response remains bounded. However, if we apply a step input, which is bounded, the response is a ramp, which has no upper bound. For this reason, we cannot accept any control system that has closed-loop poles lying on the imaginary axis. They must be in the LHP.1 The addition of a feedback control loop can stabilize or destabilize a process. We will see plenty examples of the latter. For now, we use the classic example of trying to stabilize an open-loop unstable process. Example 7.1: Consider the unstable process function G p = [K /(s в€’ a)], which may arise from a linearized model of an exothermic chemical reactor with an improper cooling design. The question is whether we can make a stable control system by using simply a proportional controller. For illustration, we consider a unity feedback loop with G m = 1. We also take the actuator transfer function to be unity, G a = 1. With a proportional controller, G c = K c , the transfer function of this closed-loop servo system is GcG p Kc K Y = = . R 1 + GcG p s в€’ a + Kc K The characteristic equation is s в€’ a + K c K = 0, which means that if we want a stable system, the closed-loop poles must satisfy s = a в€’ K c K < 0. In other words, the closed-loop system is stable if K c > a/K . For a more complex problem, the characteristic polynomial will not be as simple, and we need tools to help us. The two techniques that we will learn are the RouthвЂ“Hurwitz criterion and root locus. Root locus is, by far, the more important and useful method, especially when we can use a computer. Where circumstances allow (i.e., the algebra is not too ferocious), we can also п¬Ѓnd the roots on the imaginary axis вЂ“ the case of marginal stability. In the simple example above, this is where K c = a/K . Of course, we have to be smart enough to pick K c > a/K , and not K c < a/K . 7.2. The RouthвЂ“Hurwitz Criterion The time-honored (i.e., ancient!) RouthвЂ“Hurwitz criterion is introduced for stability testing. It is not proved here вЂ“ hardly any text does anymore. Nonetheless, two general polynomials 1 130 Do not be confused by the integral control function; its pole at the origin is an open-loop pole. This point should be cleared up when we get to the root-locus section. Furthermore, conjugate poles on the imaginary axis are BIBO stable вЂ“ a step input leads to a sustained oscillation that is bounded in time. However, we do not consider this oscillatory steady state as stable, and hence we exclude the entire imaginary axis. In an advanced class, you should п¬Ѓnd more mathematical deп¬Ѓnitions of stability. 7.2. The RouthвЂ“Hurwitz Criterion are used to illustrate some simple properties. First, consider a second-order polynomial with the leading coefп¬Ѓcient a2 = 1. If the polynomial has two real poles p1 and p2 , it can be factored as P(s) = s 2 + a1 s + a0 = (s в€’ p1 )(s в€’ p2 ). (7.3) в€љ We may observe that if a1 is zero, both roots, В± j a 0 , are on the imaginary axis. If a0 is zero, one of the two roots is at the origin. We can expand the pole form to give P(s) = s 2 в€’ ( p1 + p2 )s + p1 p2 (7.4) and compare the coefп¬Ѓcients with the original polynomial in Eq. (7.3). If both p1 and p2 are negative, the coefп¬Ѓcients a1 and a0 must be positive deп¬Ѓnite, which is the mathematiciansвЂ™ way of saying that a1 > 0 and a0 > 0. Next, consider a third-order polynomial with leading coefп¬Ѓcient a3 = 1 and the form in terms of the poles: P(s) = s 3 + a2 s 2 + a1 s + a0 = (s в€’ p1 )(s в€’ p2 )(s в€’ p3 ). (7.5) We expand the pole form to P(s) = s 3 в€’ ( p1 + p2 + p3 )s 2 + ( p1 p2 + p1 p3 + p2 p3 )s в€’ p1 p2 p3 . (7.6) Once again, if all three poles are negative, the coefп¬Ѓcients a2 , a1 , and a0 must be positive deп¬Ѓnite. The idea is that the signs of the pole are related to the coefп¬Ѓcients an , anв€’1 , . . . , a0 of an nth-order characteristic polynomial. If we require all the poles to have negative-real parts, there must be some way that we can tell from the coefп¬Ѓcients without actually having to solve for the roots. The idea is that all of the coefп¬Ѓcients in the characteristic polynomial must be positive deп¬Ѓnite. We could develop a comprehensive theory, which Routh did. The attractiveness of the Routh criterion is that without solving for the closed-loop poles, we can derive inequalities that would provide a bound for stable controller design. The complete Routh array analysis allows us to п¬Ѓnd, for example, the number of poles on the imaginary axis. Because we require that all poles lie in the LHP, we will not bother with these details (which are still in many control texts). Consider the fact that we can easily calculate the exact roots of a polynomial with MATLAB; we use the Routh criterion to the extent that it serves its purpose.2 That would be to derive inequality criteria for proper selection of controller gains of relatively simple systems. The technique loses its attractiveness when the algebra becomes too messy. Now the simpliп¬Ѓed RouthвЂ“Hurwitz recipe without proof follows. (1) Hurwitz test for the polynomial coefп¬Ѓcients. For a given nth-order polynomial P(s) = an s n + anв€’1 s nв€’1 + В· В· В· + a2 s 2 + a1 s + a0 , (7.7) all the roots are in the LHP if and only if all the coefп¬Ѓcients a0 , . . . , an are positive deп¬Ѓnite. 2 MATLAB does not even bother with a Routh function. Such an M-п¬Ѓle is provided on the Web Support for demonstration purposes. 131 Stability of Closed-Loop Systems If any one of the coefп¬Ѓcients is negative, at least one root has a positive-real part [i.e., in the right hand plane (RHP)]. If any of the coefп¬Ѓcients is zero, not all of the roots are in the LHP: it is likely that some of them are on the imaginary axis. Either way, stop. This test is a necessary condition for BIBO stability. There is no point in doing more other than to redesign the controller. (2) Routh array construction. If the characteristic polynomial passes the coefп¬Ѓcient test, we then construct the Routh array to п¬Ѓnd the necessary and sufп¬Ѓcient conditions for stability. This is one of the few classical techniques that is not emphasized and the general formula is omitted. The array construction up to a fourth-order polynomial is used to illustrate the concept. Generally, for an nth-order polynomial, we need (n + 1) rows. The п¬Ѓrst two rows are п¬Ѓlled in with the coefп¬Ѓcients of the polynomial in a column-wise order. The computation of the array entries is very much like the negative of a normalized determinant anchored by the п¬Ѓrst column. Even without the general formula, you may pick out the pattern as you read the following three examples. The Routh criterion states that, in order to have a stable system, all the coefп¬Ѓcients in the п¬Ѓrst column of the array must be positive deп¬Ѓnite. If any of the coefп¬Ѓcients in the п¬Ѓrst column is negative, there is at least one root with a positive-real part. The number of sign changes is the number of positive poles. Here is the array for a second-order polynomial, p(s) = a2 s 2 + a1 s + a0 : 1: a2 a0 2: a1 0 3: b1 = a1 a0 в€’ (0)a2 = a0 a1 In the case of a second-order system, the п¬Ѓrst column of the Routh array reduces to simply the coefп¬Ѓcients of the polynomial. The coefп¬Ѓcient test is sufп¬Ѓcient in this case. Or we can say that both the coefп¬Ѓcient test and the Routh array provide the same result. The array for a third-order polynomial, p(s) = a3 s 3 + a2 s 2 + a1 s + a0 , is 1: a3 a1 0 2: a2 a0 0 b1 = 3: 4: c1 = a2 a1 в€’ a3 a0 a2 b1 a 0 в€’ b 2 a 2 = a0 b1 b2 = (a2 )(0) в€’ (a3 )0 =0 a2 0 In this case, we have added one column of zeros; they are needed to show how b2 is computed. Because b2 = 0 and c1 = a0 , the Routh criterion adds one additional constraint in the case of a third-order polynomial: b1 = 132 a2 a1 в€’ a3 a0 > 0. a2 (7.8) 7.2. The RouthвЂ“Hurwitz Criterion We follow with the array for a fourth-order polynomial, p(s) = a4 s 4 + a3 s 3 + a2 s 2 + a1 s + a0 : 1: a4 a2 a0 2: a3 a1 0 3: b1 = a3 a2 в€’ a1 a4 a3 b2 = a3 a0 в€’ (0)a4 = a0 a3 4: c1 = b1 a 1 в€’ b2 a 3 b1 c2 = b1 (0) в€’ (0)a3 =0 b1 5: d1 = c1 b2 в€’ (0)b1 = b2 = a0 c1 0 0 The two additional constraints from the Routh array are hence b1 = a3 a2 в€’ a1 a4 > 0, a3 c1 = b1 a 1 в€’ b 2 a 3 b1 a 1 в€’ a 0 a 3 = > 0. b1 b1 (7.9) (7.10) Example 7.2: If we have only a proportional controller (i.e., one design parameter) and negative-real open-loop poles, the RouthвЂ“Hurwitz criterion can be applied to a fairly highorder system with ease. For example, for the following closed-loop system characteristic equation, 1 1 + Kc = 0, (s + 3)(s + 2)(s + 1) п¬Ѓnd the stability criteria. We expand and rearrange the equation to the polynomial form: s 3 + 6s 2 + 11s + (6 + K c ) = 0. The Hurwitz test requires that K c > в€’6 or simply K c > 0 for positive controller gains. The Routh array is 1: 1 11 2: 6 6 + Kc 3: b1 0 4: 6 + Kc 0 The Routh criterion requires, as in Eq. (7.8), that (6)(11) в€’ (6 + K c ) b1 = > 0 or 60 > K c . 6 The range of proportional gain to maintain system stability is hence 0 < K c < 60. Example 7.3: Consider a second-order process function G p = [1/(s 2 + 2s + 1)], which is critically damped. If we synthesize a control system with a PI controller, what are the stability constraints? 133 Stability of Closed-Loop Systems For illustration, we take G m = G a = 1 and the closed-loop transfer function for a servo problem is simply GcG p C . = R 1 + GcG p In this problem, the closed-loop characteristic equation is 1 + Gc G p = 1 + Kc 1 + 1 1 =0 2 П„ I s s + 2s + 1 or П„ I s 3 + 2П„ I s 2 + П„ I (1 + K c )s + K c = 0. With the RouthвЂ“Hurwitz criterion, we need immediately П„ I > 0 and K c > 0. (The s term requires that K c > в€’1, which is overridden by the last constant coefп¬Ѓcient.) The Routh array for this third-order polynomial is 1: П„I П„ I (1 + K c ) 2: 2П„ I Kc 3: b1 0 4: Kc With the use of Eq. (7.8), we require that b1 = 2П„ I2 (1 + K c ) в€’ П„ I K c Kc = П„ I (1 + K c ) в€’ > 0. 2П„ I 2 The inequality is rearranged to П„I > Kc 2(1 + K c ) 2П„ I > Kc, 1 в€’ 2П„ I or which can be interpreted in two ways. For a given K c , there is a minimum integral time constant. If the proportional gain is sufп¬Ѓciently large such that K c 1, the rough estimate for the integral time constant is П„ I > 1/2. Or if the value of П„ I is less than 0.5, there is an upper limit on how large K c could be. If the given value of П„ I is larger than 0.5, the inequality simply infers that K c must be larger than some negative number. To be more speciп¬Ѓc, if we pick П„ I = 1,3 the Routh criterion becomes Kc 2> , (1 + K c ) which of course can be satisп¬Ѓed for all K c > 0. No new stability requirement is imposed in this case. Let us try another choice of П„ I = 0.1. In this case, the requirement for the proportional gain is 0.2(1 + K c ) > K c 3 134 or K c < 0.25. Note that, with this very speciп¬Ѓc case, if П„ I = 1 is chosen, the open-loop zero introduced by the PI controller cancels one of the open-loop poles of the process function at в€’1. If we do a root-locus plot, we will see how the root loci change to that of a purely second-order system. With respect to this example, the value is not important as long as П„ I > 1/2. 7.3. Direct-Substitution Analysis The entire range of stability for П„ I = 0.1 is 0 < K c < 0.25. We will revisit this problem when we cover root-locus plots; we can make much better sense without doing any algebraic work! 7.3. Direct-Substitution Analysis The closed-loop poles may lie on the imaginary axis at the moment a system becomes unstable. We can substitute s = jП‰ into the closed-loop characteristic equation to п¬Ѓnd the proportional gain that corresponds to this stability limit (which may be called marginal unstable). The value of this speciп¬Ѓc proportional gain is called the critical or ultimate gain. The corresponding frequency is called the crossover or ultimate frequency. Example 7.2A: Apply direct substitution to the characteristic equation in Example 7.2: s 3 + 6s 2 + 11s + (6 + K c ) = 0. Substitution of s = jП‰ leads to в€’ jП‰3 в€’ 6П‰2 + 11П‰j + (6 + K c ) = 0. The real- and imaginary-part equations are Re: в€’ 6П‰2 + (6 + K c ) = 0 Im: в€’ П‰3 + 11П‰ = 0 or or K c = 6(П‰2 в€’ 1), П‰(11 в€’ П‰2 ) = 0. в€љ From the imaginary-part equation, the ultimate frequency is П‰u = 11. Substituting this value into the real-part equation leads to the ultimate gain K c,u = 60, which is consistent with the result of the Routh criterion. If we have chosen the other possibility, П‰u = 0, meaning that the closed-loop poles are on the real axis, the ultimate gain is K c,u = в€’6, which is consistent with the other limit obtained with the Routh criterion. Example 7.3A: Repeat Example 7.3 to п¬Ѓnd the condition for the ultimate gain. If we make the s = jП‰ substitution into П„ I s 3 + 2П„ I s 2 + П„ I (1 + K c )s + K c = 0, it becomes в€’П„ I П‰3 j в€’ 2П„ I П‰2 + П„ I (1 + K c )П‰j + K c = 0. We have two equations after collecting all the real and the imaginary parts and requiring both to be zero: Re: K c в€’ 2П„ I П‰2 = 0, Im: П„ I П‰[в€’П‰2 + (1 + K c )] = 0. Thus we have either П‰ = 0 or в€’П‰2 + (1 + K c ) = 0. Substitution of the real-part equation into the nontrivial imaginary-part equation leads to в€’П‰2 + 1 + 2П„ I П‰2 = 0 or П‰u2 = 1 , 1 в€’ 2П„ I 135 Stability of Closed-Loop Systems where in the second form, we have added a subscript to denote the ultimate frequency, П‰u . Substitution of the ultimate frequency back into the real-part equation gives the relation for the ultimate proportional gain: K c,u = 2П„ I . 1 в€’ 2П„ I Note that if we have chosen the other possibility of П‰u = 0, meaning where the closedloop poles are on the real axis, the ultimate gain is K c,u = 0, which is consistent with the other limit obtained with the Routh criterion. The result of direct substitution conп¬Ѓrms the inequality derived from the Routh criterion, which should not be a surprise. We may question whether direct substitution is a better method. There is no clear-cut winner here. By and large, we are less prone to making algebraic errors when we apply the RouthвЂ“Hurwitz recipe, and the interpretation of the results is more straightforward. With direct substitution, we do not have to remember any formulas, and we can п¬Ѓnd the ultimate frequency, which, however, can be obtained with a root-locus plot or frequency-response analysis вЂ“ techniques that are covered in Section 7.4 and Chap. 8. When the system has dead-time, we must make an approximation, such as the PadВґe approximation, on the exponential dead-time function before we can apply the RouthвЂ“ Hurwitz criterion. The result is hence only an estimate. Direct substitution allows us to solve for the ultimate gain and ultimate frequency exactly. The next example illustrates this point. Example 7.4: Consider a system with a proportional controller and a п¬Ѓrst-order process but with dead-time. The closed-loop characteristic equation is given as 1 + Kc 0.8eв€’2s = 0. 5s + 1 Find the stability criteria of this system. Let us п¬Ѓrst use the п¬Ѓrst-order PadВґe approximation for the time-delay function and apply the RouthвЂ“Hurwitz criterion. The approximate equation becomes 1 + Kc 0.8 (в€’s + 1) =0 5s + 1 (s + 1) or 5s 2 + (6 в€’ 0.8K c )s + (1 + 0.8K c ) = 0. The RouthвЂ“Hurwitz criterion requires that 6 в€’ 0.8K c > 0 or K c < 7.5, and K c > в€’1/0.8. When K c is kept positive, the approximate range of the proportional gain for system stability is 0 < K c < 7.5. We now repeat the problem with the s = jП‰ substitution into the characteristic equation and rewrite the time-delay function with EulerвЂ™s identity: (5П‰j + 1) + 0.8K c (cos 2П‰ в€’ j sin 2П‰) = 0. Collecting terms of the real and the imaginary parts provides the two equations: Re: 1 + 0.8K c cos 2П‰ = 0 Im: 5П‰ в€’ 0.8K c sin 2П‰ = 0. 136 or K c = в€’1/(0.8 cos 2П‰), 7.4. Root-Locus Analysis Substitution of the real-part equation into the imaginary-part equation gives 5П‰ + tan 2П‰ = 0. The solution of this equation is the ultimate frequency П‰u = 0.895, and, from the realpart equation, the corresponding ultimate proportional gain is K c,u = 5.73. Thus the more accurate range of K c that provides system stability is 0 < K c < 5.73. Note 1: This result is consistent with the use of frequency-response analysis in Chap. 8. Note 2: The iterative solution in solving the ultimate frequency is tricky. The equation has poor numerical properties вЂ“ arising from the fact that tan Оё вЂњjumpsвЂќ from inп¬Ѓnity at Оё = (ПЂ/2)в€’ to negative inп¬Ѓnity at Оё = (ПЂ/2)+ . To better see why, use MATLAB to make a plot of the function (LHS of the equation) with 0 < П‰ < 1. With MATLAB, we can solve the equation with the fzero() function. Create an M-п¬Ѓle named f.m, and enter these two statements in it: function y=f(x) y = 5*x + tan(2*x); After you have saved the п¬Ѓle, enter, at the MATLAB prompt; fzero('f',0.9) where 0.9 is the initial guess. MATLAB should return 0.8953. If you shift the initial guess just a bit, say by using 0.8, you may get a вЂњsolutionвЂќ of 0.7854. Note that (2)(0.7854) is ПЂ/2. If you blindly accept this incorrect value, K c,u will be inп¬Ѓnity according to the real-part equation. MATLAB is handy, but it is not foolproof! 7.4. Root-Locus Analysis The idea of a root-locus plot is simple вЂ“ if we have a computer. We pick one design parameter, say, the proportional gain K c , and write a small program to calculate the roots of the characteristic polynomial for each chosen value of K c as in 0, 1, 2, 3, . . . . , 100, . . . , etc. The results (the values of the roots) can be tabulated or, better yet, plotted on the complex plane. Even though the idea of plotting a root locus sounds so simple, it is one of the most powerful techniques in controller design and analysis when there is no time delay. Root locus is a graphical representation of the roots of the closed-loop characteristic polynomial (i.e., the closed-loop poles) as a chosen parameter is varied. Only the roots are plotted. The values of the parameter are not shown explicitly. The analysis most commonly uses the proportional gain as the parameter. The value of the proportional gain is varied from 0 to inп¬Ѓnity, or in practice, just вЂњlarge enough.вЂќ Now a simple example is needed to get this idea across. Example 7.5: Construct the root-locus plots of a п¬Ѓrst- and a second-order system with a proportional controller. See how the loci approach inп¬Ѓnity. (a) Consider the characteristic equation of a simple system with a п¬Ѓrst-order process and a proportional controller: 1 + Kc Kp = 0. П„ps + 1 137 Stability of Closed-Loop Systems (a) (b) вЂ“1 П„p вЂ“1 П„ 2 вЂ“1 П„ 1 Figure E7.5. The solution, meaning the closed-loop poles of the system, is s= в€’(1 + K c K p ) . П„p The root-locus plot (Fig. E7.5a) is simply a line on the real axis starting at s = в€’1/П„ p when K c = 0 and extends to negative inп¬Ѓnity as K c approaches inп¬Ѓnity. As we increase the proportional gain, the system response becomes faster. Would there be an upper limit in reality? (Yes, saturation.) (b) We repeat the exercise with a second-order overdamped process function. The closedloop characteristic equation of the closed-loop system is 1 + Kc Kp =0 (П„1 s + 1)(П„2 s + 1) or П„1 П„2 s 2 + (П„1 + П„2 )s + (1 + K c K p ) = 0. The two closed-loop poles are s= в€’(П„1 + П„2 ) В± (П„1 + П„2 )2 в€’ 4П„1 П„2 (1 + K c K p ) . 2П„1 П„2 In the mathematical limit of K c = 0, we should п¬Ѓnd two negative real poles at s= в€’(П„1 + П„2 ) В± (П„1 в€’ П„2 )2 1 =в€’ 2П„1 П„2 П„1 or 1 в€’ , П„2 which are the locations of the two open-loop poles. (This result is easy to spot if we use the п¬Ѓrst step without expanding the terms.) As K c becomes larger, we should come to a point where we have two repeated roots at s= в€’(П„1 + П„2 ) . 2П„1 П„2 If we increase further the value of K c , the closed-loop poles will branch off (or break away) from the real axis and become two complex conjugates (Fig. E7.5b). No matter how large K c becomes, these two complex conjugates always have the same real part as given by the repeated root. Thus what we п¬Ѓnd are two vertical loci extending toward positive and negative inп¬Ѓnity. In this analysis, we also see how, as we increase K c , the system changes from overdamped to become underdamped, but it is always stable. This is the idea behind the plotting of the closed-loop poles вЂ“ in other words, construction of root-locus plots. Of course, we need mathematical or computational tools when we have more complex systems. An important observation from Example 7.5 is that with simple п¬Ѓrstand second-order systems with no open-loop zeros in the RHP, the closed-loop system is always stable. 138 7.4. Root-Locus Analysis We can now state the problem in more general terms. Let us consider a closed-loop characteristic equation 1 + K c G 0 = 0, where K c G 0 is referred to as the вЂњopen-loopвЂќ transfer function, G OL . The proportional gain is K c and G 0 is вЂњeverything else.вЂќ If we have only a proportional controller, then G 0 = G m G a G p . If we have other controllers, then G 0 would contain parts of the controller function. We further denote G 0 as a ratio of two polynomials, Q(s)/P(s), and rewrite the polynomials in the pole-zero form4 : Q(s) (s в€’ z 1 )(s в€’ z 2 ) В· В· В· (s в€’ z n ) = 1 + Kc =0 (7.11) 1 + Kc G0 = 1 + Kc P(s) (s в€’ p1 )(s в€’ p2 ) В· В· В· (s в€’ pm ) or as [(s в€’ p1 )(s в€’ p2 ) В· В· В· (s в€’ pm )] + K c [(s в€’ z 1 )(s в€’ z 2 ) В· В· В· (s в€’ z n )] = 0. (7.11a) The roots of the mth-order P(s) = 0, p1 , p2 , . . . , pm , are the open-loop poles. The roots of the nth-order Q(s) = 0, z 1 , z 2 , . . . , z n , are the open-loop zeros. The roots of the entire characteristic equation (7.11) are the closed-loop poles that constitute the root loci. There will be m root loci, matching the order of the characteristic polynomial. We can easily see that when K c = 0 the poles of the closed-loop system characteristic polynomial (1 + K c G 0 ) are essentially the same as the poles of the open loop. When K c approaches inп¬Ѓnity, the poles of the closed-loop system are the zeros of the open-loop. These are important mathematical features. In other words, on a root-locus plot, we expect the вЂњtraceвЂќ of the root loci to begin at the open-loop poles and terminate at the open-loop zeros (if there is one). For real systems, m > n and n в‰Ґ 0. In these cases, the (m в€’ n) root loci will originate from an open-loop pole and extend toward inп¬Ѓnity somehow, depending on the speciп¬Ѓc problem. Before reading further, it is very important that you go through at least the п¬Ѓrst half of MATLAB Session 6 and do computations with sample numbers while reading these root-locus examples. Example 7.2B: Do the root-locus plot and п¬Ѓnd the ultimate gain of Example 7.2. The closed-loop equation from that example is 1 + Kc 1 = 0. (s + 3)(s + 2)(s + 1) We can easily use MATLAB to п¬Ѓnd that the ultimate gain is roughly 60. The statements to use are G=zpk([],[-1 -2 -3],1); k=0:1:100; % We have to use our own gain vector in this example rlocus(G,k) % because the MATLAB default plot does not cross the % Im axis rlocfind(G) 4 If you cannot follow the fancy generalization, think of a simple problem such as a unity feedback loop with a PD controller and a п¬Ѓrst-order process. The closed-loop characteristic equation is 1 + Kc K p (П„ D s + 1) = 0. (П„ p s + 1) The closed-loop pole вЂњrunsвЂќ from the point s = в€’1/П„ p at the mathematical limit of K c = 0 to the point s = в€’1/П„ D as K c approaches inп¬Ѓnity. 139 Stability of Closed-Loop Systems After we enter the rlocfind() command, MATLAB will prompt us to click a point on the root-locus plot. In this problem, we select the intersection between the root locus and the imaginary axis for the ultimate gain. Example 7.3B: Repeat Example 7.3 with a root-locus analysis. The closed-loop characteristic equation from Example 7.3 is 1 + Kc (П„ I s + 1) = 0. П„ I s(s 2 + 2s + 1) Select various values of П„ I and use MATLAB to construct the root-locus plots. Sample statements are taui=0.2; % Open-loop zero at -5 G=tf([taui 1],conv([taui 0],[1 2 1])); rlocus(G) We should п¬Ѓnd that for values of П„ I > 0.5 the system stays stable. For П„ I = 0.5, the system may become unstable, but only at inп¬Ѓnitely large K c . The system may become unstable for П„ I < 0.5 if K c is too large. Finally, for the choice of П„ I = 0.1, we should п¬Ѓnd with the MATLAB function rlocfind that the ultimate gain is roughly 0.25, the same answer from Example 7.3. How close you get depends on how accurately you can click the axis crossover point. Even as we rely on MATLAB to generate root-locus plots, it is important to appreciate how they are generated. To say the least, we need to know how to identify the open-loop poles and zeros and the direction of the loci. These hints are given in the Web Support. The following example illustrates some of the more common ones that we may encounter in control analysis. There are many other possibilities, but this exercise should be a good starting point. MATLAB Session 6 has more suggestions regarding the plots that are associated with the use of controllers. Example 7.6: Construct the root-locus plots of some of the more common closed-loop equations with numerical values. Make sure you try them yourself with MATLAB. (a) A sample п¬Ѓrst-order system and the MATLAB statement: 1+K 1 =0 (s + 2) rlocus(tf(1,[1 2])) (b) A second-order system: 1+K 1 =0 (s + 2)(s + 1) rlocus(zpk([],[-1 -2],1)) (c) Second-order system with repeated open-loop poles: 1+K 1 =0 (s + 2)2 rlocus(zpk([],[-2 -2],1)) (d) Second-order system with different open-loop zeros: 1+K 140 (s + 0.5) =0 (s + 2)(s + 1) rlocus(zpk(-0.5,[-1 -2],1)) 7.4. Root-Locus Analysis 1+K (s + 1.5) =0 (s + 2)(s + 1) rlocus(zpk(-1.5,[-1 -2],1)) 1+K (s + 4) =0 (s + 2)(s + 1) rlocus(zpk(-4,[-1 -2],1)) (e) Third-order system: 1+K 1 =0 (s + 3)(s + 2)(s + 1) rlocus(zpk(-1 -2 -3],1)) (f) Third-order system with an open-loop zero: 1+K s + 1.5 =0 (s + 3)(s + 2)(s + 1) rlocus(zpk(-1.5,[-1 -2 -3],1)) See also what the plot is like if the open-loop zero is at в€’0.5, в€’2.5, and в€’3.5. Rough sketches of what you should obtain with MATLAB are in Fig. E7.6. The root locus of the system in (a) is a line on the real axis extending to negative inп¬Ѓnity (Fig. E7.6A). The root loci in (b) approach each other (arrows not shown) on the real axis and then branch off toward inп¬Ѓnity at 90в—¦ . The repeated roots in (c) simply branch off toward inп¬Ѓnity. With only open-loop poles, examples (a) вЂ“ (c) can represent only systems with a proportional controller. In case (a), the system contains a п¬Ѓrst-order process and in (b) and (c) are overdamped and critically damped second-order processes. The sketches for (d) illustrate how an open-loop zero, say, in an ideal PD controller, may affect the root-locus plot and dynamics of a system containing an overdamped second-order process. Underdamped system behavior is expected only when the open-loop zero is large enough (i.e., П„ D sufп¬Ѓciently small). On the left-hand panel of Fig. E7.6B, one locus goes from the в€’1 open-loop pole to the open-loop zero at в€’0.5 (arrow not shown). The second locus goes from в€’2 to negative inп¬Ѓnity on the real axis. In the middle panel, one locus goes from в€’1 to the open-loop zero at в€’1.5 (arrow not shown). On the right, where the open-loop zero is at в€’4, two root loci move toward each from в€’1 and в€’2 (arrows not shown), then branch off. The branches break in later onto the real axis; one locus approaches the zero at в€’4, the other toward negative inп¬Ѓnity. Systems (e) and (f) would contain a third-order process. Of course, we can have a proportional control only in case (e), and (f) represents one probable example of using an ideal PD controller. 1 4 (a) 4 (b) (c) 0.5 2 2 0 0 0 в€’0.5 в€’2 в€’2 в€’1 в€’4 в€’2 0 в€’4 в€’3 в€’2 в€’1 0 1 в€’4 в€’3 в€’2 в€’1 0 1 Figure E7.6A. 141 Stability of Closed-Loop Systems 1 1 (d) Zero at в€’0.5 4 Zero at в€’1.5 0.5 0.5 2 0 0 0 в€’0.5 в€’0.5 в€’2 в€’1 в€’4 в€’2 0 в€’1 в€’4 в€’2 0 в€’4 Zero at в€’4 в€’8 в€’6 в€’4 в€’2 0 Figure E7.6B. The system in (e) can become unstable, whereas a proper addition of an open-loop zero, as in (f), can help stabilize the system (Fig. E7.6C). In (e), the two loci from в€’1 and в€’2 approach each other (arrows not shown). They then break away, and the closed-loop poles become unstable. The two loci approach positive inп¬Ѓnity at В±60в—¦ . In (f), the system is always stable. The dominant closed-loop pole is the locus moving from в€’1 to в€’1.5 (arrow not shown). This system is faster than if we had put the open-loop zero at, say, в€’0.5. In most classic control texts, we п¬Ѓnd plotting guidelines to help hand-sketching of rootlocus plots. After going over Example 7.6, you should п¬Ѓnd that some of them are quite intuitive. These are the simple guidelines: (1) The root-locus plot is symmetric about the real axis. (2) The number of loci equals the number of open-loop poles (or the order of the system). (3) A locus (closed-loop root path) starts at an open-loop pole and either terminates at an open-loop zero or extends to inп¬Ѓnity. (4) On the real axis, a root locus exists only to the left of an odd number of real poles and zeros. (The explanation of this point is on the Web Support.) (5) The points at which the loci cross the imaginary axis can be found by the RouthвЂ“ Hurwitz criterion or by substituting s = jП‰ in the characteristic equation. (Of course, we can also use MATLAB to do that.) To determine the shape of a root-locus plot, we need other rules to determine the locations of the so-called breakaway and break-in points, the corresponding angles of departure and arrival, and the angle of the asymptotes if the loci approach inп¬Ѓnity. They all arise from the analysis of the characteristic equation. These features, including item (4) above, are explained on the Web Support. With MATLAB, our need for them is minimal. 3 2 3 (e) 2 1 1 0 0 в€’1 в€’1 в€’2 в€’2 в€’3 в€’4 в€’3 в€’2 Figure E7.6C. 142 в€’1 0 1 в€’3 в€’4 (f) в€’3 в€’2 в€’1 0 1 7.5. Root-Locus Design Note: There are two important steps that you must follow. First, make sure you go through the MATLAB tutorial (Session 6) carefully to acquire a feel on the probable shapes of rootlocus plots. Secondly, test guidelines (3) and (4) listed previously for every plot that you make in the tutorial. These guidelines can become your most handy tool to deduce, without doing any algebra, whether a system will exhibit underdamped behavior, or, in other words, whether a system will have complex closed-loop poles. 7.5. Root-Locus Design In terms of controller design, the closed-loop poles (or now the root loci) also tell us about the system dynamics. We can extract much more information from a root-locus plot than from a Routh criterion analysis or an s = jП‰ substitution. In fact, it is common to impose, say, a time constant or a damping ratio speciп¬Ѓcation on the system when we use root-locus plots as a design tool. Example 7.5A: Consider the second-order system in Example 7.5: What should the proportional gain be if we specify the controlled system to have a damping ratio of 0.7? The second-order closed-loop characteristic equation in Example 7.5 can be rearranged as П„1 П„2 (П„1 + П„2 ) s2 + s + 1 = 0. (1 + K c K p ) (1 + K c K p ) We can compare this equation with the general form П„ 2 s 2 + 2О¶ П„ s + 1 = 0, where now П„ is the system time period and О¶ is the system damping ratio. From Example 5.2, we have derived that П„= П„1 П„2 , (1 + K c K p ) О¶ = 1 2 (П„1 + П„2 ) . П„1 П„2 (1 + K c K p ) Thus we can solve for K c with a given choice of О¶ in a speciп¬Ѓc problem. However, MATLAB allows us to get the answer with very little work вЂ“ something that is very useful when we deal with more complex systems. Consider a numerical problem with values of the process gain K p = 1 and process time constants П„1 = 2 and П„2 = 4 such that the closed-loop equation is 1 + Kc 1 = 0. (2s + 1)(4s + 1) We enter the following statements in MATLAB5 : G=tf(1,conv([2 1],[4 1])); rlocus(G) sgrid(0.7,1) % plot the 0.7 damping ratio lines [kc,cpole]=rlocfind(G) 5 The technique of using the damping ratio line Оё = cosв€’1 О¶ in Eq. (2.34) is applied to higher-order systems. When we do so, we are implicitly making the assumption that we have chosen the dominant closed-loop pole of a system and that this system can be approximated as a second-order underdamped function at sufп¬Ѓciently large times. For this reason, root locus is also referred to as dominant-pole design. 143 Stability of Closed-Loop Systems Where the root locus intersects the 0.7 damping ratio line, we should п¬Ѓnd, from the result returned by rlocfind(), the proportional gain to be 1.29 (1.2944 to be exact) and the closed-loop poles at в€’0.375 В± 0.382 j. The real and imaginary parts are not identical because cosв€’1 0.7 is not exactly 45в—¦ . We can conп¬Ѓrm the answer by substituting the values into our analytical equations. We should п¬Ѓnd that the real part of the closed-loop pole agrees with what we derived in Example 7.5 and the value of the proportional gain agrees with the expression that we derived in this example. Example 7.7: Consider installing a PI controller in a system with a п¬Ѓrst-order process such that we have no offset. The process function has a steady-state gain of 0.5 and a time constant of 2 min. Take G a = G m = 1. The system has the simple closed-loop characteristic equation: 1 + Kc 0.5(П„ I s + 1) = 0. П„ I s(2s + 1) We also want to have a slightly underdamped system with a reasonably fast response and a damping ratio of 0.7. How should we design the controller? To restrict the example for illustration, we consider (a) П„ I = 3 min and (b) П„ I = 2/3 min. To begin with, this is a second-order system with no positive zeros, and so stability is not an issue. Theoretically speaking, we could have derived and proved all results with the simple second-order characteristic equation, but we take the easy way out with root-locus plots. (a) The open-loop poles are at в€’0.5 and at the origin. If the integral time constant is П„ I = 3 min, the open-loop zero is at в€’1/3, and all we have are negative and real closed-loop poles [Fig. E7.7(a)]. The system will not become underdamped. The вЂњspeedвЂќ of the response, as we increase the proportional gain, is limited by the zero position at в€’1/3. This is where the dominant closed-loop pole will approach. (b) The situation is more interesting with a smaller integral time constant, П„ I = 2/3 min, now with the open-loop zero at в€’1.5. The root loci п¬Ѓrst approach one another (arrows not shown) before breaking away to form the circle [Fig. E7.7(b)]. As K c increases further, the loci break into the real axis. The dominant closed-loop pole will approach the zero at в€’1.5. (a) X вЂ“0.5 X B (b) A X X C вЂ“3 вЂ“2 Figure E7.7. 144 вЂ“1 7.6. Final Remark on Root-Locus Plots We use the following MATLAB statements to draw the 0.7 damping ratio line, and it intersects the root loci at two points (Fig. E7.7): at point A, в€’0.312 + 0.323 j, when K c = 0.55, and at point B, в€’1.15 + 1.17 j, when K c = 7.17: kc=1; taui=2/3; Gc=tf(kc*[taui 1], [taui 0]); Gp=tf(0.5, [2 1]); rlocus(Gc*Gp) sgrid(0.7,1) [kc,cpole]=rlocfind(Gc*Gp) If saturation is not a problem, the proportional gain K c = 7.17 (point B) is preferred. The corresponding closed-loop pole has a faster time constant. (The calculation of the time period or frequency and conп¬Ѓrmation of the damping ratio are left as homework.) Note 1: Theoretically speaking, point C on the root-locus plot is ideal вЂ“ the fastest possible response without any oscillation. We rarely can do that in practice; the proportional gain would be so large that the controller would be saturated. Note 2: As we reduce the integral time constant from П„ I = 3 min to exactly 2 min, we have the situation of pole-zero cancellation. The terms in the closed-loop characteristic equation cancel out, and the response in this example is reduced to that of a п¬Ѓrst-order system. If П„ I is only slightly less than 2 min, we have a very slightly underdamped system; the вЂњcircleвЂќ in Fig. E7.7(b) is very small (try this with MATLAB). In reality, it is very difп¬Ѓcult to have perfect pole-zero cancellation, and if we design the controller with П„ I too close to П„ p , the system response may вЂњbounce back and forth.вЂќ 7.6. Final Remark on Root-Locus Plots Instructive as root-locus plots appear to be, this technique does have its limitations. The most important one is that it cannot handle dead-time easily. When we have a system with deadtime, we must make an approximation with the PadВґe formulas. This is the same limitation that applies to the RouthвЂ“Hurwitz criterion. In this computer age, you may question why nobody would write a program that can solve for the roots with dead-time accurately. Someone did. There are even reп¬Ѓned hand-sketching techniques to account for the lag that is due to dead time. However, these tools are not as easy to apply and are rarely used. Few people use them because frequency-response analysis in Chap. 8 can handle dead-time accurately and extremely easily. A second point on root-locus plots is that they can give us only the so-called absolute stability, but not relative stability, meaning that there is no easy way to deп¬Ѓne a good, general вЂњsafety margin.вЂќ We may argue that we can deп¬Ѓne a certain distance that a closedloop pole must stay away from the imaginary axis, but such an approach is very problem speciп¬Ѓc. Recall that the proportional gain is an implicit parameter along a locus, and is very difп¬Ѓcult to tell what effects we may have with slight changes in the proportional gain (the sensitivity problem). Frequency-response analysis once again does better and allows us to deп¬Ѓne general relative stability criteria. Furthermore, frequency-response analysis can help us to understand why a certain choice of, for example, an integral time constant may destabilize a system. (Jumping ahead, it has to do with when we bring in phase lead. We shall see that in Chap. 8 and Example 10.1.) 145 Stability of Closed-Loop Systems On the other hand, frequency-response analysis cannot reveal information on dynamic response easily вЂ“ something root locus does very well. Hence controller design is always an iterative procedure. There is no one-stop shopping. There is never a unique answer. Finally, you may wonder if you can use the integral or the derivative time constant as the parameter. Theoretically, you can. You may rearrange the characteristic equation in such a way that can take advantage of prepackaged programs that use the proportional gain as the parameter. In practice, nobody does that. One main reason is that the resulting loci plot will not have this nice interpretation that we have by varying the proportional gain. Review Problems (1) Repeat Examples 7.2 and 7.4 with the general closed-loop characteristic polynomial: a3 s 3 + a2 s 2 + a1 s + a0 + K c = 0. Derive the ultimate frequency and the ultimate gain. (2) Revisit Example 5.4. What choice of П„ D should we make? (3) No additional reviews are needed as long as you go through each example carefully with MATLAB. Hint: (2) We can use Example 7.6(d) as a hint. To have a faster system response, we want to choose П„ D < П„ p such that the open-loop zero в€’1/П„ D lies to the left of the open-loop pole в€’1/П„ p on the real-axis. 146 8 Frequency-Response Analysis he response of a stable system at large times is characterized by its amplitude and phase shift when the input is a sinusoidal wave. These two quantities can be obtained from the transfer function, of course, without inverse transform. The analysis of this frequency response can be based on a simple substitution (mapping) s = jП‰, and the information is given by the magnitude (modulus) and the phase angle (argument) of the transfer function. Because the analysis begins with a Laplace transform, we are still limited to linear or linearized models. T What Are We Up to? r Theoretically, we are making the presumption that we can study and understand the dynamic behavior of a process or system by imposing a sinusoidal input and measuring the frequency response. With chemical systems that cannot be subject to frequency-response experiments easily, it is very difп¬Ѓcult for a beginner to appreciate what we will go through. So until then, take frequency response as a math problem. r Both the magnitude and the argument are functions of the frequency. The so-called Bode and Nyquist plots are nothing but graphical representations of this functional dependence. r Frequency-response analysis allows us to derive a general relative stability criterion that can easily handle systems with time delay. This property is used in controller design. 8.1. Magnitude and Phase Lag Our analysis is based on the mathematical property that, given a stable process (or system) and a sinusoidal input, the response will eventually become a purely sinusoidal function. This output will have the same frequency as the input, but with different amplitude and phase angle. The two latter quantities can be derived from the transfer function. 147 Frequency-Response Analysis The idea of frequency response is п¬Ѓrst illustrated with an inverse Laplace transform. Consider our good old familiar п¬Ѓrst-order model equation,1 dy + y = K p f (t) dt with its transfer function Kp Y (s) = . G(s) = F(s) П„ps + 1 П„p (8.1) (8.2) If the input is a sinusoidal function such that f (t) = A sin П‰t, the output Y (s) is Y (s) = Kp AП‰ . П„ p s + 1 s 2 + П‰2 (8.3) If we do the partial-fraction expansion and inverse transform, we should п¬Ѓnd, after some hard work, the time-domain solution: пЈ« пЈ¶ AK p П„ p П‰ в€’t/П„ p пЈ AK p пЈё + y(t) = 2 2 sin(П‰t + П†), (8.4) e П„pП‰ + 1 П„ 2 П‰2 + 1 p where П† = tanв€’1 (в€’П‰П„ p ) (8.5) is the phase lag.2 The algebraic details of deriving Eq. (8.4) are not important. The important aspects will be derived by means of an alternative route just a few steps ahead. For now, the crucial point is to observe that if time is sufп¬Ѓciently large (as relative to П„ p ), the exponential term in Eq. (8.4) will decay away and the time response becomes a purely sinusoidal function (Fig. 8.1). Figure 8.1. Schematic response (solid curve) of a п¬Ѓrst-order function to a sinusoidal input (dashed curve). The response has a smaller amplitude, a phase lag, and its exponential term decays away quickly to become a pure sinusoidal response. 1 2 148 Although we retain our usual notation of using a subscript p for a п¬Ѓrst-order function in illustrations, the analysis is general; it applies to an open-loop process or a closed-loop system. In other words, we can apply a sinusoidal change to the set point of a system if we need to. In these calculations, the units of frequency are radians per time. If the period T is given in seconds, then frequency П‰ = 1/T Hz (hertz or cycles per second) or П‰ = 2ПЂ/T (rad/s). 8.1. Magnitude and Phase Lag The time response at sufп¬Ѓciently large times can be normalized with respect to the amplitude of the input sine wave: пЈ« пЈ¶ K yв€ћ (t) p пЈё sin(П‰t + П†), =пЈ (8.6) A 2 2 П„ П‰ +1 p where now the amplitude of this normalized large-time response is called the amplitude ratio (AR). The response has a different amplitude and phase lag in relation to the input wave. If we further normalize the equation such that the amplitude of the sine wave is bounded by one, we obtain пЈ« пЈ¶ yв€ћ (t) пЈ 1 пЈё sin(П‰t + П†). = (8.7) AK p П„ 2 П‰2 + 1 p The amplitude of this normalized response, yв€ћ /AK p , is called the magnitude ratio.3 Next, letвЂ™s substitute s = jП‰ into the transfer function, i.e., G p (s) = G p ( jП‰), which makes G p a complex number4 : G p ( jП‰) = Kp в€’ jП‰П„ p + 1 jП‰П„ p + 1 в€’ jП‰П„ p + 1 = Kp П„ p2 П‰2 + 1 в€’j K p П‰П„ p . П„ p2 П‰2 + 1 (8.8) If we put Eq. (8.8) in polar coordinates, G p ( jП‰) = |G p ( jП‰)|e jП† , the magnitude and the phase angle are |G p ( jП‰)| = Kp П„ p2 П‰2 +1 , П† = G p ( jП‰) = tanв€’1 (в€’П‰П„ p ). (8.9) A comparison of Eqs. (8.9) with Eq. (8.6) shows that the magnitude and the phase angle of G p ( jП‰) are exactly the same as the amplitude and the phase lag of the normalized вЂњlargetimeвЂќ time-domain solution. We should note that the magnitude and the phase angle of G p ( jП‰) are functions of the input frequency. The larger the frequency, the lower the magnitude and the larger the phase lag. We can make this observation by writing П„ p П‰ = П‰/П‰ p . When the imposed frequency is large with respect to the process вЂњnaturalвЂќ frequency, П‰ p , the process cannot respond fast enough, resulting in low magnitude and large phase lag. When the imposed frequency is relatively small, the magnitude approaches the steady-state gain and the phase lag approaches zero. 8.1.1. The General Analysis We now generalize our simple illustration. Consider a general transfer function of a stable model G(s), which we also denote as the ratio of two polynomials, G(s) = Q(s)/P(s). We 3 4 This is a major source of confusion in many texts. The magnitude ratio is not the magnitude of a transfer function. It is the AR that is the same as the magnitude of G(s). To avoid confusion, we stick strictly with the mathematical property, i.e., the magnitude of G(s). We will use neither AR nor magnitude ratio in our terminology. It is much more sensible to consider the magnitude of a transfer function. If you need that, a brief summary of complex variable deп¬Ѓnitions is on the Web Support. 149 Frequency-Response Analysis impose a sinusoidal input f (t) = A sin П‰t such that the output is Y (s) = G(s) AП‰ s 2 + П‰2 = Q(s) AП‰ . P(s) (s + jП‰)(s в€’ jП‰) Because the model is stable, all the roots of P(s), whether they be real or complex, have negative-real parts and their corresponding time-domain terms will decay away exponentially. Thus if we are interested in only the time-domain response at sufп¬Ѓciently large times, we need to consider only the partial-fraction expansion of the two purely sinusoidal terms associated with the input: Yв€ћ (s) = a aв€— + . (s + jП‰) (s в€’ jП‰) (8.10) We can п¬Ѓnd their inverse transform easily. Apply the Heaviside expansion, a = (s + jП‰)Y (s)|s=в€’ jП‰ = G(в€’ jП‰) AG(в€’ jП‰) AП‰ = , в€’2 jП‰ в€’2 j and its conjugate redundantly just in case you do not believe the result is correct: a в€— = (s в€’ jП‰)Y (s)|s=+ jП‰ = G( jП‰) AG( jП‰) AП‰ = . 2 jП‰ 2j The time-domain solution (at large times) is hence yв€ћ (t) = AG(в€’ jП‰) в€’ jП‰t AG( jП‰) jП‰t e e . + в€’2 j 2j Note that G( jП‰) = |G( jП‰)|e jП† and G(в€’ jП‰) = |G( jП‰)|eв€’ jП† , and we can write eв€’ jП† eв€’ jП‰t e jП† e jП‰t yв€ћ (t) = |G( jП‰)| + . A в€’2 j 2j Apply EulerвЂ™s identity and the п¬Ѓnal result for the normalized response is yв€ћ (t) = |G( jП‰)| sin(П‰t + П†), A (8.11) where П† = G( jП‰). This is a crucial result. It constitutes the basis of frequency-response analysis, where in general, all we need are the magnitude and the argument of the transfer function G(s) after the substitution s = jП‰. 8.1.2. Some Important Properties We need to appreciate some basic properties of transfer functions when they are viewed as complex variables. They are important in performing frequency-response analysis. Consider that any given transfer function can be вЂњbroken upвЂќ into a product of simpler ones: G(s) = G 1 (s)G 2 (s) В· В· В· G n (s). (8.12) We do not need to expand the entire function into partial fractions. The functions G 1 , G 2 , etc., are better viewed as simply п¬Ѓrst-order and, at the most, second-order functions. In 150 8.1. Magnitude and Phase Lag frequency-response analysis, we make the s = jП‰ substitution and further write the function in terms of magnitude and phase angle as G( jП‰) = G 1 ( jП‰)G 2 ( jП‰) В· В· В· G n ( jП‰) = |G 1 ( jП‰)|e jП†1 |G 2 ( jП‰)|e jП† 2 В· В· В· |G n ( jП‰)|e jП†n or G( jП‰) = |G 1 ( jП‰)||G 2 ( jП‰)| В· В· В· |G n ( jП‰)|e j(П†1 +П† 2 +В·В·В·+П†n ) . The magnitude of G( jП‰) is |G( jП‰)| = |G 1 ( jП‰)||G 2 ( jП‰)| В· В· В· |G n ( jП‰)| (8.13) or log |G( jП‰)| = log |G 1 | + log |G 2 | + В· В· В· + log |G n |. The phase angle is П† = G( jП‰) = G 1 ( jП‰) + G 2 ( jП‰) + В· В· В· + G n ( jП‰). (8.14) Example 8.1: Derive the magnitude and phase lag of the following transfer function: G(s) = (П„a s + 1) . (П„1 s + 1)(П„2 s + 1) We can rewrite the function as a product of G(s) = (П„a s + 1) 1 1 . (П„1 s + 1) (П„2 s + 1) The magnitude and the phase angle of these terms with the use of Eqs. (8.9) are |G( jП‰)| = 1 + П„a2 П‰2 1 1 + П„12 П‰2 1 1 + П„22 П‰2 , П† = tanв€’1 (П‰П„a ) + tanв€’1 (в€’П‰П„1 ) + tanв€’1 (в€’П‰П„2 ). We have not covered the case of a zero term in the numerator. Here, we are just guessing that its result is the reciprocal of the п¬Ѓrst-order function result in Eqs. (8.9). The formal derivation comes later in Example 8.4. Note that П„ has units of time, П‰ is in radians per time, and П„ П‰ is in radians. We could have put G(s) in a slightly different form: G( jП‰) = G a ( jП‰)G b ( jП‰) В· В· В· G m ( jП‰) |G a ||G b | В· В· В· j(Оёa +Оёb +В·В·В·в€’Оё1 в€’Оё2 в€’В·В·В·) . = e G 1 ( jП‰)G 2 ( jП‰) В· В· В· G n ( jП‰) |G 1 ||G 2 | В· В· В· (8.15) In this case, the equivalent form of Eq. (8.13) is log |G( jП‰)| = (log |G a | + log |G b | + В· В· В· + log |G m |) в€’ (log |G 1 | + log |G 2 | + В· В· В· + log |G n |) (8.16) and the equivalent to Eq. (8.14) is П† = ( G a + G b + В· В· В· + G m ) в€’ ( G 1 + G 2 + В· В· В· + G n ). (8.17) 151 Frequency-Response Analysis With these results, we are ready to construct plots used in frequency-response analysis. The important message is that we can add up the contributions of individual terms to construct the п¬Ѓnal curve. The magnitude, of course, would be on the logarithmic scale. 8.2. Graphical Analysis Tools We know that both |G(jП‰)| and G( jП‰) are functions of frequency П‰. We certainly would like to see the relationships graphically. There are three common graphical representations of the frequency dependence. All three methods are described brieп¬‚y. The introduction relies on the use the so-called Bode plots, and more details will follow with respective examples. 8.2.1. Magnitude and Phase Plots: log G( j П‰) versus log П‰ and G( j П‰) versus log П‰ Here, we simply plot the magnitude (modulus) and phase angle (argument) individually against frequency вЂ“ the so-called Bode plots. From Eq. (8.16), we should use a log scale for the magnitude. We also use a log scale for the frequency to cover a larger range. Thus we use a logвЂ“log plot for |G( jП‰)| versus П‰ and a semilog plot for G( jП‰) versus П‰. The unit of phase angle is commonly converted to degrees, and frequency is in radians per unit time. In most electrical engineering or industrial control books, the magnitude is plotted in units of decibels (dB) as 1 dB = 20 log |G( jП‰)|. (8.18) Even with MATLAB, we should still know the expected shape of the curves and its вЂњtelltaleвЂќ features. This understanding is crucial in developing our problem-solving skills. Thus doing a few simple hand constructions is very instructive. When we sketch the Bode plot, we must identify the corner (break) frequencies, slopes of the magnitude asymptotes, and the contributions of phase lags at small and large frequencies. We will pick up the details in the examples. Another advantage of frequency-response analysis is that we can вЂњidentifyвЂќ the process transfer function with experimental data. With either a frequency-response experiment or a pulse experiment with the proper Fourier transform, we can construct the Bode plot by using the open-loop transfer functions and use the plot as the basis for controller design.5 8.2.2. Polar-Coordinate Plots вЂ“ G( j П‰) in Polar Coordinates or Re[G( j П‰)] versus Im[G( j П‰)] We can plot the real and the imaginary parts of G( jП‰) on the s plane with П‰ as the parameter вЂ“ the so-called Nyquist plot. Because a complex number can be put in polar coordinates, the Nyquist plot is also referred to as the polar plot: G( jП‰) = Re[G( jП‰)] + Im[G( jП‰)] = |G( jП‰)|e jП† . This plotting format contains the same information as the Bode plot. The polar plot is more compact, but the information on the frequency is not shown explicitly. If we do not have 5 152 The pulse experiment is not crucial for our understanding of frequency-response analysis and is provided on the Web Support, but we will do the design calculations in Section 8.3. 8.2. Graphical Analysis Tools a computer, we theoretically could read numbers off a Bode plot to construct the Nyquist plot. The use of Nyquist plots is more common in multiloop or multivariable analyses. A Bode plot, on the other hand, is easier to interpret and is a good learning tool. There are hand-plotting techniques, but we will rely on the computer. Still, we need to know the qualitative features of the plot resulting from several simple transfer functions. 8.2.3. Magnitude versus Phase Plot вЂ“ log G( j П‰) versus G( j П‰) In the third rendition of good old G( jП‰), we can plot the logarithmic magnitude against the phase lag вЂ“ the so-called Nichols chart. Generally, this plot is made with the open-loop transfer function of a unity feedback system. The magnitude and the argument contours of the closed-loop transfer function are overlaid on it. The plot allows for a frequency-response design analysis that better relates to probable closed-loop dynamic response. For now, we take a look at the construction of Bode and Nyquist plots of transfer functions that we have discussed in Chaps. 2 and 3. Keep in mind that these plots contain the same information: G( jП‰). It is important that you run MATLAB with sample numerical values while reading the following examples. Yes, you need to go through MATLAB Session 7 п¬Ѓrst. Example 8.2: What are the Bode and Nyquist plots of a п¬Ѓrst-order transfer function? We use the time-constant form of transfer functions. The magnitude and the phase angle of G(s) = Kp Y (s) = F(s) П„ps + 1 are derived in Eqs. (8.9) as |G p ( jП‰)| = Kp П„ p2 П‰2 +1 , П† = G p ( jП‰) = tanв€’1 (в€’П‰П„ p ). We may try the following MATLAB statements to generate Fig. E8.2: kp=1; % Just arbitrary values. Try different ones yourself. tau=2; G=tf(kp,[tau 1]); figure(1), bode(G); Kp slope = вЂ“1 G Kp П‰=0 П‰=в€ћ П† 0В° П‰ = 1/ П„ p вЂ“45В° вЂ“90В° 1/ П„ p П‰ Figure E8.2. 153 Frequency-Response Analysis figure(2), nyquist(G); % If the Nyquist default plot is % confusing, % follow the instructions in MATLAB Session 7 We plot, in theory, log |G p ( jП‰)| = log K p в€’ 1 log 1 + П„ p2 П‰2 . 2 To make the phase-angle plot, we simply use the deп¬Ѓnition of G p ( jП‰). As for the polar (Nyquist) plot, we do a frequency parametric calculation of |G p ( jП‰)| and G p ( jП‰), or we can simply plot the real part versus the imaginary part of G p ( jП‰).6 To check that a computer program is working properly, we need to use only the high- and the low-frequency asymptotes вЂ“ the same if we had to do the sketch by hand, as in the old days. In the limit of low frequencies, П‰ в†’ 0, |G p | = K p , П† = 0. On the magnitude plot, the low-frequency (also called zero-frequency) asymptote is a horizontal line at K p . On the phase-angle plot, the low-frequency asymptote is the 0в—¦ line. On the polar plot, the zero-frequency limit is represented by the point K p on the real axis. In the limit of high frequencies, П‰ в†’ в€ћ, |G p | = Kp , П„pП‰ П† = tanв€’1 (в€’в€ћ) = в€’90в—¦ . With the phase lag, we may see why a п¬Ѓrst-order function is also called a п¬Ѓrst-order lag. On the magnitude logвЂ“log plot, the high-frequency asymptote has a slope of в€’1. This asymptote also intersects the horizontal K p line at П‰ = 1/П„ p . On the phase-angle plot, the high-frequency asymptote is the в€’90в—¦ line. On the polar plot, the inп¬Ѓnity-frequency limit is represented by the origin as the G p ( jП‰) locus approaches it from the в€’90в—¦ angle. The frequency at which П‰ = 1/П„ p is called the corner frequency (also the break frequency). At this position, в€љ П‰ = 1/П„ p , |G p | = K p / 2, П† = tanв€’1 (в€’1) = в€’45в—¦ . We may question the signiп¬Ѓcance of the break frequency, П‰ = 1/П„ . LetвЂ™s take the п¬Ѓrstorder transfer function as an illustration. If the time constant is small, the break frequency is large. In other words, a fast process or system can respond to a large range of input frequencies without a diminished magnitude. On the contrary, a slow process or system has a large time constant and a low break frequency. The response magnitude is attenuated quickly as the input frequency increases. Accordingly, the phase lag also decreases quickly to the theoretical high-frequency asymptote. A common term used in control engineering is bandwidth, which is deп¬Ѓned as the frequency at which any given |G( jП‰)| drops to 70.7% of its low-frequency asymptotic value в€љ (Fig. 8.2). It is obvious that the 70.7% comes from the 1/ 2 of the п¬Ѓrst-order function 6 154 All comments on Nyquist plots are made without the need of formal hand-sketching techniques. Strictly speaking, the polar plot is a mapping of the imaginary axis from П‰ = 0+ to в€ћ. You will see that in texts that provide a more thorough discussion on choosing the so-called Nyquist path. 8.2. Graphical Analysis Tools 100 |G(jП‰)| 10-1 0.707K Small П„, Wide bandwidth 10-2 Large П„, Narrow bandwidth 10-3 10-4 10-2 10-1 100 101 102 Freq. Figure 8.2. Schematic illustration of two systems with wide and narrow bandwidths. calculation in Example 8.2.7 A large bandwidth is related to a fast system with a small time constant. On the other hand, slow responses or large time constants are cases with narrow bandwidths. From another perspective, the п¬Ѓrst-order function is the simplest example of a low-pass п¬Ѓlter. Signals with frequencies below the bandwidth are not attenuated, and their magnitude diminishes with higher frequencies. Example 8.3: What are the Bode and Nyquist plots of a second-order transfer function? We make the s = jП‰ substitution into the transfer function, G(s) = П„ 2s2 K , + 2О¶ П„ s + 1 to obtain G( jП‰) = K K [(1 в€’ П„ 2 П‰2 ) в€’ j2О¶ П„ П‰] . = (1 в€’ П„ 2 П‰2 ) + j2О¶ П„ П‰ (1 в€’ П„ 2 П‰2 )2 + (2О¶ П„ П‰)2 After a few algebraic steps, the resulting magnitude and phase-angle of G( jП‰) are |G( jП‰)| = K (1 в€’ П„ 2 П‰2 )2 + (2О¶ П„ П‰)2 , П† = G( jП‰) = tanв€’1 в€’2О¶ П„ П‰ 1 в€’ П„ 2 П‰2 . These are sample MATLAB statements to plot the magnitude and the phase angle as in Fig. E8.3: k=1; % Just arbitrary values. Try different ones yourself. tau=2; zeta=0.2; G=tf(k,[tau*tau 2*zeta*tau 1]); damp(G) % confirm the damping ratio figure(1), bode(G); figure(2), nyquist(G); In the limit of low frequencies, П‰ в†’ 0, |G| = K , П† = 0. In the limit of high frequencies, П‰ в†’ в€ћ, |G| = 7 K , П„ 2 П‰2 П† в‰€ tanв€’1 в€’2О¶ П„ П‰ в€’П„ 2 П‰2 = tanв€’1 2О¶ П„П‰ = tanв€’1 (0) = в€’180в—¦ . в€љ This is also referred to as the 3-dB bandwidth. The term comes from 20 log(1/ 2) = в€’3.01 dB. 155 Frequency-Response Analysis О¶ < 1/2 K G О¶в‰Ґ1 slope = вЂ“2 K О¶в‰Ґ1 0В° П† вЂ“90В° О¶ < 1/2 О¶в‰Ґ1 О¶ < 1/2 вЂ“180В° 1/ П„ П‰ Figure E8.3. We choose в€’180в—¦ (not 0в—¦ ) because we know that there must be a phase lag. On the magnitude logвЂ“log plot, the high-frequency asymptote has a slope of в€’2. This asymptote intersects the horizontal K line at П‰ = 1/П„ . At the corner frequency, П‰ = 1/П„, |G( jП‰)| = K , 2О¶ П† = tanв€’1 (в€’в€ћ) = в€’90в—¦ . For a process or system that is sufп¬Ѓciently underdamped, О¶ < 1/2, the magnitude curve will rise above the low-frequency asymptote or the polar plot will extend beyond the K -radius circle. We can take the derivative of the magnitude equation |G( jП‰)| = K / (1 в€’ П„ 2 П‰2 )2 + (2О¶ П„ П‰)2 to п¬Ѓnd the actual maximum and its associated frequency, the so-called resonant frequency,8 П‰r , П‰r = 1 в€’ 2О¶ 2 = П‰ 1 в€’ 2О¶ 2 , П„ (8.19) and the maximum magnitude M pП‰ is M pП‰ = |G( jП‰)|max = K 2О¶ 1 в€’ О¶ 2 . (8.20) в€љ From Eq. (8.19), there would be a maximum only if 0 < О¶ < 1/ 2 (or 0.707). We can design a controller by specifying an upper limit on the value of M pП‰ . The smaller the system damping ratio О¶ , the larger the value of M pП‰ and the more OS, or underdamping we expect in the time-domain response. Needless to say, excessive resonance is undesirable. 8 156 The step just before the result in Eq. (8.19) is 2(1 в€’ П„ 2 П‰2 )(в€’2П‰П„ 2 ) + 2(2О¶ П„ П‰)(2О¶ П„ ) = 0. In most texts, the derivation is based on unity gain K = 1 and so it will not show up in Eq. (8.20). Most texts also plot Eqs. (8.19) and (8.20) as functions of О¶ . However, with MATLAB, we can do that ourselves as an exercise. 8.2. Graphical Analysis Tools G slope = 1 П‰=в€ћ 1 1 П† П‰=0 90В° 45В° 0В° 1/ П„ p П‰ Figure E8.4. Example 8.4: What are the Bode and Nyquist plots of a п¬Ѓrst-order lead G(s) = (П„ p s + 1)? After the s = jП‰ substitution, we have G( jП‰) = 1 + jП‰П„ p , where |G( jП‰)| = 1 + П‰2 П„ p2 , G( jП‰) = tanв€’1 (П‰П„ p ). We may try the following MATLAB statements for Fig. E8.4: taud=2; % Just an arbitrary value G=tf([taud 1],1); figure(1), bode(G); figure(2), nyquist(G); The magnitude and the phase-angle plots are sort of вЂњupside-downвЂќ versions of п¬Ѓrst-order lag, with the phase angle increasing from 0в—¦ to 90в—¦ in the high-frequency asymptote. The polar plot, on the other hand, is entirely different. The real part of G( jП‰) is always 1 and not dependent on frequency. Example 8.5: What are the Bode and Nyquist plots of a dead-time function G(s) = eв€’Оёs ? Again, we make the s = jП‰ substitution into the transfer function to obtain G( jП‰) = eв€’Оё П‰j . The magnitude is simply unity, |G( jП‰)| = 1, and the phase angle is G( jП‰) = в€’П‰Оё. When П‰ = ПЂ/Оё, G( jП‰) = в€’ПЂ . On the polar plot, the dead-time function is a unit circle. The phase-angle plot is not a straight line because frequency is on a log scale. We need hints from our MATLAB Session to plot this function in Fig. E8.5. WeвЂ™ll do it together in the next example. The important point is that the phase lag of the dead-time function increases without bound with respect to frequency. This is what is called a nonminimum-phase system, as opposed to the п¬Ѓrst and second transfer functions, which are minimum-phase systems. 157 Frequency-Response Analysis 1 1 G П† 0В° вЂ“180В° П‰ Figure E8.5. Formally, a minimum-phase system is one that has no dead time and has neither poles nor zeros in the RHP. (See the Review Problems.) From here on, only the important analytical equations or plots of asymptotes are provided in the examples. You should generate plots with sample numerical values by using MATLAB as you read them. Example 8.6: What are the Bode and Nyquist plots of a п¬Ѓrst-order lag with dead time? This example shows us the important reason why and how frequency-response analysis can handle dead time so easily. We substitute s = jП‰ into G(s) = [(K p eв€’td s )/(П„ p s + 1)], and we have K p eв€’ jП‰td G( jП‰) = . jП‰П„ p + 1 From Example 8.5, we know that the magnitude of the dead time function is 1. Combining also with the results in Example 8.2, we п¬Ѓnd that the magnitude and the phase angle of G( jП‰) are Kp , G( jП‰) = tanв€’1 (П‰П„ p ) в€’ П‰td . |G( jП‰)| = 2 2 1 + П‰ П„p The results are exact вЂ“ we do not need to make approximations as we had to with the rootlocus or the Routh array. The magnitude plot is the same as the п¬Ѓrst-order function, but the phase lag increases without bound because of the dead-time contribution in the second term. We will see that this is a major contribution to instability. On the Nyquist plot, the G( jП‰) locus starts at K p on the real axis and then вЂњspiralsвЂќ into the origin of the s plane. This is how we may do the plots with time delay (details in MATLAB Session 7). Half of the work is taken up by the plotting statements. kp=1; % Some arbitrary values taup=10; G=tf(kp,[taup 1]); tdead=2; freq=logspace(-1,1); [mag,phase]=bode(G,freq); mag=mag(1,:); phase=phase(1,:); % Make a frequency vector % MATLAB specific step phase = phase - ((180/pi)*tdead*freq); % Add dead-time phase lag 158 8.2. Graphical Analysis Tools figure(1); subplot(211), loglog(freq,mag) ylabel('Magnitude'),title('Bode Plot'), grid subplot(212), semilogx(freq,phase) ylabel('Phase (degree)'),xlabel('Frequency'), grid figure(2) % We have to switch over to the polar plot phase=phase*pi/180; % function to do this Nyquist plot polar(phase,mag) Example 8.7: What are the Bode and Nyquist plots of a pure integrating function G(s) = K /s? The s = jП‰ substitution into the integrating function leads to a pure imaginary number: G( jП‰) = K /jП‰ = в€’ j K /П‰. The magnitude and the phase angle are |G( jП‰)| = K /П‰, G( jП‰) = tanв€’1 (в€’в€ћ) = в€’90в—¦ . Sample MATLAB statements are G=tf(1,[1 0]); figure(1), bode(G) figure(2), nyquist(G) The magnitude logвЂ“log plot is a line with slope в€’1. The phase-angle plot is a line at в€’90в—¦ . The polar plot is the negative-imaginary axis, approaching from negative inп¬Ѓnity with П‰ = 0 to the origin with П‰ в†’ в€ћ. Example 8.8: What are the Bode and Nyquist plots of a п¬Ѓrst-order lag with an integrator? Our п¬Ѓrst impulse may be an s = jП‰ substitution into the transfer function: G(s) = Kp . s(П„ p s + 1) However, the result is immediately obvious if we consider the function as the product of a п¬Ѓrst-order lag and an integrator. Combining the results from Examples 8.2 and 8.7, we п¬Ѓnd that the magnitude and the phase angle are |G( jП‰)| = Kp П‰ 1 + П„ p2 П‰2 , G( jП‰) = tanв€’1 (в€’в€ћ) + tanв€’1 (в€’П„ p П‰) = в€’90в—¦ + tanв€’1 (в€’П„ p П‰). Sample MATLAB statements are kp=1; taup=2; G=tf(kp,[taup 1 0]); figure(1), bode(G) figure(2), nyquist(G) 159 Frequency-Response Analysis Because of the integrator, the magnitude logвЂ“log plot does not have a low- or a highfrequency asymptote. The plot is a curve in which magnitude decreases with frequency. The phase-angle plot starts at в€’90в—¦ at the low-frequency asymptote and decreases to в€’180в—¦ at the high-frequency asymptote. The polar-plot curve approaches from negative inп¬Ѓnity along the vertical line в€’K p П„ p and approaches the origin as П‰ в†’ в€ћ. Example 8.9: Sketch the Bode plot of the following transfer function: G(s) = (5s + 1) . (10s + 1)(2s + 1) The MATLAB statements are G=tf([5 1],conv([10 1],[2 1])); bode(G); Formally, we would plot |G( jП‰)| = 1 + 52 П‰ 2 в€љ 1 1 , в€љ 1 + 102 П‰2 1 + 22 П‰2 G( jП‰) = tanв€’1 (5П‰) + tanв€’1 (в€’10П‰) + tanв€’1 (в€’2П‰). With MATLAB, what you п¬Ѓnd is that the actual curves are very smooth; it is quite different from hand-sketching. Nevertheless, understanding the asymptotic features is important in helping us check whether the results are correct. This is particularly easy (and important) with the phase-lag curve. To help us understand MATLAB results, a sketch of the low- and the high-frequency asymptotes is provided in Fig. E8.9. A key step is to identify the corner frequencies. In this case, the corner frequency of the п¬Ѓrst-order lead is at 1/5 or 0.2 rad/s, whereas the two п¬Ѓrst-order lag terms have their corner frequencies at 1/10 and 1/2 rad/s. The п¬Ѓnal curve is a superimposition of the contributions from each term in the overall transfer function. In addition, if you want to better see the little phase-lag вЂњhumpвЂќ that you expect from hand-sketching, change the term in the denominator from (2s + 1) to (s + 1) so that the phase lag of this term will not kick in too soon. slope = 1 G 1 slope = вЂ“1 90В° П† 0В° вЂ“90В° 0.1 Figure E8.9. 160 0.2 0.5 1 П‰ 8.3. Stability Analysis 8.3. Stability Analysis With frequency-response analysis, we can derive a general relative stability criterion. The result is applicable to systems with dead time. The analysis of the closed-loop system can be reduced to using only the open-loop transfer functions in the computation. 8.3.1. Nyquist Stability Criterion Consider the characteristic equation of a closed-loop system, 1 + G m G c G a G p = 0, (7.2) where often G OL is used to denote the open-loop transfer function: G OL = G m G c G a G p . To вЂњprobeвЂќ the property on the imaginary axis, we can make a substitution of s = jП‰ and rewrite the equation as G m ( jП‰)G c ( jП‰)G a ( jП‰)G p ( jП‰) = в€’1 or G OL ( jП‰) = в€’1. (7.2a) This equation, of course, contains information regarding stability and, as it is written, implies that we may match properties on the LHS with the point (в€’1, 0) on the complex plane. The form in Eq. (7.2a) also implies that in the process of analyzing the closed-loop stability property, the calculation procedures (or computer programs) require only the open-loop transfer functions. For complex problems, this fact eliminates unnecessary algebra. Just the Nyquist stability criterion is given here.9 Nyquist stability criterion: Given the closed-loop equation 1 + G OL ( jП‰) = 0, if the function G OL ( jП‰) has P open-loop poles and if the polar plot of G OL ( jП‰) encircles the (в€’1, 0) point N times as П‰ is varied from в€’в€ћ to в€ћ, the number of unstable closed-loop poles in the RHP is Z = N + P. [Z is named after the number of zeros to 1 + G OL ( jП‰) = 0.] Do not panic! Without the explanation in the Web Support, this statement makes little sense. On the other hand, we do not really need this full deп¬Ѓnition because we know that just one unstable closed-loop pole is bad enough. Thus the implementation of the Nyquist stability criterion is much simpler than the theory. A simpliп¬Ѓed statement of Nyquist stability criterion (Fig. 8.3): Given the closed-loop equation 1 + G OL ( jП‰) = 0, the closed-loop system is stable if the polar plot of G OL ( jП‰) does not encircle the (в€’1, 0) point in the G OL -plane. In this statement, the term polar plot of G OL has been used to replace a mouthful of words. G OL -plane has been added in the wording to emphasize that we are using an analysis based on Eq. (7.2a). The real question lies in what safety margin we should impose on a given system. This question leads to the deп¬Ѓnitions of gain and phase margins, which constitute the basis of the general relative stability criteria for closed-loop systems. When we make a Nyquist plot, we usually just map the positive-imaginary axis from П‰ = 0 to inп¬Ѓnity, as opposed to the entire axis starting from negative inп¬Ѓnity. If a system is unstable, the resulting plot will contribute only ПЂ to the (в€’1, 0) point as opposed to 9 The formal explanation is in the Web Support. For a quick idea, our result is based on writing G OL ( jП‰) = в€’1. One simple thinking of instability is that if we feed back a sinusoidal wave, it will undergo a в€’180в—¦ phase shift at the summing point of a negative-feedback loop. If the amplitude of the wave is less than one after passing through the summing point, it will die out. However, if the amplitude is larger than one, the oscillations will grow. 161 Frequency-Response Analysis вЂ“1 Unstable Stable GOL(j П‰ ) Figure 8.3. lllustration of the stable versus unstable possibilities under the Nyquist stability criterion. 2ПЂ вЂ“ what encirclement really means. However, just mapping the positive-imaginary axis is sufп¬Ѓcient to observe whether the plot may encircle the (в€’1, 0) point. 8.3.2. Gain and Phase Margins Once we understand the origin of Nyquist stability criterion, putting it to use is easy. Suppose we have a closed-loop system with the characteristic equation 1 + G c G p = 0. With the point (в€’1, 0) as a reference and the G c ( jП‰)G p ( jП‰) curve on a Nyquist plot, we can establish a relative measure on how safe we are вЂ“ that is, how far we are from the (в€’1, 0) point. There are two possibilities. They are shown in Fig. 8.4, together with their interpretations on a Bode plot. Gc G p = 1 A measure of GM вЂ“1 вЂ“1 PM вЂ“ Gc Gp at вЂ“180В° G c G p (jП‰) G c G p (jП‰) 1 GM = Gc Gp 1 Gc Gp 0В° П† PM вЂ“180В° П‰ cp П‰ cg Figure 8.4. Interpretation of the gain and the phase margins based on the Nyquist stability criterion with Nyquist and Bode plots. 162 8.3. Stability Analysis (1) On the negative-real axis (в€’180в—¦ ), п¬Ѓnd the вЂњdistanceвЂќ of |G c G p | from (в€’1, 0). This is the gain margin (GM). The formal deп¬Ѓnition is GM = 1 , |G c ( jП‰cg )G p ( jП‰cg )| (8.21) where G c ( jП‰)G p ( jП‰) is evaluated at the point at which it has a phase lag of в€’180в—¦ . The particular frequency at this point is the gain crossover frequency, П‰ = П‰cg . The smaller the magnitude of G c G p at в€’180в—¦ , the larger the GM and the вЂњsafer we are.вЂќ (2) Find the frequency where the magnitude |G c ( jП‰)G p ( jП‰)| is 1. This particular frequency is the phase crossover frequency, П‰ = П‰cp . We then п¬Ѓnd the angle between G c G p and в€’180в—¦ . This is the phase margin (PM). The formal deп¬Ѓnition is PM = П† в€’ (в€’180в—¦ ) = П† + 180в—¦ , (8.22) where the phase lag П† (a negative value) is measured at the point where G c ( jП‰cp )G p ( jП‰cp ) has a magnitude of one. The larger the angle, the larger the PM and the вЂњsafer we are.вЂќ For most control systems, we usually take a GM between 1.7 and 2 and a PM between 30в—¦ and 45в—¦ as the design speciп¬Ѓcations. The Nyquist stability criterion can be applied to Bode plots. In fact, the calculation in which the Bode plot is used is much easier. To obtain the GM, we п¬Ѓnd the value of |G c G p | that corresponds to a phase lag of в€’180в—¦ . To п¬Ѓnd the PM, we look up the phase lag corresponding to when |G c G p | is 1. Once again, recall that simple п¬Ѓrst- and second-order systems with no positive zeros are always stable, and no matter what a computer program may return, the GM and the PM have no meaning. Also, the GM crossover frequency is the same as the so-called ultimate frequency when we do the s = jП‰ substitution in the closed-loop characteristic equation or when we measure the value with the ZieglerвЂ“Nichols ultimate-cycle method. The GM and the PM are used in the next section for controller design. Before that, letвЂ™s plot different controller transfer functions and infer their properties in frequency-response analysis. Generally speaking, any function that introduces additional phase lag or magnitude tends to be destabilizing, and the effect is frequency dependent. We skip the proportional controller, which is just G c = K c . Again, do the plots by using sample numbers with MATLAB as you read the examples. Example 8.10: Derive the magnitude and the phase lag of the transfer function of a PI controller. We could make the substitution s = jП‰ into G c (s) = K c [1 + (1/П„ I s)]. However, we can obtain the result immediately if we see that the function is a product of an integrator and a п¬Ѓrst-order lead: G c (s) = K c 1 (П„ I s + 1). П„I s Thus |G c ( jП‰)| = K c 1 П‰П„ I 1 + П‰2 П„ I2 , G c ( jП‰) = в€’90в—¦ + tanв€’1 (П‰П„ I ). 163 Frequency-Response Analysis To do a demonstration plot, we may try the following MATLAB statements: kc=1; % Just some arbitrary numbers taui=2; G=tf(kc*[taui 1],[taui 0]); figure(1), bode(G); figure(2), nyquist(G); On the magnitude plot, the low-frequency asymptote is a line with slope в€’1. The highfrequency asymptote is a horizontal line at K c . The phase-angle plot starts at в€’90в—¦ at very low frequencies and approaches 0в—¦ in the high-frequency limit. On the polar plot, the G c ( jП‰) locus is a vertical line that approaches from negative inп¬Ѓnity at П‰ = 0. At inп¬Ѓnity frequency, it is at the K c point on the real axis. Integral control adds an additional phase lag (в€’90в—¦ ) at low frequencies below the corner frequency 1/П„ I . A larger value of the integral time constant will limit the frequency range in which the controller introduces phase lag. This is one reason why choosing a large П„ I tends to be more stable than choosing a system with a small П„ I .10 Example 8.11: Derive the magnitude and the phase lag of the transfer function of an ideal PD controller. The result is that of a п¬Ѓrst-order lead, as in Example 8.4. From G c (s) = K c (1 + П„ D s), we have, after s = jП‰ substitution, G c ( jП‰) = K c (1 + jП‰П„ D ), and thus |G c ( jП‰)| = K c 1 + П‰2 П„ D2 , G c ( jП‰) = tanв€’1 (П‰П„ D ). On the magnitude plot, the low-frequency asymptote is a horizontal line at K c . The highfrequency asymptote has a slope of +1. The phase-angle plot starts at 0в—¦ at very low frequencies and approaches 90в—¦ in the high-frequency limit. On the polar plot, the G c ( jП‰) locus is a vertical line that starts at the point K c on the real axis and approaches inп¬Ѓnity. Based on the phase-angle plot, the PD controller provides a phase lead and thus a stabilizing effect. At the same time, the higher magnitude at higher-frequency ranges will amplify noises. There is a practical limit as to how fast a response a PD controller can handle. The MATLAB statements are essentially the same as the п¬Ѓrst-order lead function: kc=1; % Just some numbers we pick arbitrarily taud=2; G=tf(kc*[taud 1],1); figure(1), bode(G); figure(2), nyquist(G); 10 164 Furthermore, we may want to choose П„ I such that 1/П„ I is smaller than the corner frequency associated with the slowest open-loop pole of the process function. In this way, we help to stabilize the system by reducing the phase lag that is due to the integration before the phase lag of the process function вЂњkicks in.вЂќ However, integral control will not be effective if П„ I is too large, and there will be a design trade-off when we work with very slow processes. We will test this idea in Homework Problem II.38. 8.3. Stability Analysis G Kc 90В° П† 0В° вЂ“90В° 1/ П„I 1/ П„D П‰ Figure E8.12. Only highand low-frequency asymptotes are shown here. Fill in the rest with the help of MATLAB. Example 8.12: Derive the magnitude and the phase lag of the controller transfer function G c (s) = K c 1 + П„I s (1 + П„ D s), П„I s which is a PID controller with ideal derivative action and in the so-called interacting form. We look at this function because it is a product of an integrator and two п¬Ѓrst-order leads, and we can identify the high- and the low-frequency asymptotes easily. It is not identical to the ideal (noninteracting) PID controller, but the numerical results are very similar. First, we need to generate the plots. Use Fig. E8.12 to help interpret the MATLABgenerated Bode plot11 : kc=1; taui=4; taud=1; Gi=tf(kc*[taui 1],[taui 0]); Gd=tf([taud 1],1); G=Gi*Gd; bode(G); By choosing П„ D < П„ I (i.e., corner frequencies 1/П„ D > 1/П„ I ), we п¬Ѓnd that the magnitude plot has a notch shape. How sharp it is will depend on the relative values of the corner frequencies. The low-frequency asymptote below 1/П„ I has a slope of в€’1. The high-frequency asymptote above 1/П„ D has a slope of +1. The phase-angle plot starts at в€’90в—¦ , rises to 0в—¦ after the corner frequency 1/П„ I , and п¬Ѓnally reaches 90в—¦ at the high-frequency limit. Relatively speaking, a PID controller behaves like a PI controller at low frequencies, whereas it is more like a PD controller at high frequencies. The controller is most desirable in the midrange, where it has the features of both PI and PD controllers. Also, in the notch region, the controller function has the lowest magnitude and allows for a larger GM for the system. 11 If you want to see a plot for an ideal PID controller, use G=tf(kc*[taui*taud taui 1],[taui 0]); 165 Frequency-Response Analysis Example 8.13: Derive the magnitude and the phase lag of the transfer functions of phaselead and phase-lag compensators. In many electromechanical control systems, the controller G c is built with relatively simple R-C circuits and takes the form of a leadвЂ“lag element: (s + z 0 ) . G c (s) = K (s + p0 ) Here, z 0 and p0 are just two positive numbers. There are obviously two possibilities: case (a) z 0 > p0 , and case (b) z 0 < p0 . Sketch the magnitude and the phase-lag plots of G c for both cases. Identify which case is the phase-lead and which case is the phase-lag compensation. What types of classical controllers may phase-lead and phase-lag compensations resemble? We may look at the controller transfer function in the time-constant form: G c (s) = K z 0 (s/z 0 + 1) , p0 (s/ p0 + 1) where we could further write K c = K z 0 / p0 , but we will use the pole-zero form. Either way, we should see that the corner frequencies are at П‰ = z 0 and p0 . To make a Bode plot, we theoretically should do the s = jП‰ substitution, but we can write the magnitude and the phase angle immediately if we recognize that the function is a product of a п¬Ѓrst-order lead and a п¬Ѓrst-order lag. Hence, making use of Examples 8.2 and 8.4, we can write 1 |G c ( jП‰)| = K П‰2 + z 02 П‰2 + p02 , G c ( jП‰) = tanв€’1 (П‰/z 0 ) + tanв€’1 (в€’П‰/ p0 ). Figure E8.13 is a rough hand sketch with the high- and low-frequency asymptotes. It is meant to help interpret the MATLAB plots that we will generate next. (a) With z 0 > p0 , the phase angle is always negative, and this is the phase-lag compensator. The MATLAB statements to get an illustrative plot are k=1; zo=4; % Try repeat with various choices of zo and po po=1; G=zpk(-zo,-po,k); bode(G); (a) Phase-lag G G Kc Kc 1 1 90В° 90В° П† П† 0В° 0В° вЂ“90В° вЂ“90В° po Figure E8.13. 166 (b) Phase-lead zo П‰ zo po П‰ 8.3. Stability Analysis The shape of the magnitude plot resembles that of a PI controller, but with an upper limit on the low-frequency asymptote. We can infer that the phase-lag compensator could be more stabilizing than a PI controller with very slow systems.12 The notch-shaped phase angle plot of the phase-lag compensator is quite different from that of a PI controller. The phase lag starts at 0в—¦ versus в€’90в—¦ for a PI controller. From a stability point of view, a phaselag compensator is preferred to a PI controller. On the other hand, without an integrating function, the phase-lag compensator cannot eliminate offset. (b) With z 0 < p0 , the phase-angle is positive, and this is the phase-lead compensator. First, we need an illustrative plot. Sample MATLAB statements to use are zo=1; po=4; G=zpk(-zo,-po,1); bode(G); The nice feature of the phase-lead compensator, and for that matter a real PD controller, is that it limits the high-frequency magnitude. In contrast, an ideal PD controller has no upper limit and would amplify high-frequency input noises much more signiп¬Ѓcantly. Example 8.14: Designing phase-lead and phase-lag compensators. Consider a simple unity feedback loop with the characteristic equation 1 + G c G p = 0 and with a п¬Ѓrst-order process function G p = [K p /(П„ p s + 1)]. What are the design considerations if we use either a phaselead or a phase-lag compensator? Consider the consequences by using Bode and root-locus plots. With K c = K z 0 / p0 , the closed-loop transfer function is K c K p (s/z 0 + 1) G c Gp C = = , R 1 + G c Gp (s/ p0 + 1)(П„ p s + 1) + K c K p (s/z 0 + 1) and after one more algebraic step, we will see that the system steady-state gain is K c K p / (1 + K c K p ), which means that there will be an offset whether we use a phase-lead or a phase-lag compensator. From the characteristic polynomial, it is probable that we will get either an overdamped or an underdamped system response, depending on how we design the controller. The choice is not clear from the algebra, and this is where the root-locus plot comes in handy. From the perspective of a root-locus plot, we can immediately make the decision that, no matter what, both z 0 and p0 should be larger than the value of 1/П„ p in G p . ThatвЂ™s how we may вЂњsteerвЂќ the closed-loop poles away from the imaginary axis for better system response. (If we know our root locus, we should know that this system is always stable.) (a) LetвЂ™s п¬Ѓrst consider a phase-lead compensator, z 0 < p0 . We п¬Ѓrst construct the Bode and the root-locus plots that represent a system containing the compensator and a п¬Ѓrst-order process: 12 From the perspective of a root-locus plot, a phase-lag compensator adds a large open-loop zero and a relatively small open-loop pole. And for a phase-lead compensator, we are adding a large open-loop pole. When p0 z 0 (or 1/ p0 1/z 0 ), we can also look at the phase-lead compensator as the real PD controller. How to design their locations, of course, depends on the particular process that we need to control. We will see that in Example 8.14. 167 Frequency-Response Analysis Kp=1; % Arbitrary numbers for the process function taup=1; Gp=tf(Kp,[taup 1]); zo=2; % Phase-lead, zo < po po=4; Gc=zpk(-zo,-po,1) figure(1), bode(Gc*Gp) figure(2), rlocus(Gc*Gp) The root-locus plot resembles that of a real PD controller. The system remains overdamped with no complex closed-loop poles. One root locus runs from the вЂњreal PD poleвЂќ в€’ p0 to negative inп¬Ѓnity. The other is from в€’П„ p to в€’z 0 , which limits the fastest possible system response. How large z 0 , and thus K c , can be depends on the real physical hardware and process. On the Bode plot, the corner frequencies are, in increasing order, 1/П„ p , z 0 , and p0 . The two magnitude high frequency asymptotes intersecting 1/П„ p and p0 are those of п¬Ѓrst-order lags. The high frequency asymptote at П‰ = z 0 is that of a п¬Ѓrst-order lead. The largest phase lag of the system is в€’90в—¦ at very high frequencies. The system is always stable, as displayed by the root-locus plot. (b) With a phase-lag compensator, z 0 > p0 , we can use these statements: % Gp remains the same as in part (a) zo=4; po=2; Gc=zpk(-zo,-po,1) figure(1), bode(Gc*Gp) figure(2), rlocus(Gc*Gp) The shape of the root-locus plot resembles that of a PI controller, except of course we do not have an open-loop pole at the origin anymore. The root loci approach one another from в€’П„ p and в€’ p0 , then break away from the real axis to form a circle that breaks in to the left of the open-loop zero at в€’z 0 . One locus approaches negative inп¬Ѓnity and the other goes toward в€’z 0 . We may design the controller with an approach similar to that in Example 7.7. On the Bode plot, the corner frequencies are, in increasing order, 1/П„ p , p0 , and z 0 . The individual term frequency asymptotes have the same properties as in part (a). The largest phase lag of the system is larger than в€’90в—¦ just past П‰ = p0 , but still much less than в€’180в—¦ . This system is always stable. 8.4. Controller Design The concept of GMs and PMs derived from the Nyquist criterion provides a general relative stability criterion. Frequency-response graphical tools such as Bode, Nyquist, and Nichols plots can all be used in ensuring that a control system is stable. As in root-locus plots, we can vary only one parameter at a time, and the common practice is to vary the proportional gain. 168 8.4. Controller Design R+ вЂ“ Kc G* C Figure 8.5. A simple unity feedback system. 8.4.1. How Do We Calculate Proportional Gain without Trial and Error? This is a big question when we use, for example, a Bode plot. LetвЂ™s presume that we have a closed-loop system in which we know вЂњeverythingвЂќ but the proportional gain, and we write the closed-loop characteristic equation as 1 + G OL = 1 + K c G в€— = 0, where G OL = G c G a G p G m . We further rewrite the function as K c G в€— to indicate that we would like to п¬Ѓnd K c (Fig. 8.5). The notation G в€— is more than just the product of G a G p G m ; G в€— includes the integral and the derivative terms if we use a PID controller. With the straight textbook deп¬Ѓnition and explanation, the GM and the PM of the closedloop system apply to only the magnitude and the phase-lag plots by use of the entire open-loop function, |G OL | and G OL . It means that we need to know the value for the proportional gain K c . Of course, we do not, and the whole affair appears to be a trial-and-error calculation. The question is whether we can calculate K c without guessing. The answer is yes. The next question is whether we can get the answer from the plots of |G в€— | and G в€— . This answer is also yes. From the deп¬Ѓnition of GM, we have GM = 1 |G OL ( jП‰cg )| or GM = 1 , K c |G в€— ( jП‰cg )| (8.23) where the magnitudes are evaluated at the gain crossover frequency П‰cg as deп¬Ѓned by the в€’180в—¦ phase lag. We can п¬Ѓnd |G в€— ( jП‰cg )| simply by a Bode plot of G в€— itself. The key is that the phase angle of K c is zero, G в€— is identical to G OL , and both |G в€— | and |G OL | have the same П‰cg .13 From the deп¬Ѓnition of GM, GM = 1 at marginal stability. Hence, with GM = 1 in Eq. (8.23), we can evaluate K cu = 1/|G в€— ( jП‰cg )|. Equation (8.23) can alternatively be stated as GM = K cu . Kc (8.24) Once we know the value of K cu , we can calculate the K c for a given GM in a problem. Typically, we select GM, say, 1.7, to п¬Ѓnd the proportional gain. We can also use the Bode plot of G в€— to do PM calculations. From the textbook deп¬Ѓnition, we are supposed to п¬Ѓnd the phase angle П† = G OL , where |G OL | = 1. If the PM is 45в—¦ , П† should be в€’135в—¦ . It appears that we need to know K c beforehand to calculate G OL , but we do not. We use the fact that G в€— is identical to G OL and П‰cp is the same in both plots, so we can go backward. On the G в€— Bode plot, we can п¬Ѓnd the value of |G в€— ( jП‰cp )| that corresponds to 13 We can use MATLAB to do the Bode plot of G в€— and use the margin() function, which will return the вЂњGMвЂќ of G в€— , but we now know it is really 1/|G в€— ( jП‰cg )| or, following Eq. (8.24), also the value for the ultimate gain K cu . 169 Frequency-Response Analysis a phase lag of, say, в€’135в—¦ . Now |G OL | = K c |G в€— | and because |G OL | = 1 at the PM, we can п¬Ѓnd K c = 1/|G в€— ( jП‰cp )| that will provide a phase margin of 45в—¦ without trial and error. How do we know the answer is correct? Just вЂњplugвЂќ K c back into G OL and repeat the Bode plot by using G OL . It does not take that much time to check with MATLAB. Now we are п¬Ѓnally ready for some examples. Again, run MATLAB to conп¬Ѓrm the results while you read them. Example 7.2C: LetвЂ™s revisit Example 7.2 with the closed-loop characteristic equation: 1 = 0. (s + 3)(s + 2)(s + 1) If we want to design a PI controller, how should we proceed with frequency-response methods? LetвЂ™s presume that the unit of the time constants is in minutes. The п¬Ѓrst step is to п¬Ѓnd the ultimate gain. With the given third-order process transfer function, we use the following MATLAB commands, 1 + Kc p=poly([-1 -2 -3]); G=tf(1,p); margin(G); MATLAB returns K cu as 35.6 dB (at П‰cg = 3.3 rad/min),14 which we easily recalculate as 60 (1035.6/20 ). Note that the low-frequency asymptote of the magnitude plot is not 1 (0 dB). Why? ThatвЂ™s because the transfer function is not in the time-constant form. If we factor the function accordingly, we should expect a low-frequency asymptote of 1/6 (в€’15.6 dB). If we take a GM of 1.7 in Eq. (8.24), we should use a proportional gain of K c = 60/1.7 = 35.3. This is the case if we use only a proportional controller as in Example 6.2. We repeat the Bode plot and margin calculation with K c = 35.3: G=tf(35.3,p); margin(G); Now we should п¬Ѓnd that the GM is indeed 1.7 (4.6 dB, 104.6/20 ) and the PM is 18.8в—¦ , which is a bit low according to the design rule of thumb. We have yet to tackle the PI controller. There are, of course, different ways to п¬Ѓnd a good integral time constant. With frequency response, we have the handy tool of the ZieglerвЂ“ Nichols ultimate-cycle tuning relations. So with K cu = 60 and П‰cg = 3.3 rad/min, we п¬Ѓnd by referring to Table 6.1 that if we use only a proportional controller, we should use K c = 30, and if we use a PI controller, we should use K c = 27.3 and П„ I = 1.58 min.15 Using the ZieglerвЂ“Nichols tuning parameters, we repeat the proportional controller system Bode plot: G=tf(30,p); margin(G); We should п¬Ѓnd a GM of 2 (6 dB) and a PM of 25.4в—¦ , which is deп¬Ѓnitely a bit more conservative than the 1.7 GM result. 14 15 170 MATLAB always considers time to be in seconds, but this should not concern us as long as we keep our own time units consistent. All these calculations are done with the M-п¬Ѓle recipe.m from the Web Support. 8.4. Controller Design With the PI controller, we use the following statements: kc=27.3; taui=1.58; Gc=tf(kc*[taui 1],[taui 0]); G=tf(1,p); % p was defined at the beginning of this example margin(Gc*G); We should п¬Ѓnd a GM of 1.47 (3.34 dB) and a PM of 12.3в—¦ . Both margins are a bit small. If we do a root-locus plot on each case and with the help of rlocfind() in MATLAB , we should п¬Ѓnd that the corresponding closed-loop poles of these results are indeed quite close to the imaginary axis. Where do we go from here? We may stay with the design or we may increase the margins. We also can use MATLAB to simulate the closed-loop time-domain response and from the underdamped time-response curve, estimate numerically the effective damping ratio and other quantities such as percentage of OS. If the time-domain response does not meet our speciп¬Ѓcation, we will have to tune the values of K c or П„ I . If we want to increase the margin, we either have to reduce the value of K c or increase П„ I . One possibility is to keep П„ I = 1.58 min and repeat the Bode plot calculation to п¬Ѓnd a new K c that may provide a gain margin of, say, 2 (6 dB), as in the case of using only the proportional controller. To do so, we п¬Ѓrst need to п¬Ѓnd the new ultimate gain by using the PI controller: kc=1; taui=1.58; Gc=tf(kc*[taui 1],[taui 0]); margin(Gc*G); % G remains as above MATLAB should return K cu = 40.2 (32.1 dB). Thus, following Eq. (8.24), we need to use K c = 40.2/2 = 20.1 to meet the GM speciп¬Ѓcation of 2. You can double check the result yourself with kc=20.1,taui=1.58. If so, you should п¬Ѓnd that the phase margin is now 23в—¦ вЂ“ a bit low, but we probably can accept that. After this, we may proceed to the time-domain response calculations. The ideas in this example can be applied to a PID controller. Yes, controller design is indeed an iterative process. A computer is a handy tool, but we still need to know what we are doing. Example 7.2D: Back in the last example with a proportional controller, a gain margin of 1.7 created a system with a very small PM. What proportional gain should we use to achieve a PM of at least 45в—¦ ? Following the explanation after Eq. (8.24), we should calculate the phase angle of G( jП‰) = [( jП‰ + 3)( jП‰ + 2)( jП‰ + 1)]в€’1 . Of course, weвЂ™d rather use bode() in MATLAB: p=poly([-1 -2 -3]); G=tf(1,p); [mag,phase,w]=bode(G); mag=mag(1,:); % MATLAB returns a three-dimensional % array 16 16 The MATLAB function bode() returns the actual magnitude even though the documentation says decibels. This is a detail that we can check with the function freqresp() as explained on the Web Support, especially with future upgrades of the software. 171 Frequency-Response Analysis phase=phase(1,:); tmp=[w';mag;phase]' % Be careful with the primes here From the tmp matrix of frequency, magnitude, and phase angle, we п¬Ѓnd that at П‰ = 1.74 rad/min, |G| = 0.054 and G = в€’131.4в—¦ , which provides a PM of 48.6в—¦ . Also, at П‰ = 2.21 rad/min, |G| = 0.037 and G = в€’150в—¦ , which provides a PM of 30в—¦ . (We will need to do an interpolation if the problem statement dictates, say, a PM of exactly 45в—¦ .) To achieve a PM of 48.6в—¦ , we use a proportional gain of K c = 1/0.054 = 18.5. We can repeat the Bode plot calculation with MATLAB by using K c = 18.5 and the statements G=tf(18.5,p); margin(G); The result conп¬Ѓrms that the system has a PM of 48.6в—¦ and a GM of 3.2 (10.2 dB), a much more conservative design than a GM of 1.7. If we choose to use K c = 1/0.037 = 26.96, a MATLAB calculation should conп¬Ѓrm a PM of 30в—¦ and a GM of 2.2. Example 7.4A: This time, letвЂ™s revisit Example 7.4, which is a system with dead time. We would like to know how to start designing a PI controller. The closed-loop characteristic equation with a proportional controller is (again assuming the time unit is in minutes) 1 + Kc 0.8eв€’2s = 0. 5s + 1 The п¬Ѓrst step is to п¬Ѓnd the ultimate gain. Following Example 8.6, we can add easily the extra phase lag that is due to the dead-time function: G=tf(0.8,[5 1]); tdead=2; [Mag,Phase,freq]=bode(G); Mag=Mag(1,:); Phase=Phase(1,:) - ((180/pi)*tdead*freq'); [Gm,Pm,Wcg,Wcp]=margin(Mag,Phase,freq) We should п¬Ѓnd K cu = 5.72 at П‰cg = 0.893 rad/min, which is exactly what we found in Example 7.4, but which takes a lot more work. If we take a GM of 1.7, we should use a proportional gain of K c = 5.72/1.7 = 3.36. We use G=tf(3.36*0.8,[5 1]); and repeat the calculation. We should п¬Ѓnd the corresponding PM to be 54.6в—¦ , which is plenty. These are the controller settings if we again use the ZieglerвЂ“Nichols tuning relations (or really recipe.m): with K cu = 5.73 and П‰cg = 0.895 rad/min, we should use a proportional controller with K c = 2.87, and if we use a PI controller, we should use K c = 2.61 and П„ I = 5.85 min. The tasks of checking the GM, PM, and the time-domain response are left as a Review Problem. Example 5.7D: We can now п¬Ѓnally wrap up the dye-mixing problem that we left in Example 5.7C in Chap. 6. 172 8.4. Controller Design (a) The control system can be unstable if we place the photodetector too far downstream. To cut down on the algebra, we look into the problem with a slight simpliп¬Ѓcation. At this point, we use only a proportional controller. Because the regulating valve is so much faster than the mixing process, we retain only the mixing п¬Ѓrst-order lag to obtain the approximate closed-loop characteristic equation: 1+ K c K v K p K m eв€’td s = 0. (П„ p s + 1) Again for illustration purpose, we supposedly have chosen K c such that K c K v K p K m = 5, and П„ p is the mixing-process time constant. Find, without trial and error and without further approximation, the maximum distance L that the photodetector can be placed downstream such that the system remains stable. (There are two ways to get the answer. The idea of using magnitude and phase angle and the Nyquist criterion is by far the less confusing method and less prone to algebraic mistakes.) (b) To factor in some safety margin, we install the photodetector at half the maximum distance that we found in part (a). With the same proportional gain and the same approximation used in part (a), what is the GM? (c) Now that we can install the photodetector at a well-chosen distance, we can put the dynamics of the regulating valve back into our closed-loop analysis. What is the critical proportional gain when we include the п¬Ѓrst-order lag that is due to the regulating valve? And what is the proportional gain if we want a GM of 1.7? (d) Finally we come to the controller design. All we know is that customers may be fussy about the color of their jeans. Of course, we also want to avoid the need to dispose of off-spec dye solutions unnecessarily. Despite these concerns, an old plant engineer mentioned that the actual dye tank downstream is huge and an OS of as much as 20 to 25% is acceptable as long as the mixing system settles down вЂњquickly.вЂќ So select your choice of controller, performance speciп¬Ѓcation, and controller gains. Double check your п¬Ѓnal design with time-domain simulation and frequency-response analysis to see that we have the proper controller design. (a) LetвЂ™s use the abbreviation G OL = G c G v G p G m , and thus the magnitude and phaseangle equations, from Example 5.7, are |G OL | = 5 |eв€’td s |, П„ps + 1 G OL = tanв€’1 (в€’П‰П„ p ) в€’ td П‰, where П„ p = 4s. At crossover П‰cg , G OL = в€’180в—¦ and GM = 1, meaning that |G OL | = 1. Because the magnitude of the dead-time transfer function is unity, the magnitude equation is simply 1= 5 1+ 2 42 П‰cg or П‰cg = 1.12s в€’1 . With the crossover frequency known, we now use the phase-angle equation to п¬Ѓnd the dead time: в€’180в—¦ = tanв€’1 (в€’4 Г— 1.12) в€’ td (1.12)(180/ПЂ) or td = 1.45s. Note the unit conversion to angles in the phase equation. With our arbitrary choice of proportional gain such that K c K v K p K m = 5, a dead time of td = 1.45s is associated with GM = 1. 173 Frequency-Response Analysis (b) Refer back to Example 5.7 in Chap. 5. The average п¬‚uid velocity is 400 cm/s. Thus the photodetector is located (1.45)(400) = 580 cm downstream from the mixer. To reduce the distance by half means that we now install the sensor at a location 580/2 = 290 cm downstream. The reduced transport lag is now 1.45/2 = 0.725s. To п¬Ѓnd the new GM, we need to, in theory, reverse the calculation sequence. We п¬Ѓrst use the phase equation to п¬Ѓnd the new crossover frequency П‰cg . Then we use the magnitude equation to п¬Ѓnd the new |G OL |, and the new GM is of course 1/|G OL |. However, because we now know the values of td , П„ p , and K c K v K p K m , we might as well use MATLAB. These are the statements: k=5; tdead=0.725; taup=4; % The large time constant; dominant pole is at 1/4 G=tf(k,[taup 1]); freq=logspace(-1,1)'; [Mag,Phase]=bode(G,freq); Mag=Mag(1,:); Phase=Phase(1,:) - ((180/pi)*tdead*freq'); [Gm,Pm,Wcg,Wcp]=margin(Mag,Phase,freq) We should п¬Ѓnd that the new gain margin is 1.86. (c) We now include the regulating valve and still use a proportional controller. The closed-loop equation is 1 + Kc 0.6 2.6eв€’0.725s = 0. 4s + 1 0.8 0.2s + 1 With MATLAB, the statements are k=0.8*0.6*2.6; G=tf(k, conv([0.2 1],[4 1])); [Mag,Phase]=bode(G,freq); Mag=Mag(1,:); Phase=Phase(1,:) - ((180/pi)*tdead*freq'); [Gm,Pm,Wcg,Wcp]=margin(Mag,Phase,freq) We should п¬Ѓnd that the ultimate gain is K c,u = 6.42 at the crossover frequency of П‰cg = 1.86 sв€’1 . To keep to a gain margin of 1.7, we need to reduce the proportional gain to K c = 6.42/1.7 = 3.77. (d) With K c,u = 6.42 and П‰cg = 1.86 sв€’1 , we can use the ZieglerвЂ“Nichols ultimate-gain tuning relations (with recipe.m) to п¬Ѓnd the following values for different objectives: Quarter decay Little overshoot No overshoot Kc П„I П„D 3.8 2.1 1.3 1.7 1.7 1.7 0.42 1.1 1.1 If we repeat the time-response simulations as in Example 5.7C in Chap. 6, we should п¬Ѓnd 174 8.4. Controller Design that the settings for the quarter decay lead to a 51% OS (roughly a 0.26 DR), the little OS settings have a 27% OS, and the so-called no OS settings still have в€ј8% OS. Finally, there is no unique solution for the п¬Ѓnal design. The problem statement leaves us with a lot of latitude, especially when we have seen from the time-response simulations that many combinations of controller settings give us similar closed-loop responses. In process engineering, we do not always have to be fastidious with the exact time-response speciп¬Ѓcations and hence values of the controller settings. Many real-life systems can provide acceptable performances within a certain range of response characteristics. As for controller settings, a good chance is that we have to perform п¬Ѓeld tuning to account for anything ranging from inaccurate process identiп¬Ѓcation to shifting operating conditions of nonlinear processes. For the present problem, and from all the settings provided by the different methods, we may select П„ I = 3s and П„ D = 0.5s. We next tune the proportional gain to give us the desired response. The closed-loop equation with an ideal PID controller is now 1 + Kc 1 + 1.248 1 + П„D s eв€’0.725s = 0. П„1 s (0.2s + 1)(4s + 1) First, we need MATLAB to п¬Ѓnd the ultimate gain: taui=3; taud=0.5; gc=tf([taui*taud (taui+taud) 1],[taui 0]); % ideal PID without % the Kc tdead=0.725; k=0.8*0.6*2.6; G=tf(k, conv([0.2 1],[4 1])); [Mag,Phase]=bode(gc*G,freq); Mag=Mag(1,:); Phase=Phase(1,:) - ((180/pi)*tdead*freq'); [Gm,Pm,Wcg,Wcp]=margin(Mag,Phase,freq) We should п¬Ѓnd the ultimate gain to be K cu = 5.87. And at a gain margin of 1.7, we need to use a proportional gain of K c = 5.87/1.7 = 3.45. A time-response simulation shows that the system, with respect to a unit-step change in the set point, has an OS of 23%. This tuning is slightly less oscillatory than if we had chosen П„ I = 3 s and П„ D = 0.3 s, as suggested by ITAE (Example 5.7A). In this case, K cu = 6.79 and K c = 4, which is closer to the K c from the ZieglerвЂ“Nichols tuning. Again, conп¬Ѓrm these results in the Review Problems. 8.4.2. A Final Word: Can Frequency-Response Methods Replace Root Locus? No. These methods complement each other. Very often a design decision is made only after analyses with both methods. The root-locus method gives us a good indication of the transient response of a system and the effect of varying the controller gain. However, we need a relatively accurate model for the analysis, not to mention that root locus does not handle dead time as well. Frequency methods can give us the relative stability (the GMs and PMs). In addition, we could construct the Bode plot with experimental data by using a sinusoidal or pulse input, i.e., the subsequent design does not need a (theoretical) model. If we do have a model, the 175 Frequency-Response Analysis data can be used to verify the model. However, there are systems that have more than one crossover frequency on the Bode plot (the magnitude and the phase lag do not decrease monotonically with frequency), and it would be hard to judge which is the appropriate one with the Bode plot alone. Review Problems (1) Derive Eqs. (8.19) and (8.20). Use MATLAB to plot the resonant frequency and maximum magnitude as functions of the damping ratio with K = 1. (2) What are the low- and the high-frequency asymptotes of the minimum phase function (s + z)/(s + p) versus the simplest nonminimum-phase function (s в€’ z)/(s + p) in a Bode plot? (3) What is the bandwidth of a second-order function? (4) We used П„ D < П„ I in Example 8.12. What if we use П„ D > П„ I in our PID controller design? What if we use a real PID controller? (5) Sketch the Bode plots for G(s) = s n , with n = В±1, В±2, . . . , etc. (6) In Example 8.12, we used the interacting form of a PID controller. Derive the magnitude and the phase-angle equations for the ideal noninteracting PID controller. (It is called noninteracting because the three controller modes are simply added together.) See that this function will have the same frequency asymptotes. (7) Finish the controller calculations in Example 5.7D. Hints: (1) The plotting statements can be z = 0.05:0.01:0.7; wr = sqrt(1в€’2*z.*z); dum = sqrt(1в€’z.*z); Mp = 1./(2*z.*dum); plot(z,wr, z,Mp); (2) Try to perform the Bode plots with MATLAB. The magnitude plots are identical. The phase angle of the nonminimum-phase example will go from 0в—¦ to в€’180в—¦ , whereas youвЂ™d see a minimum of the phase angle in the minimum-phase function. Thus for a transfer function that is minimum phase, we may identify the function from simply the magnitude plot. But we cannot do the same if the function is nonminimum phase. (3) We need to п¬Ѓnd the в€љ frequency П‰b when the magnitude drops from the low frequency asymptote by 1/ 2. From the magnitude equation in Example 8.3, we need to solve 1 в€’ П„ 2 П‰b2 2 + (2О¶ П„ П‰b )2 = 2 If we now вЂњsolveвЂќ this equation using П„ 2 П‰2 as a variable, we should п¬Ѓnd П„ 2 П‰b2 = 1 в€’ 2О¶2 + 4П‚ 2 (О¶ 2 в€’ 1) + 2 and the п¬Ѓnal from with П‰b explicitly on the LHS is one small step away. (4) Sketching with MATLAB should help. 176 8.4. Controller Design (5) G( jП‰) = j n П‰n . This function is real if n is even, imaginary if n is odd. Also, |G| = П‰n , and the phase angle of G( jП‰) is tanв€’1 (0) when n is even and is tanв€’1 (в€ћ) when n is odd. (6) Substitution of s = jП‰ into G c (s) = K c [1 + (1/П„ I s) + П„ D s] gives G c ( jП‰) = K c 1 + 1 + jП‰П„ D jП‰П„I = Kc 1 + j П„ I П„ D П‰2 в€’ 1 , П‰П„ I and thus |G c ( jП‰)| = K c 1 + П„ D П‰ в€’ G c ( jП‰) = tanв€’1 П„ D П‰ в€’ 1 П‰П„ I 2 , 1 . П‰П„ I The magnitude equation has slopes of в€’1 and +1 at very low and very high frequencies. In the phase-angle equation, the two limits are в€’90в—¦ and +90в—¦ , as in Example 8.12. Furthermore, from the phase-angle equation of the ideal controller, the вЂњtroughвЂќ center should be located at the frequency П‰ = (П„ I П„ D )в€’1/2 . The polar plot of the ideal PID controller is like combining the images of a PI and an ideal PD controller вЂ“ a vertical line at K c that extends from negative inп¬Ѓnity at П‰ = 0 toward positive inп¬Ѓnity at extremely high frequencies. (7) The MATLAB statements and plots are provided on the Web Support. 177 9 Design of State-Space Systems e now return to the use of state-space representation that was introduced in Chap. 4. As you may have guessed, we want to design control systems based on state-space analysis. A state feedback controller is very different from the classical PID controller. Our treatment remains introductory, and we will stay with linear or linearized SISO systems. Nevertheless, the topics here should enlighten (!) us as to what modern control is all about. W What Are We Up to? r Evaluating the controllability and observability of a system. r Designing pole placement of state feedback systems. Applying AckermannвЂ™s formula. r Designing with full-state and reduced-order observers (estimators). 9.1. Controllability and Observability Before we formulate a state-space system, we need to raise two important questions. One is whether the choice of inputs (the manipulated variables) may lead to changes in the states, and the second is whether we can evaluate all the states based on the observed output. These are what we call the controllability and the observability problems. 9.1.1. Controllability A system is said to be completely state controllable if there exists an input u(t) that can drive the system from any given initial state x0 (t0 = 0) to any other desired state x(t). To derive the controllability criterion, let us restate the linear system and its solution from Eqs. (4.1), (4.2), and (4.10): xЛ™ = Ax + Bu, (9.1) y = Cx, (9.2) t x(t) = eAt x(0) + 0 178 eв€’A(tв€’П„ ) Bu(П„ ) dП„. (9.3) 9.1. Controllability and Observability With our deп¬Ѓnition of controllability, there is no loss of generality if we choose to have x(t) = 0, i.e., moving the system to the origin. Thus Eq. (9.3) becomes t x(0) = в€’ eв€’AП„ Bu(П„ ) dП„. (9.4) 0 We next use Eq. (4.15), i.e., the fact that we can expand the matrix exponential function as a closed-form series: eAt = О±0 (t)I + О±1 (t)A + О±2 (t)A2 + В· В· В· + О±nв€’1 (t)Anв€’1 . (9.5) Substitution of Eq. (9.5) into Eq. (9.4) gives nв€’1 x(0) = в€’ t Ak B k =0 О±k (П„ )u(П„ ) dП„. 0 We now hide the ugly mess by deп¬Ѓning the (n Г— 1) vector ОІ with elements t ОІk (П„ ) = О±k (П„ )u(П„ ) dП„, 0 and Eq. (9.4) appears as пЈЇ . . . . пЈЇ Ak BОІk = в€’[B ....... AB ....... A2 B ....... В· В· В· ....... Anв€’1 B] пЈЇ пЈ° k =0 nв€’1 x(0) = в€’ пЈ® ОІ0 ОІ1 .. . пЈ№ пЈє пЈє пЈє. пЈ» (9.6) ОІnв€’1 If Eq. (9.6) is to be satisп¬Ѓed, the (n Г— n) matrix [B AB В· В· В· Anв€’1 B] must be of rank n. This is a necessary and sufп¬Ѓcient condition for controllability. Hence we can state that a system is completely controllable if and only if the controllability matrix C0 = [B AB A2 B В· В· В· Anв€’1 B] (9.7) is of rank n. The controllability condition is the same even when we have multiple inputs, u. If we have r inputs, then u is (r Г— 1), B is (n Г— r ), each of the ОІk is (r Г— 1), ОІ is (nr Г— 1), and C0 is (n Г— nr ). When we have multiple outputs y, we want to control the output rather than the states. Complete state controllability is neither necessary nor sufп¬Ѓcient for actual output controllability. With the output y = Cx and the result in Eq. (9.6), we can infer that the output controllability matrix is C0 = [CB CAB CA2 B В· В· В· CAnв€’1 B]. (9.8) If we have m outputs, y is (m Г— 1) and C is (m Г— n). If we also have r inputs, then the output controllability matrix is (m Г— nr ). From our interpretation of Eq. (9.6), we can also infer that to have complete output controllability, the matrix in Eq. (9.8) must have rank m. 9.1.2. Observability The linear time-invariant system in Eqs. (9.1) and (9.2) is completely observable if every initial state x(0) can be determined from the output y(t) over a п¬Ѓnite time interval. The 179 Design of State-Space Systems concept of observability is useful because in a given system, not all of the state variables are accessible for direct measurement. We will need to estimate the unmeasurable state variables from the output in order to construct the control signal. Because our focus is to establish the link between y and x, or observability, it sufп¬Ѓces to consider only the unforced problem: xЛ™ = Ax, (9.9) y = Cx. (9.10) Substitution of the solution of Eq. (9.9) into Eq. (9.10) gives y(t) = CeAt x(0). We again take that we can expand the exponential function as in Eq. (9.5). Thus we have пЈ№ C пЈЇ CA пЈє пЈє О±k (t)CAk x(0) = [О±0 О±1 В· В· В· О±nв€’1 ] пЈЇ пЈ° ... пЈ» x(0). пЈ® nв€’1 y(t) = k =0 (9.11) CAnв€’1 With the same reasoning that we applied to Eq. (9.6), we can infer that to have complete observability, the observability matrix1 пЈ® пЈ№ C пЈЇ CA пЈє пЈє Ob = пЈЇ пЈ° ... пЈ» (9.12) CAnв€’1 must be of rank n. When we have m outputs, y is (m Г— 1), C is (m Г— n), and Ob is (mn Г— n). Example 9.1: Consider a third-order model: пЈ№ пЈ® пЈ№ пЈ® 0 0 1 0 0 1 пЈ», B = пЈ°0пЈ», C = [1 0 0], A=пЈ° 0 1 в€’6 в€’11 в€’6 which is the controllable canonical form of the problem in Example 4.9. Construct the controllability and the observability matrices. To compute the controllability matrix, we can use the MATLAB function ctrb(): A=[0 1 0; 0 0 1; -6 -11 -6]; B=[0; 0; 1]; Co=ctrb(A,B) Or we can use the deп¬Ѓnition itself: Co=[B A*B A^2*B] 1 180 Controllability and observability are dual concepts. With C = BT and A = AT , we can see that Ob = C0T . 9.1. Controllability and Observability Either way, we should obtain пЈ№ пЈ® 0 0 1 C0 = пЈ°0 1 в€’6пЈ», 1 в€’6 25 which has a rank of 3, and the model is completely state controllable. Similarly, we can use the MATLAB function obsv() for the observability matrix: C=[1 0 0]; Ob=obsv(A,C) Or we can use the deп¬Ѓnition Ob=[C; C*A; C*A^2] We should п¬Ѓnd that Ob is the identity matrix, which of course, is of rank 3. Example 4.8A: We now revisit the fermentor example 4.8. Our question is whether we can control the cell mass and glucose concentration by adjusting only D. From Eq. (E4.38) in Example 4.8, we have пЈ® пЈ№ пЈ® пЈ№ 0 C1 Вµ в€’C1 пЈ», B = пЈ° C пЈ» . A=пЈ° Вµ C1 1 в€’ Вµ в€’Вµ в€’ Y Y Y First, we evaluate пЈ® 0 AB = пЈ° Вµ в€’ Y пЈ№ C12 Вµ в€’C1 C1 Вµ пЈЇ Y пЈє пЈє пЈ» пЈ° C1 пЈ» = пЈЇ C1 пЈ° C 2 Вµ пЈ». в€’ Вµ в€’Вµ y в€’ 12 Y Y пЈ№пЈ® The controllability matrix is пЈ® C12 Вµ в€’C 1 пЈЇ Y C0 = [B AB] = пЈЇ пЈ°C 2 Вµ C 1 в€’ 12 Y Y пЈ№ пЈ® пЈ№ пЈє пЈє. пЈ» The determinant of C0 is 0 and the rank of C0 is 1. Hence, both cell mass and substrate cannot be controlled simultaneously by just varying D. The answer is quite obvious with just a bit of intuition. If we insist on using D as the only input, we can control either C1 or C2 , but not both quantities. To effectively regulate both C1 and C2 , we must implement a system with two inputs. An obvious solution is to adjust the glucose feed concentration (C20 ) as well as the total п¬‚ow rate (dilution rate D). Now we will see what happens with two inputs. Compared with Eq. (E4.38), here A remains the same, and B in Eq. (E4.47) is now a (2 Г— 2) matrix with a rank of 2. Hence the controllability matrix C0 = [B AB] is a (2 Г— 4) matrix and it must have a rank of 2 (as at least B is), and both C1 and C2 are controllable. 181 Design of State-Space Systems u B + + вЂ“ x 1 s C y A K Figure 9.1. Closed-loop system with state feedback. 9.2. Pole-Placement Design 9.2.1. Pole Placement and AckermannвЂ™s Formula When we used root locus for controller design in Chap. 7, we chose a dominant pole (or a conjugate pair if complex). With state-space representation, we have the mathematical tool to choose all the closed-loop poles. To begin, we restate the state-space model in Eqs. (4.1) and (4.2): xЛ™ = Ax + Bu, (9.13) y = Cx. (9.14) With a control system, the input u is now the manipulated variable that is driven by the control signal (Fig. 9.1). For the moment, we consider only the regulator problem and omit changes in the set point. We state the simple control law that depends on full state feedback as u(t) = в€’Kx = в€’K 1 x1 (t) в€’ K 2 x2 (t) В· В· В· в€’K n xn (t), (9.15) where K is the state feedback gain (1 Г— n) vector. In this formulation, the feedback information requires x(t), meaning that we must be able to measure all the state variables. We now substitute Eq. (9.15) into Eq. (9.13) to arrive at the system equation xЛ™ = (A в€’ BK)x. (9.16) The eigenvalues of the system matrix A в€’ BK are called the regulator poles. What we want is to п¬Ѓnd K such that it satisп¬Ѓes how we select all the eigenvalues (or where we put all the closed-loop poles). To do that easily, we п¬Ѓrst need to put our model [Eq. (9.13)] in the controllable canonical form as in Eq. (4.19): пЈ® 0 пЈЇ 0 пЈЇ . . xЛ™ = пЈЇ пЈЇ . пЈ° 0 в€’a0 182 1 0 0 в€’a1 0 1 0 в€’a2 пЈ№ пЈ® пЈ№ 0 0 пЈЇ0пЈє 0 пЈє пЈЇ.пЈє .. пЈє пЈЇ.пЈє . пЈє пЈє x + пЈЇ . пЈє u. пЈ» пЈ°0пЈ» В·В·В· 1 1 В· В· В· в€’anв€’1 В·В·В· В·В·В· (9.17) 9.2. Pole-Placement Design After substituting for u using Eq. (9.15), we п¬Ѓnd that the system matrix in Eq. (9.16) is пЈ® 0 пЈЇ 0 пЈЇ . . A в€’ BK = пЈЇ пЈЇ . пЈ° 0 в€’a0 1 0 0 1 В·В·В· В·В·В· 0 в€’a1 0 в€’a2 В·В·В· В·В·В· пЈ№ пЈ® пЈ№ 0 0 0 пЈє пЈЇ0пЈє пЈЇ.пЈє .. пЈє пЈЇ.пЈє . пЈє пЈє в€’ пЈЇ . пЈє [K 1 K 2 В· В· В· K n ] пЈ» пЈ°0пЈ» 1 в€’anв€’1 1 or пЈ® 0 пЈЇ 0 пЈЇ .. пЈЇ A в€’ BK = пЈЇ . пЈ° 0 в€’a0 в€’ K 1 1 0 0 1 В·В·В· В·В·В· 0 в€’a1 в€’ K 2 0 в€’a2 в€’ K 3 В·В·В· В·В·В· 0 0 .. . 1 пЈ№ пЈє пЈє пЈє. пЈє пЈ» (9.18) в€’anв€’1 в€’ K n As in Eq. (4.21), the closed-loop characteristic equation |sI в€’ A + BK| = 0 will appear as s n + (anв€’1 + K n )s nв€’1 + В· В· В· + (a1 + K 2 )s + (a0 + K 1 ) = 0. (9.19) We next return to our assertion that we can choose all our closed-loop poles, or in terms of eigenvalues, О»1 , О»2 , . . . , О»n . This desired closed-loop characteristic equation is (s в€’ О»1 )(s в€’ О»2 ) В· В· В· (s в€’ О»n ) = s n + О±nв€’1 s nв€’1 + В· В· В· + О±1 s + О±0 = 0, (9.20) where we compute the coefп¬Ѓcients О±i by expanding the terms on the LHS. By matching the coefп¬Ѓcients of like powers of s in Eqs. (9.19) and (9.20), we obtain a0 + K 1 = О±0 , a1 + K 2 = О±1 , ... anв€’1 + K n = О±nв€’1 . Thus in general, we can calculate all the state feedback gains in K by K i = О±iв€’1 в€’ aiв€’1 , i = 1, 2, . . . , n. (9.21) This is the result of full state feedback pole-placement design. If the system is completely state controllable, we can compute the state gain vector K to meet our selection of all the closed-loop poles (eigenvalues) through the coefп¬Ѓcients О±i . There are other methods in pole-placement design. One of them is AckermannвЂ™s formula. The derivation of Eq. (9.21) is predicated on the fact that we have put Eq. (9.13) in the controllable canonical form. AckermannвЂ™s formula requires only that system (9.13) be completely state controllable. If so, we can evaluate the state feedback gain as 2 K = [0 0 В· В· В· 1][B AB В· В· В· Anв€’1 B]в€’1 О±c (A), 2 (9.22) Roughly, AckermannвЂ™s formula arises from the application of the CayleyвЂ“Hamilton theorem to Eq. (9.20). The details of the derivation are in the Web Support. 183 Design of State-Space Systems where О±c (A) = An + О±nв€’1 Anв€’1 + В· В· В· + О±1 A + О±0 I (9.23) is the polynomial derived from the desired eigenvalues as in Eq. (9.20), except now О±c (A) is an (n Г— n) matrix. 9.2.2. Servo Systems We now reintroduce the change in reference, r (t). We will stay with analyzing a SISO system. By a proper choice in the indexing of the state variables, we select x1 = y. In a feedback loop, the input to the process model may take the form u(t) = K r r (t) в€’ Kx(t), where K r is some gain associated with the change in the reference and K is the state feedback gain as deп¬Ѓned in Eq. (9.15). One of the approaches that we can take is to choose K r = K 1 such that u(t) is u(t) = K 1 [r (t) в€’ x1 (t)] в€’ K 2 x2 (t) в€’ В· В· В· в€’K n xn (t), (9.24) where we may recognize that r (t) в€’ x1 (t) is the error e(t). The system equation is now xЛ™ = Ax + B[K 1r в€’ Kx] or xЛ™ = (A в€’ BK)x + BK 1r. (9.25) The system matrix and thus design procedures remain the same as in the regulator problem in Eq. (9.16).3 9.2.3. Servo Systems with Integral Control You may have noticed that nothing that we have covered so far does integral control as in a PID controller. To implement integral action, we need to add one state variable, as in 3 The system must be asymptotically stable. At the new steady state (as t в†’ в€ћ), we have 0 = (A в€’ BK)x(в€ћ) + BK 1 r (в€ћ) and subtracting this equation from Eq. (9.25), we have xЛ™ = (A в€’ BK)[x в€’ x(в€ћ)] + BK 1 [r в€’ r (в€ћ)]. If we deп¬Ѓne e = x в€’ x(в€ћ), and also r (t) as a step function such that r is really a constant for t > 0, the equation is simply eЛ™ = (A в€’ BK)e. Not only is this equation identical to the form in Eq. (9.16), but we also can interpret the analysis as equivalent to a problem in which we want to п¬Ѓnd K such that the steady-state error e(t) approaches zero as quickly as possible. 184 9.2. Pole-Placement Design r + e 1 s вЂ“ x n+1 K n+1 + u вЂ“ + B y = x1 x 1 s + C A K Figure 9.2. State feedback with integral control. Fig. 9.2. Here, we integrate the error [r (t) в€’ x1 (t)] to generate the new variable xn+1 . This quantity is multiplied by the additional feedback gain K n+1 before being added to the rest of the feedback data. The input to the process model now takes the form u(t) = K n+1 xn+1 (t) в€’ Kx(t). (9.26) The differential equation for xn+1 is xЛ™ n+1 = r (t) в€’ Cx. (9.27) We have written x1 = y = Cx just so that we can package this equation in matrix form in the next step. Substituting Eq. (9.26) into state model (9.13), together with Eq. (9.27), we can write this (n + 1) system as xЛ™ .................. xЛ™ n+1 = ... ... ... n+1 ... ....................................................... ... ... ... A в€’ BK BK в€’C 0 x xn+1 .................. + 0 r. 1 ..... (9.28) In terms of dimensions, (A в€’ BK), B, and C remain, respectively, (n Г— n), (n Г— 1), and (1 Г— n). We can interpret the system matrix as ... A в€’ BK ........ BK n+1 A 0 B ....................................................... Л† в€’B Л† K, Л† ... = в€’ [K в€’K n+1 ] = A в€’C 0 0 в€’C ....... 0 (9.29) where now our task is to п¬Ѓnd the (n + 1) state feedback gains: Л† = [K в€’K n+1 ]. K (9.30) With Eq. (9.29), we can view the characteristic equation of the system as Л† +B Л† K| Л† = 0, |sI в€’ A (9.31) which is in the familiar form of the problem in Eq. (9.16). Thus we can make use of the pole-placement techniques in Subsection 9.2.1. Example 9.2: Consider the second-order model in Example 9.1. What are the state feedback gains if we specify that the closed-loop poles are to be at в€’3 В± 3 j and в€’6? With the given model in the controllable canonical form, we can use Eq. (9.21). The MATLAB statements are A=[0 1 0; 0 0 1; -6 -11 -6]; % Should find p1=poly(A) % [1 6 11 6], coefficients ai in % Eq. (9.19) 185 Design of State-Space Systems P=[-3+3j -3-3j -6]; p2=poly(P) p2-p1 % [1 12 54 108], coefficients О±i in % Eq.(9.20) % [0 6 43 102], Ki as in Eq.(9.21) To obtain the state feedback gains with Eq. (9.21), we should subtract the coefп¬Ѓcients of the polynomial p1 from p2, starting with the last constant coefп¬Ѓcient. The result is, indeed, K = (K 1 , K 2 , K 3 ) = (108 в€’ 6, 54 в€’ 11, 12 в€’ 6) = (102, 43, 6). Check 1: The same result can be obtained with the MATLAB function acker(), which uses AckermannвЂ™s formula. The statements are B=[0; 0; 1]; acker(A,B,P) % Should return [102 43 6] Check 2. We can do the AckermannвЂ™s formula step by step. The statements are M=[B A*B A^2*B]; ac=polyvalm(p2,A); [0 0 1]* inv(M)* ac % controllability matrix % Eq.(9.23) % Eq.(9.22) To evaluate the matrix polynomial in Eq. (9.23), we use the MATLAB function polyvalm(), which applies the coefп¬Ѓcients in p2 to the matrix A. Example 4.7B: Let us revisit the two CSTR-in-series problems in Example 4.7. Use the inlet concentration as the input variable and check that the system is controllable and observable. Find the state feedback gain such that the reactor system is very slightly underdamped with a damping ratio of 0.8, which is equivalent to approximately a 1.5% OS. From Eq. (E4.27) of Example 4.7, the model is d c1 в€’5 = 2 dt c2 0 в€’4 c1 4 + c 0 0 c2 and C2 is the only output. We can construct the model and check the controllability and observability with A=[-5 0; 2 -4]; B=[4; 0]; C=[0 1]; D=0; rank(ctrb(A,B)) % should find rank = 2 rank(obsv(A,C)) % for both matrices Both the controllability and the observability matrices are of rank 2. Hence the system is controllable and observable. To achieve a damping ratio of 0.8, we can п¬Ѓnd that the closed-loop poles must be at в€’4.5 В± 3.38 j (by using a combination of what we learned in Example 7.5 and Fig. 2.5), but 186 9.2. Pole-Placement Design we can cheat with MATLAB and use root-locus plots! [q,p]=ss2tf(A,B,C,D); Gp=tf(q,p); rlocus(Gp) sgrid(0.8,1) [kc,P]=rlocfind(Gp) % converts state-space to transfer % function4 % should find kc = 1.46 We now apply the closed-loop poles P directly to AckermannвЂ™s formula: K=acker(A,B,P) % should find K = [0 1.46] The state-space state feedback gain (K 2 ) related to the output variable C2 is the same as the proportional gain obtained with root locus. Given any set of closed-loop poles, we can п¬Ѓnd the state feedback gain of a controllable system by using state-space pole-placement methods. The use of root locus is not necessary, but it is a handy tool that we can take advantage of. Example 4.7C: Add integral action to the system in Example 4.7B so we can eliminate the steady-state error. To п¬Ѓnd the new state feedback gain is a matter of applying Eq. (9.29) and AckermannвЂ™s formula. The hard part is to make an intelligent decision on the choice of closed-loop poles. Following the lead of Example 4.7B, we use root-locus plots to help us. With the understanding that we have two open-loop poles at в€’4 and в€’5, a reasonable choice of the integral time constant is 1/3 min. With the open-loop zero at в€’3, the reactor system is always stable, the dominant closed-loop pole is real, and the reactor system will not suffer from excessive oscillation. Hence our п¬Ѓrst step is to use root locus to п¬Ѓnd the closed-loop poles of a PI control system with a damping ratio of 0.8. The MATLAB statements to continue with Example 4.7B are kc=1; taui=1/3; Gc=tf(kc*[taui 1],[taui 0]); rlocus (Gc*Gp); % Gp is from Example 4.7B sgrid(0.8,1) [kc, P]=rlocfind(Gc*Gp) % should find proportional gain kc=1.66 The closed-loop poles P are roughly at в€’2.15 and в€’3.43 В± 2.62 j, which we apply immeЛ† and B Л† in Eq. (9.29): diately to AckermannвЂ™s formula by using A Ah=[A zeros(2,1); -C 0]; % Eq.(9.29) Bh=[B; 0]; Kh=acker(Ah,Bh,P) % should find Kh = [0 1.66 -4.99] Л† is [0 1.66 в€’4.99]. Unlike the simple The state feedback gain including integral control K proportional gain, we cannot expect that K n+1 = 4.99 would resemble the integral time 4 Another way here is to make use of the analytical result in Example 4.7: Gp=zpk([],[-5 -4],8); % transfer function C2/Co taken from % Eq.(E4.30a) 187 Design of State-Space Systems constant in classical PI control. For doing the time-domain simulation, the task is similar to the hints that are provided for Example 7.5B in the Review Problems. The actual statements are also provided on the Web Support. Example 7.5B: Consider the second-order system in Example 7.5. What are the state feedback gains if we specify that the closed-loop poles are to be в€’0.375 В± 0.382 j as determined in Example 7.5A? The problem posed in Examples 7.5 and 7.5A is not in the controllable canonical form (unless we do the transform ourselves). Thus we use AckermannвЂ™s formula. The MATLAB statements are G=tf(1,conv([2 1],[4 1])); % Make the state--space object % from the transfer function S=ss(G); scale=S.c(2); % Rescale MATLAB model matrices S.c=S.c/scale; S.b=S.b*scale; P=[-0.375+0.382j -0.375-0.382j] % Define the closed-loop poles k=acker(S.a,S.b,P) % Calculate the feedback gains MATLAB will return the vector [0 1.29], meaning that K 1 = 0 and K 2 = 1.29, which was the proportional gain obtained in Example 7.5A. Because K 1 = 0, we feed back only the controlled variable as analogous to proportional control. In this simple example, the statespace system is virtually the classical system with a proportional controller. A note of caution is necessary when we let MATLAB generate the state-space model from a transfer function. The vector C (from S.c) is [0 0.5], which means that the indexing is reversed such that x2 is the output variable and x1 is the derivative of x2 . Second, C is not [0 1], and hence we have to rescale the matrices B and C. These two points are further covered in MATLAB Session 4. 9.3. State Estimation Design 9.3.1. State Estimator The pole-placement design is predicated on the feedback of all the state variables x (Fig. 9.1). Under many circumstances, this may not be true. We have to estimate unmeasurable state variables or signals that are too noisy to be measured accurately. One approach to work around this problem is to estimate the state vector with a model. The algorithm that performs this estimation is called the state observer or the state estimator. The estimated state xЛњ is then used as the feedback signal in a control system (Fig. 9.3). A full-order state observer r u + вЂ“ ~ KX Process (Plant) y State Estimator Figure 9.3. Concept of using a state estimator to generate an estimated state feedback signal. 188 9.3. State Estimation Design ~ x u + B + + + ~ y 1 s C y + вЂ“ A yвЂ“~ y Ke Figure 9.4. A probable model for a state estimator. estimates all the states even when some of them are measured. A reduced-order observer does the smart thing and skips these measurable states. The next task is to seek a model for the observer. We stay with a SISO system, but the concept can be extended to multiple outputs. The estimate should embody the dynamics of the plant (process). Thus one probable model, as shown in Fig. 9.4, is to assume that the state estimator has the same structure as the plant model, as in Eqs. (9.13) and (9.14) or Fig. 9.1. The estimator also has the identical plant matrices A and B. However, one major difference is the addition of the estimation error, y в€’ yЛњ , in the computation of the estimated state xЛњ . The estimated state variables based on Fig. 9.4 can be described by (details in Review Problems) xЛ™Лњ = AЛњx + Bu + Ke (y в€’ CЛњx) = (A в€’ Ke C)Лњx + Bu + Ke y. (9.32) Here, yЛњ = CЛњx has been used in writing the error in the estimation of the output, (y в€’ yЛњ ). The (n Г— 1) observer gain vector Ke does a weighting on how the error affects each estimate. In the next two subsections, we will apply the state estimator in Eq. (9.32) to a state feedback system and see how we can formulate the problem such that the error (y в€’ yЛњ ) can become zero. 9.3.2. Full-Order State Estimator System A system that makes use of the state estimator is shown in Fig. 9.5, where, for the moment, changes in the reference are omitted. What we need is the set of equations that describes this regulator system with state estimation. u + B y x 1 s + вЂ“ C A ~ Kx K вЂ“ B u + ~ x + + + 1 s ~ y C + вЂ“ A Ke Figure 9.5. A regulator system with a controllerвЂ“ estimator. 189 Design of State-Space Systems By itself, the estimator in Eq. (9.32) has the characteristic equation |sI в€’ A + Ke C| = 0. (9.33) Our intention is to use the estimated states to provide feedback information: u = в€’KЛњx. (9.34) The state-space model of Eq. (9.13) now appears as xЛ™ = Ax + Bu = Ax в€’ BKЛњx. (9.35) If we substitute y = Cx into Eq. (9.32), we can integrate Eqs. (9.32) and (9.35) simultaneously to compute x(t) and xЛњ (t). In matrix form, this set of 2n equations can be written as d x = dt xЛњ ... ... ... ........................................................................ .... ... . e .... e A в€’BK K C A в€’ K C в€’ BK x . xЛњ (9.36) 9.3.3. Estimator Design With Eq. (9.36), it is not obvious how Ke affects the choices of K. We now derive a form of Eq. (9.36) that is based on the error of the estimation and is easier for us to make a statement on its properties. We deп¬Ѓne the state error vector as e(t) = x(t) в€’ xЛњ (t). (9.37) Subtracting Eq. (9.32) from Eq. (9.35) and using y = Cx, we should п¬Ѓnd that (Л™x в€’ xЛ™Лњ ) = (A в€’ Ke C)(x в€’ xЛњ ) or eЛ™ = (A в€’ Ke C)e. (9.38) This error equation has the same characteristic equation as the estimator in Eq. (9.33). The goal is to choose eigenvalues of the estimator such that the error decays away quickly. We may note that the form of Eq. (9.38) is the same as that of the regulator problem. Thus we should be able to use the tools of pole placement for the estimator design. In fact, we can apply, without derivation, a modiп¬Ѓed form of AckermannвЂ™s formula to evaluate пЈ№в€’1 пЈ® пЈ№ пЈ® C 0 пЈЇ CA пЈє пЈЇ0пЈє пЈє пЈЇ пЈє (9.39) Ke = О±e (A) пЈЇ пЈ° ... пЈ» пЈ° ... пЈ», 1 CAnв€’1 where, as analogous to Eq. (9.20), О±e (s) = s n + О±nв€’1 s nв€’1 + В· В· В· + О±1 s + О±0 (9.40) is the polynomial derived from our own chosen estimator eigenvalues. Equation (9.39) is different from Eq. (9.22) because we are now solving the dual problem for the (n Г— 1) vector Ke . Next, we can replace xЛњ in Eq. (9.35) with the deп¬Ѓnition of the error vector, and the equation becomes xЛ™ = Ax в€’ BK(x в€’ e). 190 (9.41) 9.3. State Estimation Design Equations (9.38) and (9.41) can be put in matrix form as xЛњ = eЛ™ ... ... ... ............................................................... .... ... ... .. e A в€’ BK BK 0 Aв€’K C x . e (9.42) Now it is clear that the characteristic equation of the controllerвЂ“estimator system is |sI в€’ A + BK||sI в€’ A + Ke C| = 0. (9.43) We have the very important result that choices for the eigenvalues for the pole-placement design and the observer design can be made independently. Generally, we want the observer response to be two to п¬Ѓve times faster than the system response. We should not have to worry about saturation as the entire observer is software based, but we do have to consider noise and sensitivity problems. Example 9.3: Consider the second-order model in Example 9.1, which was used in Example 9.2 to calculate the state feedback gains. What is the observer gain vector Ke if we specify that the estimator error should have eigenvalues в€’9 repeated thrice? With eigenvalues selected at в€’9, we have chosen the estimator to be faster than the state feedback, and all the errors are to decay exponentially. We make use of the AckermannвЂ™s formula in Eq. (9.39) for observer gains. The MATLAB statements are A=[0 1 0; 0 0 1; -6 -11 -6]; % Define the model B=[0; 0; 1]; C=[1 0 0]; pe=poly([-9 -9 -9]); ae=polyvalm(pe,A); Ob=[C; C*A; C*A^2]; Ke=ae*inv(Ob)*[0; 0; 1] % Make estimator polynomial (9.40) % Eq. (9.39) We should п¬Ѓnd that Ke = (21, 106, в€’144). The estimator calculations are purely mathematical, and the values of the observer gains can be negative. Furthermore, we can check that the system of equations in Eq. (9.42) has the correct eigenvalues as suggested by Eq. (4.43). K=[102 43 6]; % Feedback gains calculated from Example 9.2 A11=A-B*K; % Submatrices in Eq.(9.42) A12=B*K; A21=zeros(3,3); A22=A-Ke*C; BIGA=[A11 A12; A21 A22]; eig(BIGA) Indeed, we should п¬Ѓnd that the big matrix BIGA has eigenvalues of в€’3 В± 3 j, в€’6, and в€’9 repeated three times. 9.3.4. Reduced-Order Estimator We should not have to estimate variables that we can measure. It is logical to design a reduced-order estimator that estimates only the states that cannot be measured or are too noisy to be measured accurately. Following our introductory practice, we consider only one 191 Design of State-Space Systems measured output. The following development assumes that we have selected x1 to be the measured variable. Hence the output is y = Cx = [1 0 В· В· В· 0]x. (9.44) Next, we partition the state vector as x= x1 , xe (9.45) where xe = [x2 В· В· В· xn ] contains the (n в€’ 1) states that have to be estimated. State model equation (9.13) is partitioned accordingly as A1e a xЛ™ 1 = 11 xЛ™ e Ae1 Aee x1 b + 1 u, xe Be (9.46) where the dimensions of A1e , Ae1 , and Aee are, respectively, (1 Г— n в€’ 1), (n в€’ 1 Г— 1), and (n в€’ 1 Г— n в€’ 1), and that of Be is (n в€’ 1 Г— 1). The next task is to make use of the full state estimator equations. Before that, we have to remold Eq. (9.46) as if it were a full state problem. This exercise requires some careful bookkeeping of notations. LetвЂ™s take the п¬Ѓrst row in Eq. (9.46) and make it constitute the output equation. Thus we make a slight rearrangement: xЛ™ 1 в€’ a11 x1 в€’ b1 u = A1e xe such that it takes the form of y = Cx. We repeat with the second row of (9.46) and put it as xЛ™ e = Aee xe + (Ae1 x1 + Be u) such that it can be compared with xЛ™ = Ax + Bu. The next step is to take the full state estimator in Eq. (9.32), xЛ™Лњ = (A в€’ Ke C)Лњx + Bu + Ke y, and substitute term by term by using the reduced-order model equations.5 The result is, п¬Ѓnally, xЛ™Лњ e = (Aee в€’ Ker A1e )Лњxe + (Ae1 x1 + Be u) + Ker (xЛ™ 1 в€’ a11 x1 в€’ b1 u), (9.47) which is the reduced-order equivalent of Eq. (9.32). Note that in this equation x1 = y. 5 192 The matching of terms for reduced-order substitution in Eq. (9.31) to derive Eqs. (9.47)вЂ“(9.49): Full-order state estimator Reduced-order state estimator xЛњ y C A Ke (n Г— 1) Bu xЛњ e xЛ™ 1 в€’ a11 x1 в€’ b1 u A1e Aee Ker (n в€’ 1 Г— 1) Ae1 x1 + Be u 9.3. State Estimation Design The computation of the (n в€’ 1) weighting factors in Ker can be based on the equivalent form of Eq. (9.38). Again, doing the substitution for the notations, we п¬Ѓnd that the error estimate becomes eЛ™ = (Aee в€’ Ker A1e )e, (9.48) which means that AckermannвЂ™s formula in Eq. (9.39) now takes the form пЈ№в€’1 пЈ® пЈ№ пЈ® A1e 0 пЈЇ A A пЈє пЈЇ0пЈє 1e ee пЈє пЈЇ пЈє пЈЇ Ker = О±e (Aee ) пЈЇ пЈє пЈЇ .. пЈє. .. пЈ» пЈ°.пЈ» пЈ° . nв€’1 1 A1e Aee (9.49) We are not quite done yet. If we use Eq. (9.47) to compute xЛњ e , it requires taking the derivative of x1 , an exercise that can easily amplify noise. Therefore we want a modiп¬Ѓed form that allows us to replace this derivative. To begin, we deп¬Ѓne a new variable xЛњ e1 = xЛњ e в€’ Ker x1 . (9.50) This variable is substituted into Eq. (9.47) to give (xЛ™Лњ e1 + Ker xЛ™ 1 ) = (Aee в€’ Ker A1e )(Лњxe1 + Ker x1 ) + (Ae1 x1 + Be u) + Ker (xЛ™ 1 в€’ a11 x1 в€’ b1 u). After cancellation of the derivative term, we have xЛ™Лњ e1 = (Aee в€’ Ker A1e )Лњxe1 + (Aee Ker в€’ Ker A1e Ker + Ae1 в€’ Ker a11 )x1 + (Be в€’ Ker b1 )u. (9.51) This differential equation is used to compute xЛњ e1 , which then is used to calculate xЛњ e with Eq. (9.50). With the estimated states, we can compute the feedback to the state-space model as x1 . xЛњ e T u = в€’ K 1 K1e (9.52) The application of Eqs. (9.50)вЂ“(9.52) is a bit involved and best illustrated as shown in Fig. 9.6. r(t) u(t) + x Process y = x1 C вЂ“ K1e + + ~ xe + ~ x e1 Reduced-order estimator + K er K1 Figure 9.6. State feedback with reduced-order estimator. 193 Design of State-Space Systems Example 9.4: Consider the estimator in Example 9.3: What is the reduced-order observer gain vector Ker if we specify that the estimator error should have eigenvalues of в€’9 repeated twice? We can use Eq. (9.49), and the MATLAB statements are A=[0 1 0; 0 0 1; -6 -11 -6]; N=size(A,1); a11=A(1,1); % Extract matrix partitions as in % Eq.(9.46) A1e=A(1,2:N); Ae1=A(2:N,1); Aee=A(2:N,2:N); pe=poly([-9 -9]); % Make estimator polynomial ae=polyvalm(pe,Aee); Ob=[A1e; A1e*Aee]; Ker=ae*inv(Ob)*[0; 1] % Eq.(9.49) for n=2 We should п¬Ѓnd that Ker = (12 в€’2). After all this fancy mathematics, a word of caution is needed. It is extremely dangerous to apply the state estimate as presented in this chapter. Why? The п¬Ѓrst hint is in Eq. (9.32). We have assumed perfect knowledge of the plant matrices. Of course, we rarely do. Furthermore, we have omitted actual terms for disturbances, noises, and errors in measurements. Despite these drawbacks, material in this chapter provides the groundwork to attack serious problems in modern control. Review Problems (1) For the second-order transfer function, 1 Y = 2 , U s + 2О¶ П‰n s + П‰n2 derive the controllable canonical form. If the desired poles of a closed-loop system are to be placed at О»1 and О»2 , what should be the state feedback gains? (2) Presume we do not know what the estimator should be other than that it has the form xЛ™Лњ = FЛњx + Gu + Hy. Find Eq. (9.32). (3) Do the time-response simulation in Example 7.5B. We found that the statespace system has a steady-state error. Implement integral control and п¬Ѓnd the new state feedback gain vector. Perform a time-response simulation to conп¬Ѓrm the result. (4) With respect to Fig. R9.4, what is the transfer function equivalent to the controllerвЂ“ estimator system in Eq. (9.32)? 194 9.3. State Estimation Design вЂ“Y R=0 + вЂ“ Y U G ec State space model Figure R9.4. Hints: (1) The controllable canonical form was derived in Example 4.1. The characteristic polynomial of (sI в€’ A + BK) should be s 2 + (2О¶ П‰n + K 2 )s + П‰n2 + K 1 = 0. The characteristic polynomial of desired poles is s 2 + (О»1 + О»2 )s + О»1 О»2 = 0. Thus K 1 = О»1 О»2 в€’ П‰n2 , K 2 = (О»1 + О»2 ) в€’ 2О¶ П‰n . (2) The Laplace transform of the given equation is Лњ = FX Лњ + GU + HY. sX Substituting Y = CX, we have Лњ = (sI в€’ F)в€’1 [GU + HCX]. X We further substitute for X = (sI в€’ A)в€’1 BU with the simple state-space model to give Лњ = (sI в€’ F)в€’1 [G + HC(sI в€’ A)в€’1 B]U. X What we want is to dictate that the transfer function of this estimator is the same as that of the state-space model: (sI в€’ F)в€’1 [G + HC(sI в€’ A)в€’1 B] = (sI в€’ A)в€’1 B. Moving the second term to the RHS and factoring out the (sI в€’ A)в€’1 B gives (sI в€’ F)в€’1 G = [I в€’ (sI в€’ F)в€’1 HC](sI в€’ A)в€’1 B. Thus we can multiply (sI в€’ F) to both sides to have G = [(sI в€’ F) в€’ HC](sI в€’ A)в€’1 B. Finally, [(sI в€’ F) в€’ HC]в€’1 G = (sI в€’ A)в€’1 B. Comparing term by term, we have F + HC = A, or F = A в€’ HC G = B. This result is what we need in Eq. (9.32) if we also set H = Ke . 195 Design of State-Space Systems (3) For the time-response simulation, we also duplicate the classical control design for comparision. The statements are G=tf(1,conv([2 1],[4 1])); S=ss(G); scale=S.c(2); S.c=S.c/scale; S.b=S.b*scale; P=[-0.375+0.382j -0.375-0.382j]; % % % % MATLAB uses reverse indexing Need to rescale B and C too % Define the closed-loop % poles K=acker(S.a,S.b,P) % Compute the system matrices for plotting A = S.a - S.b*K % Makes system matrix, % Eq.(9.25) B = S.b*K(2) C = S.c D=0; step(A,B,C,D) hold % to add the classical design result Gcl=feedback(1.29*G,1); % Kc=1.29 was the proportional gain obtained in % Example 7.5A step(Gcl) To eliminate offset, we need Subsection 9.2.3. With an added state that is due to integration, we have to add one more closed-loop pole. We choose it to be в€’1, sufп¬Ѓciently faster than the real part of the complex poles. The statements are G=tf(1,conv([2 1],[4 1])); S=ss(G); % Generates the matrices S.a, S.b, S.c, S.d Ah=[S.a zeros(2,1); -S.c 0] % A-head in Eq.(9.29) Bh=[S.b; 0] % B-head P=[-0.375+0.382j -0.375-0.382j -1]; % Add a faster pole % at -1 kh=acker(Ah,Bh,P) % K-head in (9.29) Л† = [2 3.6 в€’2.3]. To do the time-response simulation, we can We should п¬Ѓnd K use Asys=Ah-Bh*kh; % System matrix(9.29) Bsys=[0; 0; 1]; % Follows Eq.(9.28) Csys=[S.c 0]; step(Asys, Bsys,Csys,0) 196 9.3. State Estimation Design (4) For the estimator, y is the input and u is the output. With u = в€’KЛњx, the Laplace transform of Eq. (9.32) is Лњ [sI в€’ A + Ke C + BK]X(s) = Ke Y (s) or Лњ X(s) = [sI в€’ A + Ke C + BK]в€’1 Ke Y (s). Лњ back into the Laplace transform of u = в€’KЛњx to obtain We now substitute X U (s) = в€’K[sI в€’ A + Ke C + BK]в€’1 Ke Y (s) = в€’G ec (s)Y (s). 197 10 Multiloop Systems here are many advanced strategies in classical control systems. Only a limited selection of examples is presented in this chapter. We start with cascade control, which is a simple introduction to a multiloop, but essentially SISO, system. We continue with feedforward and ratio control. The idea behind ratio control is simple, and it applies quite well to the furnace problem that is used as an illustration. Finally, a multiple-input multipleoutput (MIMO) system is addressed with a simple blending problem as illustration, and the problem is used to look into issues of interaction and decoupling. These techniques build on what we have learned in classical control theories. T What Are We Up to? r Applying classical controller analysis to cascade control, feedforward control, feedforwardвЂ“feedback control, ratio control, and the Smith predictor for time-delay compensation r Analyzing a MIMO system with relative gain array and assessing the pairing of manipulated and controlled variables r Attempting to decouple and eliminate the interactions in a two-input two-output system 10.1. Cascade Control A common design found in process engineering is cascade control. This is a strategy that allows us to handle load changes more effectively with respect to the manipulated variable. To illustrate the idea, we consider the temperature control of a gas furnace, which is used to heat up a cold process stream. The fuel-gas п¬‚ow rate is the manipulated variable, and its п¬‚ow is subject to п¬‚uctuations that are due to upstream pressure variations. In a simple single-loop system, we measure the outlet temperature, and the temperature controller (TC) sends its signal to the regulating valve. If there is п¬‚uctuation in the fuel-gas п¬‚ow rate, this simple system will not counter the disturbance until the controller senses that the temperature of the furnace has deviated from the set point (T s ). 198 10.1. Cascade Control Hot process stream T Cold TT TC process stream T s FC Furnace FT Fuel gas Figure 10.1. Cascade control of the temperature of a furnace, taken to be the same as that of the outlet process stream. The temperature controller (TC) does not actuate the regulating valve directly; it sends its signal to a secondary п¬‚ow-rate control loop, which in turn ensures that the desired fuel-gas п¬‚ow is deliverd. A cascade control system can be designed to handle a fuel-gas disturbance more effectively (Fig. 10.1). In this case, a secondary loop (also called the slave loop) is used to adjust the regulating valve and thus manipulate the fuel-gas п¬‚ow rate. The TC (the master or primary controller) sends its signal, in terms of the desired п¬‚ow rate, to the secondary п¬‚ow control loop вЂ“ in essence, the signal is the set point of the secondary п¬‚ow controller (FC). In the secondary loop, the FC compares the desired fuel-gas п¬‚ow rate with the measured п¬‚ow rate from the п¬‚ow transducer (FT) and adjusts the regulating valve accordingly. This inner п¬‚ow control loop can respond immediately to п¬‚uctuations in the fuel-gas п¬‚ow to ensure that the proper amount of fuel is delivered. To be effective, the secondary loop must have a faster response time (smaller time constant) than the outer loop. Generally, we use as high a proportional gain as feasible. In control jargon, we say that the inner loop is tuned very tightly. We can use a block diagram to describe Fig. 10.1. Cascade control adds an inner control loop with secondary controller function G c2 [Fig. 10.2(a)]. This implementation of cascade L (a) Upstream pressure fluctuation GL R + E p + Gc вЂ“ e2 вЂ“ Gc 2 + Gv a + C Gp Process stream temperature Fuel gas flow L (b) G*L R + E вЂ“ Gc G*v + + C Gp Figure 10.2. (a) Block diagram of a simple cascade control system with reference to the furnace problem, (b) reduced block diagram of a cascade control system. 199 Multiloop Systems control requires two controllers and two measured variables (fuel-gas п¬‚ow and furnace temperature). The furnace temperature is the controlled variable, and the fuel-gas п¬‚ow rate remains the only manipulated variable. For cleaner algebra, we omit the measurement transfer functions, taking G m 1 = G m 2 = 1. A disturbance, such as upstream pressure, that speciп¬Ѓcally leads to changes in the fuel-gas п¬‚ow rate is now drawn to be part of the secondary п¬‚ow control loop. (A disturbance such as change in the process stream inlet temperature, which is not part of the secondary loop, would still be drawn in its usual location, as in Section 5.2). We now reduce the block diagram. The п¬Ѓrst step is to close the inner loop so the system becomes a standard feedback loop [Fig. 10.2(b)]. With hindsight, the result should be intuitively obvious. For now, we take the slow route. Using the lowercase letter locations in Fig. 10.2(a), we write the algebraic equations e2 = p в€’ a, a = G c2 G v e2 + G L L . Substitution of e2 leads to a = G c2 G v ( p в€’ a) + G L L , and the result after rearrangement is a form that allows us to draw Fig. 10.2(b): a= G c2 G v 1 + G c2 G v p+ GL L = G в€—v p + G в€—L L , 1 + G c2 G v where G в€—v = G c2 G v , 1 + G c2 G v G в€—L = GL . 1 + G c2 G v (10.1) The remaining task to derive the closed-loop transfer functions is routine. Again, slowly, we can write the relation in Fig. 10.2(b) as C = G в€—L G p L + G c G в€—v G p E, and substituting E = R в€’ C, we have, after rearrangement, C= G c G в€—v G p R+ 1 + G c G в€—v G p G p G в€—L L. 1 + G c G в€—v G p (10.2) The closed-loop characteristic polynomial of this cascade system is 1 + G c G в€—v G p = 0. (10.3) If we now substitute G в€—v from Eq. (10.1), the characteristic polynomial takes the form1 1 + G c2 G v + G c G c2 G v G p = 0. 1 200 (10.3a) If we remove the secondary loop, this characteristic equation should reduce to that of a conventional feedback system equation. It is not obvious from Eq. (10.3) because our derivation has taken the measurement function G m 2 to be unity. If we had included G m 2 in a more detailed analysis, we could get the single-loop result by setting G c2 = 1 and G m 2 = 0. 10.1. Cascade Control So far, we know that the secondary loop helps to reduce disturbance in the manipulated variable. If we design the control loop properly, we should also accomplish a faster response in the actuating element: the regulating valve. To go one step further, cascade control can even help to make the entire system more stable. These points may not be intuitive. A simple example is used to illustrate these features. Example 10.1: Consider a simple cascade system as shown in Fig. 10.2(a) with a PI controller in the primary loop and a proportional controller in the slave loop. For simplicity, consider п¬Ѓrst-order functions: 0.8 0.5 0.75 Gp = , Gv = , GL = . 2s + 1 s+1 s+1 (a) How can the proper choice of K c2 of the controller in the slave loop help to improve the actuator performance and eliminate disturbance in the manipulated variable (e.g., fuel-gas п¬‚ow in the furnace temperature control)? If we substitute G c2 = K c2 and G v = [K v /(П„v s + 1)] into G в€—v in Eq. (10.1), we should п¬Ѓnd G в€—v = K c2 K v (П„v s + 1) + K c2 K v = K vв€— , П„vв€— s + 1 (E10.1) where K vв€— = K c2 K v , 1 + K C2 K v П„vв€— = П„v . 1 + K c2 K v (E10.2) Similarly, substituting G L = [K L /(П„v s + 1)] into G в€—L should give K Lв€— = KL . 1 + K c2 K v (E10.3) Thus as the proportional gain K c2 becomes larger, K vв€— approaches unity gain, meaning that there is a more effective change in the manipulated variable, and K Lв€— approaches zero, meaning that the manipulated variable is becoming less sensitive to changes in the load. Furthermore, the effective actuator time constant П„vв€— will become smaller, meaning a faster response. (b) The slave loop affords us a faster response with respect to the actuator. What is the proportional gain K c2 if we want the slave-loop time constant П„vв€— to be only one tenth of the original time constant П„v in G v ? From the problem statement, K v = 0.5 and П„v = 1 s. Thus П„vв€— = 0.1 s, and substitution of these values into П„vв€— of Eqs. (E10.2) gives 0.1 = 1 , or K c2 = 18. 1 + 0.5 K c2 The steady-state gain is K vв€— = (18)(0.5) = 0.9. 1 + (18)(0.5) The slave loop will have a 10% offset with respect to desired set-point changes in the secondary controller. 201 Multiloop Systems П„ I = 0.05 s П„ I = 0.5 s П„I = 5 s Open-loop zero at вЂ“20 Open-loop zero at вЂ“2 Open-loop zero at вЂ“0.2 x x x x x x x x x Open-loop poles at 0, вЂ“0.5, вЂ“10 Figure E10.1. (c) So far, we have used only proportional control in the slave loop. We certainly expect offset in this inner loop. Why do we stay with proportional control here? The modest 10% offset that we have in the slave loop is acceptable under many circumstances. As long as we have integral action in the outer loop, the primary controller can make necessary adjustments in its output and ensure that there is no steady-state error in the controlled variable (e.g., the furnace temperature). (d) Now we tackle the entire closed-loop system with the primary PI controller. Our task here is to choose the proper integral time constant among the given values of 0.05, 0.5, and 5s. We can tolerate underdamped response but absolutely not a system that can become unstable. Of course, we want a system response that is as fast as we can make it, i.e., with a proper choice of proportional gain. Select and explain your choice of the integral time constant. Among other methods, root locus is the most instructive in this case. With a PI primary controller and numerical values, Eq. (10.3) becomes 1 + Kc П„I s + 1 П„I s 0.9 0.1s + 1 0.8 2s + 1 = 0. With MATLAB, we can easily prepare the root-locus plots of this equation for the cases of П„ I = 0.05, 0.5, and 5s. (You should do it yourself. Only a rough sketch is shown in Fig. E10.1. Help can be found in the Review Problems.) From the root-locus plots, it is clear that the system may become unstable when П„ I = 0.05s. The system is always stable when П„ I = 5s, but the speed of the system response is limited by the dominant pole between the origin and в€’0.2. The proper choice is П„ I = 0.5s, in which case the system is always stable but the closed-loop poles can move farther away, loosely speaking, from the origin. (e) Take the case without cascade control and, using the integral time constant that you have selected in part (d), determine the range of proportional gain that you can use (without the cascade controller and secondary loop) to maintain a stable system. How is this different from when we use cascade control? With the choice of П„ I = 0.5s, but without the inner loop or the secondary controller, the closed-loop equation is 1 + Gc Gv G p = 1 + Kc 202 0.5s + 1 0.5s 0.5 s+1 0.8 2s + 1 = 0, 10.2. Feedforward Control which can be expanded to s 3 + 1.5s 2 + (0.5 + 0.2K c )s + 0.4K c = 0. With the RouthвЂ“Hurwitz analysis in Chap. 7, we should п¬Ѓnd that, to have a stable system, we must keep K c < 7.5. (You п¬Ѓll in the intermediate steps in the Review Problems. Other techniques such as root locus, direct substitution, or frequency response in Chap. 8 should arrive at the same result.) With cascade control, we know from part (d) that the system is always stable. Nevertheless, we can write the closed-loop characteristic equation 1 + Kc 0.5s + 1 0.5s 0.9 0.1s + 1 0.8 2s + 1 =0 or 0.1s 3 + 1.05s 2 + (0.5 + 0.36K c )s + 0.72K c = 0 A RouthвЂ“Hurwitz analysis can conп¬Ѓrm that. The key point is that, with cascade control, the system becomes more stable and allows us to use a larger proportional gain in the primary controller. The main reason is the much faster response (smaller time constant) of the actuator in the inner loop.2 10.2. Feedforward Control To counter probable disturbances, we can take an even more proactive approach than cascade control and use feedforward control. The idea is that if we can make measurements of disturbance changes, we can use this information and our knowledge of the process model to make proper adjustments in the manipulated variable before the disturbance has a chance to affect the controlled variable. We will continue with the gas furnace to illustrate feedforward control. For simplicity, letвЂ™s make the assumption that changes in the furnace temperature (T ) can be effected by changes in the fuel-gas п¬‚ow rate (Ffuel ) and the cold process stream п¬‚ow rate (Fs ). Other variables, such as the process stream temperature, are constant. In Section 10.1, the fuel-gas п¬‚ow rate is the manipulated variable (M) and cascade control is used to handle its п¬‚uctuations. Now we also consider changes in the cold process stream п¬‚ow rate as another disturbance (L). LetвЂ™s presume further that we have derived diligently from heat and mass balances the corresponding transfer functions, G L and G p , and we have the process model C = G L L + G p M, (10.4) where we have used the general notation C as the controlled variable in place of furnace temperature T . 2 If you are skeptical of this statement, try to do the Bode plots of the systems with and without cascade control and think about the consequence of changing the break frequency (or bandwidth) of the valve function. If you do not pick up the hint, the answer can be found on the Web Support in the details of Example 10.1. 203 Multiloop Systems We want the controlled variable to track set-point changes (R) precisely, so we substitute the ideal case C = R, and rearrange Eq. (10.4) to M= 1 GL Rв€’ L. Gp Gp (10.5) This equation provides us with a model-based rule as to how the manipulated variable should be adjusted when we either change the set point or are faced with a change in the load variable. Equation (10.5) is the basis of what is called dynamic feedforward control because Eq. (10.4) has to be derived from a time-domain differential equation (a transient model).3 In Eq. (10.5), 1/G p is the set-point tracking controller. This is what we need if we install only a feedforward controller, which in reality, we seldom do.4 Under most circumstances, the change in set point is handled by a feedback control loop, and we need to implement only the second term of Eq. (10.5). The transfer function в€’G L /G p is the feedforward controller (or the disturbance rejection controller). In terms of disturbance rejection, we may also see how the feedforward controller arises if we command C = 0 (i.e., no change), and write Eq. (10.4) as 0 = G L L + G p M. To see how we implement a feedforward controller, we now turn to a block diagram (Fig. 10.3).5 For the moment, we omit the feedback path from our general picture. With the expectation that we will introduce a feedback loop, we will not implement the set-point tracking term in Eq. (10.5). The implementation of feedforword control requires measurement of the load variable. If there is more than one load variable, we theoretically could implement a feedforward controller on each one. However, that may not be good engineering. Unless there is a compelling reason, we should select the variable that either undergoes the most severe п¬‚uctuation or has the strongest impact on the controlled variable. L G mL GFF + GL G*v Gp + C + Figure 10.3. A feedforward control system on a major load variable with measurement function G ML and feedforward controller G FF . 3 4 5 204 In contrast, we could have done the derivation by using steady-state models. In such a case, we would arrive at the design equation for a steady-state feedforward controller. WeвЂ™ll skip this analysis. As will be shown in Eq. (10.9), we can identify this steady-state part from the dynamic approach. The set-point tracking controller not only becomes redundant as soon as we add feedback control, but it also unnecessarily ties the feedforward controller into the closed-loop characteristic equation. If the transfer functions G L and G p are based on a simple process model, we know quite well that they should have the same characteristic polynomial. Thus the term в€’G L /G p is nothing but a ratio of the steady-state gains, в€’K L /K p . 10.2. Feedforward Control Here, we use L to denote the major load variable and its corresponding transfer function is G L . We measure the load variable with a sensor, G m L , which transmits its signal to the feedforward controller G FF . The feedforward controller then sends its decision to manipulate the actuating element, or valve, G v . In the block diagram, the actuator transfer function is denoted by G в€—v . The idea is that cascade control may be implemented with the actuator, G v , as we have derived in Eq. (10.1). We simply use G в€—v to reduce clutter in the diagram. With the feedforward and load path shown, the corresponding algebraic representation is C = [G L + G m L G FF G в€—v G p ]L . (10.6) The ideal feedforward controller should allow us to make proper adjustment in the actuator to achieve perfect rejection of load changes. To have C = 0, the theoretical feedforward controller function is GL G FF = в€’ , (10.7) G m L G в€—v G p which is a slightly more complete version of what we derived in Eq. (10.5). Before we blindly try to program Eq. (10.7) into a computer, we need to recognize certain pitfalls. If we write out the transfer functions in Eq. (10.7), we should п¬Ѓnd that G FF is not physically realizable вЂ“ the polynomial in the numerator has a higher order than the one in the denominator.6 If we approximate the composite function G m L G в€—v G p as a п¬Ѓrst-order function with dead time, K eв€’Оёs /(П„ s + 1), Eq. (10.7) would appear as G FF = в€’ K L П„ s + 1 Оёs e . K П„ps + 1 Now the dead time appears as a positive exponent or an advance in time. We cannot foresee the future, and this idea is not probable either.7 The consequence is that the most simple implementation of a feedforward controller, especially with off-the-shelf hardware, is a leadвЂ“lag element with a gain: G FF = K FF 6 7 П„FLD s + 1 . П„FLG s + 1 (10.8) If we go by the book, G L and G p are the transfer functions to a process and their dynamic terms (characteristic polynomial) in Eq. (10.7) must cancel out. The feedforward transfer function would be reduced to something that looks like (в€’K L /K m L K vв€— K p )(П„m L s + 1)(П„vв€— s + 1) whereas the denominator is just 1. In the simplest case, in which the responses of the transmitter and the valve are extremely fast such that we can ignore their dynamics, the feedforward function consists of only the steady-state gains, as in Eq. (10.9). If the load transfer function in Eq. (10.7) had also been approximated as a п¬Ѓrst-order function with dead time, say, of the form K L eв€’td s /(П„ p s + 1), the feedforward controller would appear as G FF = в€’ K L П„ s + 1 в€’(td в€’Оё)s e . K П„ps + 1 Now, if td > Оё , it is possible for the feedforward controller to incorporate dead-time compensation. The situation in which we may п¬Ѓnd that the load function dead time is larger than that in the feedforward path of G m G в€—v G p is not obvious from our simpliп¬Ѓed block diagram. Such a circumstance arises when we deal with more complex processing equipment consisting of several units (i.e., multicapacity process) and the disturbance enters farther upstream than where the controlled and manipulated variables are located. 205 Multiloop Systems Based on Eq. (10.7), the gain of this feedforward controller is K FF = в€’ KL . K m L K vв€— K p (10.9) This is the steady-state compensator. The leadвЂ“lag element with lead time constant П„FLD and lag time constant П„FLG is the dynamic compensator. Any dead-time in the transfer functions in Eq. (10.7) is omitted in this implementation. When we tune the feedforward controller, we may take, as a п¬Ѓrst approximation, П„FLD as the sum of the time constants П„m and П„vв€— . Analogous to the вЂњrealвЂќ derivative control function, we can choose the lag time constant to be a tenth smaller, П„FLG в‰€ 0.1П„FLD . If the dynamics of the measurement device is extremely fast, G m = K m L , and if we have cascade control, the time constant П„vв€— is also small, and we may not need the leadвЂ“lag element in the feedforward controller. Just the use of the steady-state compensator K FF may sufп¬Ѓce. In any event, the feedforward controller must be tuned with computer simulations and subsequently п¬Ѓeld tests. 10.3. FeedforwardвЂ“Feedback Control Because we do not have the precise model function G p embedded in the feedforward controller function in Eq. (10.8), we cannot expect perfect rejection of disturbances. In fact, feedforward control is never used by itself; it is implemented in conjunction with a feedback loop to provide the so-called feedback trim [Fig. 10.4(a)]. The feedback loop handles (1) measurement errors, (2) errors in the feedforward function, (3) changes in unmeasured load variables, such as the inlet process stream temperature in the furnace that one single feedforward loop cannot handle, and of course, (4) set-point changes. Our next task is to п¬Ѓnd the closed-loop transfer functions of this feedforwardвЂ“feedback system. Among other methods, we should see that we can вЂњmoveвЂќ the G в€—v G p term, as shown in Fig. 10.4(b). (You can double check with algebra.) After this step, the rest is routine. We can almost write the п¬Ѓnal result immediately. Anyway, we should see that C = [G L + G m L G FF G в€—v G p ]L + [G c G в€—v G p ]E, E = R в€’ G m C. After substitution for E and rearrangement, we arrive at C= G L + G m L G FF G в€—v G p G c G в€—v G p L+ R. в€— 1 + Gm GcGv G p 1 + G m G c G в€—v G p L (a) GFF E вЂ“ Gc + + Gm M Gp + + G mL G FF G*v G p GL G*v L (b) G mL R (10.10) C R E вЂ“ G c G*v G p GL + + + C + Gm Figure 10.4. (a) A feedforward-feedback control system, (b) the diagram after G в€—v G p is moved. 206 10.4. Ratio Control Temperature Hot process stream T TT TT Process stream TC T s в€‘ FT Furnace Flow rate FC FFC FT Fuel gas Figure E10.2. If we do not have cascade control, G в€—v is simply G v . If we are using cascade control, we can substitute Eq. (10.1) for G в€—v , but weвЂ™ll skip this messy algebraic step. The key point is that the closed-loop characteristic polynomial is 1 + G m G c G в€—v G p = 0, (10.11) and the feedforward controller G FF does not affect the system stability. Example 10.2: Consider the temperature control of a gas furnace used in heating a process stream. The probable disturbances are in the process stream temperature and п¬‚ow rate and the fuel-gas п¬‚ow rate. Draw the schematic diagram of the furnace temperature control system and show how feedforward, feedback, and cascade controls can all be implemented together to handle load changes. The design in Fig. E10.2 is based on our discussion of cascade control. The fuel-gas п¬‚ow is the manipulated variable, and so we handle disturbance in the fuel-gas п¬‚ow with a п¬‚ow controller (FC) in a slave loop. This secondary loop remains the same as the G в€—v function in Eq. (10.1), where the secondary transfer function is denoted by G c2 . Of the other two load variables, we choose the process stream п¬‚ow rate as the major disturbance. The п¬‚ow transducer FT sends the signal to the feedforward controller (FFC, transfer function G FF ). A summer ( ) combines the signals from both the feedforward and the feedback controllers, and its output becomes the set point for the secondary fuel-gas п¬‚ow rate controller (FC). The handling of disturbance in the inlet process stream temperature is passive. Any changes in this load variable will affect the furnace temperature. The change in furnace temperature is measured by the outlet temperature transducer (TT) and sent to the feedback temperature controller (TC). The primary controller then acts accordingly to reduce the deviation in the furnace temperature. 10.4. Ratio Control We are not entirely п¬Ѓnished with the furnace. There is one more piece missing from the whole picture вЂ“ the air п¬‚ow rate. We need to ensure sufп¬Ѓcient air п¬‚ow for efп¬Ѓcient combustion. The regulation of air п¬‚ow is particularly important in the reduction of air pollutant emission. 207 Multiloop Systems Fuel gas flow entering furnace, FFG FT RS Ratio station FC Air flow, FA FT Figure 10.5. Simple ratio control of the air п¬‚ow rate. To regulate the air п¬‚ow rate with respect to the fuel-gas п¬‚ow rate, we can use ratio control. Figure 10.5 illustrates one of the simplest implementations of this strategy. LetвЂ™s say the air to fuel-gas п¬‚ow rates must be kept at some constant ratio, R= FA . FFG (10.12) What we can do easily is to measure the fuel-gas п¬‚ow rate, multiply the value by R in the so-called ratio station, and send the signal as the set point to the air п¬‚ow controller. The calculation can be based on actual п¬‚ow rates rather than on deviation variables. A more sophisticated implementation is full metering control (Fig. 10.6). In this case, we send the signals from the fuel-gas controller (FC in the fuel-gas loop) and the air п¬‚ow transmitter (FT) to the ratio controller (RC), which takes the desired п¬‚ow ratio (R) as the set point. This controller calculates the proper air п¬‚ow rate, which in turn becomes the set point to the air п¬‚ow controller (FC in the air п¬‚ow loop). If we take away the secondary п¬‚ow control loops on both the fuel-gas and the air п¬‚ow rates, what we have is called parallel positioning control. In this simpler case, of course, the performance of the furnace is subject to п¬‚uctuations in fuel- and air-supply lines. We are skipping the equations and details because the air п¬‚ow regulation should not affect the stability and system analysis of the fuel-gas controller and ratio control is best implemented with Simulink in simulation and design projects. From the feedforward-feedback loops в€‘ FC FT To furnace Fuel gas To furnace FC RC R FT Air flow, FA Figure 10.6. Full metering ratio control of fuel and air п¬‚ows. 208 10.5. Time-Delay Compensation вЂ“ The Smith Predictor R + Y G(s)e вЂ“ t ds Gc (s) вЂ“ Figure 10.7. System with inherent dead time. 10.5. Time-Delay Compensation вЂ“ The Smith Predictor There are different schemes to handle systems with a large dead time. One of them is the Smith predictor. It is not the most effective technique, but it provides a good thought process. Consider a unit feedback system with a time delay in its process function (Fig. 10.7). The characteristic polynomial is 1 + G c (s)G(s)eв€’td s = 0. (10.13) We know from frequency-response analysis that time lag introduces extra phase lag, reduces the gain margin, and is a signiп¬Ѓcant source of instability. This is mainly because the feedback information is outdated. If we have a model for the process, i.e., we know G(s) and td , we can predict what may happen and feed back this estimation. From the way the dead-time compensator (or predictor) is written (Fig. 10.8), we can interpret the transfer function as follows. Assuming that we know the process model, we feed back the вЂњoutputвЂќ calculation based on this model. We also have to subtract out the вЂњactualвЂќ calculated time-delay output information. Now the error E also includes the feedback information from the dead-time compensator: E = R в€’ Y в€’ E[G c G(1 в€’ eв€’td s )], and substituting Y = G c Geв€’td s E, we have E = R в€’ E[G c Geв€’td s + G c G(1 в€’ eв€’td s )], where the exponential terms cancel out and we are left with simply E = R в€’ E G c G. (10.14) The time-delay effect is canceled out, and this equation at the summing point is equivalent to a system without dead time (in which the forward path is C = G c G E). With simple Deadtime Compensator G(s)[1 вЂ“ e R+ вЂ“ E вЂ“ вЂ“ tds Gc(s) ] G(s)e вЂ“tds ] Y Figure 10.8. Implementation of the Smith predictor. 209 Multiloop Systems R + вЂ“ Gc G eвЂ“ tds Y Figure 10.9. An interpretation of the compensator effect. block-diagram algebra, we can also show that the closed-loop characteristic polynomial with the Smith predictor is simply 1 + G c G = 0. (10.15) The time delay is removed. With the delay compensator included, we can now use a larger proportional gain without becoming unstable. Going back to the fact that the feedback information is G c G R, we can also interpret the compensator effect as in Fig. 10.9. The Smith predictor is essentially making use of state feedback as opposed to output feedback. Just like feedforward control (or any other model-based control), we have perfect compensation only if we know the precise process model. Otherwise, the effectiveness of the compensator (or predictor) is diminished. Assume that we have an imperfect model approximation H (s) and dead-time estimation Оё (H = G and Оё = td ); the feedback information is now Y + [H (1 в€’ eв€’Оё s )]G c R = [G c Geв€’td s + G c H (1 в€’ eв€’Оёs )]R = G c [Geв€’td s + H (1 в€’ eв€’Оё s )]R, where the RHS becomes G c G R if and only if H = G and Оё = td . Note that the time-delay term is an exponential function. Error in the estimation of the dead time is more detrimental than error in the estimation of the process function G. Because few things are exact in this world, we most likely have errors in the estimation of the process and the dead time. Therefore we only have partial dead-time compensation, and we must be conservative in picking controller gains based on the characteristic polynomial 1 + G c G = 0. In a chemical plant, time delay is usually a result of a transport lag in the pipe п¬‚ow. If the п¬‚ow rate is fairly constant, the use of the Smith predictor is acceptable. If the п¬‚ow rate varies for whatever reason, this compensation method will not be effective. 10.6. Multiple-Input Multiple-Output Control In this section, we analyze a MIMO system. There are valuable insights that can be gained from using the classical transfer function approach. One decision that we need to appreciate is the proper pairing of manipulated and controlled variables. To do that, we also need to know how strong the interaction is among different variables. The key points will be illustrated with a blending process. Here, we mix two streams with mass п¬‚ow rates m 1 and m 2 , and both the total п¬‚ow rate F and the composition x of a solute A are to be controlled (Fig. 10.10). With simple intuition, we know changes in both m 1 and m 2 will affect F and x. We can describe the relations with the block diagram in Fig. 10.11, where interactions are represented by the two yet to be derived transfer functions G 12 and G 21 . Given Fig. 10.11 and classical control theory, we can infer the structure of the control system, which is shown in Fig. 10.12. That is, we use two controllers and two feedback 210 10.6. Multiple-Input Multiple-Output Control m1 FC CC F x FT m2 CT Figure 10.10. A blending system with manipulated and controlled variable pairings yet to be determined. loops, in which, for simplicity, the measurement and the actuator functions have been omitted. Simple reasoning can illustrate now interactions may arise in Fig. 10.12. If a disturbance (not shown in diagram) moves C1 away from its reference R1 , the controller G c1 will respond to the error and alter M1 accordingly. The change in M1 , however, will also affect C2 by means of G 21 . Hence C2 is forced to change and deviate from R2 . Now the second controller G c2 kicks in and adjusts M2 , which in turn also affects C1 by means of G 12 . With this case the system may eventually settle, but it is just as likely that the system in Fig. 10.12 will spiral out of control. It is clear that loop interactions can destabilize a control system, and tuning controllers in a MIMO system can be difп¬Ѓcult. One logical thing that we can do is to reduce loop interactions by proper pairing of manipulated and controlled variables. This is the focus of the analysis in the following subsections. 10.6.1. MIMO Transfer Functions We now derive the transfer functions of the MIMO system. This sets the stage for the more detailed analysis that follows. The transfer functions in Fig. 10.11 depend on the process that we have to control, and we derive them in the next subsection for the blending process. Here, we consider the general system shown in Fig. 10.12. With the understanding that the errors are E 1 = R1 в€’ C1 and E 2 = R2 в€’ C2 in Fig. 10.12, we can write immediately M1 M1 = G c1 (R1 в€’ C1 ), (10.16) M2 = G c2 (R2 в€’ C2 ). (10.17) G 11 C1 = x + + G 12 G 21 M2 + G 22 C2 = F + Figure 10.11. Block diagram of an interacting 2 Г— 2 process, with the output x and F referring to the blending problem. 211 Multiloop Systems R1 вЂ“ + Gc M1 1 G 11 C1 + + G 12 G 21 R2 + вЂ“ Gc 2 M2 G 22 + C2 + Figure 10.12. Block diagram of a 2 Г— 2 servo system. The pairing of the manipulated and controlled variables is not necessarily the same as that shown in Fig. 10.11. The process (also in Fig. 10.11) can be written as C1 = G 11 M1 + G 12 M2 , (10.18) C2 = G 21 M1 + G 22 M2 . (10.19) Substituting for M1 and M2 by using Eqs. (10.16) and (10.17), and factoring C1 and C2 to the left, we п¬Ѓnd that Eqs. (10.18) and (10.19) become 1 + G 11 G c1 C1 + G 12 G c2 C2 = G 11 G c1 R1 + G 12 G c2 R2 , (10.20) G 21 G c1 C1 + 1 + G 22 G c2 C2 = G 21 G c1 R1 + G 22 G c2 R2 . (10.21) Making use of KramerвЂ™s rule, we should identify (derive!) the system characteristic equation, p(s) = 1 + G 11 G c1 1 + G 22 G c2 в€’ G 12 G 21 G c1 G c2 = 0, (10.22) which, of course, is what governs the dynamics and the stability of the system. We may recognize that when either G 12 = 0 or G 21 = 0, the interaction term is zero.8 In either case, the system characteristics analysis can be reduced to those of two single-loop systems: 1 + G 11 G c1 = 0, 1 + G 22 G c2 = 0. Now back to п¬Ѓnding the transfer functions with interaction. To make the algebra appear a bit cleaner, we consider the following two cases. When R2 = 0, we can derive, from Eqs. (10.20) and (10.21), C1 G 11 G c1 + G c1 G c2 (G 11 G 22 в€’ G 12 G 21 ) = . R1 p(s) (10.23) When R1 = 0, we can п¬Ѓnd G 12 G c2 C1 = . R2 p(s) 8 212 (10.24) When G 12 = G 21 = 0, the system is decoupled and behaves identically to two single loops. When either G 12 = 0 or G 21 = 0, the situation is referred to as one-way interaction, which is sufп¬Ѓcient to eliminate recursive interactions between the two loops. In such a case, one of the loops is not affected by the second whereas it becomes a source of disturbance to this second loop. 10.6. Multiple-Input Multiple-Output Control If both references change simultaneously, we just need to add their effects in Eqs. (10.23) and (10.24) together. (What about C2 ? YouвЂ™ll get to try that in the Review Problems.) It is apparent from Eq. (10.22) that, with interaction, the controller design of the MIMO system is different from a SISO system. One logical question is under what circumstances may we make use of SISO designs as an approximation? Or in other words, can we tell if the interaction may be weak? This takes us to the next two subsections. 10.6.2. Process Gain Matrix We come back to derive the process transfer functions for the blending problem.9 The total mass п¬‚ow balance is F = m1 + m2, (10.25) where F is the total п¬‚ow rate after blending, and m 1 and m 2 are the two inlet п¬‚ows that we manipulate. The mass balance for a solute A (without using the subscript A explicitly) is x F = x1 m 1 + x2 m 2 , (10.26) where x is the mass fraction of A after blending and x1 and x2 are the mass fractions of A in the two inlet streams. We want to п¬Ѓnd the transfer functions as shown in Fig. 10.11: X (s) G 11 (s) = G 21 (s) F(s) G 12 (s) G 22 (s) M1 (s) . M2 (s) (10.27) We take stream m 1 to be pure solute A and stream m 2 to be pure solvent. In this example, x1 = 1 and x2 = 0, and Eq. (10.26) is simpliп¬Ѓed to m1 m1 x= . (10.28) = F m1 + m2 Because xi and m i are functions of time, we need to linearize Eq. (10.26). A п¬Ѓrst-order Taylor expansion of x is xв‰€ m1 F + s m2 (m 1 + m 2 )2 (m 1 в€’ m 1,s ) в€’ s m1 (m 1 + m 2 )2 (m 2 в€’ m 2,s ), s where the subscript s to the brackets denotes terms evaluated at steady state. The п¬Ѓrst term on the RHS is really the value of x at steady state, xs , which can be moved to the LHS to make the deviation variable in x. With that, we take the Laplace transform to obtain the transfer functions of the deviation variables: X (s) = G 11 (s)M1 (s) + G 12 (s)M2 (s), (10.29) where G 11 (s) = 9 m2 (m 1 + m 2 )2 = K 11 , s G 12 (s) = в€’ m1 (m 1 + m 2 )2 = K 12 . (10.30) s Because the point is to illustrate the analysis of interactions, we are using only steady-state balances, and it should not be a surprise that the transfer functions end up being only steady-state gains in Eq. (10.32). For a general dynamic problem in which we have to work with the transfer functions G i j (s), we can still apply the results here by making use of the steady-state gains of the transfer functions. 213 Multiloop Systems The transfer functions are constants, and hence we denote them with the gains K 11 and K 12 . If the solvent п¬‚ow rate m 2 increases, the solute will be diluted. Hence K 12 is negative. The functions G 21 and G 22 are much easier. From Eq. (10.25), we can see immediately that G 21 (s) = K 21 = 1, G 22 (s) = K 22 = 1. (10.31) Thus, in this problem, the process transfer function matrix of Eq. (10.27) can be written in terms of the steady-state gain matrix: X (s) K 11 = F(s) K 21 K 12 K 22 M1 (s) . M2 (s) (10.32) In more general terms, we replace the LHS of Eq. (10.32) with a controlled variable vector: C(s) = KM(s), (10.33) where C = [X F]T . If there is a policy such that the manipulated variables can regulate the controlled variables, we must be able to п¬Ѓnd an inverse of the gain matrix such that M(s) = Kв€’1 C(s). (10.34) Example 10.3: If m 1 = 0.1 g/s and m 2 = 10 g/s, what is the process gain matrix? What are the interpretations? Making use of Eqs. (10.30), we can calculate K 11 = 9.8 Г— 10в€’2 and K 12 = в€’9.8 Г— 10в€’4 . With Eqs. (10.31), the process gain matrix is K= 9.8 Г— 10в€’2 1 в€’9.8 Г— 10в€’4 . 1 Under the circumstances of the particular set of numbers given, changing either m 1 or m 2 has a stronger effect on the total п¬‚ow rate F than on x. With respect to the composition x, changing the solute п¬‚ow m 1 has a much stronger effect than changing the solvent п¬‚ow. The situation very much resembles a one-way interaction. We may question other obvious examples of the process gain matrix. The sweetest is an identity matrix, meaning no interaction among the manipulated and controlled variables. Here is a quick summary of several simple possibilities10 : K= K= K= 10 214 1 0 1 Оґ 1 0 0 1 Оґ 0 1 1 : No interaction. Controller design is like that of single-loop systems. : Strong interaction if Оґ is close to 1; weak interaction if Оґ , 1 0 1 1 1. : One-way interaction There is more to вЂњlookingвЂќ at K. We can, for example, make use of its singular value and condition number, which should be deferred to a second course in control. 10.6. Multiple-Input Multiple-Output Control 10.6.3. Relative Gain Array You may not п¬Ѓnd observing the process gain matrix satisfactory. That takes us to the relative gain array (RGA), which can provide for a more quantitative assessment of the effect of changing a manipulated variable on different controlled variables. We start with the blending problem before coming back to the general deп¬Ѓnition. For the blending process, the relative gain parameter of the effect of m 1 on x is deп¬Ѓned as О»x,m 1 = в€‚ x/в€‚m 1 |m 2 . в€‚ x/в€‚m 1 | F (10.35) It is the ratio of the partial derivative evaluated under two different circumstances. On top, we look at the effect of m 1 while holding m 2 constant. The calculation represents an open-loop experiment, and the value is referred to as an open-loop gain. In the denominator, the total п¬‚ow rate, the other controlled variable, is held constant. Because we are varying (in theory) m 1 , F can be constant only if we have a closed-loop with perfect control involving m 2 . The partial derivative in the denominator is referred to as some closed-loop gain. How do we interpret the relative gain? The idea is that if m 2 does not interfere with m 1 , the derivative in the denominator should not be affected by the closed-loop involving m 2 , and its value should be the same as the open-loop value in the numerator. In other words, if there is no interaction, О»x,m 1 = 1. Example 10.4: Evaluate the RGA matrix for the blending problem. The complete RGA matrix for the 2 Г— 2 blending problem is deп¬Ѓned as О›= О»x,m 1 О» F,m 1 О»x,m 2 . О» F,m 2 (E10.4) For the п¬Ѓrst element, we use Eq. (10.28) to п¬Ѓnd в€‚x в€‚m 1 = m2 m2 , (m 1 + m 2 )2 в€‚x в€‚m 1 = F Hence, with the deп¬Ѓnition in Eq. (10.35), m2 О»x,m 1 = = 1 в€’ x. m1 + m2 1 1 . = F m1 + m2 (E10.5) We proceed to п¬Ѓnd the other three elements (see Review Problems), and we have the RGA for the blending problem: О›= 1в€’x x . x 1в€’x (E10.6) There are several general points regarding this problem that should be noted, i.e., without proving them formally here. The sum of all the entries in each row and each column of the RGA О› is 1. Thus, in the case of a 2 Г— 2 problem, all we need is to evaluate one element. Furthermore, the calculation is based on only open-loop information. In Example 10.4, the derivation is based on Eqs. (10.25) and (10.26). 215 Multiloop Systems We can now state the general deп¬Ѓnition of the relative gain array О›. For the element relating the ith controlled variable to the jth manipulated variable, О»i, j = в€‚ci /в€‚m j |m k,k= j , в€‚ci /в€‚m j |ck,k=i (10.36) where the (open-loop) gain in the numerator is evaluated with all other manipulated variables held constant and all the loops open (no loops!). The (closed-loop) gain in the denominator is evaluated with all the loops вЂ“ other than the ith loop вЂ“ closed. The value of this so-called closed-loop gain reп¬‚ects the effect from other closed loops and the open loop between m j and ci . The RGA can be derived in terms of the process steady-state gains. Making use of gain matrix equation (10.32), we can п¬Ѓnd (not that hard; see Review Problems) О»x,m 1 = 1 1в€’ K 12 K 21 K 11 K 22 , (10.37) which can be considered a more general form of Eq. (E10.5) and hence (E10.6).11 The next question comes back to the meaning of the RGA and how that may inп¬‚uence our decision in pairing manipulated with controlled variables. Here is the simple interpretation, making use of Eqs. (10.36) and (10.37): О»i, j = 1: Requires K 12 K 21 = 0. вЂњOpen-loopвЂќ gain is the same as the вЂњclosed-loopвЂќ gain. The controlled variable (or loop) i is not subject to interaction from other manipulated variables (or other loops). Of course, we know nothing about whether other manipulated variables may interact and affect other controlled variables. Nevertheless, pairing the ith controlled variable to the jth manipulated variable is desirable. О»i, j = 0: The open-loop gain is zero. The manipulated variable j has no effect on the controlled variable i. Of course m j may still inп¬‚uence other controlled variables (by means of one-way interaction). Either way, it makes no sense to pair m j with ci in a control loop. 0 < О»i, j < 1: No doubt there are interactions from other loops, and from Eq. (10.37), some of the process gains must have opposite signs (or act in different directions). When О»i, j = 0.5, we can interpret that the effect of the interactions is identical to the open-loop gain вЂ“ recall the statement after Eq. (10.36). When О»i, j > 0.5, the interaction is less than the main effect of m j on ci . However, when О»i, j < 0.5, the interactive effects predominate and we want to avoid pairing m j with ci . О»i, j > 1: There are interactions from other loops as well, but now with all the process gains having the same sign. Avoid pairing m j with ci if О»i, j is much larger than 1. О»i, j < 0: We can infer by using Eq. (10.36) that the open-loop and the closed-loop gains have different signs or opposing effects. The overall inп¬‚uence of the other loops is in opposition to the main effect of m j on ci . Moreover, from Eq. (10.37), the interactive product K 12 K 21 must be larger than the direct terms K 11 K 22 . Undesirable interaction is strong. The overall multiloop system may become unstable easily if we open up one of its loops. We deп¬Ѓnitely should avoid pairing m j with ci . 11 216 For your information, RGA can be computed as the so-called Hadamard product, О»i j = K i j K в€’1 ji , which is the element-by-element product of the gain matrix K and the transpose of its inverse. You can conп¬Ѓrm this by repeating the examples with MATLAB calculations. 10.7. Decoupling of Interacting Systems To sum up, the key is to pair the manipulated and the controlled variables such that the relative gain parameter is positive and as close to one as possible. Example 10.5: If m 1 = 0.1 g/s and m 2 = 10 g/s, what is the proper pairing of manipulated and controlled variables? What if m 1 = 9 g/s and m 2 = 1 g/s? In the п¬Ѓrst case, in which m 1 is very small, it is like a dosing problem. From Eq. (10.28), x = 0.0099. Because x 1, О»x,m 1 is very close to 1 by Eq. (E10.5). Thus interaction is not signiп¬Ѓcant if we pair x with m 1 and F with m 2 . Physically, we essentially manipulate the total п¬‚ow with the large solvent п¬‚ow m 2 and tune the solute concentration by manipulating m 1 . In the second case, x = 0.9. Now О»x,m 1 = 0.1 by Eq. (E10.5). Because О»x,m 1 1, we do not want to pair x with m 1 . Instead, we pair x with m 2 and F with m 1 . Now we regulate the total п¬‚ow with the larger solute п¬‚ow m 1 and tune the concentration with the solvent m 2 . 10.7. Decoupling of Interacting Systems After proper pairing of manipulated and controlled variables, we still have to design and tune the controllers. The simplest approach is to tune each loop individually and conservatively while the other loop is in manual mode. At a more sophisticated level, we may try to decouple the loops mathematically into two noninteracting SISO systems with which we can apply single-loop tuning procedures. Several examples applicable to a 2 Г— 2 system are offered here. 10.7.1. Alternative Deп¬Ѓnition of Manipulated Variables We seek choices of manipulated variables that may decouple the system. A simple possibility is to pick them to be the same as the controlled variables. In the blending problem, the two new manipulated variables can be deп¬Ѓned as12 Вµ1 = F, Вµ2 = x. (10.38) (10.39) Once the controller (a computer) evaluates these two manipulated variables, it also computes on the п¬‚y the actual signals necessary for the two mass п¬‚ow rates m 1 and m 2 . The computation follows directly balance equations (10.25) and (10.28). Figure 10.13 is a schematic diagram of how this idea may be implemented. 10.7.2. Decoupler Functions In this subsection, we add the so-called decoupler functions to a 2 Г— 2 system. Our starting point is Fig. 10.12. The closed-loop system equations can be written in matrix form, virtually by visual observation of the block diagram, as C1 G 11 = C2 G 21 12 G 12 G 22 G c1 0 0 G c2 R1 в€’C1 . R2 в€’C2 (10.40) The blending problem can be reduced to one-way interaction if we use m 1 instead of x as the new manipulated variable Вµ2 . We will do that in the Review Problems. 217 Multiloop Systems m1 вЂ“ + + m1 x m1 m1 + m 2 m1 + m 2 FC CC x F FT m2 CT Figure 10.13. A decoupled control scheme. The controller outputs are the manipulated variables in Eqs. (10.38) and (10.39), and they are rewritten based on their deп¬Ѓnitions in Eqs. (10.25) and (10.28). In matrix form, this equation looks deceptively simple, but if we expand the algebra, we should arrive at Eqs. (10.20) and (10.21) again. In a system with interactions, G 12 and G 21 are not zero, but we can manipulate the controller signal such that the system appears (mathematically) to be decoupled. So letвЂ™s try to transform the controller output with a matrix D, which will contain our decoupling functions. The manipulated variables are now M1 d = 11 M2 d21 d12 d22 G c1 0 R1 в€’C1 , R2 в€’C2 0 G c2 and the system equations become C1 G 11 = C2 G 21 G 12 G 22 d11 d21 d12 d22 G c1 0 0 G c2 R1 в€’C1 R в€’C1 = GDGc 1 . R2 в€’C2 R2 в€’C2 (10.41) To decouple the system equations, we require that GDGc be a diagonal matrix. Deп¬Ѓne G0 = GDGc , and the previous step can be solved for C: R C1 = [I + G0 ]в€’1 G0 1 . C2 R2 (10.42) Because G0 is diagonal, the matrix [I + G0 ]в€’1 G0 is also diagonal, and happily, we have two decoupled equations in Eq. (10.42). Now we have to п¬Ѓnd D. Because Gc is already diagonal, we require that GD be diagonal: G 11 G 21 G 12 G 22 d11 d21 H1 d12 = 0 d22 0 . H2 (10.43) With a little bit of algebra to match term by term, we should п¬Ѓnd (see Review Problems) 218 d11 = G 22 H1 , G 11 G 22 в€’ G 12 G 21 d21 = в€’G 21 d11 , G 22 d12 = d22 = G 11 H2 , G 11 G 22 в€’ G 12 G 21 в€’G 12 d22 . G 11 (10.44) (10.45) 10.7. Decoupling of Interacting Systems We have six unknowns (four di j and two Hi ) but only four equations. We have to make two (arbitrary) decisions. One possibility is to choose (or deп¬Ѓne) H1 = G 11 G 22 в€’ G 12 G 21 , G 22 H2 = G 11 G 22 в€’ G 12 G 21 G 11 (10.46) such that d11 and d22 become 1. (We can also think in terms of choosing both d11 = d22 = 1 and then derive the relations for H1 and H2 .) It follows that d21 = в€’G 21 , G 22 d12 = в€’G 12 . G 11 (10.47) Now the closed-loop equations are C1 H1 = 0 C2 0 H2 0 G c2 G c1 0 R1 в€’C1 H1 G c1 = 0 R2 в€’C2 0 H2 G c2 R1 в€’C1 , R2 в€’C2 (10.48) from which we can easily write, for each row of the matrices, G c1 H1 C1 = , R1 1 + G c1 H1 C2 G c2 H2 = , R2 1 + G c2 H2 (10.49) and the design can be based on the two characteristic equations 1 + G c1 H1 = 0, 1 + G c2 H2 = 0. (10.50) Recall from Eqs. (10.46) that H1 and H2 are deп¬Ѓned entirely by the four plant functions G i j . This is another example of model-based control. With the deп¬Ѓnitions of H1 and H2 given in Eqs. (10.46), the calculations are best performed with a computer. 10.7.3. Feedforward Decoupling Functions A simpler approach is to use only two decoupler functions and implement them as if they were feedforward controllers that may reduce the disturbance arising from loop interaction. As implemented in Fig. 10.14, we use the function D12 to вЂњforeseeвЂќ and reduce the interaction that is due to G 12 . Likewise, D21 is used to address the interaction that is due to G 21 . To п¬Ѓnd these two decoupling functions, we focus on how to cancel the interaction at the points identiп¬Ѓed as a and b in Fig. 10.14. R1 вЂ“ Gc + M1 + + 1 D21 G11 + 2 M2 + C2 G21 D12 Gc C1 G12 R2 + вЂ“ + b + + G22 + a Figure 10.14. A decoupling scheme in which two feedforwardlike decoupler functions are used. 219 Multiloop Systems LetвЂ™s pick point a п¬Ѓrst. If the signal from M1 through G 21 can be canceled by the compensation through D21 , we can write G 21 M1 + G 22 D21 M1 = 0. Cancel out M1 and we have D21 = в€’G 21 /G 22 . (10.51) Similarly, if D12 can cancel the effect of G 12 at point b, we have G 12 M2 + G 11 D12 M2 = 0 or D12 = в€’G 12 /G 11 . (10.52) We may note that Eqs. (10.51) and (10.52) are the same as d21 and d12 in Eqs. (10.47). The strategy of implementing D12 and D21 is similar to the discussion of feedforward controllers in Section 10.2, and typically we remove the time-delay terms and apply a leadвЂ“ lag compensator as in Eq. (10.8). If the time constant of the п¬Ѓrst-order lead is similar to the time constant of the п¬Ѓrst-order lag, then we just need a steady-state compensator. Example 10.6: A classic example of a MIMO problem is a distillation column.13 From open-loop step tests, the following transfer functions are obtained: пЈ№ пЈ® 0.07eв€’3s в€’0.05eв€’s пЈЇ 12s + 1 X D (s) 15s + 1 пЈє пЈє L(s) . =пЈЇ пЈ° в€’4s X B (s) 0.1e в€’0.15eв€’2s пЈ» V (s) 11s + 1 10s + 1 In this model, x D and x B are the distillate and the bottom compositions, respectively; L is the reп¬‚ux п¬‚ow rate, and V is the boil-up rate. Design a 2 Г— 2 MIMO system with PI controllers and decouplers as in Fig. 10.14. Before we design the MIMO system, we need to check the paring of variables. The steady-state gain matrix is K= 0.07 в€’0.05 . 0.1 в€’0.15 With Eqs. (10.37) and (E10.6), the RGA is О›= 1.91 в€’0.91 . в€’0.91 1.91 The relative gain parameter О»x D в€’L is 1.91. It is not 1 but at least it is not negative. Physically, it also makes sense to manipulate the distillate composition with the more neighboring reп¬‚ux 13 220 Pardon me if you have not taken a course in separation processes yet, but you do not need to know what a distillation column is to read the example. In a simple-minded way, we can think of making moonshine. We have to boil a dilute alcohol solution at the bottom and we need a condenser at the top to catch the distillate. This is how we have the V and L manipulated variables. Furthermore, the transfer functions are what we obtain from doing an experiment, not from any theoretical derivation. 10.7. Decoupling of Interacting Systems 1.2 1 SISO 0.8 MIMO xd 0.6 MIMO with decouplers 0.4 0.2 0 0 10 20 30 t 40 50 Figure E10.6. п¬‚ow. Therefore we pair x D в€’ L and x B в€’ V . Next, with Eqs. (10.51) and (10.52), the two decoupling functions are D12 = K d,12 12s + 1 , 15s + 1 D21 = K d,21 10s + 1 в‰€ K d,21 . 11s + 1 To do the tuning, we can use the initial values K d,12 = в€’0.05/0.07 в‰€ в€’0.7 and K d,21 = в€’0.1/0.15 в‰€ в€’0.7. We will have to skip the details for the remainder of the exercise. You may try to generate a plot similar to that of Fig. E10.6 in the Review Problems. This is roughly how it is done. All the simulations are performed with Simulink. First, we use G 11 and G 22 as the п¬Ѓrst-order with dead-time functions and apply them to the ITAE tuning relations in Table 6.1. With that, we have the PI controller settings of two SISO systems. The single-loop response to a unit-step change in the set point of x D is labeled SISO in Fig. E10.6. We retain the ITAE controller settings and apply them to a Simulink block diagram constructed as in Fig. 10.12. The result is labeled MIMO in the п¬Ѓgure. Finally, we use Fig. 10.14 and the two decouplers, and the simulation result with the initial setting is labeled вЂњMIMO with decouplers.вЂќ In this illustration, we do not have to detune the SISO controller settings. The interaction does not appear to be severely detrimental mainly because we have used the conservative ITAE settings. It would not be the case if we had tried CohenвЂ“Coon relations. The decouplers also do not appear to be particularly effective. They reduce the oscillation, but also slow down the system response. The main reason is that the leadвЂ“lag compensators do not factor in the dead times in all the transfer functions. Review Problems (1) Derive Eqs. (10.3) and (10.3a) with measurement transfer functions G m 1 and G m 2 in the primary and the secondary loops. Conп¬Ѓrm footnote 1 that this equation can be reduced to that of a single-loop system. (2) Do the root-locus plots in Example 10.1(d). Conп¬Ѓrm the stability analysis in Example 10.1(e). 221 Multiloop Systems L GFF R + e + Gc + вЂ“ GL Gv M + Gp C + Figure R10.4. (3) Draw the block diagram of the system in Example 10.2. Label the diagram with the proper variables. (4) Attempt a numerical simulation of a feedforwardвЂ“feedback system in Fig. R10.4. Consider the simpliп¬Ѓed block diagram with Gv = (5) (6) (7) (8) (9) 0.5 , s+1 Gp = 0.8 , 2s + 1 GL = (a) The load function has a negative gain. What does it mean? (b) Omit for the moment the feedback loop and controller G c , and consider only G FF as deп¬Ѓned in Eq. (10.8). Use MATLAB functions (or Simulink) to simulate the response in C when we impose a unit step change to L. Experiment with different values of the gain and the time constants in the leadвЂ“lag element. (c) Consider a PI controller for the feedback loop with an integral time of 1.5 s; п¬Ѓnd the proportional gain such that the system has an underdamped behavior equivalent to a damping ratio of 0.7. (d) With the feedback loop and PI controller in part (c), use MATLAB to simulate the response of C to a unit-step change in L. Repeat the different values of the feedforward controller as in part (b). Consider the simpler problem in Fig. R10.5 based on Fig. 10.12. If we implement only one feedback loop and one controller, how is the transfer function C1 /M1 affected by the interaction? Derive the transfer functions C2 /R1 and C2 /R2 from Eqs. (10.20) and (10.21). Fill in the details and derive the RGA [Eq. (E10.6)] in Example 10.4. Derive Eq. (10.37). Show that we also can obtain Eq. (E10.6) by applying Eq. (10.37) to the blending problem. M1 G11 C1 + + G12 G21 R2 + + вЂ“ Gc 2 Figure R10.5. 222 в€’0.4 . 2s + 1 M2 G22 + C2 10.7. Decoupling of Interacting Systems (f) Repeat Subsection 10.7.1 by replacing the second manipulated variable in Eq. (10.39) with Вµ2 = m 1 . Find the gain matrix and show that the relative gain parameter is 1. Show how this partially decoupling scheme can be implemented as analogous to Fig. 10.13. (10) Derive Eqs. (10.44) and (10.45). (11) Try to do the Simulink simulations in Example 10.6. If you need help, the Simulink п¬Ѓle is on the Web Support. Hints: (2) The MATLAB statements can be Part (d) Gp=tf(0.8,[2 1]); Gv=tf(0.9,[0.1 1]); % With cascade control taui=0.05; % Just one example Gi=tf([taui 1],[taui 0]) rlocus(Gi*Gv*Gp) Part (e) Gvo=tf(0.5,[1 1]); rlocus(Gi*Gvo*Gp) (4) (a) If L is the inlet process stream п¬‚ow rate, the furnace temperature will decrease if the process п¬‚ow increases. (b) Use Eq. (10.9) and the comments that follow to select the parameters for the feedforward controller. Compare with the case when we do not have a feedforward controller by setting K FF = 0. You should observe that the major compensation to the load change is contributed by the steady-state compensator. (c) The proportional gain is в€ј1.4. The feedforward controller does not affect the system stability and we can design the controller G c with only G v , G p , and the feedback loop. We have to use, for example, the root-locus method in Chap. 6 to do this part. Root locus can also help us to determine if П„ I = 1.5 s is a good choice. (d) You should п¬Ѓnd that the feedback loop takes over much of the burden in load changes. The system response is rather robust even with relatively large errors in the steady-state compensator. (5) C2 = G 22 G c2 (R2 в€’ C2 ) + G 21 M1 , C1 = G 11 M1 + G 12 G c2 (R2 в€’ C2 ). Setting R2 = 0, C2 = G 21 M1 . 1 + G c2 G 22 223 Multiloop Systems Substituting C2 into the C1 equation, we can п¬Ѓnd, after two algebraic steps, C1 = G 11 в€’ G 12 G 21 G c2 M1 . 1 + G c2 G 22 The second term in the parentheses is due to interaction. (6) We apply CramerвЂ™s rule to п¬Ѓnd C2 just as we had with C1 . The solution has the same characteristic polynomial in Eq. (10.22). The transfer functions are, with R1 = 0, G 22 G c2 + G c1 G c2 (G 11 G 22 в€’ G 12 G 21 ) C2 , = R2 p(s) and with R2 = 0, C2 G 21 G c1 . = R1 p(s) (7) We still use Eq. (10.28) as in Example 10.4. To п¬Ѓnd О»x,m 2 , в€‚x в€‚m 2 = m1 О»x,m 2 в€’m 1 в€‚x , 2 (m 1 + m 2 ) в€‚m 2 m1 = = x. m1 + m2 = F F в€’ m2 F в€‚ в€‚m 2 =в€’ 1 , F To п¬Ѓnd О» F,m 1 , в€‚F в€‚m 1 = 1, using Eq. (10.25), m2 в€‚F в€‚m 1 = x 1 , using F = m 1 /x, x О» F,m 1 = x. To п¬Ѓnd О» F,m 2 : в€‚F в€‚m 2 в€‚F в€‚m 2 = 1 в€’ x. = 1, m1 О» F,m 2 = x 1 , using x F = F в€’ m 2 , F = m 2 /(1 в€’ x), 1в€’x (8) We may just as well use Eq. (10.32) in its time-domain form: K 11 x = K 21 F K 12 K 22 m1 , m2 where now x, F, m 1 , and m 2 are deviation variables. From the п¬Ѓrst row, it is immediately obvious that в€‚x в€‚m 1 = K 11 . m2 We next substitute for m 2 by using the second row to get x = K 11 m 1 + K 12 224 (F в€’ K 21 m 1 ) . K 22 10.7. Decoupling of Interacting Systems вЂ“ + m1 + m1 + m 2 FC m1 CC x F m2 FT CT Figure R10.10. Now we can п¬Ѓnd K 12 K 21 в€‚x = K 11 в€’ . в€‚m 1 F K 22 From here on, getting Eq. (10.37) is a simple substitution step. To derive Eq. (E10.6) by using K . This is just a matter of substituting the values of the K i j from Eqs. (10.30) and (10.31) into Eq. (10.37). We should п¬Ѓnd once again that О»x,m 1 = 1 в€’ x, as in Eq. (E10.5), and Eq. (E10.6) follows. (9) We need to п¬Ѓnd how Вµ1 and Вµ2 affect F and x. With Вµ1 = F and Вµ2 = m 1 , we can rewrite the deп¬Ѓnition of x = m 1 /F as x = Вµ1 /Вµ2 . This is the form that we use to take a п¬Ѓrst-order Taylor expansion as we have done with the step after Eq. (10.28). The result in matrix form of the Laplace transform of the deviation variables is пЈ® пЈ№ 1 0 F(s) Вµ (s) 1 пЈ» 1 = пЈ° Вµ2 . x(s) Вµ2 (s) в€’ 2 Вµ1 ss Вµ1 ss By putting F in the п¬Ѓrst row, it is clear that we have a one-way interaction system. By Eq. (10.37), О» = 1. With F = m 1 + m 2 and m 1 as the output of the controllers, we can implement this scheme as in Fig. R10.10. (10) We will п¬Ѓnd d11 and d21 as an illustration. The п¬Ѓrst column on the RHS of Eq. (10.43) is rewritten as two equations: G 11 d11 + G 12 d21 = H1 , G 21 d11 + G 22 d21 = 0. Solving them simultaneously will lead to d11 and d21 in Eqs. (10.44) and (10.45). When d11 = 1 is chosen, Eqs. (10.44) can be rewritten as Eqs. (10.46). 225 MATLAB Tutorial Sessions ATLAB is a formidable mathematics analysis package. An introduction to the most basic commands that are needed for our use is provided. No attempts are made to be comprehensive. We do not want a big, intimidating manual. The beauty of MATLAB is that we need to know only a tiny bit to get going and be productive. Once we get started, we can pick up new skills quickly with MATLABвЂ™s excellent on-line help features. We can learn only with hands-on work; these notes are written as a вЂњwalk-throughвЂќ tutorial вЂ“ you are expected to enter the commands into MATLAB as you read along. M Session 1. Important Basic Functions For each session, the most important functions are put in a table at the beginning of the section for easy reference or review. The п¬Ѓrst one is on the basic commands and plotting. Try the commands as you read. You do not have to enter any text after the вЂњ%вЂќ sign. Any text that follows вЂњ%вЂќ is considered a comment and can be ignored. To save some paper, the results generated by MATLAB are omitted. If you need to see that for help, they are provided on the Web Support. There is also where any new MATLAB changes and upgrades are posted. Features in our tutorial sessions are based on MATLAB Version 6.1 and Control System Toolbox 5.1. Important basic functions General functions: cd demo (intro) dir (what) help, helpwin load lookfor print quit save who, whos 226 Change subdirectory Launch the demo (introduction) List of п¬Ѓles in current directory (or only M-п¬Ѓles) Help! Help window Load workspace Keyword search Print graph; can use pull-down menu Quit! Save workspace List of variables in workspace Session 1. Important Basic Functions Calculation functions: conv size, length Convolution function to multiply polynomials Size of an array, length of a vector Plotting functions: axis grid hold legend plot text (gtext) title xlabel, ylabel Override axis default of plot Add grid to plot Hold a п¬Ѓgure to add more plots (curves) Add legend to plot Make plots Add text (graphical control) to plot Add title to plot Add axis labels to plot M1.1. Some Basic MATLAB Commands The following features are covered in this session: r r r r using help creating vectors, matrices, and polynomials simple matrix operations multiplying two polynomials with conv() To begin, we can explore MATLAB by using its demonstrations. If you are new to MATLAB, it is highly recommended that you take a look at the introduction: intro demo % launch the introduction % launch the demo program It is important to know that the MATLAB on-line help is excellent, and there are different ways to get that: help helpbrowser % old-fashioned help inside the Command Window % launch the help browser window; also available % from the Help pull-down menu and toolbar We should make a habit of using the on-line help. The user interface of the help browser, which also works as a Web browser, is extremely intuitive, and it is highly recommended. When the help command is mentioned, that is just a general comment; it does not mean that you have to use the old-style help. To use help in the Command Window, turn the page display mode on п¬Ѓrst. HereвЂ™s an example of seeking help on the print command with the old-style help: more on help print lookfor print which print % turn the page mode on % general key-word search % list the path name of print.m The help features and the Command Window interface tend to evolve quickly. For that reason, the Web Support is used to provide additional hints and tidbits so that we can quickly 227 MATLAB Tutorial Sessions post the latest MATLAB changes. For now, a few more basic commands are introduced: who whos dir what cd pwd % % % % % % list the variables that are currently defined whos is a more detailed version of who list the files in the current subdirectory list only the M-files change the subdirectory list the present working directory For fun, we can try why fix(clock) MATLAB is most at home in dealing with arrays, which we refer to as matrices and vectors. They are all created by enclosing a set of numbers in brackets, [ ]. First, we deп¬Ѓne a row vector by entering, in the MATLAB Command Window, x = [1 2 3 4 5 6 7 8 9 10] If we add a semicolon at the end of a command, as in x = [1 2 3 4 5 6 7 8 9 10]; we can suppress the display of the result. We can check what we have later by entering the name of the variable. To generate a column vector, we insert semicolons between numbers (a more speciп¬Ѓc example is given with a matrix below). The easier route is to take the transpose of x: x = x' Keep in mind that in MATLAB variables are case sensitive. Small letter x and capital X are two different variables. We can also generate the row vector with the colon operator: x = 1:10 y = 0:0.1:2 % same as 1:1:10 % just another example The colon operator is very useful when we make longer vectors for plotting or calculations. With this syntax, the increment is squeezed between the beginning and the ending values of the vector and they are separated by colons. If the increment value is missing, the default increment is 1. Remember to add a semicolon at the end of each statement to suppress the display of a long string of numbers. This is skipped in the illustration just so you may see what is generated. When we do calculations based on vectors, MATLAB will vectorize the computation, which is much faster than if we write a loop construct as in the for loop in c or the do loop in fortran. To create a matrix, we use a semicolon to separate the rows: a = [1 2 3 ; 4 5 6 ; 7 8 9] In place of the semicolons, we can also simply hit the return key as we generate the matrix. 228 Session 1. Important Basic Functions There are circumstances in which we need the size of an array or the length of a vector. They can be found easily: size(y) length(y) % find the size of an array % find the length of a vector In MATLAB, polynomials are stored exactly the same as vectors. Functions in MATLAB will interpret them properly if we follow the convention that a vector stores the coefп¬Ѓcients of a polynomial in descending order вЂ“ it begins with the highest-order term and always ends with a constant, even if it is zero. Some examples: p1 =[1 -5 4] p2 =[1 0 4] p3 =[1 -5 0] % defines p1(s) = s^2 - 5*s + 4 % defines p2(s) = s^2 + 4 % defines p3(s) = s^2 - 5*s We can multiply two polynomials together easily with the convolution function conv(). For example, to expand (s 2 в€’ 5s + 4)(s 2 + 4), we can use conv(p1,p2) % this multiplies p1 by p2 or conv([1 -5 4], [1 0 4]) MATLAB supports every imaginable way that we can manipulate vectors and matrices. We need to know only a few of them, and we will pick up these necessary ones along the way. For now, weвЂ™ll do a couple of simple operations. With the vector x and matrix a that weвЂ™ve deп¬Ѓned above, we can perform simple operations such as y1 y2 b y3 c = = = = = 2*x sqrt(x) sqrt(a) y1 + y2 a*b % % % % % multiplies x by a constant takes the square root of each element in x takes the square root of each element in a adds the two vectors multiplies the two matrices Note that all functions in MATLAB, such as sqrt(), are smart enough that they accept scalars, vectors, and, where appropriate, matrices.1 When we operate on an element-by-element basis, we need to add a period before the operator. Examples based on the two square matrices a and b: d a3 e f = = = = a.^3 a^3 a.*b a*b % % % % takes the cube of each element versus the cube of the matrix multiplies each element a(i,j)*b(i,j) versus matrix multiplication a*b Of course, we can solve the matrix equation Ax = b easily. For example, we can try A = [ 4 -2 -10; 2 10 -12; -4 -6 16]; b = [-10; 32; -16]; x = A\b % Bingo! 1 In computer science, this is referred to as polymorphism. The fact that mathematical operators can work on different data types is called overloading. 229 MATLAB Tutorial Sessions LetвЂ™s check the solution by inverting the matrix2 A: C = inv(A); x = C*b We can п¬Ѓnd the eigenvalues and eigenvectors of A easily: [X,D] = eig(A) Finally, we do a simple polynomial п¬Ѓt illustration. LetвЂ™s say we have a set of (x, y) data: x = [ 0 1 2 4 6 10]; y = [ 1 7 23 109 307 1231]; To make a third-order polynomial п¬Ѓt of y = y(x), all we need to enter is c = polyfit(x,y,3) % should obtain c = [1 2 3 1] The returned vector c contains the coefп¬Ѓcients of the polynomial. In this example, the result should be y = x 3 + 2x 2 + 3x + 1. We can check and see how good the п¬Ѓt is. In the following statements, we generate a vector xfit so that we can draw a curve. Then we calculate the corresponding yfit values and plot the data with a symbol and the п¬Ѓt as a line: xfit=1:0.5:10; yfit=xfit. Л† 3 + 2*xfit. Л† 2 + 3*xfit +1; plot(x,y,'o', xfit,yfit) % explanation on plotting title('3rd order polynomial fit') % is in the next subsection legend('data','3rd order fit') Speaking of plotting, this is what we get into next. M1.2. Some Simple Plotting The following features are covered in this session: r Using the plot() function r Adding titles, labels, and legends LetвЂ™s create a few vectors п¬Ѓrst: x = 0:0.5:10; y1= 2*x; y2= sqrt(x); Now we plot y1 versus x and y2 versus x together: plot(x,y1, x,y2) 2 If you have taken a course on numerical methods, you would be pleased to know that MATLAB can do LU decomposition: [L,U] = lu(A); 230 Session 1. Important Basic Functions We have a limited selection of line patterns or symbols. For example, we can try3 plot(x,y1,'-.') hold plot(x,y2,'--') hold % or use "hold on" % or use "hold off" We can п¬Ѓnd the list of pattern selections with on-line help. The command hold allows us to add more plots to the same п¬Ѓgure, and hold works as a toggle. That is why we do not have to state вЂњonвЂќ and вЂњoffвЂќ explicitly. We can add a title and axis labels too: title('A boring plot') xlabel('The x-axis label'), ylabel('The y-axis label') We can issue multiple commands on the same line separated by commas. What makes MATLAB easy to learn is that we can add goodies one after another. We do not have to worry about complex command syntax. We can also do logarithmic plots. Try entering help semilogx, semilogy, or loglog. WeвЂ™ll skip them because they are not crucial for our immediate needs. We can add a grid and a legend with grid legend('y1','y2') A box with the п¬Ѓgure legend will appear in the Graph Window. Use the mouse to drag the box to where you want it to be. We can also add text to annotate the plot with text(1,9,'My two curves') % starting at the point (1,9) The text entry can be interactive with the use of gtext('My two curves') Now click on the Graph Window, and a crosshair will appear. Move it to where you want the legend to begin and click. Presto! Repeat for additional annotations. In rare cases, we may not like the default axis scaling. To override what MATLAB does, we can deп¬Ѓne our own minimum and maximum of each axis with axis([0 15 0 30]) % the syntax is [xmin xmax ymin ymax] We need the brackets inside because the argument to the axis function is an array. Plotting for Fun We do not need to do three-dimensional (3-D) plots, but then itвЂ™s too much fun not to do at least a couple of examples. However, we will need to use a few functions that we do not need otherwise, so do not worry about the details of these functions that we will not use 3 To do multiple plots, we can also use plot(x,y1,'-.', x,y2,'--') 231 MATLAB Tutorial Sessions again. We will get a pretty 3-D picture: [x,y]=meshgrid(-10:0.5:10, -10:0.5:10); % meshgrid transforms the specified domain % where -10 < x < 10, and -10 < y < 10 % into a grid of (x,y) values for evaluating z r=sqrt(x.^2 + y.^2) + eps; z=sin(r)./r; mesh(z) title('The Sinc Sombrero') % We add the machine epsilon eps % so 1/r won't blow up So you say wow! But MATLAB can do much more and fancier than that. We try one more example with Bessel functions, which you can come across in heat and mass transfer problems with cylindrical geometry: % Here we do a 3-D mesh plot of Jo(sqrt(x^2+y^2)) % The x and y grids remain the same as in the previous plot r=sqrt(x.^2+y.^2); z=bessel(0,r); mesh(z) M1.3. Making M-п¬Ѓles and Saving the Workspace The following features are covered in this session: r Executing repeated commands in a script, the so-called M-п¬Ѓle r Saving a session For tasks that we have to repeat again and again, it makes sense to save them in some kind of a script and execute them. In MATLAB, these scripts are called M-п¬Ѓles. The name came from the use of macros in the old days of computing. We can use M-п¬Ѓles to write unstructured scripts or user-deп¬Ѓned functions. MATLAB now refers to both as programs. You may want to keep in mind that a scripting interpretive language is not the same as a compiled language like C. For our needs, a simple script sufп¬Ѓces in most circumstances. To use an M-п¬Ѓle4 (1) save all the repetitious MATLAB statements in a text п¬Ѓle with the ".m" extension, (2) execute the statements in that п¬Ѓle by entering the п¬Ѓle name without the ".m" extension. 4 232 There is another easy way to вЂњcheat.вЂќ On unix/Linux workstations, open up a new text editor and enter your frequently used statements there. In Windows, you can use the really nice MATLAB Editor. You can copy and paste multiple commands back and forth between the text editor window and the MATLAB window easily. If you want to save the commands, you certainly can add comments and annotations. You can consider this text п¬Ѓle as a вЂњfree-format notebookвЂќ without having to launch the Microsoft Word Notebook for MATLAB. Session 1. Important Basic Functions Here is one simple example. We need to plot x versus y repeatedly and want to automate the task of generating the plots. The necessary statements are % ---------------- M-file script: plotxy.m ---------------% A very simple script to plot x vs y and add the labels % . . . the kind of things we don't want to repeat typing % again and again. . . plot(x,y) grid xlabel('Time [min]') ylabel('Step Response') title('PID Controller Simulation') % End of plotxy.m. An "end" statement is not needed. Save these statements in a п¬Ѓle named, say, plotxy.m. Anything after the вЂњ%вЂќ sign is regarded as a comment, which you do not have to enter if you just want to repeat this exercise. After we have deп¬Ѓned or updated the values of x and y in the Command Window, all we need is to enter "plotxy" at the prompt and MATLAB will do the rest. The key is to note that the M-п¬Ѓle has no вЂњreadвЂќ or вЂњinputвЂќ for x and y. All statements in an M-п¬Ѓle are simply executed in the Command Window. If you have an M-п¬Ѓle, MATLAB may not п¬Ѓnd it unless it is located within its search path. LetвЂ™s see where MATLAB looks п¬Ѓrst. On unix/Linux machines, MATLAB by default looks in the subdirectory from which it is launched. A good habit is to keep all your work in one subdirectory and change to that speciп¬Ѓc subdirectory before you launch MATLAB. On Windows machines, MATLAB looks for your п¬Ѓles in the Work folder buried deep inside the Program Files folder. A good chance is that you want to put your п¬Ѓles in more convenient locations. To coax MATLAB to п¬Ѓnd them, you need to change the directory or the search path. So the next question is how to do that, and the answer applies to both unix/Linux and Windows machines. The formal way is to learn to use the "cd" and "path" commands. The easy way is to use point-and-click features that can be found under pull-down menus, on toolbars, or in subwindows. Because these graphical interface features tend to change with each MATLAB upgrade, please refer to the Web Support, from which you can п¬Ѓnd updates of new changes and additional help. If we want to take a coffee break and save all the current variables that we are working with, enter save before we quit MATLAB. When we launch MATLAB again, we type load and everything will be restored. Do not save the workspace if you are going to be away any longer because the old workspace is not very useful if you have all these variables п¬‚oating around and you forget what they mean. 233 MATLAB Tutorial Sessions As a п¬Ѓnal comment, we can use load and save to import and export arrays of data. Because we do not really need this feature in what we do here, this explanation is deferred to the Web Support. Session 2. Partial-Fraction and Transfer Functions This tutorial is to complement our development in Chap. 2. You may want to go over the tutorial quickly before you read the text and come back later a second time for the details. Partial-fraction and transfer functions poly residue roots tf2zp zp2tf Construct a polynomial from its roots Partial-fraction expansion Find the roots to a polynomial Transfer function to zero-pole form conversion Zero-pole form to transfer function conversion Object-oriented functions: tf get pole zpk Create a transfer function object List the object properties Find the poles of a transfer function Create a transfer function in pole-zero-gain form M2.1. Partial Fractions The following features are covered in this session: r Finding the roots of a polynomial with roots() r Generating a polynomial from its roots with poly() r Doing partial fractions with residue() Of secondary importance: r Transfer function to zero-pole form, tf2zp() r Zero-pole form to transfer function, zp2tf() LetвЂ™s п¬Ѓrst deп¬Ѓne a polynomial: p = [1 5 4] % makes p(s) = s Л† 2 + 5*s + 4 We can п¬Ѓnd the roots of p(s) = 0 with the function roots(): poles = roots(p) MATLAB should return в€’4 and в€’1. That means the polynomial can be factored as p(s) = (s + 4)(s + 1).5 5 234 MATLAB has the function fzero() to п¬Ѓnd a root of a given function. Session 2. Partial-Fraction and Transfer Functions We can go backwards. Given the roots (or pole positions), we can get the polynomial with p2 = poly(poles) MATLAB returns the results in a column vector. Most functions in MATLAB take either row or column vectors, and we usually do not have to worry about transposing them. We can do partial fractions with the residue() function. Say we have a transfer function G(s) = 1 q(s) = 2 , p(s) s + 5s + 4 where q(s) = 1 and p(s) remains [1 5 4], as previously deп¬Ѓned. We can enter q = 1; residue(q,p) MATLAB returns the numbers в€’0.3333 and 0.3333. That is because the function can be factored as s2 в€’1/3 1/3 1 = + . + 5s + 4 s+4 s+1 How can we be sure that it is the в€’0.3333 coefп¬Ѓcient that goes with the root at в€’4? We can use the syntax [a,b,k]=residue(q,p) MATLAB will return the coefп¬Ѓcients in a, the corresponding poles in b, and whatever is left over in k, which should be nothing in this case. Note that [] denotes an empty matrix or vector. LetвЂ™s try another transfer function with poles at 0, в€’1, в€’2, and в€’3: G(s) = 1 . s(s + 1)(s + 2)(s + 3) To п¬Ѓnd the partial fractions, this is what we can do6 : poles=[0 -1 -2 -3]; p=poly(poles); q=1; [a,b,k]=residue(q,p) One more example. Find the partial fractions of the nasty-looking function G(s) = 6 s 2 + 4s + 3 . s 4 в€’ 7s 3 + 11s 2 + 7s в€’ 12 If we need to write complex-conjugate roots, make sure there are no spaces withinв€љ a complex number. For example, enter [-3+4*j -3-4*j]. Either i or j can be used to denote в€’1. 235 MATLAB Tutorial Sessions q=[1 4 3]; zeros=roots (q) % should return -3, -1 p=[1 -7 11 7 -12]; poles=roots (p) % should return 4, 3, 1, -1 [a,b,k]=residue (q,p) See that MATLAB returns the expansion: 2.33 3 0.67 s 2 + 4s + 3 = в€’ + . s 4 в€’ 7s 3 + 11s 2 + 7s в€’ 12 sв€’4 sв€’3 sв€’1 Note that the coefп¬Ѓcient associated with the pole at в€’1 is zero. That is because it is canceled by the zero at в€’1. In other words, the (s + 1) terms cancel out. It is nice to know that the program can do this all by itself. We do not need to know the roots to use residue(), but it is a good habit to get a better idea of what we are working with. A transfer function can be written in terms of its poles and zeros. For example, в€љ в€љ 6s 2 в€’ 12 6(s в€’ 2)(s + 2) F(s) = 3 = . (s + s 2 в€’ 4s в€’ 4) (s + 1)(s + 2)(s в€’ 2) The RHS is called the pole-zero form (or zero-pole form). MATLAB provides two functions, tf2zp() and zp2tf(), to do the conversion. For instance, q=[6 0 -12]; p=[1 1 -4 -4]; [zeros, poles, k]=tf2zp(q,p) Of course, we can go backward with [q,p]=zp2tf(zeros,poles,k) Note: The factor k is 6 here, and in the MATLAB manual it is referred to as the вЂњgain.вЂќ This factor is really the ratio of the leading coefп¬Ѓcients of the two polynomials q(s) and p(s). Make sure you understand that the k here is not the steady-state gain, which is the ratio of the last constant coefп¬Ѓcients. (In this example, the steady-state gain is в€’12/ в€’ 4 = 3.) MATLAB actually has a function called dcgain to do this. One more simple example: zero= -2; % generate a transfer function poles=[-4 -3 -1]; % with given poles and zeros k=1; [q,p]=zp2tf(zero,poles,k) Double check that we can recover the poles and zeros with [zero,poles,k]=tf2zp(q,p) We can also check with roots(q) roots(p) Try zp2tf or tf2zp on your carвЂ™s license plate! 236 Session 2. Partial-Fraction and Transfer Functions M2.2. Object-Oriented Transfer Functions The following features are covered in this session: r Deп¬Ѓning a transfer function object with tf() or zpk() r Determining the poles with pole() r Using overloaded operators MATLAB is object oriented. Linear-time-invariant (LTI) models are handled as objects. Functions use these objects as arguments. In classical control, LTI objects include transfer functions in polynomial form or in pole-zero form. The LTI-oriented syntax allows us to better organize our problem solving; we no longer have to work with individual polynomials that we can identify only as numerators and denominators. We will use this syntax extensively starting in Session 3. Here, we see how the objectoriented syntax can make the functions tf2zp() and zp2tf() redundant and obsolete. To deп¬Ѓne a transfer function object, we use tf(), which takes the numerator and denominator polynomials as arguments. For example, we deп¬Ѓne G(s) = [s/(s 2 в€’ 5s + 4)] with G1 = tf([1 0], [1 -5 4]) We deп¬Ѓne G(s) = [(6s 2 в€’ 12)/(s 3 + s 2 в€’ 4s в€’ 4)] with G2 = tf([6 0 -12], [1 1 -4 -4]) We can also use the zero-pole-gain function zpk() which takes as arguments the zeros, poles, and gain factor of a transfer function. Recall the comments after zp2tf(). This gain factor is not the steady-state (or dc) gain. For example, we deп¬Ѓne G(s) = {4/[s(s + 1)(s + 2)(s + 3)]} with G3 = zpk([],[0 -1 -2 -3], 4) % the [] means there is no zero The tf() and zpk() functions also serve to perform model conversion from one form to another. We can п¬Ѓnd the polynomial form of G3 with tf(G3) and the pole-zero form of G2 with zpk(G2) The function pole() п¬Ѓnds the poles of a transfer function. For example, try pole(G1) pole(G2) You can check that the results are identical to the use of roots() on the denominator of a transfer function. 237 MATLAB Tutorial Sessions We may not need to use them, but it is good to know that there are functions that help us extract the polynomials or poles and zeros back from an object. For example, [q,p]=tfdata(G1,'v') [z,p,k]=zpkdata(G3,'v') % option 'v' for row vectors The addition and multiplication operators are overloaded, and we can use them to manipulate or synthesize transfer functions. This capability will come in handy when we analyze control systems. For now, letвЂ™s consider one simple example. Say we are given G1 = 1 , s+1 G2 = 2 . s+2 We can п¬Ѓnd G 1 + G 2 and G 1 G 2 easily with G1=tf(1,[1 1]); G2=tf(2,[1 2]); G1+G2 % or we can use zpk(G1+G2) G1*G2 % or we can use zpk(G1*G2) This example is simple enough to see that the answers returned by MATLAB are correct. With object-oriented programming, an object can hold many properties. We п¬Ѓnd the associated properties with get(G1) Among the MATLAB result entries, we may п¬Ѓnd the properties InputName, OutputName, and Notes. We can set them with 7 G1.InputName = 'Flow Rate'; G1.OutputName = 'Level'; G1.Notes = 'My first MATLAB function'; You will see the difference if you enter, from now on, G1 get (G1) MATLAB can use symbolic algebra to do the Laplace transform. Because this skill is not crucial to solving control problems, we skip it here. You can п¬Ѓnd a brief tutorial on the Web Support, and you are encouraged to work through it if you want to know what symbolic algebra means. Session 3. Time-Response Simulation This tutorial is to complement our development in Chap. 3. You may want to go over the tutorial quickly before you read the text and come back later a second time for the details. 7 238 We are using the typical structure syntax, but MATLAB also supports the set() function to perform the same task. Session 3. Time-Response Simulation Time-response simulation functions damp impulse lsim step pade Find damping factor and natural frequency Impulse response Response to arbitrary inputs Unit-step response Time-delay PadВґe approximation ltiview Launch the graphics viewer for LTI objects M3.1. Step- and Impulse-Response Simulations The following features are covered in this session: r Using step() and impulse() r Time response to any given input, lsim() r Dead-time approximation, pade() Instead of spacing out in the Laplace domain, we can (as we are taught) guess how the process behaves from the pole positions of the transfer function. However, wouldnвЂ™t it be nice if we could actually trace the time proп¬Ѓle without having to do the reverse Laplace transform ourselves? Especially the response with respect to step and impulse inputs? Plots of time-domain dynamic calculations are extremely instructive and a useful learning tool.8 The task of time-domain calculation is easy with MATLAB. LetвЂ™s say we have Y (s) 1 = 2 , X (s) s + 0.4s + 1 and we want to plot y(t) for a given input x(t). We can easily do q=1; p=[1 0.4 1]; G=tf(q,p) % poles at -0.2 + - 0.98j step(G) % plots y(t) for unit step input, X(s)=1/s impulse(G) % plots y(t) for impulse input, X(s)=1 What a piece of cake! Not only does MATLAB perform the calculation, but it automatically makes the plot with a properly chosen time axis. Nice! 9 As a habit, п¬Ѓnd out more about a function with help as in help step 8 9 % better yet, use helpwin or helpbrowser If you are interested, see the Web Support for using the Runge--Kutta integration of differential equations. How could we guess what the time axis should be? It is not that difп¬Ѓcult if we understand how to identify the dominant pole, the signiп¬Ѓcance behind doing partial fractions, and that the time to reach 99% of the п¬Ѓnal time response is approximately п¬Ѓve time constants. 239 MATLAB Tutorial Sessions The functions also handle multiple transfer functions. LetвЂ™s make a second transfer function in pole-zero form: H (s) = 2 ; (s + 2)(s 2 + 2s + 2) H=zpk([], [-2 -1+j -1-j], 2) We can compare the unit-step responses of the two transfer functions with step(G,H) We can, of course, choose our own axis, or rather, time vector. Putting both the unit-step and impulse-response plots together may also help us understand their differences: t=0:0.5:40; % don't forget the semicolon! ys=step(G,t); yi=impulse(G,t); plot(t,ys,t,yi) Note: In the text, the importance of relating pole positions of a transfer function to the actual time-domain response was emphasized. We should get into the habit of п¬Ѓnding what the poles are. The time-response plots are teaching tools that reafп¬Ѓrm our conп¬Ѓdence in doing analysis in the Laplace domain. Therefore we should п¬Ѓnd the roots of the denominator. We can also use the damp() function to п¬Ѓnd the damping ratio and the natural frequency. pole(G) % same result with roots(p) damp(G) % same result with damp(p) One more example. Consider the transfer function Y (s) 2s + 1 = G(s) = . X (s) (4s + 1)(s + 1) We want to plot y(t) if we have a sinusoidal input x(t) = sin(t). Here we need the function lsim(), a general simulation function that takes any given input vector: q=[2 1]; p=conv([4 1],[1 1]); G=tf(q,p) t=0:0.5:30; u=sin(t); y=lsim(G,u,t); plot(t,y,t,u,'-.'), grid % a zero at -1/2 % poles at -1/4 and -1 % (can use zpk instead) % response to a sine function input Keep this exercise in mind. This result is very useful in understanding what is called frequency response in Chap. 8. We can repeat the simulation with higher frequencies. We can also add what we are familiar with: hold ys=step(G,t); yi=impulse(G,t); 240 Session 3. Time-Response Simulation plot(t,ys,t,yi) hold off For fun, try one more calculation with the addition of random noise: u=sin(t)+rand(size(t)); y=lsim(G,u,t); plot(t,y,'r',t,u,'b'), grid % Color lines red and blue For useful applications, lsim() is what we need to simulate a response to, say, a rectangular pulse. This is one simple example that uses the same transfer function and time vector that we have just deп¬Ѓned: t=0:0.5:30; u=zeros(size(t)); u(3:7)=1; y=lsim(G,u,t); yi=impulse(G,t); % % % % t = [0 .5 1 1.5 2 2.5 3 ... ] make a vector with zeros make a rectangular pulse from t=1 to t=3 % compare the result with impulse % response plot(t,u, t,y, t,yi,'-.'); Now we switch gears and look into the dead-time transfer function approximation. To do a PadВґe approximation, we can use the MATLAB function 10 [q,p]=pade(Td,n) where Td is the dead time, n is the order of the approximation, and the results are returned in q(s)/p(s). For example, with Td = 0.2 and n = 1, entering [q,p]=pade(0.2,1) % first-order approximation will return q = -1 s + 10 p = 1 s + 10 We expected q(s) = в€’0.1s + 1 and p(s) = 0.1s + 1. Obviously MATLAB normalizes the polynomials with the leading coefп¬Ѓcients. On second thought, the PadВґe approximation is so simple that there is no reason why we cannot do it ourselves as in a textbook. For the п¬Ѓrst-order approximation, we have Td=0.2; q = [-Td/2 1]; p = [ Td/2 1]; 10 When we use pade() without the left-hand argument [q,p], the function automatically plots the step and the phase responses and compares them with the exact responses of the time delay. A PadВґe approximation has unit gain at all frequencies. These points will not make sense until we get to frequency-response analysis in Chap. 8. For now, keep the [q,p] on the LHS of the command. 241 MATLAB Tutorial Sessions We can write our own simple-minded M-п¬Ѓle to do the approximation. You may now try [q,p]=pade(0.2,2) % second-order approximation and compare the results of this second-order approximation with the textbook formula. M3.2. LTI Viewer The following feature is covered in this session: r Graphics viewer for LTI objects, ltiview11 We can use the LTI Viewer to do all the plots, not only step and impulse responses, but also more general time-response and frequency-response plots in later chapters. If we know how to execute individual plot statements, it is arguable whether we really need the LTI Viewer. Nonetheless, that would be a personal choice. Here the basic idea and some simple instructions are provided. To launch the LTI Viewer, enter in the MATLAB Command Window ltiview A blank LTI window will pop up. The п¬Ѓrst task would be to poke into features supported under the File and Tools pull-down menus and see what we can achieve by point and click. There is also a Help pull-down menu, which activates the Help Window. The LTI Viewer runs in its own workspace, which is separate from the MATLAB workspace. The Viewer also works with only LTI objects generated by functions such as tf() and zpk(), and after Chap. 4, state-space objects, ss(). So letвЂ™s generate a couple of objects in the MATLAB Command Window п¬Ѓrst: G=tf(1,[1 0.4 1]) H=zpk([], [-2 -1+j -1-j], 2) Now, go to the LTI Viewer window and select Import under the File pull-down menu. A dialog box will pop out to help import the transfer function objects. By default, a unit-step response will be generated. Click on the axis with the right mouse button to retrieve a pop-up menu that will provide options for other plot types, for toggling the object to be plotted, and for other features. With a step-response plot, the Characteristics feature of the pop-up menu can identify the peak time, rise time, and settling time of an underdamped response. The LTI Viewer was designed to do comparative plots, either comparing different transfer functions or comparing the time-domain and (later in Chap. 8) frequency-response properties of a transfer function. Therefore a more likely (and quicker) case is to enter, for example, ltiview('step',G,H) The transfer functions G and H will be imported automatically when the LTI Viewer is launched, and the unit-step response plots of the two functions will be generated. 11 242 The description is based on Version 5.1 of the MATLAB control toolbox. If changes are introduced in newer versions, they will be presented on the Web Support. Session 4. State-Space Functions Another useful case is, for example, ltiview({'step';'bode'},G) In this case, the LTI Viewer will display both the unit-step response plot and the Bode plot for the transfer function G. We will learn about Bode plots in Chap. 8, so donвЂ™t panic yet. Just keep this possibility in mind until we get there. Session 4. State-Space Functions This tutorial is to complement our development in Chap. 4. You may want to go over the tutorial quickly before you read the text and come back later a second time for the details. State-space functions canon eig ss2ss ss2tf tf2ss printsys Canonical state-space realization Eigenvalues and eigenvectors Transformation of state-space systems Conversion from state-space to transfer function Conversion from transfer function to state-space Slightly prettier looking display of model equations ltiview ss Launch the graphics viewer for LTI objects Create state-space object M4.1. Conversion between Transfer Function and State-Space The following features are covered in this session: r Using ss2tf() and tf2ss() r Generating object-oriented models with ss() We need to revisit Example 4.1 with a numerical calculation. LetвЂ™s use the values О¶ = 0.5 and П‰n = 1.5 Hz to establish the transfer function and п¬Ѓnd the poles: z=0.5; wn=1.5; q=wn*wn; p=[1 2*z*wn wn*wn] roots(p) % % % % Should find q=2.25 p=[1 1.5 2.25] -0.75 + - 1.3j From the results in Example 4.1, we expect to п¬Ѓnd A= 0 в€’2.25 1 0 , B= , C = [1 в€’1.5 2.25 0], D = 0. Now letвЂ™s try our hands with MATLAB by using its transfer function to state-space conversion function: [a,b,c,d]=tf2ss(q,p) 243 MATLAB Tutorial Sessions MATLAB returns with a= в€’1.5 1 в€’2.25 , 0 b= 1 , c = [0 0 2.25], d = 0, which are not the same as those in Example 4.1. You wonder whatвЂ™s going on? Before you kick the computer, a closer look should reveal that MATLAB probably uses a slightly different convention. Indeed, MATLAB п¬Ѓrst вЂњsplitвЂќ the transfer function into product form: 1 1 Y X2 Y 2.25. П‰2 = 2 = 2 = U U X2 (s + 1.5s + 2.25) s + 2О¶ П‰n s + П‰n2 n From X 2 /U = 1/(s 2 + 2О¶ П‰n s + П‰n2 ) and with the state variables deп¬Ѓned as x1 = dx2 , dt x2 = x2 (i.e., same), we should obtain the matrices a and b that MATLAB returns. From Y/ X 2 = П‰n2 , it should be immediately obvious how MATLAB obtains the array c. In addition, we should be aware that the indexing of state variables in MATLAB is in reverse order of textbook examples. Despite these differences, the inherent properties of the model remain identical. The most important of all is to check the eigenvalues: eig(a) % should be identical to the poles A conversion from state-space back to a transfer function should recover the transfer function: [q2,p2]=ss2tf(a,b,c,d,1) % same as q/p as defined earlier The last argument in ss2tf() denotes the ith input, which must be 1 for our SISO model. To make sure we cover all bases, we can set up our own state-space model as in Example 4.1, a=[0 1; -2.25 -1.5]; b=[0; 2.25]; c=[1 0]; d=0; and check the results with eig(a) [qs,ps]=ss2tf(a,b,c,d,1) % still the same! The important message is that there is no unique state-space representation, but all model matrices should have the same eigenvalues. In addition, the number of state variables is the same as the order of the process or system. The fact that the algorithm used by MATLAB does not return a normalized output matrix C can create problems when we do feedback calculations in Chap. 9. The easy solution is to rescale the model equations. The output equation can be written as y = [О± 0]x = [1 0]ВЇx, where xВЇ = О±x. Substitution for x by xВЇ in dx/dt = Ax + Bu will lead to dВЇx ВЇ = AВЇx + О±Bu = AВЇx + Bu, dt 244 Session 4. State-Space Functions ВЇ = О±B. In other words, we just need to change C to the normalized vector and where B multiply B by the scaling factor. We can see that this is correct from the numerical results of Example 4.1. (Again, keep in mind that the MATLAB indexing is in reverse order of textbook examples.) We will use this idea in Chap. 9. We now repeat the same exercise to show how we can create object-oriented state-space LTI models. In later chapters, all control toolbox functions take these objects as arguments. We п¬Ѓrst repeat the statements above to regenerate the state matrices a, b, c, and d. Then we use ss() to generate the equivalent LTI object. q=2.25; p=[1 1.5 2.25]; [a,b,c,d]=tf2ss(q,p); sysв€’ obj=ss(a,b,c,d) We should see that the LTI object is identical to the state-space model. We can retrieve and operate on individual properties of an object. For example, to п¬Ѓnd the eigenvalues of the matrix a inside sysв€’ obj, we use eig(sysв€’ obj.a) % find eigenvalue of state matrix a We can obtain the transfer function, as analogous to using ss2tf(), with tf(sysв€’ obj) Now you may wonder if we can generate the state-space model directly from a transfer function. The answer is, of course, yes. We can use sys2=ss(tf(q,p)) eig(sys2.a) % should be identical to the poles MATLAB will return with matrices that look different from those given previously: a= в€’1.5 в€’1.125 1 , b= , c = [ 0 1.125], d = 0. 2 0 0 With what we know now, we bet ss() uses a different scaling in its algorithm. This time, MATLAB factors the transfer function into this product form: 2 Y X2 Y = 2 = 1.125. U U X2 (s + 1.5s + 2.25) From X 2 /U = 2/(s 2 + 1.5s + 2.25) and with the state variables deп¬Ѓned as x1 = dx2 1 dx2 i.e., = 2x1 , 2 dt dt x2 = x2 , we should obtain the new state matrices. Again, the key result is that the state matrix a has the same eigenvalue. This exercise underscores one more time that there is no unique way to deп¬Ѓne state variables. Because our objective here is to understand the association between transfer function and state-space models, the introduction continues with the ss2tf() and the tf2ss() functions. 245 MATLAB Tutorial Sessions Two minor tidbits before we move on. First, the printsys() function displays the model matrices or polynomials in a slightly more readable format. Sample usage: printsys(a,b,c,d) printsys(q,p,'s') Second, with a second-order transfer function, we can generate the textbook state-space matrices, given a natural frequency wn and damping ratio z: [a,b,c,d]=ord2(wn,z) % good for only q=1 If we examine the values of b and c, the result is restricted to a unity numerator in the transfer function. M4.2. Time-Response Simulation To begin with, we can launch the LTI Viewer with ltiview as explained in MATLAB Session 3. The graphics interface is designed well enough so that no further explanation is needed. The use of step() and impulse() on state-space models is straightforward as well. Here just a simple example is provided. LetвЂ™s go back to the numbers that we have chosen for Example 4.1 and deп¬Ѓne a=[0 1; -2.25 -1.5]; b=[0; 2.25]; c=[1 0]; d=0; sys=ss(a,b,c,d); The step() function also accepts state-space representation, and generating the unit-step response is no more difп¬Ѓcult than using a transfer function: step(sys) Now we repeat the calculation in the transfer function form and overlay the plot on top of the last one: G=tf(2.25,[1 1.5 2.25]); hold step(G,'x') hold off Sure enough, the results are identical. We would be in big trouble if it were not! In fact, we should get the identical result with other state-space representations of the model. (You may try this yourself with the other set of a,b,c,d returned by tf2ss() when we п¬Ѓrst went through Example 4.1.) Many other MATLAB functions, for example, impulse(), lsim(), etc., take both transfer function and state-space arguments (what can be called polymorphic). There is very little reason to do the conversion back to the transfer function once you can live in state-space with peace. 246 Session 4. State-Space Functions M4.3. Transformations The following features are covered in this session: r Similarity and canonical transforms r Using functions canon() and ss2ss() First a similarity transform is demonstrated. For a nonsingular matrix A with distinct eigenvalues, we can п¬Ѓnd a nonsingular (modal) matrix P such that the matrix A can be transformed into a diagonal made up of its eigenvalues. This is one useful technique in decoupling a set of differential equations. Consider the matrix A from Example 4.6. We check to see if the rank is indeed 3, and compute the eigenvalues for reference later: A=[0 1 0; 0 -1 -2; 1 0 -10]; rank(A) eig(A) % -0.29, -0.69, -10.02 We now enter [P,L] = eig(A) a = inv(P)*A*P % % % % L is a diagonal matrix of eigenvalues P is the modal matrix whose columns are the corresponding eigenvectors Check that the results are correct Indeed, we should п¬Ѓnd a to be the diagonal matrix with the eigenvalues. The second route is to diagonalize the entire system. With Example 4.6, we further deп¬Ѓne B=[0; 2; 0]; C=[1 0 0]; D=[0]; S=ss(A,B,C,D); SD=canon(S) % Generates the system object The canon() function by default will return the diagonalized system and, in this case, in the system object SD. For example, we should п¬Ѓnd SD.a to be identical to the matrix L that we obtained a few steps back. The third alternative to generate the diagonalized form is to use the state-space to statespace transform function. The transform is based on the modal matrix that we obtained earlier: SD=ss2ss(S,inv(P)) To п¬Ѓnd the observable canonical form of Example 4.6, we use SO=canon (S,'companion') In the returned system SO, we should п¬Ѓnd SO.a and SO.b to be пЈ№ пЈ® пЈ® пЈ№ 0 0 в€’2 1 Bob = пЈ°0пЈ». Aob = пЈ°1 0 в€’10пЈ», 0 1 в€’11 0 247 MATLAB Tutorial Sessions Optional reading: The rest of this section requires material on the Web Support and is better read together with Chap. 9. Using the supplementary notes on canonical transformation, we п¬Ѓnd that the observable canonical form is the transpose of the controllable canonical form. In the observable canonical form, the coefп¬Ѓcients of the characteristic polynomial (in reverse sign) are in the last column. The characteristic polynomial is, in this case, P(s) = s 3 + 11s 2 + 10s + 2. We can check that with roots([1 11 10 2]) poly(A) % Check the roots % Check the characteristic polynomial of A We can п¬Ѓnd the canonical forms ourselves. To evaluate the observable canonical form Aob , we deп¬Ѓne a new transformation matrix based on the controllability matrix: P=[B A*B A^2*B]; inv(P)*A*P inv(P)*B % Should be Aob as found by canon() % Shoud be Bob (Bob!) To п¬Ѓnd the controllable canonical form, пЈ® Actr 0 =пЈ° 0 в€’2 пЈ№ 1 0 0 1 пЈ», в€’10 в€’11 Bctr пЈ® пЈ№ 0 = пЈ°0пЈ», 1 we use the following statements based on the Web Support supplementary notes. Be very careful when constructing the matrix M: poly(A); %To confirm that it is [1 11 10 2] M=[10 11 1; 11 1 0; 1 0 0]; T=P*M; inv(T)*A*T inv(T)*B We now repeat the same ideas one more time with Example 4.9. We п¬Ѓrst make the transfer function and the state-space objects: G=zpk([],[-1 -2 -3],1); S=ss(G); As a habit, we check the eigenvalues: eig(S) % Should be identical to eig(G) To п¬Ѓnd the modal matrix, we use [P,L]=eig(S.a) inv(P)*S.a*P 248 % Just a check of L Session 5. Feedback Simulation Functions The observable canonical form is SD=canon(S) The component SD.a is, of course, the diagonalized matrix L with eigenvalues. We can check that SD.b and SD.c are respectively computed from inv(P)*S.b S.c*P % Identical to SD.b % Identical to SD.c Finally, the observable canonical form is SO=canon(S, 'companion') The matrix SO.a is пЈ® Aob 0 0 = пЈ°1 0 0 1 пЈ№ в€’6 в€’11пЈ», в€’6 meaning that P(s) = s 3 + 6s 2 + 11s + 6, which is the characteristic polynomial poly([-1 -2 -3]) as expected from the original transfer function. Session 5. Feedback Simulation Functions This tutorial is to complement our development in Chaps. 5 and 6. You may want to go over the tutorial quickly before you read the text and come back later a second time for the details. Feedback simulation functions feedback simulink Generate feedback-system transfer function object Launch Simulink M5.1. Simulink Comments with Respect to Launching Simulink Simulink is a user-friendly simulation tool with an icon-driven graphics interface that runs within MATLAB. The introduction here is more conceptual than functional for two reasons. One, the Simulink interface design is very intuitive and you may not need help at all! Second, for a thorough introduction, we need to reproduce many of the graphics windows. To conserve 249 MATLAB Tutorial Sessions Figure M5.1. paper (and trees), these print-intensive and detailed explanations have been moved to the Web Support. Furthermore, the Helpbrowser of MATLAB is extremely thorough and should serve as our main guide for further applications. To launch Simulink, enter in the Command Window simulink and MATLAB will launch the Simulink Block Library window with pull-down menus. A few sample block library icons are shown in Fig. M5.1. Each icon represents a toolbox and contains within it a set of models, which will make themselves available if we double-click on the toolbox icons. For example, we can п¬Ѓnd within the Sources toolbox (Fig. M5.1) a model for generating a step input function and within the Sinks toolbox a model for graphing results. Within the Continuous toolbox are the important models for generating transfer functions and state-space models (Fig. M5.2). All we need is to drag and drop the icons that we need from the toolboxes into a blank model window. If this window is not there, open a new one with the File pull-down menu. From here on, putting a feedback loop together to do a simulation is largely a point-and-click activity. An example of what Simulink can generate is shown in Fig. M5.3. Simulink is easy to learn, fun, and instructive, especially with more complex MIMO systems. For systems with time delays, Simulink can handle the problem much better than the classical control toolbox. Simulink also has ready-made objects to simulate a PID controller. A few quick pointers: r These are some of the features that we use most often within the Simulink Block Library: Sources: Step input; clock for simulation time Sinks: Plotting tools; output to MATLAB workspace or a п¬Ѓle Continuous: Transfer functions in polynomial or pole-zero form; state-space models; transport delay Math: Sum; gain or gain slider Nonlinear: Saturation; dead zone Blocksets: From the Blocksets and Toolboxes, choose вЂњSimulink Extras,вЂќ and then вЂњAdditional Linear.вЂќ In there are the PID and the PID with approximate derivative controllers. Figure M5.2. 250 Session 5. Feedback Simulation Functions Figure M5.3. r All Simulink simulation block diagrams are saved as ascii п¬Ѓles with the вЂњmdlвЂќ extension. r Before we start a simulation, choose Parameters under the Simulation pull-down menu to select the time of simulation. If we are using the XY Graph, we need to double-click its icon and edit its parameters to make the time information consistent. r Simulink shares the main MATLAB workspace. When we enter information into, say, the transfer function block, we can use a variable symbol instead of a number. We then deп¬Ѓne the variable and assign values to it in the MATLAB Command Window. This allows for a much quicker route for doing parametric studies than does changing the numbers within the Simulink icons and dialog boxes. r We can build our own controllers, but two simple ones are available: an ideal PID and a PID with approximate derivative action. For curious minds: The п¬Ѓrst time you use the PID controllers, drag the icon onto a new simulation window, select the icon, and then Look under mask under the Edit pull-down menu. You will see how the controllers are put together. The simple PID controller is KI + K D s, s and the PID with approximate derivative controller is G c (s) = K c + G c (s) = K c + KI K Ds + 1 + . s s/N + 1 We also see the transfer functions used by each icon when we double-click on it and open up the parameter entry dialog window. Therefore, in terms of notation, we have K I = K c /П„ I , K D = K c П„ D , and N = 1/О±П„ D . M5.2. Control Toolbox Functions The following feature is covered in this session: r Synthesizing a closed-loop transfer function with feedback() The closed-loop transfer function of a servo problem with proper handling of units is Eq. (5.11) in text: Km Gc G p C = . R 1 + Gm GcG p It can be synthesized with the MATLAB function feedback(). As an illustration, we use a simple п¬Ѓrst-order function for G p and G m and a PI controller for G c . When all is done, we test 251 MATLAB Tutorial Sessions the dynamic response with a unit-step change in the reference. To make the reading easier, we break the task up into steps. Generally, we would put the transfer function statements inside an M-п¬Ѓle and deп¬Ѓne the values of the gains and time constants outside in the workspace. Step 1: Deп¬Ѓne the transfer functions in the forward path. The values of all gains and time constants are arbitrarily selected: km=2; % Gc is a PI controller kc=10; taui=100; Gc=tf(km*kc*[taui 1], [taui 0]); kp=1; taup=5; Gp=tf(kp, [taup 1]); % Gp is the process function In the deп¬Ѓnition of the controller G c , we have included the measurement gain K m , which usually is in the feedback path and the reference (Fig. 5.4). This is a strategy that helps to eliminate the mistake of forgetting about K m in the reference. One way to spot whether you have made a mistake is if the system calculation has an offset when in theory you know that it should not. Step 2: Deп¬Ѓne the feedback path function. LetвЂ™s presume that our measurement function is п¬Ѓrst order too. The measurement gain has been taken out and implemented in Step 1: taum=1; Gm=tf(1, [taum 1]); % Gm is the measurement function % Its s.s. gain km is in Gc Step 3: Deп¬Ѓne the closed-loop function: Gcl=feedback(Gc*Gp,Gm); % Gcl is the closed-loop function C/R Comments: r By default, feedback() uses negative feedback. r With unity feedback, i.e., G m = K m = 1, we would simply use Gcl=feedback(Gc*Gp,1) to generate the closed-loop function. r We could generate a closed-loop function with, for example, Gc*Gp/(1 + Gc*Gp), but this is not recommended. In this case, MATLAB simply multiplies everything together with no reduction and the resulting function is very unclean. Step 4: We can now check (if we want to) the closed-loop poles and do the dynamic simulation for a unit-step change in R: disp('The closed-loop poles & s.s. gain:') pole(Gcl) dcgain(Gcl) step(Gcl) % Of course, we can customize the plotting This is the general idea. You can now put it to use by writing M-п¬Ѓles for different kinds of processes and controllers. 252 Session 5. Feedback Simulation Functions When we have a really simple problem, we should not even need to use feedback(). Yes, we can derive the closed-loop transfer functions ourselves. For example, if we have a proportional controller with G c = K c and a п¬Ѓrst-order process, all we need are the following statements, which follow Example 5.1 and Eq. (E5.1) in text: kc=1; kp=0.8; taup=10; Gcl=tf(kc*kp,[taup 1+kc*kp]); pole(Gcl) step(Gcl); % Again for unit-step change in R Try a proportional controller with a second-order process as derived in Example 5.2 in text. This is another simple problem for which we do not really need feedback(). We now п¬Ѓnish up with what we left behind in Session 4. LetвЂ™s revisit Example 4.6. For checking our results later, we п¬Ѓrst п¬Ѓnd the poles of the closed-loop transfer function with q=2*[1 10]; p=[1 11 10 2]; roots(p) % -0.29, -0.69, and -10.02 Next, we deп¬Ѓne each of the transfer functions in the example: G1=tf(1,[1 0]); G2=tf(2,[1 1]); H=tf(1,[1 10]); Note that the numbering and notation are entirely arbitrary. We now generate the closed-loop transfer function and check that it has the same closed-loop poles: Gcl=feedback(G1*G2,H); pole(Gcl) We can also easily obtain a state-space representation and see that the eigenvalues of the state matrix are identical to the closed-loop poles: ssm=ss(Gcl); eig(ssm.a) For fun, we can recover the closed-loop transfer function Gcl with: tf(ssm) One п¬Ѓnal check with our own derivation. We deп¬Ѓne the coefп¬Ѓcient matrices with Eqs. (E4.23) and (E4.24) and then do the conversion: a=[0 1 0; 0 -1 -2; 1 0 -10]; b=[0; 2; 0]; c=[1 0 0]; d=0; eig(a) [q3,p3]=ss2tf(a,b,c,d,1) % should return the same % eigenvalues and transfer % function 253 MATLAB Tutorial Sessions If this is not enough to convince you that everything is consistent, try step() on the transfer function and different forms of the state-space model. You should see the same unit-step response. Session 6. Root-Locus Functions This tutorial is to complement our development in Chap. 7. You may want to go over the tutorial quickly before you read the text and come back later a second time for the details. Root-locus functions rlocus rlocfind sgrid Root-locus plot Find the closed-loop gain graphically Draw the damping and natural frequency lines sisotool Launch the SISO system design graphics interface M6.1. Root-Locus Plots The following features are covered in this session: r Root-locus calculation and plots, rlocus() r Frequency and damping factor grid, sgrid() r Obtaining gain of chosen closed-loop pole, rlocfind() In simple terms, we want to solve for s in the closed-loop equation 1 + G 0 (s) = 1 + kG(s) = 0, where we further write G 0 = kG(s) and G(s) is the ratio of two polynomials, G(s) = q(s)/ p(s). In the simplest case, we can think of the equation as a unity feedback system with only a proportional controller (i.e., k = K c ) and G(s) as the process function. We are interested in п¬Ѓnding the roots for different values of the parameter k. We can either tabulate the results or we can plot the solutions s in the complex plane вЂ“ the result is the root-locus plot. LetвЂ™s pick an arbitrary function such that q(s) = 1 and p(s) = s 3 + 6s 2 + 11s + 6. We can generate the root-locus plot of the system with: p=[1 6 11 6]; roots(p) G=tf(1,p); rlocus(G) % Check the poles % Bingo! For the case in which q(s) = s + 1, we use G=tf([1 1],p); rlocus(G) % Try an open-loop zero at -1 % to cancel the open-loop pole at -1 MATLAB automatically selects a reasonable vector for k, calculates the roots, and plots them. The function rlocus() also adds the open-loop zeros and poles of G(s) to the plot. 254 Session 6. Root-Locus Functions LetвЂ™s try two more examples with the following two closed-loop characteristic equations: 1+K 1 = 0, (s + 1)(s + 3) 1+K 1 = 0; (s + 1)(s + 2)(s + 3) G=zpk([],[-1 -3],1) rlocus(G) % The second-order example G=zpk([],[-1 -2 -3],1) rlocus(G) % The third-order example The point of the last two calculations is that a simple second-order system may become extremely underdamped, but it never becomes unstable. Reminder: We supply the polynomials q(s) and p(s) in G(s), but do not lose sight that MATLAB really solves for s in the equation 1 + kq(s)/ p(s) = 0. In the initial learning stage, it can be a bad habit to rely on MATLAB too much. Hence the following two exercises take the slow way in making root-locus plots, which, it is hoped, may make us more aware of how the loci relate to pole and zero positions. The п¬Ѓrst thing, of course, is to identify the open-loop poles: q=[2/3 1]; p=[1 6 11 6]; poles=roots(p)' zeros=roots(q)' G=tf(q,p); k=0:0.5:100; rlocus(G,k); % Redefine q(s) and p(s) % display poles and zeros as row vectors % define our own gains; may need % 0:0.1:10 to better see the break-off point % MATLAB will plot the roots with '+' Until we have more experience, it will take some trial and error to pick a good range and increment for k, but then that is the whole idea of trying it ourselves. This manual approach makes us better appreciate the placements and changes of closed-loop poles as we vary the proportional gain.12 We may also want to override the MATLAB default format and use little dots: r=rlocus(G,k); plot(r,'.') hold pzmap(G) hold off % % % % % Save loci to array "r" first Now use plot() to do the dots hold the plot to add goodies pzmap() draws the open-loop poles and zeros Be careful to read where the loci are on the real axis because pzmap() also traces the axis with little dots, which can be confusing. We may want to п¬Ѓnd the ultimate gain when the loci cross the imaginary axis. Again there are many ways to do it. The easiest method is to estimate with the MATLAB function rlocfind(), which is introduced next. 12 The gain vector generated automatically by MATLAB is not always instructive if we want to observe the region close to the imaginary axis. We can use вЂњtricks,вЂќ like making two gain vectors with different increments, concatenating them, and using the result in rlocus(). However, we should not get bogged down with п¬Ѓne details here. Certainly for day-to-day routine calculations, we can omit the gain vector and let MATLAB generate it for us. 255 MATLAB Tutorial Sessions There are two very useful MATLAB features. First, we can overlay onto the root-locus plot lines of the constant damping factor and the natural frequency. These lines help us pick the controller gain if the design speciп¬Ѓcation is in terms of the frequency or the damping ratio, sgrid % use the default grid or, better yet, sgrid(zeta,wn) % plot only lines with given damping ratio % and natural frequency Example: sgrid(0.7,1) % add the approx. 45в—¦ line for zeta=0.7 and % the unit circle (frequency=1) The second feature is the function rlocfind(), which allows us to п¬Ѓnd the gain associated with a closed-loop pole. We can enter [ck,cpole]=rlocfind(G) or rlocfind(G) % this simple form will not return % the values of the closed-loop poles MATLAB will wait for us to click on a point (the chosen closed-loop pole) in the rootlocus plot and then return the closed-loop gain (ck) and the corresponding closed-loop poles (cpole). MATLAB does the calculation with the root-locus magnitude rule, which is explained on the Web Support. What if we click a point not exactly on a root locus? When we select a point s в€— , MATLAB calculates the value k в€— = в€’ p(s в€— )/q(s в€— ), which will be a real positive number only if s в€— satisп¬Ѓes the closed-loop equation. Otherwise, k в€— is either complex or negative if the pole is a real number. In this case, MATLAB calculates the magnitude of k в€— , uses it as the gain, and computes the corresponding closed-loop poles. Thus we п¬Ѓnd that the chosen points are always right on the root loci, no matter where we click. We may also want to use the zoom feature of MATLAB to zoom in and out of a plot to get a better picture of, say, the break-off point of two loci. Make sure you enter вЂњzoom offвЂќ when you are done. M6.2. SISO System Design Graphics Interface The following feature is covered in this session: r Graphics user interface for designing SISO systems, sisotool 13 The control toolbox supports an extremely nice SISO system design tool that is ideal for experimentation. This graphics interface is even more intuitive and self-explanatory than that of Simulink. The same approach is taken as that of our introduction to Simulink, and 13 256 The description is based on Version 5.1 of the MATLAB control toolbox. If changes are introduced in newer versions, they will be presented on the Web Support. Session 6. Root-Locus Functions F + G C +/вЂ“ H Figure M6.1. the not-so-necessary and print-intensive window display and instructions have been moved to the Web Support. Only a very brief conceptual introduction is provided here. To launch the SISO system design tool, enter in the MATLAB Command Window sisotool % default view sisotool('rlocus') % root-locus view only or A graphics window with pull-down menus and tool buttons will pop out, slowly. The default view displays both the root-locus and the Bode editors. Because we have not learned Bode plots yet, the second option with rlocus is less intimidating for the moment. Here are some pointers on the usage of the tool: r The SISO design tool supports a п¬‚exible block diagram, as shown in Fig. M6.1. The feedback can be either positive or negative. Similar to the LTI Viewer, the tool runs in its own functional space. We have to import the transfer functions under the File pull-down menu. By default, the transfer functions F, C, G, and H are all assigned the value вЂњ1,вЂќ so we have to import at least a transfer function for G to do meaningful calculations. r The default compensator C in Fig. M6.1 is a proportional controller, but it can be changed to become a PI, PD, or PID controller. The change can be accomplished many ways. One is to retrieve the compensator-editing window by clicking on the C block or by using the Compensator pull-down menu. We can also use the set of button on the toolbar to add or move open-loop poles and zeros associated with the controller. r Once a root-locus plot is generated, we can interactively change the locations of the closed-loop poles and the tool will compute the corresponding controller gain for us. r For a given system and chosen closed-loop poles displayed in the root-locus plot, we can generate its corresponding time-response and frequency-response plots with features under the Tools pull-down menu. In the next section, you can use the SISO design tool if you prefer, but the explanation is given with commands. It is easier to get the message across with commands, and in the beginnerвЂ™s learning stage, entering your own command can give you a better mental imprint of the purpose of the exercise. M6.3. Root-Locus Plots of PID Control Systems The following feature is covered in this session: r Making root-locus plots that model situations of PID control systems 257 MATLAB Tutorial Sessions Here are some useful suggestions regarding root-locus plots of control systems. In the following exercises, we consider only the simple unity feedback closed-loop characteristic equation: 1 + G c G p = 0. We ignore the values of any gains. We focus on only the probable open-loop pole and zero positions introduced by a process or by a controller, or, in other words, the shape of the root-locus plots. LetвЂ™s begin with a п¬Ѓrst-order process G p = 1/(s + 1). The root-locus plot of a system with this simple process and a proportional controller, G c = K c , is generated as follows: Gp=tf(1,[1 1]); subplot(221), rlocus(Gp) % open-loop pole at -1 % Gc = Kc To implement an ideal PD controller, we will have an additional open-loop zero. Two (of inп¬Ѓnite) possibilities are taud=2; Gc=tf([taud 1],1); subplot(222), rlocus(Gc*Gp) % open-loop zero at -1/2 and taud=1/2; Gc=tf([taud 1],1); subplot(223), rlocus(Gc*Gp) % open-loop zero at -2 What are the corresponding derivative time constants? Which one would you prefer? We next turn to a PI controller. We п¬Ѓrst make a new п¬Ѓgure and repeat proportional control for comparison: figure(2) subplot(221), rlocus(Gp) % Gc = Kc Integral control will add an open-loop pole at the origin. Again, we have two regions where we can put the open-loop zero: taui=2; Gc=tf([taui 1],[taui 0]); subplot(222), rlocus(Gc*Gp) % open-loop zero at -1/2 and taui=1/2; Gc=tf([taui 1],[taui 0]); subplot(223), rlocus(Gc*Gp) % open-loop zero at -2 Once again, what are the corresponding integral time constants? Which one would you prefer? Finally, letвЂ™s take a look at the probable root loci of a system with an ideal PID controller, which introduces one open-loop pole at the origin and two open-loop zeros. For illustration, we will not use the integral and derivative time constants explicitly, but refer to only the two zeros that the controller may introduce. We will also use zpk() to generate the 258 Session 6. Root-Locus Functions transfer functions: figure(3) subplot(221), rlocus(Gp) % redo Gc = Kc opв€’ pole=[0]; % open-loop pole at 0 opв€’ zero=[-0.3 -0.8]; Gc=zpk(opв€’ zero,opв€’ pole,1); subplot(222),rlocus(Gc*Gp) % both zeros larger than -1 opв€’ zero=[-1.5 -3]; Gc=zpk(opв€’ zero,opв€’ pole,1); subplot(223),rlocus(Gc*Gp) % both zeros less than -1 opв€’ zero=[-0.5 -1.8]; Gc=zpk(opв€’ zero,opв€’ pole,1); subplot(224),rlocus(Gc*Gp) % one zero in each region Yes, you know the question is coming. Which case would you prefer? We can use the rule of thumb that the derivative time constant is usually approximately one fourth the value of the integral time constant, meaning that the zero farther away from the origin is the one associated with the derivative time constant. Note that the system remains stable in all cases, as it should for a simple п¬Ѓrst- or secondorder system. One п¬Ѓnal question: Based on the design guidelines by which the system should respond faster than the process and the system should be slightly underdamped, what are the ranges of derivative and integral time constants that you would select for the PD, PI, and PID controllers? And in what region are the desired closed-loop poles? WeвЂ™ll п¬Ѓnish with implementing the P, PI, and PD controllers on a second-order overdamped process. As in the previous exercise, try to calculate the derivative or integral time constants and take a minute to observe the plots and see what may lead to better controller designs. LetвЂ™s consider an overdamped process with two open-loop poles at в€’1 and в€’2 (time constants at 1 and 0.5 time units). A system with a proportional controller would have a root-locus plot as follows. We stay with tf(), but you can always use zpk(). figure(1) p=poly([-1 -2]); Gp=tf(1,p); subplot(221),rlocus(Gp) % open-loop poles -1, -2 % proportional control To implement an ideal PD controller, we now have three possible regions in which to put the zero: taud=2; Gc=tf([taud 1],1); subplot(222), rlocus(Gc*Gp) % open-loop zero at -1/2 taud=2/3; Gc=tf([taud 1],1); subplot(223), rlocus(Gc*Gp) % open-loop zero at -1.5 259 MATLAB Tutorial Sessions taud=1/3; Gc=tf([taud 1],1); subplot(224), rlocus(Gc*Gp) % open-loop zero at -3 We will put the PI controller plots on a new п¬Ѓgure: figure(2) subplot(221),rlocus(Gp) % redo proportional control The major regions in which to place the zero are the same, but the interpretation as to the choice of the integral time constant is very different. We now repeat, adding the open-loop zeros: taui=2; Gc=tf([taui 1],[taui 0]); subplot(222), rlocus(Gc*Gp) % open-loop zero at -1/2 taui=2/3; Gc=tf([taui 1],[taui 0]); subplot(223), rlocus(Gc*Gp) % open-loop zero at -1.5 taui=1/3; Gc=tf([taui 1],[taui 0]); subplot(224), rlocus(Gc*Gp) % open-loop zero at -3 You may want to try some sample calculations using a PID controller. One way of thinking: We need to add a second open-loop zero. We can limit the number of cases if we assume that the value of the derivative time constant is usually smaller than the integral time constant. Session 7. Frequency-Response Functions This tutorial is to complement our development in Chap. 8. You may want to go over the tutorial quickly before you read the text and come back later a second time for the details. Frequency-response functions 260 bode freqresp logspace margin nichols, ngrid nyquist Bode plots Frequency response of a transfer function Logarithmically spaced vector Gain margin and crossover frequency interpolation Nichols plots Nyquist plots sisotool Launch the SISO system design graphics interface Session 7. Frequency-Response Functions M7.1. Nyquist and Nichols Plots The following feature is covered in this session: r Nyquist plots, nyquist() The SISO system design tool sisotool, as explained in Session 6, can be used to do frequency-response plots. Now we want to use the default view, so we just need to enter sisotool Hints to make better use of the design tool are on the Web Support. We use commands here because they give us a better idea behind the calculations. This section is brief as our main tool will be Bode plots, which will be explained in the next section. LetвЂ™s say we have a simple open-loop transfer function G 0 of the closed-loop characteristic equation, 1 + G 0 = 0, and we want to п¬Ѓnd the proportional gain that will give us an unstable system. For this simple exercise, we take G 0 (s) = K G(s): p=poly([-1; -2; -3]); G=tf(10,p); nyquist(G); % Open-loop poles at -1, -2, -3 % Arbitrary value K=10 % Bingo! WeвЂ™ll see two curves. By default, MATLAB also maps and plots the image of the negative imaginary axis. That can make the plot too busy and confusing, at least for a beginner. So weвЂ™ll stay away from the default in the following exercises: [re,im]=nyquist(G); plot(re(1,:),im(1,:)) % Only the positive Im-axis image Of course, we can deп¬Ѓne our own frequency vector: w=logspace(-1,1); [re,im]=nyquist(G,w); plot(re(1,:),im(1,:)) % Generate numbers between [10^-1, 10^1] The function logspace() generates a vector with numbers nicely spaced on the logarithmic scale. Its use is optional. The default of the function gives 50 points and is usually adequate. For a smoother curve, use more points. For example, this command will use 150 points: logspace(-1,1,150). hold % to add the (-1,0) point and the axes % on the plot x=-1; y=0; xh=[-2 2]; yh=[0 0]; % the axes xv=[0 0]; yv=[-2 1]; plot(x,y,'o',xh,yh,'-',xv,yv,'-') 261 MATLAB Tutorial Sessions We can increase the gain K and repeat the calculation with, for example, two more trials14 : G=tf(50,p); [re,im]=nyquist(G,w); plot(re(1,:),im(1,:)) % try again G=tf(60,p); [re,im]=nyquist(G,w); plot(re(1,:),im(1,:)) hold off % and again We do not use the Nichols plot (log magnitude versus phase) much, but it is nice to know that we can do it just as easily: p=poly([-1; -2; -3]); G=tf(10,p); nichols(G) ngrid zoom % need to zoom into the meaningful region The plot with default settings is quite useless unless we use ngrid to superimpose the closed-loop gain and phase grid lines. Instead of zooming in, we can reset the axes with axis([-360 0 -40 20]) M7.2. Magnitude and Phase-Angle (Bode) Plots The following features are covered in this session: r Bode plot calculation, bode() r Finding the gain and phase margins, margin() r Bode plots for transfer functions with dead time We begin with one simple example. LetвЂ™s say we want to analyze the closed-loop characteristic equation 1 = 0. + 0.4s + 1 We generate the Bode plot with 1+ s2 G=tf(1,[1 0.4 1]); bode(G) % Done! The MATLAB default plot is perfect! That is, except when we may not want decibels as the unit for the magnitude. We have two options. One, learn to live with decibels, the 14 All functions like nyquist(), bode(), etc., can take on multiple LTI objects, as in nyquist(G1,G2,G3) but only when we do not use LHS arguments. 262 Session 7. Frequency-Response Functions convention in the control industry, or two, we do our own plots. This is a task that we need to know when we analyze systems with dead time. This is how we can generate our own plots: w=logspace(-1,1); [mag,phase]=bode(G,w); mag=mag(1,:); phase=phase(1,:); % required since MATLAB v.5 subplot(211), loglog(w,mag) ylabel('Magnitude'), grid subplot(212), semilogx(w,phase) ylabel('Phase, deg'), grid xlabel('Frequency (rad/time)') As an option, we can omit the subplot command and put the magnitude and phase plots in individual п¬Ѓgures. This is how we can make a Bode plot with decibels as the scale for the magnitude. dB=20*log10(mag); % converts magnitude to decibels Now we do the plotting. Note that the decibel unit is already a logarithmic scale: subplot(211), subplot(212), semilogx(w,dB) % Use semilogx for dB ylabel('Magnitude(dB)') grid semilogx(w,phase) ylabel('Phase angle (degree)') xlabel('Frequency, w') grid We most often use radians per second as the unit for frequency. In the case in which cycles per second or hertz are needed, the conversion is f=w/(2*pi); % Converts w [rad/s] to [Hz] After using the subplot() command and before doing any other plots, we should make it a habit to reset the window with clf % clear figure We now п¬Ѓnd the gain margin with its crossover frequency (Gm, Wcg) and phase margin with its crossover frequency (Pm, Wcp) with either one of the following options: [Gm,Pm, Wcg,Wcp]=margin(mag,phase,w) % option 1 where mag and phase are calculated with the function bode() beforehand. We can skip the bode() step and use the transfer function directly as the argument, [Gm,Pm, Wcg,Wcp]=margin(G) % option 2 263 MATLAB Tutorial Sessions or simply margin(G) % option 3, Gm in dB In the last option without any LHS arguments, MATLAB will do the Bode plot, and display the margin calculations on the plot. Two important comments: (1) With G=tf(1,[1 0.4 1]), i.e., a simple second-order system, it is always stable. The gain margin calculation is meaningless. Nevertheless, MATLAB returns a set of results anyway. Again, a computer is not foolproof. All margin() does is an interpolation calculation. (2) If you use option 1 or 2 above, margin() returns the linear-scale gain margin in the variable Gm. With option 3, however, the gain margin displayed in the plot is in decibels, you need to convert it back with 10dB/20 . To handle dead time, all we need is a simple modiп¬Ѓcation using the fact that the time delay transfer function has magnitude 1 and phase angle в€’td П‰. We need one single statement to вЂњtag onвЂќ the lag that is due to dead time, and we do it after the bode() function call. So letвЂ™s start with the second-order function, which is always stable: G=tf(1,[1 0.4 1]); freq=logspace(-1,1); [mag,phase]=bode(G,freq); mag=mag(1,:); phase=phase(1,:); % freq is in rad/time Now letвЂ™s say we also have dead time: tdead=0.2; % [time unit] The following statement is the only addition needed to introduce the phase lag that is due to dead time: phase = phase - ((180/pi)*tdead*freq); % phase is in degrees We can now proceed with the plotting and phase/gain margin interpolation: subplot(211), subplot(212), loglog(freq,mag) ylabel('Magnitude'),title('Bode Plot') grid semilogx(freq,phase) ylabel('Phase(degree)'),xlabel('Frequency') grid % now using new phase variable that includes dead-time phase lag [Gm,Pm,Wcg,Wcp]=margin(mag,phase,freq) The whole idea of handling dead time applies to other types of frequency-domain plots, but the Bode plot is the easiest to learn from. 264 Session 7. Frequency-Response Functions There is no magic in the functions nyquist() or bode(). We could have done all our calculations by using the more basic freqresp() function. What it does essentially is make the s = jП‰ substitution numerically in a given transfer function G(s). A sample usage is w=logspace(-1,1); gjw=freqresp(G,w); % does the s=jw calculation for each % value in w After that, we can use the result to do frequency-response analysis. If you are interested in the details, they are provided in the Session 7 Supplement on the Web Support. 265 Homework Problems Part I. Basic Problems Where appropriate, use MATLAB to solve the problem or to check your derivation. (1) For the given transfer function Y (s) 2 = . X (s) (s + 2)(s 2 + 9) (a) Derive y(t) with respect to a unit-step input and an impulse input. (b) What are the values of y(t) at t = 0 and t = в€ћ with respect to a unit-step input? (2) For the given ODE, y + y + y = sin П‰t, with initial condition y(0) = y (0) = 0. What are the characteristic features of y(t)? What is y(t) when t в†’ в€ћ? (3) For the given ODE, d2 y dy + 5y = f (t), with y(0) = y (0) = 0, +4 2 dt dt sketch the probable time-domain response if f (t) is a unit-step function. (4) Derive time-domain function y(t) of the following Laplace transforms 10 (a) Y (s) = (s + 1)2 (s + 3) s+3 (b) Y (s) = 2 (s + 2s + 5) eв€’4s (c) Y (s) = s(2s 2 + 3s + 2) eв€’2s (d) Y (s) = s(s 2 + 9) 267 Homework Problems (5) Find the partial fractions of the following transfer function: G(s) = s+1 . + 1) s 2 (10s (6) For the following transfer function, 3s(s + 2)(s в€’ 2) Y (s) = 4 , X (s) 5s + 6s 3 + 2s 2 + 3s and given that the input x(t) is a unit-step function, what is y(t) as t в†’ в€ћ? Under what condition, as related to the property of the transfer function, is this result valid? (7) For the following transfer function and a unit-step input, sketch qualitatively the time response y(t): Y (s) (s + 1)eв€’0.5s = . X (s) (s + 2)(s 2 в€’ 2s + 5) Explain how each вЂњtermвЂќ in the transfer function may contribution to your sketch. (8) Plot the zeros and poles of the following transfer function on the complex plane: Y (s) 10s(s + 1) = . X (s) (s + 2)(s 2 + 2s + 2) Now, ignore the zeros, i.e., consider only 10 Y (s) = . 2 X (s) (s + 2)(s + 2s + 2) If X (s) represents a unit-step function, what is qualitatively the expected time response y(t)? What is the steady-state gain and what are some of the characteristic parameters that we can associate with (s 2 + 2s + 2)? What about the case for the following function? Y (s) 10 = . X (s) (s + 2)(s 2 + 2) (9) The dynamic behavior of a process is modeled by the transfer function 18 Y (s) = 2 . F(s) (s + 3s + 9) (a) With a given step change input of f (t) = 3u(t), where u(t) is the unit-step function, what is the new steady-state value of the step response? (b) If it is required that the output must never exceed an upper limit of 10, what is the largest step input that we can tolerate without exceeding this limit? (c) Consider the following transfer function: Y (s) 5 = 2 . F(s) s +s+9 Sketch qualitatively the output y(t) in response to when F(s) represents a unitstep input and an impulse input. In each case, label clearly the п¬Ѓnal value at 268 Part I. Basic Problems large times. With the step response, label the value of one characteristic feature (of your choice) of the response. (10) (a) Consider two stirred-tanks-in-series. The п¬Ѓrst one has a volume of 4 m3 and the second is 3 m3 . The п¬‚ow rate and the inlet concentration of an inert species into the п¬Ѓrst stirred-tank under normal circumstances are 0.02 m3 /s and 1 gmol/m3 , respectively. Consider that the inlet (deviation) concentration is being perturbed and is described by an impulse function as c(t) = 6Оґ(t). (b) (c) (11) (a) (b) What is the time response, i.e., c2 (t) in the second tank? What is c2 (t) as time is sufп¬Ѓciently large? Consider п¬Ѓve stirred-tanks-in-series. They are identical with a volume of 4 m3 , and the steady inlet п¬‚ow rate is 2 m3 /min. Consider now the change in inlet concentration as an impulse function; plot the responses in each of the tanks in the same п¬Ѓgure. Repeat with a unit-step input change. On what basis would you choose the range of the time axis, i.e., how long should the simulation be in order for you to observe the п¬Ѓnal steady state? From standard step-response measurements, the poles of a process are determined to be located at в€’4.5 В± 2.5 j. The same process is subjected to a rectangular-pulse input. The pulse has a value of 1.3 between the period t = 0 and t = 3 (time unit). What are the pole positions of the transfer function now? We are given the following transfer function: Y (s) 12 = . F(s) (6s 2 + 3s + 9) We are asked to п¬Ѓnd the steady-state gain and time constant. If someone asks вЂњIs that for the case when the input is a unit-step function,вЂќ what is your response? (c) We are asked to п¬Ѓnd the differential equation that represents the transfer function in part (b). If someone asks вЂњCan I assume zero initial condition,вЂќ what is your response? (12) From a standard step-input experiment, you determined that a process under investigation was п¬Ѓrst order with a steady-state gain of 8.5 (unit) and a process time constant of 3.5 (time unit). Your colleague did a different step-response experiment with the same equipment and determined that the steady-state gain was 5.5 and the process time constant was 2.5. The units being correct and the discrepancies being statistically signiп¬Ѓcant, who is right? Your colleague is a very careful experimentalist, and so are you. (13) For the given transfer function Y (s) K1 K2 = + , X (s) П„1 s + 1 П„2 s + 1 what is the steady-state gain? What are the poles and zeros? (14) For the given Laplace transform Y (s), derive y(t) in terms of constants and sine function: 1 Y (s) = . 2 s(s + 2s + 3) 269 Homework Problems What is the value of y(t) as time в†’ в€ћ? (15) Consider the transfer function Y 3 = . 2 F s(s + 2s + 4) (a) Qualitatively, what is the time response y(t) if f (t) represents a unit-step input? What is the value of y(t) when time is sufп¬Ѓciently large? What is the time constant that we may use to evaluate the вЂњspeedвЂќ of response? (b) Repeat step (a) if f (t) represents an impulse input. What is y(t) when time is sufп¬Ѓciently large? (16) We have two identical isothermal CSTRs-in-series. They both have space time П„ = 4 min, a п¬Ѓrst-order reaction rate constant k = 1.5 minв€’1 , and both parameters are constants. (a) Write the mass balances of the reactant concentration in the two CSTRs. What is the transfer function relating the concentration of the second CSTR, C2 , to the concentration entering the п¬Ѓrst one, C0 ? What are the steady-state gain and the process time constant of the transfer function? (b) LetвЂ™s say the reaction rate constant is zero. We now subject the two stirred-tanks to a unit-step response experiment by increasing C0 and measuring C2 . Someone suggested that we can measure the space time by the 63.2% time response for C2 to reach the new steady state. We, of course, know better, and say that it is plain wrong. What is the percent response when time is equal to the process time constant in this case? (17) We have a п¬Ѓrst-order process with the transfer function Kp Y = . F (П„ p s + 1) Because the use of a temperature ramp is common in gas chromatographs, it is suggested that an experiment with a ramp input be used. If the ramp input is f (t) = О±t, show how K p and П„ p can be determined from y(t) when time is sufп¬Ѓciently large. (18) Linearize the following two functions. Put the results in terms of deviation variables. (a) VaporвЂ“liquid equilibrium: y(x) = 1 + (О±О±xв€’ 1)x about the steady state xs . (b) Antoine equation: Pis (T ) = Pc exp(A1 в€’ T +A2A3 ) about the steady state Ts . (19) Consider the balances of the cell density x and food concentration S in a simple fermentor model: dx = в€’Dx + Вµ(S)x, dt 1 dS = D(Sin в€’ S) в€’ Вµ(S)x, dt Y where Вµ(S) = 270 Вµm S . Km + S Part I. Basic Problems R + вЂ“ 1C s K + 1 s C Kv + Figure PI.21. Coefп¬Ѓcients Y , Вµm , and K m are constants. Quantities that can vary with time are inlet concentration Sin and dilution rate D, which really is q/V , the reciprocal of space time. Derive the transfer functions that relate changes in Sin and D to changes in cell density x. (20) In heterogeneous catalysis, the reaction kinetics takes on the so-called LangmuirвЂ“ Hinshelwood form. In a catalytic reactor, the mass balance for reactant A may take the following form: dC A 1 = (C A0 в€’ C A ) в€’ r (C A , C B ), dt П„ where r (C A , C B ) = [(kC A C B )/(1 + K A C A + K B C B )], and C A (0) = C AS . In the balance, C A0 is the inlet concentration, C B is the concentration of reactant B, and both are functions of time. The rate constant is k. The adsorption equilibrium constants are K A and K B , and the space time is П„ . All these quantities are constants. Linearize the reactant A mass balance. (21) Feedback compensation can be achieved with what is called rate feedback. One example is shown in Fig. PI.21. (a) Derive the transfer function relating changes in R to those in C, i.e., C/R. (b) If K = 4 and if we desire the system response c(t) to have a damping ratio of 0.7, what should be the value of K v ? (22) We want to relate the temperature in the center of an oven to what we can measure on the outside. The oven has two insulating layers as shown in Fig. PI.22. The thermal resistance in both layers is modeled with heat transfer coefп¬Ѓcients, h 1 and h 2 , and the heat balance equations are dTm = h 1 A1 (Ti в€’ Tm ), dt dTi C p,2 M2 = h 2 A2 (To в€’ Ti ), dt C p,1 M1 Layer 1 Layer 2 h1 h2 To Ti oven measured Tm Figure PI.22. 271 Homework Problems X X X X X X (a) X X (b) X (c) Figure PI.23. where the heat capacity C p,i , mass Mi , transfer area Ai , and heat transfer coefп¬Ѓcients h i are all constants. (a) Derive the transfer function relating changes in the oven temperature To to the measured temperature Tm . (b) If the change in oven temperature To is a unit step, what is the response we would expect (i.e., qualitatively) in Tm ? What is (are) the time constant(s)? (c) Will the measured temperature Tm ever be identical to the oven temperature? (23) We have a third-order process. Three possible sets of pole positions are depicted in Fig. PI.23. Sketch the time-response characteristics of the three cases with respect to an impulse input and a step input. (24) Consider the LotkaвЂ“Volterra equations that are used to model predatorвЂ“prey problems in ecology: dy1 = (1 в€’ О±y2 )y1 , dt dy2 = (в€’1 + ОІy1 )y2 . dt The initial conditions are y1 (0) = y1s and y2 (0) = y2s . Derive the linearized equations in terms of deviation variables using y1s and y2s as the references. What is the characteristic polynomial of the linearized model? (25) The poles and zeros of a process are represented by the Г— and в—¦ symbols, respectively, in Fig. PI.25. It was determined that the steady-state gain of the process is 2 (units). (a) First, refer to Fig. PI.25(a). What is the order of the transfer function? What is the transfer function of the process? (b) What is (are) the time constant(s) of the process? (c) Sketch qualitatively the expected time response of the process with respect to a unit-step input. What is a reasonable (approximation of) settling time? (d) If the pole-zero plot is changed to that shown in Fig. PI.25(b), what does the time response look like? Does the steady-state gain remain the same as before? вЂ“2 + j (a) X вЂ“4 вЂ“1 вЂ“2 вЂ“ j X Figure PI.25. 272 (b) X вЂ“2 + j X X вЂ“4 X вЂ“1 вЂ“2 вЂ“ j X Part I. Basic Problems вЂ“6 + 2j X 2j X X вЂ“2 X вЂ“2j вЂ“6 вЂ“ 2j X Figure PI.26. (26) For a given transfer function with the pole positions shown in Fig. PI.26, reconstruct what the time-domain terms should be. (27) For the given transfer function, Q2 1 = . Q0 (6s + 1)2 If q0 (t) is an impulse function (and without doing partial fractions), what is the most probable expression for q2 (t)? What is the value of q2 at large times? What is the value of q2 at large times if we have instead 1 Q2 = ? Q0 s(6s + 1) (28) A task expected of us in the Chap. 3 Review Problems: Derive y(t) for the case in which П„1 = П„2 and Y (s) = 1 . s(П„1 s + 1)(П„2 s + 1) (29) Find the transfer function C/R for the block diagram in Fig. PI.29. (30) Consider the closed-loop system in Fig. PI.30. (a) Derive the closed-loop transfer function C/R. (b) What is the steady-state error if we impose a unit-step change in R? (c) What should be the value of K if we desire the system response (C/R) to have a 10% overshoot? (31) In process applications, there are different ways in which we can ramp up a variable such as temperature. As shown in Fig. PI.31(a), the variable is ramped up and kept constant. In Fig. PI.31(b), it is ramped down immediately to form a triangular pulse, and in Fig. PI.31(c), we do something in between. Derive the Laplace transform of these three input functions. Make use of the principle of superposition as to how we generate a rectangular pulse. Do not try to apply the integral deп¬Ѓnition. a F R + b e G вЂ“ + C + H Figure PI.29. 273 Homework Problems R + вЂ“ C 2 s (s + 1) K Figure PI.30. (32) Consider the block diagram in Fig. PI.32(a) that represents a system used to handle dead time. Find C/R for a servo problem and also the equivalent controller transfer function G c in Fig. PI.32(b). (33) The model for the change in the liquid level of a gravity п¬‚ow vessel in a textbook, A dh = q0 в€’ q, dt is в€љ an oversimpliп¬Ѓcation. The dependence of q on the level is more than just simply h. Moreover, the п¬‚ow rate also depends on the valve coefп¬Ѓcient and characteristic. For an equal-percentage valve, the proper equation for the outlet п¬‚ow rate is a function of the liquid level h and the valve opening (the so-called lift) l, where 0 в‰¤ l в‰¤ 1, such that q = q(l, h) is deп¬Ѓned as q = Cv R lв€’1 P(h) , ПЃs P(h) = (P0 + ПЃgh) в€’ P1 . gc The valve coefп¬Ѓcient Cv , rangeability R, п¬‚uid speciп¬Ѓc gravity ПЃs , density ПЃ, gravity g, gravitational constant gc , ambient pressure P0 , and valve outlet pressure P1 are all constants. The inlet п¬‚ow is a function of time, q0 = q0 (t). Derive the transfer functions that describe this liquid-level model. Identify the steady-state gain(s) and time constant(s). (34) A typical exercise is to linearize the heat and mass balances of a CSTR. LetвЂ™s say we have a chemical reaction 2A в†’ B and the reaction is second order in A. The CSTR balances (i.e., the model equations) are dC A = C A,i в€’ C A в€’ П„ 2k0 eв€’E/RT C 2A , dt dT (в€’ H )П„ U Ac П„ 2k0 eв€’E/RT C 2A в€’ = Ti в€’ T + (T в€’ Tc ), dt ПЃC p ПЃC p Q П„ where П„ is the space time, Q is the volumetric п¬‚ow rate, U is the overall heat transfer coefп¬Ѓcient, Ac is the heat transfer area, and the other notations carry their usual meanings in chemical kinetics. Derive the transfer functions that account for y (a) y y 2 4 2 1 2 0 2 4 Figure PI.31. 274 (b) 4 t 0 2 4 t 0 (c) 2 4 6 8 t Part II. Intermediate Problems L (a) R + E1 + вЂ“ E2 M G GL + Gp вЂ“ ~ G* ~ C1 C + ~ C2 в€’ Оёs e вЂ“ + ~ C вЂ“ C2 L (b) R + вЂ“ Gc M Gp GL + C + Figure PI.32. the effects of changes in the inlet temperature Ti and the cooling jacket temperature Tc on the reactor concentration C A and temperature T . You can consider the inlet concentration C A,i , space time, and all other parameters as constants. Part II. Intermediate Problems (1) After adding a real PD controller as in Eq. (5.7) to a п¬Ѓrst-order process function, we essentially have a second-order system. (a) Derive the system steady-state gain, natural time period, and damping ratio. (b) How does the choice of О± affect the natural time period? (c) If we now have a regulator problem and the load change is a unit step, what is the п¬Ѓnal steady-state value of the controlled variable? (d) For a servo system, hand-sketch the expected time responses of the systems with an ideal PD controller and the real PD controller with respect to a unit-step change in set point. (e) Will this system with the real PD controller become underdamped? (Hint: With the proper technique, you can arrive at the answer without doing any algebra.) (2) For the following given process transfer function, G p (s) = (в€’s + 1) , (s + 1)(s + 2) and the closed-loop system characteristic polynomial 1 + G c G p = 0, if we want to use a simple proportional controller, answer the following questions: (a) What is the range of K c that provides stable closed-loop responses? (b) From this exercise, can you tell the rationale behind, for example, the use of the 1/1 PadВґe approximation, and the general effect of time delay in a given system? (c) Sketch the Bode plot of the system with proportional control. Identify the contribution of magnitude and phase angle from each of the terms. 275 Homework Problems (3) Consider the process transfer function Gp = 2(s + 1) s(s + 2)(s 2 + 1) and the closed-loop system characteristic polynomial 1 + G c G p = 0: (a) Show, with an analytical technique of your choice, that the closed-loop system cannot be stabilized with a proportional controller at all. (b) Suppose we can use a PI or a PD controller with a positive proportional gain; write the appropriate system characteristic equations for each controller. Without further analysis, can you make a judgment whether a PI or a PD controller is more feasible in stabilizing the system? (c) Use root-locus plots to illustrate your answer. (4) Consider a unity feedback system with characteristic polynomial 1 + G c G p = 0. The process function is Gp = K , s(П„ p s + 1) where the steady-state gain K is 0.6 (unit) and the time constant П„ p is 3 (time unit). (a) If we use a proportional controller, what is the offset? What statement can you make regarding the selection of controllers? (b) With the proportional controller, sketch the root locus of the system. (c) With a proportional controller, в€љ what is the proportional gain if the system is to have a damping ratio of 1/ 2? What are the locations of the closed-loop poles? (d) What is the time constant of the system in part (c)? (e) What is the time constant of the system when the damping ratio is to be 0.2? (5) Consider a system with either an ideal PD [Fig. PII.5(a)] or a PI controller [Fig. PII.5(b)]. (a) What is the steady-state error of the system in each case? (b) With the PD controller, derive the stability criteria with respect to K c with an appropriate method of your choice. What statement can you make regarding the proportional gain and the stability of the system? (c) What are the different probable ranges of derivative time constants that we may have with the PD controller? Explain with sketches of root-locus plots. (d) From part (c), which case would allow for an underdamped-type response? How would you evaluate the damping ratio? Which one would you choose as the basis of your controller design? R + вЂ“ K c (1 + П„ Ds) 1 s + вЂ“ C R + Kc вЂ“ (1 + П„ I s) П„I s 3 1 s + 10 (a) 276 вЂ“ 3 1 s + 10 Figure PII.5. 1 s + (b) C Part II. Intermediate Problems R + вЂ“ 1 K C (s + 2)(s + 4) Figure PII.6. (e) With the PI controller, what are the different probable ranges of integral time constants that we may have? (f) LetвЂ™s say we have П„ I = 0.3 min. What is the proportional gain such that the dominant poles of the system are equivalent to a damping ratio of 0.9? Can the system behave with a damping ratio of 0.2? (g) What happens to the system when we choose П„ I = 1/3 min? Explain with a hand-sketch. (6) Consider the simple unity feedback loop with a proportional gain K in Fig. PII.6: (a) Sketch the root-locus plot of the system. (b) What is the value of K when the system exhibits critically damped behavior? (c) What is the value of K when the desired system damping ratio is 0.707? Identify the corresponding closed-loop pole(s) on the root-locus plot. (d) What is the steady-state error with the value of K in part (c)? (e) What is the settling time with the value of K in part (c)? (f) What is the value of K if the required system overshoot with respect to a unit set-point change is 10%? (g) If there is to be a load function to the system, what property must the load function have? (7) Consider a simple unity feedback system with the closed-loop characteristic equation: 1+ K c eв€’0.35s = 0. (5.1s + 1)(1.2s + 1) (a) Calculate analytically the exact values of the magnitude and the phase lag on the Bode plot for the speciп¬Ѓc case in which K c = 7.5 and П‰ = 0.8 rad/min. (b) What is the critical gain of the system? What would it be if there is no dead time? (c) What is the proportional gain when the desired gain margin is 1.7? (d) What is the proportional gain when the desired phase margin is 30в—¦ ? (8) Consider a simple unity feedback system with the closed-loop characteristic equation 1 + G c G p = 0 and in which G p is a п¬Ѓrst-order function with a steady-state gain of 2 (unit) and a time constant of 2 min. We now want to implement a PI controller. We are also given the choices of using integral time constants with values of 1, 2, or 4 min. Our task is to design a system with a damping ratio of 3/4. (a) Make a rational choice of the integral time constant. Provide a brief explanation. (b) With your chosen integral time constant, what is the proportional gain that you would use? (9) Consider the closed-loop system in Fig. PII.9. What is the restriction on the value of ОІ (a positive number) if we want an underdamped system? What is the restriction if the system must be stable? What is the offset of this system? 277 Homework Problems R + вЂ“ 2 s(s + 2) Y 1 + Гџs Figure PII.9. (10) Feedback compensation can be achieved with what is called rate feedback, as shown in Fig. PII.10. (a) Derive the closed-loop transfer function of the block diagram. (b) For simplicity, letвЂ™s take K = 1 and П„ = 1. If we desire the system response to have a damping factor of 0.7, what is a proper choice of the value of b? (c) Based on the transfer function (1 + bs), can you say why the term вЂњrate feedbackвЂќ is used? (d) In terms of the damping ratio, what is the general effect of b on the system response? What if b is zero? (11) We have learned that it helps to avoid derivative kick if the derivative control is on the measured variable only. LetвЂ™s do a simple exercise to understand why. Consider the two situations with ideal PD control; Fig. PII.11(a) is the textbook ideal PD control of a п¬Ѓrst-order process, and Fig. PII.11(b) is the so-called rate control in the feedback loop. (a) Sketch the root-locus plots when we use an ideal PD controller, as in Fig. PII.11(a). Should you choose П„ p > П„ D or П„ p < П„ D ? Use the proper choice for parts (b)вЂ“(f). (b) Redo the root-locus plot in which now the controller is now a real PD controller. (c) Sketch the root-locus plot for Fig. PII.11(b). (d) What is the difference in the characteristic polynomial between Figs. PII.11(a) and PII.11(b)? (e) Which system conп¬Ѓguration can eliminate offset? (f) What are the closed-loop transfer functions of the two cases in a servo problem? With what we have learned about the behavior of leadвЂ“lag elements, what is your expected time-domain responses of these two cases with respect to a unit-step change in the reference? (12) We have a system with the closed-loop characteristic equation 1 + Kc П„I s + 1 1 = 0, П„I s (s + 1)(2s + 1) where the time constants are in minutes. We want to design a slightly underdamped system. We face the possibilities of using integral time constants of 0.5, 1.5, or 2.5 min. Any oscillation of the system should exhibit behavior that resembles a R + K вЂ“ s ( П„ s + 1) 1 + bs Figure PII.10. 278 C Part II. Intermediate Problems (a) R+ вЂ“ (b) K c (1 + П„Ds) Kp C П„p s + 1 R + вЂ“ Kp Kc C П„p s + 1 (1 + П„Ds) Figure PII.11. damping ratio of 0.7, i.e., the oscillation may be due to complex poles that are not the dominant closed-loop poles. (a) For each choice of the integral time constant, п¬Ѓnd the proportional gain that may meet the speciп¬Ѓcation with the use of root-locus plots. (b) Among the three choices, which case would you use in your controller design? State your reason. (13) LetвЂ™s consider a unity feedback system with an ideal PD controller and the closedloop characteristic equation 1 + G c G v G p = 0, where G c = K c (1 + П„ D s), G v = 1 1 , Gp = . (12s + 1) (s + 1) The problem seeks to п¬Ѓnd a system with a damping ratio of О¶ = 0.7. We have three probable regions to select our derivative time constant: (1) П„ D < 1, (2) 1 < П„ D < 12, or (3) П„ D > 12. (a) Which region must we choose for the derivative time constant? (b) What is the stability region for different ranges of the derivative time constant and the proportional gain in this problem? We now consider the same problem with a PI controller, G c = K c (1 + П„1I s ). (a) If we seek a system that can provide underdamped behavior, which are the probable region(s) that we should select for the integral time constant? (1) П„ I < 1, (2) 1 < П„ I < 12, or (3) П„ I > 12? You must give a proper explanation to your answer. (b) LetвЂ™s say we have selected П„ I = 0.5. Use the RouthвЂ“Hurwitz criterion to determine the range of K c that provides a stable system. (c) Find the ultimate gain and the ultimate frequency for the case in (b). (d) What is the probable offset in this system? (14) LetвЂ™s consider a system with inherent dead time. We will use only a proportional controller. The closed-loop characteristic equation is 1 + G c G p = 0. The process transfer function is Gp = 3eв€’s . (4s + 1) (a) If you approximate the dead-time function with a п¬Ѓrst-order PadВґe formula, use any analytical method of your choice to п¬Ѓnd the approximate range of proportional gain such that the system is stable. (b) Show how you would п¬Ѓnd the exact ultimate gain with the direct substitution method. (Hint for an initial guess: The nontrivial ultimate frequency lies between ПЂ/2 and 1.8.) (c) Repeat, using a Bode plot. 279 Homework Problems (15) Consider a simple unity feedback system with the equation 1 + G c G p = 0. The process transfer function is G p = K /s 2 . (a) Before we can really design the controller, we need to measure the value of K . Why is it difп¬Ѓcult to evaluate K with a common method such as the open-loop step test? (b) We can measure K in a closed-loop by using a proportional controller with a known value of K c . Show how we may measure K with a step change in the reference. (c) Between a PI and a PD controller, which one should we use? Show, with rootlocus plots (can be hand-sketched), how you would proceed to design the controller of your choice. (16) Controller design based on direct synthesis is not unique. The result depends on our speciп¬Ѓcation of the closed-loop response. The result is also dependent on the assumptions that we make along the way. Take the case of a process function with dead time, G p (s) = K p eв€’td s , (П„ p s + 1) and the desired closed-loop function (C/R) = [eв€’Оёs /(П„c s + 1)]. In Chap. 6, we used a п¬Ѓrst-order Taylor expansion to approximate the exponential term in G c , which we know is not very good unless the dead time Оё is small. (a) If we use the п¬Ѓrst-order PadВґe approximation instead, show that the resulting G c function resembles a real PID controller. Hint: Do not multiply the terms in the numerator. One of them will become the integral and the other the real derivative controller terms. The choice of the integral time constant is similar to other case studies in direct synthesis. (b) The same can be said with the use of IMC in Example 6.5. If we choose the low-pass п¬Ѓlter such that G в€—c is a semiproper function (i.e., r = 2), the resulting G c again resembles a real PID controller. Derive the parameters of the real PID controller. (c) In both parts (a) and (b), how are the parameters associated with the real derivative control term different from what we are familiar with? (17) Consider a unity feedback system with the closed-loop characteristic equation 1 + G c G v G p = 0. Two of the functions are Gp = 2 , (20s + 1) Gv = 0.5 . (s + 1) A root-locus plot of the system has loci that originate from four points, 0, в€’0.05, в€’1, and в€’5, on the real axis, and two of the loci terminate at points в€’0.1 and в€’0.5. (a) What is the speciп¬Ѓc controller being used? Provide proper explanations by sketching the probable root-locus plot of the system. What can you say about the values, where possible, of the controller parameters? (b) What is the order of this system? (c) Illustrate with a root-locus plot the probable locations of the dominant closedloop poles in this system. 280 Part II. Intermediate Problems L (a) L (b) Gf Gf GL GL R + вЂ“ R + вЂ“ Gc 1+ 1 П„I s + + Gp + C вЂ“ Kc + вЂ“ + C + R + + Gp R + вЂ“ П„D s (c) 1 П„I s Gc Gp + C + C + вЂ“ Kc Gp П„D s + 1 (d) Figure PII.19. (18) Consider a unity feedback system with the closed-loop characteristic equation 1 + G c G p = 0. The process and controller functions are Gp = 10 (s + 5) , Gc = K . s(s + 5) (s + 20) (a) Determine the ranges of the gain K that will provide overdamped, critically damped, and underdamped behavior of the closed-loop dynamic response. (b) Based on the closed-loop characteristic polynomial, sketch the probable shape of the root-locus plot. (c) What is the value of the gain K that provides a closed-loop response that has a damping factor О¶ of 0.7? What is the steady-state error of this system with respect to changes in the set point? (19) To implement feedforward control, two schemes in Figs. PII.19(a) and PII.19(b) are proposed to eliminate the effect of disturbances with the transfer function Gf : (a) Derive the closed-loop transfer functions relating changes in R and L to changes in C in both cases. (b) What should Gf be if we want perfect elimination of any load changes, again in both cases? (c) Which case is a more sensible or probable design? And why? (d) We now look at two alternative schemes of implementing PID control: the socalled PI-D and I-PD controllers as shown in Figs. PII.19(c) and PII.19(d), respectively. Derive the closed-loop servo function C/R in each case. (e) How have these two systems changed the characteristic equation as compared with when we use a standard textbook ideal PID controller? (20) A classical example in stability analysis is that a control system can stabilize an open-loop unstable process. Consider Gp = 0.5 sв€’2 and a PI controller with an integral time constant П„ I = 1. What is the range of proportional gain for which the closed-loop system is stable? What is the frequency of 281 Homework Problems oscillation when the system is marginally stable? What is the associated proportional gain? (21) Consider the unstable process transfer function Gp = K sв€’2 and the characteristic equation of a simple control system: 1 + G c G p = 0. (a) What is the stability criterion if we use a PD controller? How is it different from a proportional controller? (b) What is the stability criterion if we use a PI controller? (c) Sketch the root-locus plots for parts (a) and (b). (d) Derive the ultimate gain and the ultimate frequency in parts (a) and (b). (22) For the given open-loop unstable process and load functions, G p (s) = 2 , sв€’4 G d (s) = 0.5 , sв€’4 we know that a proportional controller can stabilize the process in a servo problem. Would you expect the result to be different with respect to load changes? Consider a unity feedback system with the closed-loop characteristic equation 1 + G c G p = 0, but now the regulating problem. (a) For a simple proportional controller, п¬Ѓnd the range of K c that yields stable system responses with respect to load changes. (b) Draw the root-locus plot of the system. (c) What is the ultimate gain of the system? At this position, will the response be stable with respect to a step input and an impulse load input? (d) If we apply the direct synthesis method to design the controller, what is the analytical result and what is your recommendation? (23) Consider a closed-loop system with only a proportional controller; the characteristic equation is given as 1 + Kc 1.2eв€’0.7s = 0. (0.2s + 1)(4s + 1) (a) What is the ultimate gain of this system? The unit of the time constants is in minutes. (b) If you want to use a PID controller, choose the controller settings based on an appropriate empirical tuning relation of your choice. With this setting, what is the gain margin of the system with the PID controller? (c) If you want a gain margin of 1.7 instead, what proportional gain should you use (presumably you keep the same П„ I and П„ D )? (24) Consider the system with cascade control in Fig. PII.24. In this case, we implement only proportional control in both the primary and the secondary loops. We are given R + вЂ“ Kc Figure PII.24. 282 + вЂ“ Kc2 Gv Gp C Part II. Intermediate Problems that Gv = 1 , 5s + 1 Gp = 1 . (s + 1)(0.1s + 1) The cascade controller is designed to speed up the time constant of the actuating valve 10 times. What is the ultimate gain of this cascade system? Explain the stability with and without cascade control with Bode plots. What is the proportional gain that we should use if we desire a gain margin of 2 in the system? (25) We have the following transfer function: G(s) = 18(2s + 1) . (s 2 + 3s + 9)(s + 4) (a) Identify all the corner frequencies and frequency asymptotes on the magnitude and the phase-angle plot of G( jП‰). (b) Write the analytical equations with which we can calculate the magnitude and the phase angle of G(s). (c) If G(s) = Y (s)/ X (s) and X (s) denotes a unit-step input, what is the steady-state value of y(t)? Is there a dominant pole in G(s)? (d) If the transfer function is in a closed-loop system with the characteristic equation 1 + K c G(s) = 0, what should K c be if we want the gain margin to be 2? (26) Consider a system with the closed-loop characteristic equationв€љ 1 + GcG p = в€љ 0. We are given that the open-loop poles are located at в€’0.5, в€’3 В± j 3, в€’6 В± j 5, and в€’9, and there is an open-loop zero at в€’0.5. The steady-state gain of the process function G p is 2. (a) What kind of controller could G c be? What is a probable justiп¬Ѓcation for the zero at в€’0.5? (b) Suppose we want to design the system such that it has a decay ratio of 0.25. What dominant pole(s) must we use as the basis of our calculation? What are the controller settings? (c) Suppose the preceding design is too underdamped and we want a decay ratio of 0.1. What are the controller settings? (d) Sketch the root-locus plot of this system. (e) With respect to system stability, do we have enough information to design a controller that can handle disturbances? (27) A chemical reactor can be cooled by either a cooling jacket or a condenser. The transfer functions are measured as в—¦ T в€’1 C = , Qc s + 1 gpm в—¦ в€’5 C T = , Qj 10s + 1 gpm where Q c and Q j are the cooling water п¬‚ow rates to the condenser and the jacket, respectively, and the time unit is in minutes. The range of the temperature transmitter is 100вЂ“200 в—¦ C, with a corresponding output of 0вЂ“10 mV. Control valves have linear trim and constant-pressure drop and are half-open under normal operating conditions. Normal condenser п¬‚ow is 30 gpm. Normal jacket cooling water п¬‚ow is 283 Homework Problems 20 gpm. The temperature measurement through a thermowell is a п¬Ѓrst-order function with a time constant of 12 s. Our objective is to determine which is the better method to manipulate the temperature. (a) Which cooling method do you prefer? Consider п¬Ѓrst the case of a proportional controller. What are the proportional gains for a closed-loop damping coefп¬Ѓcient of 0.707? (b) With your chosen manipulated variable, design a PID controller with your own design objectives and choice of methods. (28) Consider a system with the closed-loop characteristic equation 1 + G c G v G p G m = 0 and the following transfer functions: G v (s) = 1 1 1 , G p (s) = , G m (s) = . 10s + 1 20s + 1 0.5s + 1 The controller was selected to be G c (s) = K c 1 + 2 1 + s . 3s 3 (a) Sketch the root-locus plot for this system. Find the range of K c that provides stable closed-loop responses. Hint: This problem is tricky. The system is stable for very small and very large proportional gains, but not in between. This is called conditional stable. Being a good control engineer, you, of course, dislike the arbitrary choice of the controller. So letвЂ™s design our own controller. We also want to take advantage of empirical tuning relations. (b) First, п¬Ѓnd a п¬Ѓrst-order with dead-time function that may serve as the process reaction curve function. Compare the time response (with respect to a step input) of the approximation with the original functions. (c) With only a proportional controller, what is the approximate range of the proportional gain that provides us with a stable system? (d) Now we can design a more complete controller for the system. Use either the CohenвЂ“Coon or the ITAE tuning relations. State your reason. For your controller design, provide the controller settings and a time-response sketch with respect to a unit set-point change. (e) How does the addition of derivative control affect the controller settings and the overall system response? (29) A process was pulsed tested and, by use of a fast Fourier transform, the open-loop frequency-response data in Fig. PII.29 were obtained: Identify the open-loop transfer function. If we had performed the frequency-response experiment on a closed-loop system by a pulse input into the set point, what would we be measuring? (30) Consider the model of the liquid level in a stirred-tank, A dh = q0 в€’ q, dt where h(0) = 0. The outlet п¬‚ow rate is kept constant. WeвЂ™ll look into a couple of problems with this integrating process. 284 101 0 100 -50 -100 10-1 -150 10-2 -200 10-3 Magnitude Phase angle Magnitude Part II. Intermediate Problems -250 Phase Angle -4 10 0.1 -300 100 1 10 Frequency, rad/min Figure PII.29. (a) A simple rate feedback scheme is shown in Fig. PII.30. What is the value of K v that provides a critically damped behavior? The cross-sectional area of the tank, A, is 2 (area units). (b) If we increase the value of K v , will the system be underdamped or overdamped? (c) If the set point h sp is increased by 10 (height units), what is the value of K v such that the liquid level h will also change exactly by 10 (height units)? (31) Consider the transfer function that contains integrating action: Y (s) K = . U (s) s(П„ s + 1) We take a sinusoidal input u(t) = sin П‰t, with frequency П‰ = 0.5 rad/s. The input and the output are shown in Fig. PII.31. Estimate the values of K and П„ from the data. Why is it that the mean of the sinusoidal output is 0.5 and not zero? Hint: There is only one term in Y (s) that we really need for this last part. (32) Consider two stirred-tanks with electrical heaters that are connected in series. The system block diagram is shown in Fig. PII.32, where we manipulate the voltage V1 of the heater in the п¬Ѓrst stirred vessel. The controlled variable is the temperature of the second tank. The load variables are inlet п¬‚ow rate Q 0 and voltage to the second tank heater V2 . The transfer functions that we found from previous experiments are K m = 0.15 V/в—¦ C, td = 0.5 min , G 2 = G4 = 2.5 в—¦ C/V 1 в—¦ C/в—¦ C , G5 = . 10s + 1 10s + 1 4 в—¦ C/V , s+1 G3 в€’ в€’5eв€’0.5s в—¦ C/gpm , (10s + 1)(s + 1) All the time constants have units in minutes. H sp 0.5 + вЂ“ + + 2s + 1 Qo Gp H Kv Figure PII.30. 285 Homework Problems 1 y u, y 0.5 u 0 -0.5 -1 0 5 10 15 20 25 Time (s) 30 35 40 Figure PII.31. (a) We п¬Ѓrst use a proportional controller. What is the ultimate gain of the system? What would it be if there were no transport lag between G 2 and G 5 ? (b) What is the proportional gain when the desired gain margin is 2? (c) What is the proportional gain when the desired phase margin is 40в—¦ ? (d) If we want a system that exhibits вЂњslight overshoot,вЂќ what are the PID controller settings that we may select based on the ZieglerвЂ“Nichols ultimate-gain tuning relation? (e) Provide the PID settings based on the process reaction curve and the ZieglerвЂ“ Nichols tuning relation. How different is this result compared with that of part (d)? (f) Provide on the same plot the time-domain responses of the system subject to a unit-step change in the set point when we use PID controller settings based on the CohenвЂ“Coon empirical tuning relation, the settings from part (d), and the settings from part (e). (g) If you have to design a feedforward controller to handle п¬‚uctuations in п¬‚ow rate Q 0 , what is an appropriate function for G f ? (h) Repeat part (g) with a feedforward controller to handle п¬‚uctuations in the variable V2 . (33) Consider the process function Gp = 0.01s 2 0.5 , + 0.04s + 1 where time is in the unit of seconds. We need to design a control system that cannot permit steady-state error and excessive underdamped behavior. On this note, we aim for a slightly underdamped system that, if it exhibits oscillations, is at Qo V2 G3 R + Km вЂ“ Gc G2 Km Figure PII.32. 286 G4 V1 вЂ“t d s e G5 + + + + C Part II. Intermediate Problems Magnitude 10 1 0.1 0.1 0 1 10 100 10 100 Phase -100 -200 -300 -400 -500 0.1 1 Freq (rad/s) Figure PII.34. most equivalent to a damping ratio of 0.707. We use a simple unity feedback system with the characteristic equation 1 + G c G p = 0. Also, for the sake of this exercise, we have our choice of using П„ I = 1/8 or 1/4 s and, similarly, П„ D = 1/8 or 1/4 s. (a) With respect to the function G p , what is the time constant? (b) With root-locus plots, illustrate how the choices of P, PI, and ideal PD controllers may inп¬‚uence the system time response. (Identify clearly the open-loop poles and zeros and the direction of root loci.) Provide reasons for controllers that you want to eliminate for this system design. (c) Demonstrate your choice of an ideal PID controller design with a root-locus plot. What are the settings of the controller? What is (are) the dominant pole(s) in this system? (34) A transfer function is used to generate the Bode plot in Fig. PII.34. Identify the transfer function and evaluate all parameters that are associated with this transfer function. It is also given that the corner frequency is 2.5 rad/s and, at a frequency of 1 rad/s, the magnitude is 4.9 and the phase lag is в€’33в—¦ . (35) With respect to Fig. PII.35, which represents a unity feedback system with the closed-loop equation 1 + G c G p = 0, derive the state-space model given that G c = K c (1 + 1/П„ I s) and G p = K p /(П„ p s + 1). Find the characteristic polynomial of the closed-loop system matrix A. The PI controller function can be rearranged in the system block diagram, as shown in Fig. PII.35. The locations for the two state variables are also identiп¬Ѓed as shown. (36) Derive the state-space model of the unity feedback system in Fig. PII.36, where now G c = K c (s + z/s + p) is a leadвЂ“lag compensator and G p = [K /(s + a)]. R + вЂ“ Kp 1 П„I s X2 + + Kp П„p s + 1 X1 = Y Figure PII.35. 287 Homework Problems R + вЂ“ zвЂ“p s+p Kc X2 + X1 = Y K s+a + Figure PII.36. The leadвЂ“lag element can be rearranged as zв€’p s+z =1+ , s+p s+p as shown in Fig. PII.36, such that the function is represented by proper fractions. (37) Derive the state-space model for a system with a second-order process and a PID controller, as shown in Fig. PII.37. (38) In Example 8.10, we discussed how the integral time constant must be chosen such that it is larger (slower) than the process time constants if we are to stabilize the system. This is an exercise to test this hypothesis. Consider a simple unity feedback system with the closed-loop characteristic equation 1 + G c G p = 0, where Gp = 2 (4s + 1)(5s + 1) and G c is a PI controller with either (1) П„ I = 1 or (2) П„ I = 10. All time constants have the unit of minutes. (a) Derive the stability criterion with the Routh array and using general notation for П„I . (b) Provide the stability criterion with two speciп¬Ѓc values of the integral time constant: П„ I = 1 and П„ I = 10. (c) Provide the root-locus plots for both cases of integral time constants. Indicate clearly if the system may become unstable or not. (d) When П„ I = 1, what is the ultimate gain? If we choose a proportional gain such that the gain margin is 2, what is the phase margin under this controller setting? (e) Repeat part (c) with П„ I = 10. (f) When П„ I = 1, what is the proportional в€љ gain that we must choose such that the system has a damping ratio of 1/ 2? What is the dominant closed-loop pole? R + 1 П„I s Kc вЂ“ 1/ О± Figure PII.37. 288 X4 + + X3 + + 1/ П„ D вЂ“ 1/ О±П„ D s + 1/ О±П„ D 1 П„2 s + 1 X2 Kp П„1s + 1 X1= Y Part II. Intermediate Problems (g) Repeat part (e) with П„ I = 10. (h) Provide the Bode plots for both cases of П„ I = 1 and П„ I = 10. Make use of corner frequencies and the high- and low-frequency asymptotes to explain the stability features of this problem. (39) In Subsection 3.4.3, we gave up further analysis of the model of two interacting tanks-in-series, saying that state-space representation is better for handling such problems. This is our chance to give that a try now. LetвЂ™s say we have two tanks such that A1 = 5 m2 , A2 = 2 m2 , and R1 = R2 = 1 min/m2 . (a) Compute state-space model matrices in which h 2 is the only output. (b) Conп¬Ѓrm that the eigenvalues of A are the same as the poles of characteristic equation (3.46). (c) Compute the transfer function and use that to п¬Ѓnd for a classical proportional control system such that the damping ratio is 0.7. (d) With the closed-loop poles taken from part (c), п¬Ѓnd the state feedback gain. You should п¬Ѓnd it to be identical to classical control. (e) With classical PI control, use root-locus plots to show that П„ I = 0.5 min is a reasonable choice for the control system. Find the proportional gain such that the damping ratio is 0.7. (f) Find the corresponding state feedback gain by using state-space integral control. (g) Do the time-domain simulation for the state-space system with and without integral control. (40) A classic example of MIMO problem is a distillation column. From open-loop step tests, the following transfer functions are obtained: пЈ® пЈЇ X D (s) пЈЇ =пЈЇ пЈ° X B (s) 0.6 (7s + 1)2 в€’0.5eв€’0.5s (7s + 1)2 0.3eв€’0.5s (16s + 1)(0.5s + 1) в€’0.4 (14s + 1)(0.4s + 1) пЈ№ пЈє L(s) пЈє . пЈє пЈ» V (s) In this model, x D and x B are the distillate and the bottom compositions, respectively, L is the reп¬‚ux п¬‚ow rate, and V is the boil-up rate. We want to design a 2 Г— 2 MIMO system with only PI controllers. (a) From the relative gain array, determine the proper pairing of variables. Draw the block diagram and set up the system with Simulink. (b) To begin the design, we need the controllers based on a simple SISO system п¬Ѓrst. For each of the two controllers, what are the proportional gains if the gain margin is to be 2? (c) What is the proportional gain if the phase margin is to be 45в—¦ ? (d) What are the PI controller settings based on the ZieglerвЂ“Nichols tuning relations? (e) How should we detune the controllers when we implement them in the MIMO system? (f) Determine how effective the use of decouplers is, as in Fig. 10.14, in reducing loop interactions. 289 Homework Problems Amplifier Valve Plant chamber R Km + (ppm) вЂ“ Gc K amp Gv Gm вЂ“t ds Gp CO2 (ppm) e Gas analyzer Sampling line Figure PIII.1. Part III. Extensive Integrated Problems (1) To address one aspect of the accumulation of greenhouse gases, we want to build a chamber that can be used to study the level of CO2 on the growth of plants on a space shuttle. In the system design, there is a time delay of 0.75 min in the sampling line. The sensor is a gas analyzer that can measure CO2 content from 0 to 600 ppm linearly and the output is 0 to 5 V. The gas analyzer can be modeled as a п¬Ѓrst-order function with a time constant of 0.1 min. The plant chamber itself can be modeled as a well-mixed reactor (CSTR) with a pseudo-п¬Ѓrst-order reaction. This process function has a steady-state gain of 0.23 (ppm min)/ml and a time constant of 5 min. Under normal circumstances, we would use a solenoid valve. For the present problem, we take a simpler approach. We use instead a regulating valve that can be modeled as a п¬Ѓrst-order function with a time constant of 0.02 min and a steady-state gain of 0.2 ml/(min mV). An ampliп¬Ѓer is needed to drive the valve, and it has a gain of 10 mV/mV. The block diagram is shown in Fig. PIII.1. (a) Design a proportional controller with a gain margin of 1.7. (b) What if we want a PID controller? Provide a system design that gives us an acceptable underdamped behavior, say an apparent (because the system is not second order) damping ratio of 0.45. (2) (a) We are assigned to identify a process that is believed to be second order. LetвЂ™s presume that the magnitude and the phase plots in Fig. PIII.2(a) are obtained from a pulse experiment. 100 Magnitude 10 1 0.1 0.01 0.1 0 1 10 Phase -50 -100 -150 -200 0.1 (a) Figure PIII.2a. 290 1 Frequency (rad/s) 10 Part III. Extensive Integrated Problems Magnitude 100 10 1 0.1 0 0.1 1 10 0.1 1 Frequency (rad/s) 10 Phase -50 -100 -150 -200 (b) Figure PIII.2b. We are also given the information that from a unit-step response experiment, the process exhibited a 25% overshoot. Estimate the parameters K p , П„ , and О¶ of the transfer function: Kp . G p (s) = 2 (П„ s + 2О¶ П„ s + 1) (b) The second-order process in part (a) is put in a closed-loop system with a controller. The closed-loop characteristic equation is a simply 1 + G c G p = 0, and the Bode plot of the open-loop transfer function of this system is given in Fig. PIII.2(b). What is the controller (P, PI, or PD) being used? Explain your decision with hand-sketched Bode plots with features such as low- and highfrequency asymptotes and corner frequencies. It would be difп¬Ѓcult to read the controller time constants from the п¬Ѓgure, but we should be able to estimate the order of magnitude of the time constants, especially if they are smaller or larger than that in the process transfer function. (c) Repeat part (b) with Fig. PIII.2(c). 104 Magnitude 1000 100 10 1 0.1 0.01 0.01 -80 0.1 1 10 1 10 -100 Phase -120 -140 -160 -180 -200 -220 0.01 0.1 Frequency (rad/s) (c) Figure PIII.2c. 291 Homework Problems (3) We want to design a control system for a gas absorber to reduce SO2 emission by manipulating the water п¬‚ow rate. The gas absorber can be modeled by the transfer function C(s) в€’0.05 = , F(s) (2s + 1) where, in the time domain, c(t) is the efп¬‚uent SO2 concentration as deviated from its nominal operating value of 100 ppm, and f (t) is the water п¬‚ow rate as deviated from its nominal operating value of 250 gpm. Time is measured in minutes. The system has the closed-loop equation 1 + G c G a G p G m = 0. (a) For the moment, we are told to ignore the dynamics of the actuator and measurement transfer functions, hence taking G m = K m = 1 mV/ppm. However, we have the choice of G a = В±1 gpm/mV at this moment. We are using a controller with positive proportional gain. Should G a be +1 or в€’1? Explain your choice. (b) We will stay with the choice of G a in part (a) and G m = 1 mV/ppm, but now a PI controller, still with positive proportional gain, is to be used. The integral time constant is 0.5 min. What must be the choices of the proportional gain such that the closed-loop system is stable? (c) Repeat part (b), but now we have an ideal PD controller instead. The derivative time constant is also 0.5 min. What must be the choices of the proportional gain such that the closed-loop system is stable? (d) We want a closed-loop system response that is underdamped. The system should have a dynamic response equivalent to a damping ratio of 0.7. Select the proper controller from either part (b) or part (c). Estimate the proportional gain that can conform to the damping ratio speciп¬Ѓcation. If you have more than one choice, select one and explain why. For the given closed-loop poles that you have selected, what is the corresponding natural frequency? (e) Now reality strikes! We can no longer ignore the measurement function. The voltage output of the gas analyzer was previously calibrated, and the result in actual measurement variables is v = 0.45 + 1.2 [SO2 ], where the voltage output is in millivolts and the concentration of SO2 is in parts per million. The gas analyzer is extremely fast, but the sampling line has a dead time of 0.2 min. The actuator, which really is a valve, has a steady-state gain of В±0.9 gpm/mV and exhibits a п¬Ѓrst-order response with a time constant of 0.3 min. [The sign should of course be the same as the one determined in part (a).] Now the EPA is tightening up its regulation, and we need to lower the outlet SO2 concentration. Use the controller settings that we have obtained in part (d) and provide a time-response simulation plot representative of the system subject to a step change of в€’10 ppm in the set point. (4) LetвЂ™s demonstrate our understanding and command of our control theory with the following experiment, which is on a torsion disk, but may just as well be a system 292 Position, count/2000 Part III. Extensive Integrated Problems 1.6 1.4 1.2 (a) 1 0.8 0.6 0.4 0.2 0 0 0.5 1 Time (s) 1.5 2 0.12 (b) 0.1 u, y 0.08 y(t) 0.06 0.04 0.02 u(t) 0 0 0.5 1 Time (s) 1.5 2 Figure PIII.4. with a п¬‚ow vessel with п¬Ѓxed-rate pumps. The position (angle rotation) of the disk is measured in the arbitrary unit of counts. If you need an introductory explanation of mechanical models, see the Web Support. r An open-loop step test resulted in a response that was ramplike. A closed-loop servo experiment with only a proportional controller and K c = 0.02 and a set-point change of 2000 counts resulted in an underdamped system response, and there is no offset. The response is shown in Fig. PIII.4(a). (a) What is your interpretation at this point with respect to the two experiments? r Based on the п¬Ѓrst two experiments, a suggestion was made that an open-loop frequency-response experiment be performed to identify the process function. We applied a sinusoidal input with an amplitude of 0.1 V and different frequencies to the actuator (a dc motor), and the application software returned the response in counts. We gathered from the responses that, at very high frequencies, the phase lag appeared to be approximately в€’180в—¦ . There appeared to have been a phase lag even at low frequencies. A simulated plot in which the input frequency was 2 Hz is shown in Fig. PIII.4(b). The output of the computation is scaled arbitrarily in this plot. Note that the mean of the large time response is not zero, as opposed to the input sine wave. (b) Now, our immediate task is to identify the open-loop function. Write the probable form of the transfer function G p (really the lumped G a G p G m ). Show that the mean of the sinusoidal time response is not zero and that it can be used to estimate the process gain. (c) Sketch a block diagram of the system. We can assume very fast dynamics in the dc motor (the actuator) and the encoder (the position sensor). Derive the closed-loop transfer function for servo control. (d) Because of limitations in the application software, we cannot estimate the measurement gain and actuator gain easily. Rearrange the block diagram to a form 293 Homework Problems (e) (f) (g) (h) such that we can lump the K m together with the process and actuator steady-state gains. For your proposed (lumped) G p , п¬Ѓnd the time constant of the process based on the data of the open-loop frequency-response experiment in Fig. PIII.4(b). Show how you could have used the magnitude of the proposed (lumped) G p in helping to estimate the (lumped) steady-state gains from the open-loop frequency-response experiment. Now with the closed-loop step-response data [Fig. PIII.4(a)], estimate the lumped steady-state gain (K m K a K p ). Show how, in the closed-loop step-response data, the overshoot, settling time, and time to peak are consistent with the time period, damping ratio, and lumped steady-state gain of your proposed closed-loop transfer function. r At this point, it was apparent that, because of the difп¬Ѓculty in analyzing data accurately, the estimated values of the open-loop and the closed-loop experiments were not in perfect agreement. Two more experiments were proposed and performed to resolve the discrepancies. (i) (j) (k) (l) 294 First, we used a PD controller in a closed-loop experiment. We picked a reasonable proportional gain and varied the derivative time constant. What we observed was that when П„ D was roughly 0.2 s or larger, the servo system did not exhibit any oscillations even as we increased the proportional gain further. When П„ D was roughly 0.1 s or less, the servo system exhibited underdamped behavior. (Yes, you could have guessed that the вЂњanswerвЂќ lies somewhere in between.) Second, we did a closed-loop frequency response experiment вЂ“ we imposed a sinusoidal wave (with an amplitude of 1000 counts) on the set point and observed the response of the controlled variable. We used a proportional controller with K c = 0.02. When the input sine wave had a frequency as low as 0.2 Hz, the amplitude of the closed-loop response (the amplitude ratio) relative to the input amplitude was approximately one and the phase lag was virtually zero. When the input sine wave had a frequency of 4 Hz, the amplitude ratio was only approximately 0.7. However, when we used an input frequency of 2 Hz, the ratio was almost 2! With root-locus plots, show how varying П„ D with a PD controller could have helped to resolve the discrepancies. What time constant could we have been able to identify and estimate in this experiment? What is the probable explanation of the three closed-loop frequency-response experiments? With the numbers that you have estimated regarding the system with a proportional controller, show that, at a frequency of 2 Hz, the observed result is indeed probable. If we had also measured the phase lag in the closed-loop frequency-response experiments, how could we have made used of the data? Write the appropriate magnitude and phase-angle equations and explain how we could apply them. Finally, we are ready to design the controller. What controller should we use? Can we use direct synthesis? Part III. Extensive Integrated Problems air/ammonia water feed air/ammonia feed water/ ammonia Figure PIII.5. (m) For the proper choice of controller, sketch Bode plots of the open-loop transfer function of your closed-loop system. How important is the application of the Nyquist stability criterion in your design? Of course we do not know the values of П„ I and/or П„ D yet. We can consider only the relative value with respect to П„ p . (n) Consider a design speciп¬Ѓcation of a system damping ratio of 0.7; provide an acceptable controller design, assuming saturation is not a problem. (o) I took my result of K c and П„ D = 0.05 s and tried it for real. The data showed approximately a 10% overshoot or the equivalent of a damping ratio of 0.6. How should I п¬Ѓeld tune the controller to achieve О¶ = 0.7? (5) Consider a gas absorber that is designed to remove NH3 from an air stream (Fig. PIII.5). The gas stream entering the absorber contains air and NH3 . The design speciп¬Ѓcation is such that the outlet NH3 concentration in the vapor phase must be 50 ppm or less. Available to us is an electronic concentration transducer that is calibrated to be linear over the range of 0вЂ“200 ppm NH3 and with an output range of 4вЂ“20 mA. This fancy device has a negligible time lag. There is also an air-actuated valve that, when fully opened and for the 10-psi pressure change that is available, will allow for a throughput of 500 gpm of water. You have your choice of an air-to-open or an air-toclose valve. The time constant of the valve is 5 s. Also available is an I/P transducer with which, for a change of 4 to 20 mA, the output changes from 3 to 15 psi. If you need more instrumentation to complete the design, you have permission from your manager to purchase whatever you need. (a) Draw the instrument diagram of a control loop that can maintain the outlet NH3 concentration at a set point of 50 ppm. (b) Draw the block diagram for the closed-loop system. Identify your manipulated, controlled, measured, and disturbance variables and their units. With the exception of the absorber itself, provide the transfer function for each block. We now need to п¬Ѓnd the process transfer function for the absorber. Several open-loop step tests were performed, and after averaging the data, we obtained following table: 295 Homework Problems Data for the NH3 concentration in the outlet air stream in response to a step change of the water п¬‚ow rate that dropped from 250 to 200 gpm Time (s) NH3 (ppm) Time (s) NH3 (ppm) 0 20 30 40 50 60 70 80 50.00 50.00 50.12 50.30 50.60 50.77 50.90 51.05 90 100 110 130 160 220 51.20 51.26 51.35 51.55 51.70 51.77 (c) Identify the absorber process transfer function with an appropriate model of your choice. (d) Specify the action of the control valve and the controller (i.e., positive or negative gains). (e) With proper assumptions or justiп¬Ѓcations, п¬Ѓnd the PID controller settings, using an empirical tuning relation of your choice. (f) From a time-domain simulation of the control system (using a unit-step change in the set point), п¬Ѓnd the percentage of overshoot of the system. (g) If the design speciп¬Ѓcation is such that no offset is acceptable and the percentage of overshoot should not exceed 10%, provide one new controller setting that can meet the requirement. Provide justiп¬Ѓcation of your design and a conп¬Ѓrmation with a time-response simulation. (6) The temperature of a chemical reactor with a very exothermic organic synthesis is monitored by a thermocouple with an ampliп¬Ѓer. The reactor can be cooled by a water jacket and/or a condenser that traps the vapor and returns the condensed liquid reп¬‚ux (Fig. PIII.6). Our task is to design a temperature controller system for the reactor. This is some of the information that we have. r From an experiment involving the chemical reactor and the cooling jacket, we found that changes in the reactor temperature with respect to changes in the cooling Vapor Water for condenser TT Water for cooling jacket Figure PIII.6. 296 Part III. Extensive Integrated Problems r r r r water п¬‚ow rate is a п¬Ѓrst-order function. The steady-state gain is в€’5 в—¦ C/gpm. The process can reach a new steady state with respect to a step change in п¬‚ow rate in в€ј10 min. From another experiment involving the chemical reactor and the condenser, we found the change in reactor temperature with respect to changes in the condenser water п¬‚ow rate to be п¬Ѓrst order as well. The steady-state gain is в€’0.8 в—¦ C/gpm. The time constant is в€ј2 min. The span of the temperature transmitter is calibrated within the range 70вЂ“120 в—¦ C for a full-scale output of 0вЂ“5 V. The thermowell introduces a п¬Ѓrst-order lag of 15 s. The regulating valves are similar in design. They have a linear trim and constantpressure drop. They can be described by a п¬Ѓrst order function with a time constant of 30 s. They are half open under normal operation conditions: The normal water jacket п¬‚ow rate is 10 gpm, and the normal condenser water п¬‚ow rate is 5 gpm. The controller has a full-scale output of 0вЂ“5 V, which is designed to drive the entire range of the regulating valve. The bias signal is 2.5 V. (a) Make a proper decision as to how you need to control the reactor temperature with a SISO system. Draw the instrumental diagram and the corresponding block diagram. (b) With the understanding that the reaction is extremely exothermic, what are your choices of instrumentation (controller and regulating valve) in terms of their actions? (c) Make sure you remember to identify the units on the block diagram. Determine the necessary transfer functions. Identify the probable disturbances. (d) If you use only a proportional controller, use an analytical method of your choice to п¬Ѓnd the stability limit on the proportional gain. (e) With a root-locus plot, conп¬Ѓrm your result in part (d). In addition, п¬Ѓnd the proportional gain that you need to use if you desire the system to have a dynamic response with a 5% overshoot. (f) If you want to use an empirical tuning relationship for a PID controller setting, provide a reasonable approximation of the process reaction curve function. Provide the controller settings based on the tuning relations of CohenвЂ“Coon, ZieglerвЂ“Nichols, and ITAE. (g) Can we make use of the results of IMC? If so, what is the corresponding PID controller setting? (h) Back to part (d). We know there are other methods to arrive at the same answer. Conп¬Ѓrm your value of ultimate gain with a proportional controller by using frequency-response techniques. (i) With the ultimate gain known, what would be probable PID controller settings for our system if we desire a system response with a вЂњslightвЂќ overshoot? (j) LetвЂ™s use an ideal PID controller and the integral and derivative time constants that we have chosen in part (i). What proportional gain should we use if the design speciп¬Ѓcation is to have a gain margin of 1.7? (Hint: This is a trick question.) (k) Provide the time-response simulations of the servo system with a unit-step change in the set point. Use three different, but in your opinion most reasonable, 297 Homework Problems Intake manifold Catalytic converter Air Exhaust Engine Actuator Fuel Sensor (a) Figure PIII.7a. PID controller settings based on previous results. Put all simulations on the same plot. Select the most вЂњdesirableвЂќ closed-loop system response with which you may do the п¬Ѓnal п¬Ѓeld tuning. State your reason. (7) To meet emission standards, the fuel-to-air (F/A) ratio of a combustion engine must be metered precisely. Fuel-injection systems are designed not only to accommodate engine combustion performance, but also to achieve a desired F/A ratio for the catalytic converter to function properly. The catalyst is ineffective when the F/A ratio is more than 1% different from the stoichiometric ratio of 1:14.7. A feedback control loop must be implemented. The F/A ratio is monitored by a sensor before the catalytic converter, and, based on the measurement, the fuel п¬‚ow rate is adjusted accordingly [Fig. PIII.7(a)]. The most difп¬Ѓcult task in designing the control system is due to the nonlinearity of the sensor. Made of zirconium oxide, almost all the change in output voltage occurs near the F/A value at which the feedback system must operate. The gain of the sensor falls off quickly for F/A excursions just slightly away from 1:14.7 [Fig. PIII.7(b)]. In addition to the very nonlinear gain, the sensor exhibits a п¬Ѓrst-order lag with a time constant of 0.1 s. There is also a time delay in the system from the air intake manifold to where the sensor is located. The value depends on driving conditions, but on average, we can take the transport time delay to be 0.2 s. Physically, the engine could be driven by a fast fuel п¬‚ow in the form of vapor or droplets and a slow fuel п¬‚ow in the form of a liquid п¬Ѓlm on the manifold wall. The air manifold and the engine can be approximated altogether with the linear model in Fig. PIII.7(c). The time constants can change considerably as a function of engine load and speed, but for this exercise, we will take П„1 = 0.02 s and П„2 = 1 s. (a) Draw the block diagram of the control system with proper labels. Sensor output, V 0.9 0.5 0.1 1:18 1:14.7 1:12 Fuel-air ratio, F/A (b) Figure PIII.7b. 298 Part III. Extensive Integrated Problems Fast fuel Fuel flow rate 1 0.5 П„1 s + 1 F/A ratio + 1 0.5 + П„2 s + 1 Slow fuel (c) Figure PIII.7c. (b) Based on the information provided, select a proper mode of controller and state your reasons. (c) Design the controller. Use at least two different methods in your analysis. State your assumptions and design speciп¬Ѓcations. (8) Consider the temperature control of a furnace in which we manipulate the fuel-gas п¬‚ow rate. The temperature transmitter with a 0вЂ“1 V analog output is calibrated for 300вЂ“500 в—¦ C. The open-loop step-test data of the furnace temperature (Fig. PIII.8) were measured with a step change on the manipulated variable that is equivalent to a 5% increase of the biased output of the controller. (a) Draw the block diagram of the system with proper labels and units. Based on the information given, provide probable transfer functions of each block or of blocks with transfer functions lumped together. (b) Design an appropriate controller using IMC. If in theory we dictate a п¬Ѓrst-order response, why is the system response underdamped? (c) If the design criterion is a phase margin of 30в—¦ , what controller settings would you adjust and what is the п¬Ѓnal result? (d) Finally, select a different controller setting using an empirical tuning relation. (e) Perform time-response simulations. Which design do you prefer? What is your criterion? (9) An exothermic chemical reactor (CSTR) can be open-loop unstable. The chemical reaction rate increases as the temperature shoots up, and subsequently more heat is given off. This heats the reactor to an even higher temperature such that the reaction rate is still faster and even more heat is evolved. If this process continues, what we have is an explosion. Thus open-loop instability means that the reactor temperature will take off when there is no feedback control of the cooling rate on the reactor. We do a simpliп¬Ѓed analysis on this problem. We must п¬Ѓrst assume that any changes in the reactant concentration can be neglected, i.e., we take the reactant 450 T, В°C 445 440 435 430 425 420 0 5 10 Time, min 15 20 Figure PIII.8. 299 Homework Problems concentration C A as a constant. Hence we consider only the energy balance in this simpliп¬Ѓed reactor model: dT (в€’ H R ) в€’E/RT U At F k0 e CA в€’ = (Ti в€’ T ) + (T в€’ Tc ), dt V ПЃC p ПЃC p V where Tc is the cooling jacket temperature that we can manipulate. Other notations take on their usual meanings. (a) Derive the process transfer function that relates the reactor temperature to the cooling jacket temperature, assuming that we have no п¬‚uctuations in the inlet temperature Ti . (b) Derive the criterion that provides the condition under which the chemical reactor is open-loop stable. What is the physical signiп¬Ѓcance (interpretation) of the result? (c) For the set of conditions given below, is the reactor open-loop stable? (d) Open-loop stable or not, with a proportional controller, is there a criterion on the controller gain that you must satisfy to stabilize the system? If yes, what is it? (e) Discuss qualitatively how you would choose a controller (P, PI, PD, or PID) for the reactor. What would you consider as important dynamic criteria for the control system? (f) Suppose we want to raise the temperature of the reactor by 5в—¦ . Provide the design and time response of a feedback control system that meets your design criteria stated in part (d). Reactor parameters F = 0.1 m3 /min, V = 1.4 m3 , At = 230 m2 k0 = 7 Г— 1010 sв€’1 E = 32 Kcal/gmol R = 2 cal/gmol K U = 0.062 Kcal/m2 K s (в€’ H R ) = 16,000 Kcal/kgmol ПЃ = 800 kg/m3 C p = 0.95 Kcal kgв€’1 Kв€’1 Reactor steady-state values T = 600 K C A = 0.8 kgmol/m3 (10) We consider here a simpliп¬Ѓed fermentor classical control system (as opposed to that of Example 4.8). We all agree that it would be best to use the sugar concentration S as the controlled variable, and use the volumetric п¬‚ow rate Q as the manipulated variable. Any change in inlet concentration Si is taken as disturbance. To make this problem a SISO system, we have to assume that the cell mass is a constant. This is possible only if we somehow trap the cells in the reactor and if they somehow stop growing. The simpliп¬Ѓed fermentor model is now dS Вµm C S = D(Si в€’ S) в€’ , dt Yc/s K m + S 300 Part III. Extensive Integrated Problems with the initial condition at t = 0 and S = S0 . The dilution rate D (deп¬Ѓned as Q/V ) is the reciprocal of space time. Because we are manipulating the п¬‚ow rate, it actually helps to clean up our algebra by deп¬Ѓning a variable that is directly proportional to Q. The fermentor volume is constant. Our job is to design a controller that can maintain the sugar concentration at a speciп¬Ѓed value without steady-state errors. We can tolerate only slight oscillations. After any change in operating conditions, we want the fermentor to reach a new steady-state in less than 7 h. (Fermentation is a slow process, but still we want the job done before the plant operator has to go home.) The regulating valve is rather slow. The sugar sensor is worse. Their transfer functions with proper units for their steady-state gains are Gv = 3 , 0.06s + 1 Gm = eв€’0.15s . 0.24s + 1 As for the fermentor, we have the following data: Si (steady-state) = 10 g sugar/L Ys = 0.4 g cell/g sugar Вµm = 0.4 hв€’1 K m = 0.05 g/L In developing the process model, we must pick some nominal steady-state value about which the fermentor is being operated. A good choice is to specify 95% conversion of the inlet sugar concentration. Thus we can calculate the steadystate sugar concentration. At steady state, D and C are given by the steady-state solutions D= Вµm S , Km + S as it is also true that C = Yc/s (Si в€’ S). (We could not have derived these equations without the cell mass balance. If you really want to know how they came about, revisit Example 4.8.) (11) Consider a rectilinear system that is naturally underdamped. In simple terms, it is a mass and spring system with a fraction. Using an input of 0.2 V to the actuator, we obtained the open-loop data in Fig. PIII.11. We also performed a closed-loop Change in position, cm 3 2.5 2 1.5 1 0.5 0 0 0.5 1 1.5 2 t, sec Figure PIII.11. 301 Homework Problems experiment with a proportional controller and K c = 1 and found the offset to be 40%. (a) Estimate the approximate G a G p G m as a second-order function. The sensor is extremely fast and we can take G m = K m . (b) The function in part (a) is only an estimate because the actuator dynamics is so fast that it is masked by the slower process in the open-loop step test. One strategy to п¬Ѓnd the actuator function is to п¬Ѓnd the ultimate gain and frequency with a closed-loop experiment and from that back out what the actuator should be. Here we may assume that the actuator (a dc motor) has the function G a (s) = 1 . П„a s + 1 The steady-state gain in G a is written as 1, but its actual value is really contained in the data that we used to obtain the result in part (a). We did the closed-loop ultimate-gain experiment and found K cu = 1.65 and w cg = 36.6 rad/s. Find what П„a should be. (c) Do an open-loop step-test simulation to show that indeed the dynamics of the actuator was masked by the process in Fig. PIII.11. (d) Use a method of your choice and design an ideal PID controller. Can the openloop zeros be complex numbers? State your design speciп¬Ѓcation and conп¬Ѓrm the design with time-response simulations. (12) We have three CSTRs-in-series (Fig. PIII.12), and we want to control the concentration of the third CSTR. In terms of reaction engineering, a mole fraction is the more sensible variable, but we stay with concentration so that the model equations are easier to interpret in this problem. In general, the balance of a reactant in a CSTR with a second order reaction is V dC = Q(C0 в€’ C) в€’ V kC 2 . dt The rate constant k is 1.5 m3 gmolв€’1 minв€’1 and volume V at 2 m3 is the same in all three reactors. The inlet steady-state п¬‚ow rate Qs is 1 m3 /min. In your work, use the notation П„ = V /Q s . (a) Consider only one CSTR and derive the transfer functions associated with Q and C0 . What are the time constants and steady-state gains? (b) Derive the transfer functions of all three CSTRs and put them in numerical form. (c) You want to design a unity feedback servo system with which you can control the outlet concentration in the third CSTR, c3 (t). The characteristic equation is Q, c 0 c1 c2 V1 c3 V2 V3 Figure PIII.12. 302 Part III. Extensive Integrated Problems (d) (e) (f) (g) (h) (i) simply 1 + G c G p = 0. Make your proper choice of manipulated, controlled, and load variables and derive the process transfer function G p . If you use only a proportional controller, п¬Ѓnd the range of proportional gain that is necessary for a stable system. Find the proportional gain if the gain margin is 2. You want to design a PI controller, and you are given the choice of П„ I = 0.5 min or П„ I = 2 min. Which choice of integral time constant is more likely to lead to a stable system? Provide proper explanation. You use either root-locus plots or frequency-response analysis. Choose the integral time constant (П„ I = 0.5 or 2 min) that can give us a more stable system. Find the proportional gain of this particular system if the gain margin is 2. If we want this system with a PI controller to have a damping ratio of 0.7, what is the proportional gain? Show on a plot that the process function may be approximated with a п¬Ѓrst-order with dead-time function that has a dead time of 0.5 min and a time constant of 2 min. Calculate the PI controller settings using the ZieglerвЂ“Nichols, ITAE, and CianconeвЂ“Marlin tuning relations. Compare the time responses with a unit-step input for the set point. We look into the rationale of the choice of integral and derivative time constants in this problem. Use the following so-called interacting PID controller function for the remainder of the problem: G c (s) = K c 1 + (j) (k) (l) (m) (n) (o) 1 (1 + П„ D s). П„I s To begin, use the ZieglerвЂ“Nichols ultimate-gain tuning relation to п¬Ѓnd the PID controller settings for the case of one-quarter decay ratio system response. You want to п¬Ѓnd out if it is desirable to use this integral time constant as opposed to using an integral time constant of 0.5 min. Use a derivative time constant of 0.3 min in both cases. Use Bode plots to explain why one of the two choices can stabilize the system. State clearly and explain the crucial corner frequencies and, for the function G c G p , the overall low- and high-frequency asymptotes, and the one singularly most important feature that distinguishes the stability of the two cases. Sketch the root-locus plots for the two cases. In terms of dynamic response, identify what is considered as the dominant pole(s) of the system. Identify the ultimate gain where appropriate. The one-quarter decay ratio setting is of course too underdamped. If we insist on using the same integral and derivative time constants,в€љ what is the proportional gain that provides a system with a damping ratio of 1/ 2? Conп¬Ѓrm your calculations with a time-response simulation of a unit-step change in the set point. Put the three CSTRвЂ™s in a state-space model. Consider the case of only one input, c0 . Show that the step responses of c3 with respect to a step change in c0 are identical whether we use a transfer function or state-space representation calculations. 303 Homework Problems (p) Show that this model is both controllable and observable. (q) Compute the state feedback gains if the closed-loop poles are identical to what we obtained in part (d). Do a time-response simulation to show that the design is identical to that of classical control. (r) Design the state feedback controller you obtain if you add integral action. (s) Find the estimator gains if you want to implement a full-state estimator. 304 References and Suggested Reading Only a handful of introductory control textbooks in chemical processes is given here. A more thorough reference section with annotation is available on the Web Support. 1. A. B. Corripio. Design and Application of Process Control Systems, 2nd ed. Research Triangle Park, NC: ISA, 1998. 2. D. R. Coughanowr. Process Systems Analysis and Control, 2nd ed. New York: McGraw-Hill, 1991. 3. M. L. Luyben and W. L. Luyben. Essentials of Process Control. New York: McGraw-Hill, 1997. 4. T. E. Marlin. Process Control: Designing Processes and Control Systems for Dynamic Performance. New York: McGraw-Hill, 1995. 5. B. A. Ogunnaike and W. H. Ray. Process Dynamics, Modeling and Control. New York: Oxford University Press, 1994. 6. J. B. Riggs. Chemical Process Control, Lubbock, TX: Ferret Publishing, 1999. 7. D. E. Seborg, T. F. Edgar, and D. A. Mellichamp. Process Dynamics and Control. New York: Wiley, 1989. 8. C. A. Smith and A. B. Corripio. Principles and Practice of Automatic Process Control, 2nd ed. New York: Wiley, 1997. 9. G. Stephanopoulos. Chemical Process Control. An Introduction to Theory and Practice. Englewood Cliffs, NJ: Prentice-Hall, 1984. 305 Index Accumulation. See Time-derivative term AckermannвЂ™s formula, 178, 190 Cayley-Hamilton theorem and, 183n2 controllability and, 183 MATLAB and, 186, 187 pole placement and, 182вЂ“184 state feedback gain, 187 Actuators, 2, 3, 94вЂ“95 Adaptive control, 5 Air-to-close valve, 103 Air-to-open valve, 103 Amplitude ratio, 149 Antiwindup, 87 Antoine equation, 270 Bandwidth, 154, 203n2 low-pass п¬Ѓlter, 155 3-dB bandwidth, 155n7 Bias signal, 84 BIBO. See Bounded-input bounded-output Bioreactors, 2 Blending problem, 198, 213вЂ“215 Block diagrams, 1, 37вЂ“40. See also speciп¬Ѓc components, systems Bode plots, 112, 126 cascade control and, 203n2 dead time, 264 dead-time function and, 157 decibel scale, 263 п¬Ѓrst-order effects, 153, 157вЂ“159 frequency response and, 147, 168 of integrating function, 159 MATLAB and, 153, 155, 157, 159, 165, 171, 262, 264 Nyquist plot and, 152, 159 Nyquist stability and, 163 phase-lag compensators, 168 phase-lead compensators, 167 proportional gain, 169вЂ“171 second-order effects, 155 Bounded-input bounded-output (BIBO) stability, 129, 131 Break frequency, 154, 203n2 Cancellation compensation. See Pole-zero compensation Canonical forms, 79вЂ“82, 247вЂ“248 Capacitive process, 47вЂ“48 Cascade control, 53, 198вЂ“203 stability, 283 Cascade controller time constant, 283 Characteristic equation roots of, 23 Characteristic polynomial closed-loop, 92n17 poles, 24 roots of, 45 time-domain function, 24 Characteristic polynomials, 7, 212 closed-loop functions, 70, 83, 90вЂ“92, 97 coefп¬Ѓcient test, 22, 32 differential equation, 4 of matrix, 66 poles of. See Poles Chemical reactors, 130, 271 Ciancone-Marlin relations, 110, 113, 115, 127, 303 Classical control, 3, 8, 23, 86 closed-loops. See Closed-loop systems controller functions, 84 differential equation model, 7 fermentor model, 300вЂ“302 state-space models, 71, 78 See also Laplace transforms; speciп¬Ѓc concepts 307 Index Closed-loop systems, 3 characteristic polynomial, 70, 83, 90вЂ“92 eigenvalues in, 182, 183 MATLAB and, 70 poles, 97, 182вЂ“183, 253 proportional controller, 282 SISO system, 90вЂ“95 stability of, 129вЂ“146 system response, 95вЂ“102 transfer functions, 38, 39, 90вЂ“95, 98 CohenвЂ“Coon relations, 109вЂ“110, 113вЂ“114, 284, 284вЂ“286 Command tracking function, 91 Companion form, 80 Compartmental models, 53 Complex-conjugate poles, 24 Complex roots, 98. See also Root locus Constants Laplace transforms of, 9 Continuous-п¬‚ow stirred tanks (CSTR), 26вЂ“30 in series, 68вЂ“72, 186, 270 time constant of, 61 Control laws, 2, 84 Control problems, transforms, 12вЂ“14 Controllability, 178вЂ“179 AckermannвЂ™s formula and, 183 canonical form, 79вЂ“80 criterion for, 178 MATLAB and, 180 matrix for, 179вЂ“180 observability and, 180 state-space representation, 79вЂ“80 variables and, 1, 83, 92 Controller, 94вЂ“95 bias signal, 83вЂ“84 choice of, 104вЂ“107 design of, 168вЂ“176 mode of, 125 process function, 116n6 tuning, 108вЂ“115 See also speciп¬Ѓc types Controllers. See speciп¬Ѓc types Convolution integrals, 7n2, 78, 82 CramerвЂ™s rule, 58 Critical gain, 135 Critically damped systems, 49вЂ“50 Crossover, 135, 174 CSTR. See Continuous-п¬‚ow stirred-tanks Current-to-pressure transducer, 94 Damping ratio, 292 deп¬Ѓned, 24 feedback and, 186 gain and, 124, 143, 292 308 MATLAB and, 61 natural period, 49 overshoot and, 98 resonance and, 156 Dead-time function, 52, 55вЂ“57, 109вЂ“110 approximation for, 241 Bode plots and, 157, 264 compensation, 209вЂ“210 п¬Ѓrst-order function with, 55вЂ“56, 110n3, 158 Laplace transform of, 13 MATLAB and, 57, 241 minimum-phase system, 158 Nyquist plots and, 157 PadВґe approximation, 52, 56, 122, 136 phase lag, 157, 172 PI controller and, 118 PID controller and, 118 polynomials and, 22n14 root-locus plots, 145 second-order function with, 56 Smith predictor, 198, 209вЂ“210 stability criteria, 136 system response, 118 tuning, 113 Decay ratio, 51, 98 Decibel units, 152, 263 Decoupling, 217вЂ“221 Delta function. See Dirac function Derivative control, 284 Derivative kick, 88, 90, 278 Derivative time constant, 88, 118, 146 Derivatives, transforms of, 11 Deviation variables, 45n1 initial condition and, 45n1 linearized form, 36 perturbation variables, 7 transfer function form, 21 Diagonal canonical forms, 80вЂ“82 Difference equations, 5 Differential equation models characteristic polynomials, 4 classical control and, 7 п¬Ѓrst-order, 45 linearized, 36, 64 matrix form, 64вЂ“65 poles and, 4 second-order, 48вЂ“51, 65 transfer functions and, 96 Digital control systems, 5 Dirac function, 14 Direct action, 85n3 Direct-substitution analysis, 135вЂ“137 Direct synthesis, 115вЂ“119 Discrete-time systems, 5 Distributed-parameter models, 3 Index Disturbances, 3, 92 deп¬Ѓned, 83 rejection controller, 204 Dominant-pole design. See Pole-zero cancellation; Root-locus design Dual problem, 190 Dye-mixing, 101, 112, 123, 172вЂ“174 Dynamic compensator, 206 Dynamic response, 44вЂ“63 Eigenvalues, 66, 81 Empirical tuning, 108, 124 process-reaction curve, 109вЂ“110 ZieglerвЂ“Nichols ultimate-cycle method, 112 Equal-percentage valve, 274 Error, 2. See also Speciп¬Ѓc controls constant, 98 deп¬Ѓned, 84 derivative kick, 88 integrals, 110 minimized, 111 steady-state, 97, 98 EulerвЂ™s identity, 19, 150 Exponential function, 78 Laplace transform of, 9 MATLAB and, 63 sinusoidal functions and, 11 Fail-close valve, 103 Fail-open valve, 103 Feedback, 276. See also Error AckermannвЂ™s formula, 187 block diagrams, 37вЂ“38 compensation, 271 feedforward and, 198, 206вЂ“207 integral control, 187 MATLAB and, 249вЂ“254 PID controller, 178 trim, 206 See also Error; speciп¬Ѓc controllers Feedforward, 198, 203вЂ“206 decoupling functions, 219вЂ“221 feedback and, 198, 206вЂ“207 lead-lag element, 205 Fermentor models, 75, 181, 270, 300вЂ“302 Final-value theorem, 15, 46, 99 First-order derivative, Laplace transform of, 11 First-order differential equation models, 45вЂ“48 First-order п¬Ѓlter, 122n8 First-order function, 32, 154 Bode plots, 157вЂ“159 dead time and, 55вЂ“56, 158 integrator, 159 MATLAB and, 157 Nyquist plots, 157вЂ“159 First-order model, 46вЂ“48, 93, 148 First-order transfer function. See First-order function Flow control, 104 Forcing function, 7 Frequency-response analysis, 122вЂ“8, 145вЂ“176 deп¬Ѓned, 147 MATLAB and, 260 root-locus method, 175вЂ“176 sinusoidal input, 147 See also speciп¬Ѓc methods, parameters Full metering control, 208 Full-order state estimator system, 189вЂ“190 Fundamental matrix. See State-transition matrix Fuzzy-logic control, 5 Gain crossover frequency, 163 margin, deп¬Ѓned, 163 matrix, 214 Nyquist criterion, 162вЂ“168 Gas furnace, 198 Graphical analysis, 152вЂ“160 Heaviside function, 16 expansion, 150 generalized, 18 Laplace transform, 12 MATLAB and, 17 Higher-order processes, 53вЂ“58 Hurwitz test, 131, 133 IAE. See Integral of absolute error IMC. See Internal model control Impulse function, 14 Impulse-response simulations, 239 Initial-value theorem, 15 Integral control, 185 function of, 130n1 state feedback gain, 187 steady-state error, 187 Integral of absolute error (IAE) method, 111 Integral of time-weighted absolute error (ITAE) method, 111 controller, 221 criterion, 114, 124 settings, 123 tuning relations, 284 Integral square error (ISE) method, 111 Integral time constant, 86, 113, 118, 202 derivative, 146 PI controller and, 170 reset time and, 86 root-locus plot and, 145 stability and, 145 309 Index Integral tranform, 12 Integrating function, 47вЂ“48, 158, 158вЂ“159 Interacting systems, 90, 217вЂ“221 Internal model control (IMC), 115вЂ“127 Inverse transform polynomials, 16 Inverse transform deп¬Ѓned, 6 ISE. See Integral square error ITAE. See Integral of time-weighted absolute error Iterative design PID controller, 171 Jordan canonical form, 80 Laplace transform, 4, 8вЂ“12, 22 characteristic polynomials. See Characteristic polynomials common transforms, 10 in control problems, 12вЂ“14 deп¬Ѓned, 8 inverse, 6вЂ“8 as linear operator, 9, 44 linear systems and, 6 MATLAB and, 17вЂ“18 notations for, 8n5 partial fractions. See Partial fractions poles. See Poles Taylor series expansions, 6 time-constant form, 10 time-domain analysis, 8, 44 transfer functions. See Transfer functions Lead-lag element, 22, 58вЂ“59, 88 compensators, 221 feedforward controller, 205 MATLAB and, 61 Lead-lag transfer function transfer function, 68 Level control, 104 Lift, 274 Linear lag, 32n20 Linear operator, 9 Linear systems, 6 deviation variables in, 36 nonlinear equations and, 33вЂ“37 state-space models, 64 Taylor-series expansion, 35 Linear time invariant, 3 Load changes. See Disturbances Load, deп¬Ѓned, 83 LotkaвЂ“Volterra equations, 272 Low-pass п¬Ѓlter, 155 Lower-order functions, 55 Lumped-parameter models, 3 310 Magnitude, 149 amplitude and, 149 PD controller and, 164 PI controller and, 163вЂ“164 plot, 154 ratio, 149 Manipulated variables, 90, 92, 198, 217 MATLAB canonical transformation, 248 closed-loop function, 252 feedback, 252 state-space conversion function, 243 MATLAB program, 194 AckermannвЂ™s formula, 186, 187 basic commands, 226вЂ“230 Bode plots in, 153, 155, 157, 159, 165, 262, 264 canonical forms in, 80, 81, 185, 246 cascade control, 202 case-sensitivity and, 228 closed-loop functions in, 70, 95 closed-loop poles, 255вЂ“256 colon operator, 228 control toolbox functions, 251вЂ“254 controllability matrix, 180 damping and, 61, 256 dead time in, 57, 241, 264 diagonalized form in, 246 eigenvalues in, 191, 244вЂ“245 feedback simulation, 249вЂ“254 п¬Ѓrst-order lead, 157 frequency-response functions, 260 gain vector, 255n12 Heaviside function and, 17 impulse-response simulations, 239 index of coefп¬Ѓcients in, 79n7 indexing of state variables, 244 lead-lag element, 61 LTI viewer, 242, 246 M-п¬Ѓles, 232, 233 matrix creation, 228 multicapacity models, 61 Nichols plots, 261вЂ“262 Nyquist plots, 153, 155, 157, 159, 261 object-oriented functions, 237вЂ“238 observability matrix, 181 observer gain, 191 on-line help, 226 PadВґe approximation, 241 partial fractions in, 17, 234вЂ“236 PD controller and, 164, 258 phase-angle plots, 262вЂ“265 phase-lag compensators, 166вЂ“167 PI controller and, 258, 260 PID controller and, 165, 175, 258 Index plotting in, 230вЂ“232 polynomials in, 229 proportional gain and, 255 rectangular impulse, 241 root-locus plots, 140, 142, 202, 254, 256 Routh criterion, 131 similarity transform, 246 Simulink and, 100, 250, 251 SISO systems, 256 state-space models, 66, 68, 70, 246 time constant in, 62 time delay in, 158 time-response simulation, 95, 123вЂ“124, 238, 246 time units, 170n14 tutorial sessions, 226вЂ“265 unit-step response, 62 variables in, 228 See also speciп¬Ѓc applications, parameters Matrix methods, 64вЂ“65, 71 Measured variables, 83 Measurement function, 292 Measurement gain, 94 MIMO. See Multiple-input multiple-output Minimum error-integral criteria, 110вЂ“112 Minimum-phase system, 158 Mixing system, 101 Model-based design, 115 Monod constant, 75 Multicapacity processes, 44, 55, 61 Multiloop systems, 198вЂ“225 Multiple-input multiple-output (MIMO) systems, 68, 198, 210вЂ“212, 220вЂ“221 Multivariable systems. See Multiple-input multiple-output Natural period, 49 Negative feedback, 2, 37вЂ“39, 84 Nichols plots, 126, 153, 261вЂ“262 Noisy signals, 89 Nonself-regulating process. See Integrating processes Nonlinear equations, 5, 33вЂ“37. See also Linear systems Nonminimum-phase system, 60n7, 157вЂ“158 Nyquist plots, 126, 147, 154, 168 Bode plots and, 152 dead-time function, 157 п¬Ѓrst-order effects, 153, 157вЂ“159 integrating function, 158 MATLAB and, 153, 155, 157, 159, 261 polar coordinates, 152 second-order transfer function, 155 Nyquist stability criterion, 112, 161вЂ“162 Bode plots and, 163 gain and, 162вЂ“168 phase margins, 162вЂ“168 Object-oriented functions, 237вЂ“238 Observability, 179вЂ“181 Observable canonical form, 80 Observer gains, 191 Offset, 86, 87, 99, 100, 111n4 On-line tuning, 112 Open-loop instability, 299 Open-loop poles underdamped, 144 Open-loop response, 109вЂ“110 Open-loop zeros, 138 Optimal control, 5 Oscillations, 48вЂ“51, 112 Output controllability matrix, 179 Overdamped systems, 49вЂ“50, 117вЂ“119 Overshoot, 50, 87, 98, 286 PadВґe approximation, 52, 53, 118, 145, 280 dead-time and, 52, 56, 122 MATLAB and, 241 time-delay function, 136 Parallel functions, 60 Partial fractions, 234 expansion by, 16вЂ“21, 44 Laplace transforms and, 6 MATLAB and, 17, 234вЂ“236 PD. See Proportional-derivative controller Peak time, 51 Period of oscillation, 51 Perturbation variables, 7n2 Phase-angle plots, 149, 154, 262вЂ“265 Phase crossover frequency, 163 Phase lag, 148, 153 compensators, 166вЂ“167 curve, 160 dead-time function and, 172 MATLAB and, 166вЂ“167 PD controller and, 164 PI controller and, 163вЂ“164 Phase lead, 145, 166вЂ“167 Phase margins, 162вЂ“168 Phase-variable canonical form, 79вЂ“80 PI controller. See Proportional integral controller ZieglerвЂ“Nichols relations, 289 PID. See Proportional-integral-derivative control PID controller tuning methods, 108вЂ“127 tuning relations, 297, 303 Plotting guidelines, 142 Polar plot. See Nyquist plot 311 Index Poles, 21 AckermannвЂ™s formula, 182вЂ“184 characteristic polynomial, 24 characteristics and, 24вЂ“26 complex roots, 18вЂ“20, 24 dominant, 25 eigenvalues, 66 Hurwitz test, 131вЂ“132 on imaginary axis, 25 LHP and, 129, 212 negative real part, 25 pole-zero cancellation, 59, 116, 119вЂ“120 pole-zero form, 23, 32 real distinct, 24 repeated roots, 20вЂ“21, 24 RHP and, 161, 210, 212 Routh criterion, 131вЂ“133, 135, 143 time-domain function, 24 time-response characteristics, 6 of transfer function, 22 transfer functions, 32 zeros of, 23, 116 Polynomial coefп¬Ѓcient test. See Hurwitz test Polynomials. See Characteristic polynomials inverse transform, 16 transfer functions, 22 Position error constant, 97 Position-form algorithm, 128 Predator-prey problems, 272 Pressure control, 104 Process function, 116n6 Process gain, 23n16, 213вЂ“214 Process-reaction curve, 109вЂ“110, 124 Proportional band, 85 Proportional control, 84вЂ“86, 111 general features of, 86 Proportional controller closed-loop systems, 282 Proportional-derivative (PD) controller, 88вЂ“89 derivative kick, 88 п¬Ѓrst-order process, 99 magnitude, 164 MATLAB and, 164, 258 offset, 100 phase lag, 164 root-locus plots, 278 Proportional gain, 255 Bode plots, 169вЂ“171 root-locus plots, 145 Proportional-integral-derivative (PID) control, 83вЂ“90, 115, 124 dead time, 118 derivative kick, 90 dye-mixing and, 112, 123 312 ideal, 165 interacting, 90, 165 iterative design, 171 MATLAB and, 165, 175, 258 noninteracting, 176 overshoot and, 286 PI controller and, 101 root-locus plots and, 257вЂ“260 series, 90 Simulink and, 250вЂ“251 state feedback controller, 178 tuning methods, 108 Proportional-integral (PI) controller, 86вЂ“87, 117, 171, 202 dead time, 118 design of, 172 п¬Ѓrst-order process and, 99 integral time constant, 170 magnitude, 163вЂ“164 MATLAB and, 258, 260 offset in, 87, 99, 144 phase lag, 163вЂ“164 PID controller and, 101 root locus and, 187 Pulse experiment, 152 Ramp function, 9, 47 Rate-before-reset, 90 Rate control, 278 Rate feedback, 271 Rate time, 88 Ratio control, 198, 207вЂ“208 Rectangular impulse, 13, 241 Reduced-order estimator, 55, 191вЂ“194 Reduced-order model approximation, 55 Reference. See Set point Regulator poles, 182 Regulator problems, 92 Relative gain array (RGA), 215вЂ“217 Reset-feedback, 87n10 Reset time. See Integral time constant Reset windup, 87 Residues, 16n10, 17 Resistance temperature detector (RTD), 93 Resonance, 156 Response time, 117 Reverse acting, 85n3 RGA. See Relative gain array Rise time, 51 Robustness, 3 Root-locus plots, 100, 129вЂ“130, 137вЂ“146, 168, 276 dead time, 145 frequency-response methods, 146, 175вЂ“176 integral time constant, 145 Index limitations, 145 MATLAB and, 140, 142, 202, 254 PD control, 278 PD controller, 280 phase-lag compensators, 167 PI controller and, 168 PID control systems, 257вЂ“260 proportional gain, 145 shape of, 142 stability and, 145 Routh criterion, 131вЂ“133, 143 Routh-Hurwitz criterion, 125, 129вЂ“136, 142, 145, 279 RTD. See Resistance temperature detector RungeвЂ“Kutta integration, 239n8 Safety, 94 Saturation, 86, 116, 145, 191 Second-order systems, 143 AckermannвЂ™s formula, 188 Bode plots, 155 with dead time, 56 differential equations for, 11, 48вЂ“51, 65 Nyquist plots, 155 Secondary loop, 199 Self-regulation, 46вЂ“48 Sensitivity, 84, 145 Sensor lag, 104вЂ“105 Servo systems, 92, 184 integral control, 184вЂ“188 Set point, 83 changes in, 92, 114 tracking, 104, 204 Settling time, 51, 87 Signal transmitter, 93вЂ“94 Similarity transform, 79, 246 Simulink, 208, 249вЂ“250 block diagrams, 251 MATLAB and, 100, 250вЂ“251 PID controller and, 250вЂ“251 Single-input single-output (SISO) models, 2, 189 block diagram of, 93 closed-loop system, 90вЂ“95 load variables in, 92 manipulated variables, 90 measured variables, 83 notations for, 65 regulator problems, 92 servo problems, 92 Single-loop systems, 3 analysis of, 83вЂ“107 design of, 108вЂ“127, 256 feedback in, 92вЂ“95 tuning of, 108вЂ“127 Sinusoidal functions exponential decay, 11 frequency response, 147 Laplace transforms of, 10вЂ“11 stability and, 147 SISO. See Single-input single-output Slave loop, 199, 201 Sluggishness, 87n11 Smith predictor, 198, 209вЂ“210 Stability, 16 analysis, 24, 112, 125, 136, 161вЂ“168 BIBO systems, 129вЂ“131 cascade control, 283 closed-loop systems, 129вЂ“146 conditional, 284 dead time and, 136 deп¬Ѓnition of, 129вЂ“130 integral time constant, 145 marginal, 129 Nyquist stability criterion, 161 proportional controller, 130 root-locus plots, 145 Routh array construction, 132 Routh-Hurwitz criterion, 129 sinusoidal input and, 147 Staged processes, 53 State estimator, 188вЂ“190, 194 State feedback control, 178, 182, 185, 187 State observer. See State estimator State-space models, 5, 64вЂ“71, 64вЂ“82, 69, 243, 243вЂ“248 block diagram of, 66 canonical form, 79вЂ“80 classical control techniques, 71, 78 design of, 178вЂ“197 linearized models, 64 MATLAB and, 66, 68, 70 modern control and, 64 properties of, 78вЂ“82 time-domain solution, 78вЂ“79 transfer function models, 71вЂ“78 State-transition matrix, 71 Static gain, 23n16 Steady-state compensator, 206 Steady-state error, 86, 98, 187 Steady-state gain, 23, 32, 46 Step function, 12, 13 Step response, 49вЂ“51 Stirred-tank heater, 30вЂ“33, 92вЂ“93 Stirred tanks, 26вЂ“30, 269вЂ“270 Stochastic control, 5 Stochiometric ratio, 298 Summing point, 91 Superposition principle, 60 System response, deп¬Ѓned, 118 313 Index Tanks-in-series, 54вЂ“55 interacting, 57вЂ“58 MATLAB and, 55 state-space representation, 289 Taylor series expansion, 34n22 Laplace transforms and, 6, 79 linearization and, 33, 35 Temperature control, 104вЂ“105 Thermocouples, 104 Three-mode controller. See Proportional-integral-derivative control Time constant, 23, 154 cascade controller, 283 deп¬Ѓned, 42 derivative, 118 integral, 118, 146 MATLAB and, 62 Time-constant form transfer functions and, 32 Time-delay function. See Dead time Time-derivative term, 3, 87вЂ“89. See also Transient problems Time-domain analysis п¬Ѓnal-value theorem, 15 initial-value theorem, 15 Laplace transforms and, 8, 44 state-space models, 78вЂ“79 Time-domain function characteristic polynomial, 24 poles, 24 Time response methods, 113 MATLAB and, 246 poles and, 6 zeros in, 58вЂ“60 Time-shift function. See Dead time Time-weighting function, 111 Transfer functions, 4, 21, 234 basic properties of, 150 differential equations, 96 lower-order approximations of, 55вЂ“57 in parallel, 59вЂ“60 pole-zero form, 32 polynomials, 22 products of, 150 as ratio of polynomials, 22 representation of, 64 state-space models, 71вЂ“78 time-constant form, 32 See also Laplace transforms; Poles 314 Transient models, 3, 26вЂ“33 Transmitters, 3 Transport-lag function. See Dead time Tuning methods PID controller, 108вЂ“127 Tuning relations, 115 dead time and, 113 empirical, 109вЂ“110 empirical relations, 108вЂ“115 PID controller, 108 response time, 117 single-loop systems, 108вЂ“127 table of relations, 114 See also speciп¬Ѓc methods, relations Two-input models, 76 Ultimate-cycle method, 112вЂ“115 Ultimate frequency, 135 Ultimate gain, 112, 115 Ultimate period, 112 Underdamped open-loop poles, 144 Underdamped systems, 49вЂ“51, 87, 98, 119, 156 Unit-rectangular pulse-function, 14 Unit-step function, 12, 62, 267 Unity gain, 116 Valve characteristic curve, 95 Vapor п¬‚ow rate, 104 Vapor-liquid equilibrium, 270 Velocity error constant, 97 Velocity-form algorithm, 128 Windup, reset, 87 Z transforms, 5 Zero initial condition, 34 Zero terms, 21 frequency plot, 154 pole-zero cancellation, 116 of polynomials, 23вЂ“25 time response and, 58вЂ“60 in transfer functions, 151 ZieglerвЂ“Nichols relations, 110вЂ“115, 124вЂ“125, 127, 170 PI controller, 289

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