null null

C ONTROLLABILITY
ANALYSIS FOR
PROCESS AND CONTROL SYSTEM
DESIGN
by
Audun Faanes
A Thesis Submitted for the Degree of Dr. Ing.
Department of Chemical Engineering
Norwegian University of Science and Technology
Submitted August 2003
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Abstract
Controllability is the ability of a process to achieve acceptable performance,
and in this thesis we use controllability analysis in the design of buffer tanks,
feedforward controllers, and multivariable controllers such as model predictive
control (MPC).
There is still an increasing pressure on the process industry, both from competitors (prize and quality) and the society (safety and pollution), and one important contribution is a smooth and stable production. Thus, it is important to
dampen the effect of uncontrolled variations (disturbances) that the process may
experience.
The process itself often dampens high-frequency disturbances, and feedback
controllers are installed to handle the low-frequency part of the disturbances, including at steady state if integral action is applied. However, there may be an intermediate frequency range where neither of these two dampens the disturbances
sufficiently. In the first part of this thesis we present methods for the design of
buffer tanks based on this idea. Both mixing tanks for quality disturbances and
surge tanks with “slow” level control for flow-rate variations are addressed.
Neutralization is usually performed in one or several mixing tanks, and we
give recommendations for tank sizes and the number of tanks. With local PI or
PID control, we recommend equal tanks, and give a simple formula for the total volume. We also give recommendations for design of buffer tanks for other
types of processes. We propose first to determine the required transfer function to
achieve the required performance, and thereafter to find a physical realization of
this transfer function.
Alternatively, if measurements of the disturbances are available, one may apply feedforward control to handle the intermediate frequency range. Feedforward
control is based mainly on a model, and we study the effect of model errors on
the performance. We define feedforward sensitivities. For some model classes we
provide rules for when the feedforward controller is effective, and we also design
robust controllers such as -optimal feedforward controllers.
Multivariable controllers, such as model predictive control (MPC), may use
both feedforward and feedback control, and the differences between these two
also manifest themselves in a multivariable controller. We use the class of serial processes to gain insight into how a multivariable controller works. For one
specific MPC we develop a state space formulation of the controller and its state
estimator under the assumption that no constraints are active. Thus, for example
the gains of each channel of the MPC (from measurements to the control inputs)
can be found, which gives further insight into to the controller. Both a neutralization process example and an experiment are used to illustrate the ideas.
ii
Acknowledgments
I want to thank Sigurd Skogestad for pointing out the directions in which to
proceed, and for his support along the way. He is an excellent supervisor. In
addition to his knowledge and skills, from which I have learned a lot, I appreciate
his focus on reporting and publishing of results. This improves the research and
is beneficial for progress.
I thank my fellow students at the institute in Trondheim, my former colleges
at the Norsk Hydro Research Centre in Porsgrunn, and my colleges at Statoil
Research Centre in Trondheim for always pleasant and often fruitful discussions.
The Norwegian Research Council and my former employer, Norsk Hydro
ASA, are gratefully acknowledged for financing this work. I also want to thank
my present employer, Statoil ASA, for allowing me time to finish it.
Finally, I want to thank my wife Sophie and our daughters Sarah and Mathilde.
Our family has come into being during the same period as this thesis. My parents
can be thanked for many things, and of particular relevance is their hospitality
when I needed a home in Trondheim during this work.
Contents
1 Introduction
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 pH-Neutralization: Integrated Process and Control Design
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Motivating example . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Time delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 A simple formula for the volume and number of tanks . . . . . . .
2.6 Validation of the simple formula: Improved sizing . . . . . . . .
2.7 Equal or different tanks? . . . . . . . . . . . . . . . . . . . . . .
2.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8.1 Measurement noise and errors . . . . . . . . . . . . . . .
2.8.2 Feedforward elements . . . . . . . . . . . . . . . . . . .
2.8.3 pH set-points in each tank . . . . . . . . . . . . . . . . .
2.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A Modelling . . . . . . . . . . . . . . . . . . . . . . . . . .
A.1 Single tank . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Linear model for multiple tank in series . . . . . . . . . . . .
A.3 Non-linear model for multiple tank in series . . . . . . . . . .
A.4 Representation of delays . . . . . . . . . . . . . . . . . . . .
Appendix B The effect of pH measurement errors on the scaled excess
concentration, . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix C On the optimization problem (2.32) subject to (2.35) . . .
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3 Buffer Tank Design for Acceptable Control Performance
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Introductory example . . . . . . . . . . . . . . . . . . . .
3.3 Step 2: Physical realization of with a buffer tank . .
I
Mixing tank for quality disturbance ( ) . . .
II
Surge tank for flow-rate disturbance ( ) . . .
3.4 Step 1: Desired buffer transfer function . . . . . . .
3.4.1 given (existing plant) . . . . . . . . . . . . . . .
3.4.2 not given . . . . . . . . . . . . . . . . . . . . .
3.5 Before or after? . . . . . . . . . . . . . . . . . . . . . . .
3.6 Further discussion . . . . . . . . . . . . . . . . . . . . .
3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A Surge tank with level dependent flow . . . . . . .
Appendix B Capital investments . . . . . . . . . . . . . . . . .
Appendix C Surge tank: Required volume with n-th order .
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4 Control Design for Serial Processes
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Model structure of serial processes . . . . . . . . . . . . . . . . .
4.3 Control structures for serial processes . . . . . . . . . . . . . . .
4.3.1 Local control (diagonal control) . . . . . . . . . . . . . .
4.3.2 Pure feedforward from upstream units . . . . . . . . . . .
4.3.3 Lower block triangular controller . . . . . . . . . . . . .
4.3.4 Full controller . . . . . . . . . . . . . . . . . . . . . . .
4.3.5 Final control only in last unit (input resetting) . . . . . . .
4.4 Case study: pH neutralization . . . . . . . . . . . . . . . . . . . .
4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.3 Model uncertainty . . . . . . . . . . . . . . . . . . . . .
4.4.4 Local PID-control (diagonal control) . . . . . . . . . . . .
4.4.5 Feedforward control (control elements below the diagonal)
4.4.6 Combined local PID and feedforward control (lower block
triangular control) . . . . . . . . . . . . . . . . . . . . .
4.4.7 Multivariable control . . . . . . . . . . . . . . . . . . . .
4.4.8 MPC with input resetting . . . . . . . . . . . . . . . . . .
4.4.9 Conclusion case study . . . . . . . . . . . . . . . . . . .
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Appendix A State space MPC used in case study . . . . . . . . . . . . 102
Appendix B Derivation of equations (4.20) and (4.31) . . . . . . . . . 104
5 On MPC without active constraints
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Derivation of equivalent controller from receding horizon controller without active constraints . . . . . . . . . . . . . . . . . .
5.3 The steady-state solution . . . . . . . . . . . . . . . . . . . . . .
5.4 Generalization with tracking of inputs . . . . . . . . . . . . . . .
5.5 State and disturbance estimator . . . . . . . . . . . . . . . . . . .
5.6 State-space representation of the overall controller . . . . . . . . .
5.7 On the number of estimated disturbances . . . . . . . . . . . . . .
5.8 Closed loop model . . . . . . . . . . . . . . . . . . . . . . . . .
5.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Feedforward Control under the Presence of Uncertainty
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 The characteristics of feedforward control . . . . . . . . .
6.3 Feedforward sensitivity functions . . . . . . . . . . . . .
6.4 The effect of model error with feedforward control . . . .
6.5 Some classes of model uncertainty . . . . . . . . . . . . .
6.6 Example: Two tank process . . . . . . . . . . . . . . . . .
6.7 When is feedforward control needed and when is it useful?
6.8 Design of feedforward controllers under uncertainty . . .
6.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . .
6.10 Acknowledgements . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A Modelling of the two tank process . . . . . . . . .
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7 Offset free tracking with MPC under uncertainty: Experimental verification
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7.1 Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.1.1 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.1.2 Instrumentation and logging . . . . . . . . . . . . . . . . 158
7.1.3 Basic control . . . . . . . . . . . . . . . . . . . . . . . . 158
7.2 Process model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.3 Identification of process parameters . . . . . . . . . . . . . . . . 160
7.4 Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7.5 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . 165
7.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
vi
7.7 Discussion . . . . .
7.8 Conclusions . . . .
7.9 Acknowledgements
References . . . . . . . .
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8 Conclusions and directions for future work
8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.1 Buffer tank design . . . . . . . . . . . . . . . . . . . . .
8.1.2 Feedforward control under the presence of uncertainty . .
8.1.3 Multivariable control under the presence of uncertainty . .
8.2 Directions for further work . . . . . . . . . . . . . . . . . . . . .
8.2.1 Serial processes: Selection of manipulated inputs and measurements . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.2 MIMO feedforward controllers under the presence of uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.3 Effect of model uncertainty on the performance of multivariable controllers . . . . . . . . . . . . . . . . . . . . .
8.2.4 MPC with integral action . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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A Control Structure Selection for Serial
pH-Neutralization
A.1 Example: pH neutralization . . . .
A.2 Conclusion . . . . . . . . . . . .
References . . . . . . . . . . . . . . . .
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Processes with Application to
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B A Systematic Approach to the Design of Buffer Tanks
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
B.2 Transfer functions for buffer tanks . . . . . . . . . .
B.2.1 Quality disturbance . . . . . . . . . . . . . .
B.2.2 Flow rate disturbance . . . . . . . . . . . . .
B.3 Controllability analysis . . . . . . . . . . . . . . . .
B.3.1 Additional requirements due to high order B.4 Quality variations . . . . . . . . . . . . . . . . . . .
B.5 Flow variations . . . . . . . . . . . . . . . . . . . .
B.5.1 First-order filtering . . . . . . . . . . . . . .
B.5.2 Second-order filtering . . . . . . . . . . . .
B.6 Conclusions . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 1
Introduction
We start with some words on the title of the thesis, or more precisely with a definition of what we mean by controllability and controllability analysis (Skogestad
and Postlethwaite, 1996, Chapter 5):
Definition 1.1 (Input-output) controllability is the ability to achieve acceptable
control performance; that is, to keep the outputs ( ) within specified bounds or
displacements from their references ( ), in spite of unknown but bounded variations, such as disturbances ( ) and plant changes, using available inputs ( ) and
available measurements ( or ).
A plant is controllable if there exists a controller (connecting plant measurements
and plant inputs) that yields acceptable performance for all expected plant variations. From this, controllability is independent of the controller, and a property of
the process alone. Further, controllability analysis is applied to a plant to find out
what control performance can be expected.
The definition above is in accordance with the definition given by Ziegler and
Nichols (1943) “the ability of the process to achieve and maintain the desired
equilibrium value”, but must not be confused with the more narrow state controllability definition of Kalman from the 60’s.
In particular, in this thesis we will apply controllability analysis in the design
of processes, namely such processes that are designed for dynamic and control
purposes, and in the design and understanding of feedforward and multivariable
controllers.
1.1 Motivation
High degree of competition in all branches of the process industry put pressure
on each single site and plant to stay competitive. Even within a company there is
Chapter 1. Introduction
2
an internal competition of being the most productive and effective, and delivering
the best quality products. The second best risks that investment plans are rejected
by the central management, or even that the plant is closed.
There are many important requirements that must be met by a plant organisation
(1) On-site and off-site safety
(2) Discharge shall be below certain limits, both on a long term basis, e.g., total
over a year, or on a shorter the period of time, such as on an hourly basis.
(3) Requirements for certain quality parameters to stay within given limits (to
obtain maximal prizes)
(4) Minimal production costs, such as energy consummation
(5) Maximal production
Running smoothly without abrupt changes of any kind, will be an important
contribution to meet all the above-mentioned requirements. The risk of accidents
and undesired discharge is reduced, and a natural consequence is also a more constant product quality. Finally, production cost can be reduced and the production
rate increased, because the risk of unplanned stops is reduced and because it is
possible to move the operating point closer to the constraints.
On the other hand, within a process, there are many sources that introduce
variations of all kinds, namely disturbances. This can be such as variations in
the quality of the raw materials or the incomming flow rates, inaccurate charging
equipment, sticky vales, or badly tuned control loops. Some of these things are, at
least in principle, easy to handle, others are more difficult or costly to avoid, and
must therefore be treated by other process parts.
It is our experience that the Norwegian process and oil industry has increased
the focus on smooth production in recent years, and therefore puts pressure on
process control. This is because of the increased competition in the process industry in general (the competitors focus on this), and also because of changes in
the oil production in the North Sea, which lead to more disturbances and “new”
bottle-necks (primal reasons are increased water and gas production and longer
pipes between the wells and the processing units).
In this thesis two basic ideas are elaborated. The first is that high-frequency
disturbances are dampened by the process itself (e.g., by inventories like reactor
volumes, and liquid hold-ups in distillation columns) whereas low-frequency disturbances can be dampened out with effective single-loop feedback controllers.
To handle intermediate frequencies, we look into the design of buffer tanks and
1.2 Thesis overview
3
more sophisticated controllers (like traditional feedforward control and multivariable control).
As far as we have found in the literature, even though buffer tanks are introduced for control purposes, control theory has not been applied. Further, feedforward control theory is treated by most textbooks on control, but often very
briefly, and even a simple analysis of the effect of model errors is often missing
(exceptions are Balchen (1968), and the work of Scali and co workers (Lewin and
Scali, 1988; Scali et al., 1989)).
Based on our experience from industrial processes, we assume that sinusoidal
disturbances of varying frequency are the most important. The disturbances may
be caused by oscillations in other parts of the process, for example, from aggressive control, valve stiction etc. However, in the simulations we also consider step
disturbances.
The second idea is that within multivariable feedback controllers there may be
controller blocks or elements that are similar to feedforward control. Like traditional feedforward controllers, such elements may nominally improve the performance to a large extent. Unfortunately, feedforward controllers rely heavily on a
model of the process, and this drawback also applies to the feedforward elements
within the multivariable controller.
1.2 Thesis overview
The thesis is composed of six chapters written as independent articles, each with
a separate bibliography, and most of them also have their own appendices. In
the end of the thesis there is a concluding chapter (Chapter 8) and in addition
there are two appendices, A and B, referred to by “Thesis’ Appendix A (or B)” to
distinguish from the appendices within each chapter.
Chapters 2 and 3 give rules regarding the design of buffer tanks, especially
regarding tank sizes (the first specializes on pH-neutralization). Also Chapter 4
can be useful for readers with interest in this, since it looks into different control
strategies for serial processes, and one or more buffer tanks are usually placed in
series with other process units. In particular, pH neutralization is included as a
case study.
Chapters 4 - 7 focus on control design. One may say that Chapters 5 and 6 are
theoretical foundations for Chapters 4 and 7.
If the interest is how to handle disturbances, our basic idea is that when neither
the process itself, or a simple feedback control system can handle them, either
buffer tanks (Chapters 2 and 3) or feedforward controllers (Chapter 6) may be
used to handle the resting frequencies.
Chapter 1. Introduction
4
In Chapter 2 we provide a simple rule for the size of mixing tanks for pH neutralization processes ensuring that incoming disturbances are dampened such that
the outlet pH is kept within given limits. We assume traditional single-loop feedback control, and that the efficiency of the feedback loops are limited by delays
and other high order dynamics. Neutralization processes often have large process
gains, and it is therefore often convenient to use several stages.
In Chapter 3 we extend the mixing tank design from Chaper 2 to the design of
a broader class of buffer tanks. The aim of the buffer tank is disturbance dampening in the frequency range where neither the process itself nor any feedback loop
dampen the disturbances sufficiently. We consider disturbances in both quality
and flow rates, for which mixing tanks and surge tanks with slow level control are
used, respectively.
Chapter 4 discusses control design for serial processes. As a case study we
consider neutralization in several stages, which we also discuss in Chapter 2. We
use the structure of serial processes to identify different classes of control blocks
of a multivariable controller, and comment, in particular, on feedforward effects
and how to obtain integral action.
The multivariable controller we use in Chapter 4 is a model predictive controller (MPC). In Chapter 5 we assume that no constraints are active, in which
case the MPC can be considered as a linear quadratic controller (LQ), and derive
a state-space formulation of the resulting controller, including the state estimator.
Chapter 5 is mainly a tool for Chapters 4 and 7, but also include a result on how
to choose input biases to gain integral action.
One of the control block classes discussed Chapter 4 is feedforward control,
and in Chapter 6 we discuss feedforward control under model uncertainty. In
accordance with the sensitivity function defined for feedback control, we introduce feedforward sensitivities, and discuss how this can be used to determine the
usefulness of a feedforward controller (or of a feedforward control block).
Chapter 7 verifies some of the results from Chapters 4 and 5 through an experiment. We show that even if simulations indicate that a specific controller gives
integral action, when applied to the actual process, steady-state offset is obtained.
Chapter 8 sums up the conclusions from the thesis, and tries to propose some
directions for further work.
The thesis’ Appendixes A and B are “older” published versions of Chapters 4
(only a part) and 3, respectively. They are included since they contain material that
has been removed from the chapters now included (Chapters 4 and 3). Appendix
A contains an example where
control has been applied (in Chapter 4 model
predictive control (MPC) is used). Appendix B is more focused on the short-cut
method for buffer tank design than Chapter 3, and contains some more details
regarding this.
Preliminary versions or parts of the following chapters have been or will be
REFERENCES
5
presented at the following conferences, and versions nearly identical to the chapters have been either submitted to, accepted by or published in the following journals1 :
Chapter 2: Adchem 2000, June 14-16, 2000, Pisa, Italy (preprints: 1, pp. 75-80)
Preliminary accepted for publication in Computers and Chemical
Engineering
Chapter 3: Nordic Process Control Workshop 9, January 13-15, 2000, Lyngby,
Denmark
PSE’2000, 16-21 July, 2000, Keystone, Colorado, USA (Supplement
to Computers and Chemical Engng., 24, pp.1395-1401)
Ind. Eng. Chem. Res., 42, 10, pp. 2198-2208
Chapter 4: Nordic Process Control Workshop 8, August 23-25, 1998, Stockholm,
Sweden
European Control Conference, ECC’99, Aug. 31-Sept. 3, 1999, Karlsruhe, Germany
Submitted to Journal of Process Control
Chapter 5: Submitted to Modeling, Identification and Control, MIC
Chapter 6: Nordic Process Control Workshop 11, January 9-11, 2003, Trondheim
Accepted for presentation at European Control Conference, ECC’03,
Sept. 1-4, 2003, Cambridge, UK
Preliminary accepted for publication in European Journal of Control
Chapter 7: Accepted for presentation (poster session) at AIChE, Annual Meeting,
Nov. 2003, San Francisco, US
References
Balchen, J. G. (1968). Reguleringsteknikk Bind 1 (In Norweigan) 1. Ed.. Tapir. Trondheim,
Norway.
Lewin, D. R. and C. Scali (1988). Feedforward control in presence of uncertainty. Ind.
Eng. Chem. Res. 27, 2323–2331.
Scali, C., M. Hvala and D. R. Lewin (1989). Robustness issues in feedforward control..
ACC-89 pp. 577–581.
Skogestad, S. and I. Postlethwaite (1996). Multivariable Feedback Control. John Wiley &
Sons. Chichester, New York.
Ziegler, J. G. and N. B. Nichols (1943). Process lags in automatic-control circuits. Trans.
ASME 65, 433–444.
1
The difference between the chapters and their corresponding journal article is indicated on the
front page of each chapter.
Chapter 2
pH-Neutralization: Integrated
Process and Control Design
Audun Faanes and Sigurd Skogestad
Department of Chemical Engineering
Norwegian University of Science and Technology
N–7491 Trondheim, Norway
Preliminary accepted for publication in Computers and Chemical Engineering
Extended version of a paper presented at Adchem 2000, June 14-16, 2000, Pisa,
Italy, Preprints Vol. I, pp.75-80.
also affiliated with Statoil ASA, TEK, Process Control, N-7005 Trondheim, Norway
Author to whom all correspondence should be addressed. E-mail: [email protected],
Tel.: +47 73 59 41 54, Fax.: +47 73 59 40 80
Extensions compared to paper to be published: Appendix A.3
8
Chapter 2. pH-Neutralization: Integrated Process and Control Design
Abstract
The paper addresses control related design issues for neutralization plants. Mainly for control
reasons, the neutralization is usually performed in several steps (mixing tanks) with gradual change
in the concentration. The aim is to give recommendations for issues like tank sizes and number of
tanks. Assuming strong acids and bases, we derive linearized relationships from the disturbance
variables (e.g. inlet concentration and flow rate) to the output (outlet concentration), including
the scaled disturbance gain, . With local PI or PID control in each tank, we recommend to use
identical tanks with total volume , where we give as a function of , the time delay in
each tank , the flow rate , and the number
of tanks . For , which is common in pH
neutralization, this gives .
Keywords: pH control, Process control, Processs design, PID control
2.1 Introduction
9
2.1 Introduction
The pH-neutralization of acids or bases has significant industrial importance. The
aim of the process is to change the pH in the inlet flow, the influent (disturbance,
), by addition of a reagent (manipulated variable, ) so that the outflow or effluent
has a certain pH. This is illustrated in Figure 2.1 as a simple mixing, but normally
it takes place in one or more tanks or basins, see Figure 2.3. Examples of areas where pH control processes are in extensive use are water treatment plants,
many chemical processes, metal-finishing operations, production of pharmaceuticals and biological processes. In spite of this, there is little theoretical basis for
designing such systems, and heuristic guidelines are used in most cases.
Textbooks on pH control include (Shinskey, 1973) and (McMillan, 1984).
General process control textbooks, such as (Shinskey, 1996; Balchen and Mummé,
1988), have sections on pH control. A critical review on design and control of neutralization processes that emphasizes chemical waste water treatment is given by
Walsh (1993).
Our starting point is that the tanks are installed primarily for dynamic and
control purposes. In our paper process design methods using control theory are
proposed. We focus on the neutralization of strong acids or bases, which usually
is performed in several steps. The objective is to find methods to obtain the total
required volume for a given number of tanks, and discuss whether they should be
identical or not. Design of surge (buffer) tanks is generalized to other processes in
Chapter 3. Clearly, the required tank size depends on the effectiveness of the control system, and especially with more than one tank, there are many possibilities
with respect to instrumentation and control structure design. This is discussed in
Chapter 4.
Section 2.2 motivates the problem. Since time delays are important design
limitations, Section 2.3 contains a discussion on delays. From the models presented in Section 2.4, in Section 2.5 we follow Skogestad (1996) and derive a
simple formula for the required tank volume, denoted . In Section 2.6 the validity of the simple formula for is checked numerically, and improved rules for
sizing are proposed. Whether equal tanks is best or not is discussed in Section 2.7.
Discussions on measurement noise, feedforward control and the pH set-point to
each tank are found in Section 2.8. The main conclusions are summarized in
Section 2.9.
2.2 Motivating example
We use a simple neutralization process to illustrate the ideas.
10
Chapter 2. pH-Neutralization: Integrated Process and Control Design
Example 2.1 We want to neutralize of a strong acid (disturbance) of ( ) using a strong base (input) with to obtain a
product of ( "!# %$ $ ""&'( ).
We derive a model for the process in Section 2.4, and we find that it is convenient to work with the excess
-concentration, ) *#+ (( ). In
terms of this variable, the product specification is , , and the variation requirement .-'/ corresponds to a concentration deviation 0 21.354 "& 2 . We assume that the maximum expected disturbance is 0 1.354 , corresponding to a pH variation from 67 to 879: .
We first try to simply mix the acid and base, as illustrated in Figure 2.1 (no
tank). The outlet concentration is measured (or calculated from a pH measurement), and the base addition is adjusted by a feedback PI- controller assuming a
time delay of in the feedback loop. A step disturbance in the inlet concentration of 2 , results in an immediate increase in the product of ; 7 (to
pH 67=< ), since the total flow is half the acid flow. After a while the PI controller
brings the pH back to , but for a period of about > the product is far outside
its limits. This can be seen from the simulation in Figure 2.2 (solid line).
This is clearly not acceptable, so, next, we install one mixing tank to dampen
the disturbances. For a tank with residence time ? , the response is (for the case
with no control):
[email protected] B; 7 DC FE HGJI5KML
(2.1)
Now the pH of the product does not respond immediately, and provided ? is sufficiently large, the controller can counteract the disturbance before the pH has
crossed its limit of N . Solving for [email protected] "& , we get
@P <RQ8 S ?
(2.2)
For example, ? 8> gives @ <TQU HV , that is, for a tank with a residence
time of 8> the pH goes outside its limits after 7=<W . However, no control
system can respond this fast. With a time delay of (typical value), the feedback
controller needs at least @ to counteract the disturbance, which gives a
minimum required residence time of ? <XQM S B; 7 QY S . In practice, a
larger tank is required, and in Figure 2.2 we also show the closed-loop response
for the case with ? :ZQ% S (dashed line). With a flow rate of this
corresponds to a tank size of :>>F>8,\[ . This is of course unrealistic, but in
Section 2.5 we will see that the total tank size can be reduced considerably by
adding several tanks in series as illustrated in Figure 2.3.
2.2 Motivating example
11
∆cinfl,max=
±5 mol / l
Base
Acid
q=5 l / s
pHC
pH - 1
pHI
kd ~ 106
pH 7 ± 1
∆cmax =±10-6 mol / l
Figure 2.1: Neutralization of strong acid with strong base (no tank)
8
7
6
One tank with
5
residence time
8 × 107s
pH
4
3
Pure mixing (no tank)
2
1
0
−1
0
100
200
300
400
500
Time [s]
600
700
800
900
1000
Figure 2.2: Mixing capacity is required to dampen the disturbance. Closed-loop responses
in outlet pH to a step change in inlet acid concentration from to with
time delay of in the PI-control loop. (Controller: PI with Ziegler-Nichols tuning.)
∆cinfl,max= Acid
±5 mol / l q=5 l / s
Base
pH - 1
pHC
Base
pHI
pHC
Base
V1
pHI
pHC
pHI
V2
V3
pH 7 ±1
kd ~ 106
∆cmax =±10-6 mol / l
Figure 2.3: Neutralization in three stages.
12
Chapter 2. pH-Neutralization: Integrated Process and Control Design
2.3 Time delays
Time delays provide fundamental limitations on the achievable response time,
and thereby directly influence the required volumes. The delays may result from
transport delays or from approximations of higher order responses for mixing or
reaction processes and from the instrumentation. For pH control processes, the
delays arise from
(1) Transport of species into and through the tank, in which the mixing delay is
included ( )
(2) Transport of the solution to the measurement and approximation of measurement dynamics ( )
(3) Approximation of actuator and valve dynamics ( )
(4) Transport of the solution to the next tank ( G )
In this paper, we mainly consider local feedback control, and the total effective
delay is the sum of the contributions from the process and instrumentation . If the influent (disturbance) and the reagent addition (manipulated
variable) are placed close, they will have about the same delay , but for feedback
control only the delay for manipulated variables matters.
Both the volume and the mixing speed determine the mixing delay, which
is the most important contribution to . If the volume is increased, one would
also usually increase the mixing speed, and these two effects are opposing. Walsh
S . Since the
(1993) carried out calculations for one mixer type and found exponent of 7 is close to zero he concludes that is constant (typically about
), independent of the tank size. On the other hand, Shinskey (1973, 1996)
assumes that the overall delay is proportional to the tank volume (this is not
stated explicitly, but he assumes that the ultimate or natural period of oscillation,
which here is < , varies proportionally with the volume). In this paper, we follow
Walsh and assume that the overall effective delay is in each tank.
2.4 Model
The model is derived in Appendix A. pH-control involving strong acids and bases
is usually considered as a strongly “nonlinear” process. However, if we look
at the underlying model written in terms of the excess
concentration *#,+ :
>@
3 3 (2.3)
2.4 Model
13
qinfl
cinfl
qreag
creag
V, c
pHc
c, q
Figure 2.4: Neutralization tank with pH control.
then we find that it is linear in composition (the overall model is bilinear due to
the product of flow rate ( ) and concentration ( )). The fact that the excess concentration will vary over many orders of magnitude (e.g. we want #$ "& $ : , whereas 2 for a strong acid with ), shows
to have N $ the strong sensitivity of the process to disturbances (with ; see below), but
has nothing to do with non-linearity in a mathematical sense.
In Appendix A we have derived a Laplace transformed, linearized, and scaled
model for the process illustrated in Figure 2.4:
(2.4)
0 2 1.354 is a scaled value of the effluent excess concentration, 0
3 3 1.354 is a scaled value of the reagent flow rate, and 0 1.354 0 1.354 0 3 3 1.354 0 3 3 1.354 is a disturbance vec denote the maximum tolerated ( ), possible ( ) or extor. The subscripts where
pected ( ) variation; see also Table 2.1. Note that we have included a reagent flow
rate, 3 , as a disturbance, since it may also have uncontrolled variations due
to e.g. inaccuracies in the valve or upstream pressure variations. is the transfer
function from the control input, and a vector of transfer functions from the
disturbances. Normally it is convenient to consider the effect of one disturbance
at a time, so from now on we consider as a scalar and as a (scalar) transfer
function. The reason for the scaling is to make it easier to state criteria for sufficient dampening, and we scale the model so that the output, control input and the
expected disturbances all shall lie between -1 and 1.
For a single tank the transfer functions and are represented as
E ? where ? is the nominal residence time in the tank (? ? E (2.5)
where is the
14
Chapter 2. pH-Neutralization: Integrated Process and Control Design
nominal volume and the total flow rate), and is effective time delay, due to
mixing, measurement and valve dynamics (see Section 2.3).
In Appendix A.2 we derive a linear model for a series of tanks. Neglecting
reagent disturbances (except in the first tank) and changes in outlet flow-rates of
each tank, we obtain for any disturbance entering in the first tank,
K E where ? is the total residence time . is the total volume and flow rate through the tanks, and we here assume 7277 .
(2.6)
is the
With the above-mentioned scalings, the gain from the control input is (Appendix A.1)
3 3 M 1.354
(2.7)
1.354
while for various disturbances is given in Table 2.1. We will assume that (typically is 8[ or larger for pH systems).
Table 2.1: Steady-state gain for different disturbances. Superscript denotes nominal
values, and subscript denotes maximum tolerated ( 1.354 ) or expected (the other variables) variation. 3 1.354 is maximal expected uncontrolled variation in reagent flow.
Influent
Reagent
Concentration disturbance
!# " $
! "
3 ! &(" ' *)
&(' * )
+
! "
Flow disturbance
2 ! 3 ! " " ! % ! "
&(" ' *) " ! &(' *#
) ,+ ! "
Example 2.1 (continued from page 9): We consider the influent disturbances.
Nominally, 7 (acid flow rate is half the total flow rate), 1.354 "& ,
, and - 1.354 2 (maximum inlet concentration variation).
This gives +/. QM 7 B; 7 10 & (as found earlier).
Furthermore, 1.3542 7 Q2 7 (maximum variation in acid flow rate is
2 ) so 2 ! +/ . 7 QM 7 B; 7 Q & .
2.5 A simple formula for the volume and number of
tanks
The motivating example in Section 2.2 showed that the control system is able to
reject disturbances at low frequencies (including at steady state), but we need design modifications to take care of high-frequency variations. Based on (Skogestad,
1996) a method for tank design using this basic understanding is presented.
2.5 A simple formula for the volume and number of tanks
15
The basic control structure is local control in each tank, as illustrated in Figure 2.3 (flow sheet) and Figure 2.5 (block diagram). We assume no reference
777 ), and the closed-loop response of each tank then
changes ( becomes
(2.8)
where , and for , . is the sensitivity function for tank
. Combining this into one transfer function from the external disturbance to the
final output leads to
where SISO systems.
. The factorization of d
r1
-
(2.9)
(2.10)
is possible since the tanks are
Gd1
K1
y1
G1
r2
Gd2
K2
-
y2
G2
r3
-
K3
Gd3
y3
G3
Figure 2.5: Block diagram corresponding to Figure 2.3 with local control in each tank.
We assume that the variables ( and ) have been scaled such that for disturbance rejection the performance requirement is to have for all at
all frequencies, or equivalently
(2.11)
16
Chapter 2. pH-Neutralization: Integrated Process and Control Design
Combining (2.11) and the scaled model of in (2.6) yields an expression for the
required total volume with equal tanks:
I (2.12)
Assuming I (since and the design is most critical at
frequencies where is close to 1) this may be simplified to
I I (2.13)
We see that enters into the expression in the power of . This is because
is of the same order as . This gives the important insight that a “resonance”
peak in due to several tanks in series, will not be an important issue. Specifi-
cally if the tanks are identical and the controllers are tuned equally, the expression
is
(2.14)
I where is the sensitivity function for each locally controlled tank. This condition
must be satisfied at any frequency and in particular at the bandwidth frequency
. This gives the
, here defined as the lowest frequency for which minimum requirement (Skogestad, 1996)
Since
I (2.15)
decreases as increases, this volume guarantees that
(2.16)
In words, the tank must dampen the disturbances at high frequencies where control is not effective. With only feedback control, the bandwidth (up to which
feedback control is effective), is limited by the delay, , and from (Skogestad and
Postlethwaite, 1996, p.174) we have (the exact value depends on the
controller tuning), which gives
(2.17)
where (Skogestad, 1996)
I is a “reference value” we will compare with throughout the paper. For we have
P
I
(2.18)
,
(2.19)
2.5 A simple formula for the volume and number of tanks
17
(2.19) gives the important insight that the required volume in each tank, ,
is proportional to the total flow rate, , the time delay in each tank, , and the
disturbance gain raised to the power . Table 2.2 gives as a function of for Example 2.1. With one tank the size of a supertanker ( ;> >8> [ ) is required
(as we got in the motivating example). The minimum total volume is obtained
with 18 tanks (Skogestad, 1996), but the reduction in size levels off at about 3-4
tanks, and taking cost into account one would probably choose 3 or 4 tanks. For
example, Walsh (1993) found the following formula for the capital cost in of a
stirred tank reactor
B; >8>
; 8> S
From this we obtain the following total cost for :> , > , , ; , i.e. lowest cost is for three tanks.
Table 2.2: Total tank volume,
.
from (2.18). Data:
Number of tanks, 1
2
3
4
5
(2.20)
777
[
in
,
Total volume [ ;> 8>>
N
< 7 7
7 8> : ; 8>
[
&
,
and
]
Remark 1 Conditions (2.15) and (2.17) are derived for a particular frequency
and other frequencies may be worse. However, we will see that is “flat”
around the frequency if the controller tuning is not too aggressive, and is
close to the worst frequency in many cases.
Remark 2 In (2.6) we neglected the variation in the outlet flow rate from each
tank. The outlet flow rate is determined by the level controller (see (2.59) and
(2.60)). With more than one tank and a different pH in each tank, a feed flow
rate variation (disturbance) into the first tank will give a parallel effect in the
downstream concentration variations since both the inlet flow rate and inlet concentration will vary. Also, variations in the reactant flow rate will influence the
level and thereby outlet flow rate. Perfect level control is worst since then outlet
flow rate equals inlet flow rate. With averaging level control (surge tank), the outlet flow variations are dampened, but extra volume is required also for this, which
is not taken into account in the analysis presented in this paper.
18
Chapter 2. pH-Neutralization: Integrated Process and Control Design
2.6 Validation of the simple formula: Improved sizing
In (2.18) we followed Skogestad (1996) and derived the approximate value for
the total volume. This is a lower bound on due to the following two errors:
(E1) The assumed bandwidth (e.g. PI or PID).
(E2) The maximum of
is too high if we use standard controllers
occurs at another frequency than .
In this section we compute numerically the necessary volume when these
two errors are removed. We assume first sinusoidal disturbances, and later step
changes. Each tank (labeled ) is assumed to be controlled with a PI or PID con
troller with gain , integral time ? and for PID derivative time ? :
O? ? O? 7J ? (2.21)
(cascade form of the PID controller). We consider four different controller tuningrules for PI and PID controllers: Ziegler-Nichols, IMC, SIMC and optimal tuning.
For the case with Ziegler-Nichols, IMC or SIMC tunings the controller parameters are fully determined by the process parameters , ? and , and an optimization problem for finding the minimum required tank volumes may be formulated
as:
(2.22)
subject to
is stable
(2.23)
(2.24)
To get a finite number of constraints, we define a vector containing a number of
frequencies covering the relevant frequency range (from [ to 8[ ).
It is assumed that if the constraints are fulfilled for the frequencies in , they
are fulfilled for all frequencies. The stability requirement is that the real part of
the poles of are negative. The poles are calculated using a 3rd order Padé
approximation for the time delays in , but this is not critical since the stability
constraint is never active at the optimum.
Ziegler and Nichols (1942) tunings are based on the ultimate gain and
ultimate period . For our process the resulting PI controller has gain 2.6 Validation of the simple formula: Improved sizing
7<
19
P 7 ?# and integral time ? >7 ;DP 7 . The corresponding
“ideal” PID tunings are: 7 N P 7 < ?' , ? >; ; and ? 7 < Y?# and ? ? : 7 , which correspond to >;P
for our cascade controller.
The IMC-tunings derived by Rivera et al. (1986) have a single tuning parameter , which we select according to the recommendations for a first order process
with delay as >7 for PI control and 7 : for PID control. We get a PI
controller with gain 7 > : ?' and integral time ? ? . For the cascade
form IMC-PID controller, we get 7 >M?# , ? ? and ? 7 .
However, the IMC tuning is for set-point tracking, and for “slow processes”
with ? this gives a very slow settling for disturbances. Skogestad (2003)
therefore suggests to use ? A? : , which for our process gives ? : . The controller gain is 7 Y?# . We denote this tuning SIMC PI.
For a SIMC-PID controller (on cascade form), the gain and integral time are left
unchanged, and we have chosen to set the derivative time ? to 7 .
For optimal tunings, the controller parameters are optimized simultaneously
with the volumes:
K K (2.25)
subject to
is stable
777 (2.26)
(2.27)
(2.28)
To assure a robust tuning, a limit,
1.354 ; , is put on the peak of the gain of
the individual sensitivity functions . (For PID control we also let ? 7727 ? vary in the optimization.)
In the following we apply this numerical approach to the process in Example 2.1. For multiple tanks in series, is distributed equally between the tanks,
so that for tank we get I . The results for the four different controllers (ZN, IMC, SIMC and optimal) are given in Table 2.3 for PI control and in
Table 2.4 for PID control.
The optimal controller PI-tunings (last column in Table 2.3) give a large in
tegral time, so that we in effect have obtained P-control with >Y? equal to
7 N (one tank), 7 and 7 8N (two tanks), 7 : and twice 7 (three tanks) and
7 and three times 7 (four tanks). The optimal PID-tuning (last column in
Table 2.4) also gave a large integral time (PD control) with 7 : ?' and
derivate time ? 7 < for all tanks.
20
Chapter 2. pH-Neutralization: Integrated Process and Control Design
on From Tables 2.3 and 2.4 we find that the “correction factor”,
(2.29)
is in the range >7 ; to 7 ; . The correction factor is independent of the number
of tanks in most cases, which is plausible since the combination of (2.14) and
(2.19) gives
(2.30)
where is close to independent of the number of tanks involved. To see
this, insert the tuning rules into the controller transfer function and calculate
. For the IMC tuning depends only on , so that when it is
scaled with it will independent of the process parameters. for ZN and
SIMC depends on ? , but only for low frequencies (when ? is small compared
to ). For up to three tanks, only depends on at the relevant frequencies.
Recall, however, that this analysis is not exact since (2.30) is an approximation.
Frequency-plots for 3 tanks with PI control are given in Figure 2.6. In all
four cases the bandwidth is lower that (error E1). is the worst frequency, with exception of the Ziegler-Nichols tunings (which due to the high peak
in give error E2). The optimal controller makes constant
for a wide frequency range.
Table 2.3: PI controllers: Volume requirements obtained from (2.22) (for Ziegler
Nichols, IMC, SIMC) and from (2.25) (optimal tuning). (Data: & , )
n
1
2
3
4
79 N
97 N
97 <
7 ZN
;
<
>7 :
>7 :
>7 :
>7 :
IMC
5;
<
SIMC
7=<
=7 <
=7 <
=7 <
; : ; : 5;
; N ; ; <
Optimized
87 :
87 >
87 87 N :
;
<
Table 2.4: PID controllers: Volume requirements obtained from (2.22) (for Ziegler
Nichols, IMC, SIMC) and from (2.25) (optimal tuning). (Data: & , )
n
1
2
3
4
ZN
; 7 ; 7 ; 7; ; 7 ;8
IMC
5 ;
<
>7 >7 >7 ;
>7 ; :
SIMC
5;
<
; 7J
; 7J
; 7J<
; 7J
Optimized
;
<
>7 ;8;
>7 ;8;
>7 ; >79 5 ;
<
2.6 Validation of the simple formula: Improved sizing
21
Gd
Gd
Si
Si
1/θ
θ
0
Magnitude
Magnitude
0
10
ωB
10
ωB
1/θ
θ
SGd
SGd
−1
10
−1
−2
−1
10
10
0
10
10
−2
−1
10
(a) ZN settings
Gd
Si
1Si
0
Magnitude
Magnitude
2Si’s
Gd
0
1/θ
θ
10
ωB
1/θ
θ
SGd
SGd
−1
10
10
(b) IMC settings
10
ωB
0
10
Frequency [rad/s]
Frequency [rad/s]
−1
−2
10
−1
10
Frequency [rad/s]
(c) SIMC settings
0
10
10
−2
10
−1
10
0
10
Frequency [rad/s]
(d) Optimal PI settings
Figure 2.6: Frequency-magnitude-plots corresponding to results for PI-control of 3 tanks
in Table 2.3
Chapter 2. pH-Neutralization: Integrated Process and Control Design
22
Next consider in Figure 2.7(a) the response to a step disturbance in inlet concentration ( ) for the different controller tunings and tank volumes for the case
with three tanks in series. As stated before, the optimal PI controller is actually a
P controller, and the controller with IMC tuning also has a “slow” integral action
and this is observed by the slow settling. We see that for the other two tunings,
and especially for the Ziegler-Nichols tuning, the frequency domain result is conservative when considering the step response. This is because the peak in is sharp so that exceeds 1 only for a relatively narrow frequency
range, and this peak has only a moderate effect on the step response. This means
that we can reduce the required tank volume if step disturbances are the main concern. For the step response, we find that a total tank size of >7 keeps the output
within for PI controllers tuned both with Ziegler-Nichols and SIMC. For PID
control we find that >7 and >7 N are necessary for these two tuning rules (1-4
tanks).
In conclusion, for PI control we recommend to select tanks with size P
; , whereas with PID control )P >7 N is sufficient. These recommendations
are confirmed in Figure 2.7(b) where we use ; , and we see that after a
unit disturbance step the output is within .
1.2
1.2
Optimal controller
Vtot= 1.73V 0
1
1
0.8
IMC
0.8
0.6
0.6
0.4
IMC
Vtot= 1.81V 0
0.4
0
Ziegler−Nichols
Vtot= 3.14V 0
0.2
SIMC
0.2
−0.2
0
−0.4
−0.2
−0.6
SIMC
Vtot= 2.46V 0
−0.4
0
50
100
150
200
250
Time [s]
300
Ziegler−Nichols
−0.8
350
400
450
(a) Volumes and tunings from Table 2.3
500
0
50
100
150
200
250
Time [s]
300
(b) Equal tanks with
)
Figure 2.7: Response to step disturbance in 350
400
450
500
( for 3 tanks using PI-control.
Remark 1 We have specified that in each tank ; , where is the (open
loop) disturbance gain in each tank, but the results are independent of this choice,
since the controller gains are adjusted relative to the inverse of .
2.7 Equal or different tanks?
23
Remark 2 The sensitivity functions, , are independent of the pH set-points
in each tank (see Remark 1). is determined by its time constants and
delays, which are independent of the pH-values, and its steady state overall gain,
. is defined by the inlet and outlet pH. The fundamental requirement (2.11),
and thereby the results of this and the previous section, are therefore independent
of the pH set-points in intermediate tanks.
2.7 Equal or different tanks?
In all the above optimizations (Tables 2.3 and 2.4) we allowed for different tank
sizes, but in all cases we found that equal tanks were optimal. This is partly
because we assumed a constant delay of 10 seconds in each tank, independent of
tank size.
This confirms the findings of Walsh (1993) who carried out calculations showing that equal tanks is cost optimal with fixed delay. We present here a derivation
that confirms this. We assume that the cost of a tank of volume is proportional to
where is a scaling factor. To minimize the total cost we then must minimize
QQQ
(2.31)
which provided the flow rate through all tanks are equal (which is true for example
if most of the reagent is added into the first tank) is equivalent to
O ?
K K ?
QQQ
?
(2.32)
This cost optimization is constrained by the demand for disturbance rejection
(2.11). The expression for for arbitrary sized tanks is:
E O? QQQ A? (2.33)
Combining (2.33) with the inequality (2.11) yields
C O ?
L QQQ C A ?
L (2.34)
which constraints the optimization in (2.32). We assume again that the peak in
occurs at the frequency where . (2.34) then simplifies to
C O ? L QQQ C O ? L (2.35)
and it can easily be proved (e.g. using Lagrange multipliers, see Appendix C) that
equal tanks minimizes cost.
24
Chapter 2. pH-Neutralization: Integrated Process and Control Design
This result contradicts Shinskey (1973, 1996) who assumed that the delay
varies proportionally with the volume, and found that the first tank should be about
one fourth of the second. McMillan (1984) also claims that the tanks should have
different volume. Let us check this numerically. We assume a minimum fixed
delay of and let our previous
. To get consistency with
results with constant delay of , we let for , where is
the total volume required with constant delay (see the final column of Table 2.3).
The results of the optimization with PI control are presented in Table 2.5. We see
that in this case it is indeed optimal with different sizes, with a ratio of about 1.5
between largest and smallest tank. However, if we with the same expression for
, require equal tanks and equal controller tunings in each tank, the incremental
volume is only 14% or less for up to 4 tanks (see the last column in Table 2.5).
Table 2.5: Optimal PI design with volume dependent delay:
n
2
3
4
Volume each tank
217, 326
18.4, 18.4, 30.7
5.36, 5.36, 5.36, 9.14
544
67.6
25.2
Volume ratio
1.50
1.67
1.71
.
increase with equal tanks
+4%
+9%
+14%
With a smaller fixed part in , the differences in size are larger. For example with a fixed delay of only we get a optimal ratio of up to 7 (for 3 tanks).
However, if we allow for PID-controllers the ratio is only >7 .
These numerical results seem to indicate that our proof in (2.35), which allows
for different delays in each tank, is wrong. In the proof we assumed that at the frequency where has its peak. This will hold for a complex controller,
where due to the constraint (2.26) we expect to remain flat over a large frequency region, but not necessarily for a simple controller, like PI. The frequency
plots for the resulting PI-controllers in Table 2.5 confirm this.
In conclusion, it is optimal, in terms of minimizing cost, to have identical
tanks with identical controllers, provided there are no restrictions on the controller.
With PI-control, there may be a small benefit in having different volumes, but
this benefit is most likely too small to offset the practical advantages of having
identical units. This agrees with the observations of Proudfoot (1983) from 6
neutralization plants with two or three tank in series. In all cases equal tanks had
been chosen.
2.8 Discussion
25
2.8 Discussion
2.8.1 Measurement noise and errors
In this paper, we have focused on the effect of disturbances. Another source of
control errors is errors and noise in the measurements. Normally the accuracy of
pH instruments is considerable better than the requirement for the pH variation,
which we as an example has given as pH units in the present paper. However,
due to impurities, the measured value may drift during operation. In one of Norsk
Hydro’s fertilizer plants, the probes are cleaned and recalibrated once a week, and
during this period, the pH measurement may drift up to 1 pH unit. This drifting is,
however, very slow compared to the process, and will not influence the dynamic
results from this paper, except that the controller cannot make the pH more correct
than its measurement.
The worst error type is steady-state offset in the measurement of the product.
This can lead to a product outside its specifications, and can only be avoided by
regular calibration (possibly helped by data reconciliation).
Measurement errors in upstream tanks may lead to disturbances at later stages,
since the controller using this measurement will compensate for what it believe to
be a change in the concentration. Such errors can be handled at later stages.
To study the effects of measurement errors in the setting of this paper, one
must convert the expected errors in the pH measurement to a corresponding error
in the scaled concentration variable, . Tools for such conversion is provided in
Appendix B. Often the error in becomes larger than the pH error (as seen in the
example of Appendix B).
The conclusion is that small and slowly appearing measurement errors do not
cause problems, provided frequent maintenance is performed, whereas higher frequency variation with amplitude close to allowed pH variation must be converted
into variation in and treated as disturbances.
2.8.2 Feedforward elements
In this section, we discuss the implications for the tank size of introducing feedforward control. Feedforward from an influent pH measurement is difficult since
an accurate transition from pH to concentration is needed. An indication of this
is that Shinskey removed the section “Feedforward control of pH” in his fourth
edition (compare (Shinskey, 1988) with (Shinskey, 1996)). Feedforward from the
influent flow rate is easier, and McMillan (1984) states that one tank may be saved
with effective feedforward from influent flow rate and pH.
Skogestad (1996) show for an example with three tanks that use of a feedforward controller that reduced the disturbance by 80%, reduced the required total
Chapter 2. pH-Neutralization: Integrated Process and Control Design
26
volume from <>7 to ; 7 : [ .
Previous work has considered feedforward from external disturbances. We
will in the following analyze the situation with tanks in series and “feedforward” to downstream tanks from upstream measurements. In this way, no extra
measurements are required. As is discussed in Chapter 4, a multivariable controller may give this kind of feedforward action. We assume no feedforward to
the first tank, and assume that the feedforward controllers reduce the disturbance
to each of the next tanks by a factor of , 7727 (where hopefully
$ ). The effective gain from an inlet disturbance to the concentration in the
last tank then becomes
(2.36)
To calculate the required volumes for this case, we insert (2.36) into (2.18), and
get
If
Q QQ I
(2.37)
, (2.19) and (2.37) yield:
P
+
(2.38)
For example, if
each feedforward effect reduces the disturbance by :>2 ( 7 ; ), we get ( tank), 7 < ( ; tanks), etc.; see Table 2.6 for more
details.
Table 2.6: The volume requirement with feedforward from each tank to next assuming that
the feedforward reduces the disturbance by ( ) and with perfect feedforward
control ( ). is given by (2.17).
No. of tanks
<
;
:>2
reduction Perfect feedforward control
7=<
7 <
7 7 ;>
7; ;
<
To have perfect feedforward from one tank to another one need, in addition to
a perfect model, an invertible process. With a delay in the measurement or a larger
delay for the control input than for the disturbance this is not possible. Feedforward and multivariable controllers may actually benefit from transportation delay
as will be illustrated in the following example.
2.8 Discussion
27
Example 2.2 We have three tanks with (at least) measurement of pH in tank 1
and reagent addition in at least tank 3. The transport delay is in each tank,
and the measurement delay is also (or less). If an upset occurs in tank 1 at
time , the upset reaches tank 2 at time and tank 3 at . It is “discovered”
in the measurement in tank 1 at time or before (the sum of the transport delay
and the measurement delay). With a multivariable controller or a feedforward
controller from tank 1 to 3, action can be taken in tank 3 at the same time the
upset reaches the tank. For control of tank 2, however, the measurement in tank 1
will show the upset too late. The example is illustrated in Figure 2.8.
Inlet tank 1
Outlet tank 1
Measured tank 1
Inlet tank 2
Outlet tank 2/
Inlet tank 3
0s
5s
10s
Figure 2.8: With three tanks in series, an upset entering tank 1 reaches tank 3 at the same
time the upset is seen in the measurement of tank 1. We assume the measurement and
transport delays are equal.
From the feedback analysis in the previous sections, the smaller the total time
delay the better. Example 2.2 shows, however, that if feedforward or multivariable
control is used, one may benefit from a transport delay in intermediate tanks that
is not shorter than the measurement delays. One should always seek to minimize
the measurement delay.
2.8.3 pH set-points in each tank
We have already noted that the analysis in the previous sections is independent
of the pH set-point in each tank (Remark 2, Section 2.6). Here we discuss some
issues concerning the set-points or equivalently the distribution of reagent addition
between the tanks.
For some processes e.g. in fertilizer plants, the pH in intermediate tanks is important to prevent undesired reactions. Such requirements given by the chemistry
of the process stream shall be considered first.
Next, instead of adjusting the set-points directly, one may use the set-points
in upstream tanks to slowly adjust the valves in downstream tanks to ideal resting
28
Chapter 2. pH-Neutralization: Integrated Process and Control Design
positions. But also in this case, one must have an idea of the pH levels in the tanks
when designing the valves.
Whenever possible, we prefer to add only one kind of reagent, for example
only base, to save equipment (see Figure 2.3). To be able to adjust the pH in both
directions as we have assumed, one then needs a certain nominal flow of reagent
in each tank. This implies that the pH nominally needs to be different in each
tank.
On the other hand, equal set-points in each tank minimizes the effect of flow
rate variations. In addition, more reagent is added early in the process, so that
reagent disturbances enter early.
One common solution is to distribute the pH set-points so that the disturbance
gain is equal in each tank. In this way one may keep the pH within where is
the same in each tank.
In conclusion, it is preferable to choose the set points as close as possible, but
such that we never get negative reagent flow.
2.9 Conclusions
Buffer and surge tanks are primarily installed to smoothen disturbances that cannot be handled by the control system. With this as basis, control theory has been
used to find the required number of tanks and tank volumes. We recommend identical tank sizes with a total volume of ; where is given in (2.18) as a function
of the overall disturbance gain, , time delay in each tank, the flow rate and
number of tanks . The disturbance gain can be computed from Table 2.1.
Typically, the mixing and measurement delay is about or larger.
2.10 Acknowledgements
Financial support from the Research Council of Norway (NFR) and the first author’s previous employer Norsk Hydro ASA is gratefully acknowledged.
References
Balchen, J. G. and K. I. Mummé (1988). Process Control. Structures and Applications.
Van Nostrand Reinhold. New York.
McMillan, G. K. (1984). pH Control. Instrument Society of America. Research Triangle
Park, NC, USA.
REFERENCES
29
Proudfoot, C. G. (1983). Industrial Implementation of On-Line Computer Control of pH.
PhD thesis. Univ. of Oxford, UK.
Rivera, D. E., M. Morari and S. Skogestad (1986). Internal model control. 4. PID controller design. Chem. Engn. 25, 252–265.
Shinskey, F. G. (1973). pH and pIon Control in Process and Waste Streams. John Wiley
& Sons. New York.
Shinskey, F. G. (1988). Process Control Systems - Application, Design, and Tuning, 3rd
Ed.. McGraw-Hill Inc.. New York.
Shinskey, F. G. (1996). Process Control Systems - Application, Design, and Tuning, 4th
Ed.. McGraw-Hill Inc., New York.
Skogestad, S. (1996). A procedure for SISO controllability analysis - with application to
design of pH neutralization processes. Computers Chem. Engng. 20(4), 373–386.
Skogestad, S. (2003). Simple analytic rules for model reduction and PID controller tuning.
J. Proc. Contr. 13(4), 291–309.
Skogestad, S. and I. Postlethwaite (1996). Multivariable Feedback Control. John Wiley &
Sons. Chichester, New York.
Walsh, S. (1993). Integrated Design of Chemical Waste Water Treatment Systems. PhD
thesis. Imperial College, UK.
Ziegler, J. G. and N. B. Nichols (1942). Optimum settings for automatic controllers. Trans.
ASME 64, 759–768.
Chapter 2. pH-Neutralization: Integrated Process and Control Design
30
Appendix A Modelling
A.1 Single tank
We first consider one single tank with volume , see Figure 2.4. Let ( )
concentradenote the concentration of
-ions, *#,+ ( ) denote the
tion, and denote flow rate. Let further subscript denote influent, subscript
denote reagent and no subscript denote the outlet stream. Material balances
yield:
for
and
) @
) 3 3 ) (2.39)
*#,+ * + * + 3 3 *#+ >@
( .
) is the rate of the reaction (2.40)
. For strong,
where
i.e. completely dissociated, acids and bases this is the only reaction in which
participate, since the ionization reaction already has taken place (for
and
weak acids and bases, also the ionization reaction must be included in the model).
can be eliminated from the equations by taking the difference. In this way we
get a model for the excess of acid, i.e. the difference between the concentration of
ions (Skogestad, 1996):
and
) * +
(2.41)
The component balance is then given by
>@
(2.42)
3 3 Making use of the total material balance ( > @ 3 ) the component
balance simplifies to
3 3
>@
(2.43)
Linearization of (2.43) around a steady-state nominal point (denoted with an asterisk) and Laplace transformation yields:
"
"
!+ " ! "
! &(" ' *)
! " 3 " $
! "
"
&" ' *)
! "
"
3 (2.44)
where 3 (steady-state mass balance) and the Laplace variables ,
, , 3 , and 3 now denotes deviations from their nominal point. Note
that the dynamics of have no effect on the linearized quality response.
Appendix A Modelling
31
The nominal excess acid concentration are found from the nominal C V L values:
(2.45)
The composition balance is used to obtain the nominal reagent flow rate.
The reagent flow rate, 3 , may be divided into 3 which is determined by
the controller, and a disturbance term, 3 , which is due to leakages and other
uncertainties in the dosing equipment. Thus 3 3 3 .
We introduce scaled variables, where subscript denotes maximum allowed or expected variation:
1. 354
(2.46)
1.354 2 ! 2 1. 354
(2.47)
(2.48)
3 33 1.354 3 ! 33 1. 354
3 (2.49)
3 M 1.354
Thus , 2 , ! , 3 , 3 ! and all shall stay within . We obtain
1.354 2 1.354 ! 1.354 1.354
! " 3 1.354 3 3 3 3 1.354 3 ! (2.50)
1.354 1.354
3 3 1.354 1.354
The scaling factor 1.354 is found from the given allowed variation in pH ( ):
1. 354 C V L (2.51)
1.354 C V L
(2.52)
1.354 C 1. 354 1.354 L
(2.53)
"
If we consider one disturbance at a time, the model is on the form
where ble 2.1.
? &(" ' * ) " ! &(' * )+ ! "
and ? (2.54)
(2.55)
for different disturbances are given by Ta-
Chapter 2. pH-Neutralization: Integrated Process and Control Design
32
A.2 Linear model for multiple tank in series
We will now extend the model to include tank in series, and label the tanks
777 . For the first tank we get the same expression as for the single tank
(2.50) (except for the labeling):
1.354 2 1.354 ! / 1.354 1.354 3 1.354 3 3 3 3 1.354 3 ! / 1.354 1.354
/
"
! "
3 / 1.354
(2.56)
3 1.354
For the following tanks, the inflow is equal to the outflow from previous tank, so
that
1.354 1.354 1.354 !+ " 3 1.354 3 3 3 3 1.354 3 ! 1.354 1.354
3 3 1.354 1.354
"
(2.57)
is the deviation from nominal value for the flow rate from previous tank
and is determined by the level controller in previous tank, . For tank ,
the outlet flow rate becomes
(2.58)
where is the variation in the volume set-point. We assume that
and express as a function of the total inlet flow:
3 ! 3 If a P controller is used, we get where ,
(2.59)
is the controller gain, and
(2.60)
Appendix A Modelling
33
F? A? , where
Alternatively a PI-controller can be used, is controller gain, and ?
is the integration time, but if ? , we may
ignore the integral effect in the model.
Often we may assume that the level controller is very slow, which leads to
WP (recall that denotes the deviation from the nominal value). With
the additional simplification that the disturbances from the reagent can be neglected, we get the following model for tanks:
1
..
.
where
? (2.61)
? 7727 (2.62)
From (2.61) and (2.62) we get for the scaled output of the last tank
? ? ? (2.63)
(2.64)
In the present paper we use (2.63) and (2.64) to represent the tanks.
A.3 Non-linear model for multiple tank in series
We label the tanks with and get by using (2.43):
/ 3 3 >@
/ 3 3 >@
(2.65)
..
.
3 3 >@
34
Chapter 2. pH-Neutralization: Integrated Process and Control Design
The dynamic behaviour of the volumes are given by the mass balances:
@ 3 . @ 3 (2.66)
..
3 >@
The flow rates from each tank, , are given by the flow controllers (2.59). As in
the linear case, 3 may be divided into a manipulable part and a disturbance.
A.4 Representation of delays
In section 2.3 we discuss the delays the are present in this process. In the linearized
transfer function model the total delay, , may be represented by the term
E (2.67)
For models of multiple tanks in series, the different type of delay must be considered differently. Figure 2.9 illustrates this. The total delay in the control loop
d
Delay θp
Delay θn
u
Delay θm
Delay θv
Figure 2.9: The delays in a neutralization process
is
%
(2.68)
whereas the total delay related to the transportation and mixing through a tank and
to the next is
3 (2.69)
G
2.10 Appendix B The effect of pH measurement errors on the scaled...
35
Appendix B The effect of pH measurement errors
on the scaled excess
concentration, In a real plant we measure the pH, and not the scaled excess
concentration
variable, , that we have used in this paper. The pH measurement must be transformed into if the controller shall use and not the pH value. In this appendix
we study the effect of errors and noise in the pH measurement on the scaled excess
variable .
The scaling in this paper is chosen in such a way that as long as
we are
sure that the variation in actual pH value, , around a nominal pH value, ,
is less than 1 pH units:
(2.70)
However, the implication does in general not go in the opposite direction.
concentration is *# + , or expressed by the correThe excess
sponding pH value:
V (2.71)
We denote the actual pH for , and the measurement error for 0 . Then,
what we measure is F0 . The corresponding error in the excess
acid concentration is
0
0 (2.72)
From (2.70) we obtain for the scaled variable, :
1.354 (2.73)
where corresponds to . Provided the acceptable pH variation is the maximum accepted value for the excess concentration is
1.354 0
,
(2.74)
:
(2.73) and (2.74) yield for the error in the scaled variable, 0
" " " " (2.75)
Chapter 2. pH-Neutralization: Integrated Process and Control Design
36
(2.75) can be used to find 0
corresponding to a pH measurement error or noise
of 0 .
We will now consider some special cases. As in the paper, we specify , and let the actual value equal the nominal value. We consider first F $
. Then
0 0 For L " C L V C
67 Q " >Q V "
"
(2.76)
we obtain
0 " "
C " L H C L " V Q " 7 Q V "
L 7 (since then " V " )
For X$ N we get 0 P C L 7 (since then V " and for : we get 0 P C "
). This yields the following simple formula (when ):
Y 0 $ N F :
(2.77)
7 0
Example 2.3 We have made a model of a neutralization process (as described in
Appendix A) and have chosen and . The pH measurement
may have a measurement noise of 7 pH units, and we want to determine
the corresponding noise in the scaled concentration variable . We consider an
actual pH value equal to- the nominal, and since $ N , we can use
7 7J< .
(2.77): 0
1.354
Appendix C On the optimization problem (2.32) subject to (2.35)
37
Appendix C On the optimization problem (2.32) subject to (2.35)
Here we prove that the solution to
O ?
K K C O ? QQQ
subject to
?
L QQQ C O ? ?
L
(2.78)
is to have ? ? QQQ ? . The solution will not be at an interior point so we
take the limiting of the constraint. We introduce ? , and get the following
optimization problem with the same solution as the original:
subject to
QQQ
C
L (2.79)
The Lagrange function, , for this problem is, denoting the Lagrange multiplier :
QQQ
C
L and in the constrained optimum we have
;
This implies, using the constraint, that
C
L
7277 (2.80)
(2.81)
; 7727 (2.82)
In equation (2.82), , and are independent of the index , and the value
of is therefore the same for all ’s. So QQQ , which implies that
Q
Q
Q
? ? ? .
Chapter 3
Buffer Tank Design for Acceptable
Control Performance
Audun Faanes and Sigurd Skogestad
Department of Chemical Engineering
Norwegian University of Science and Technology
N–7491 Trondheim, Norway
Ind. Eng. Chem. Res. 2003 42(10),2198-2208
Abstract
This paper provides a systematic approach for the design of buffer tanks. We consider mainly
the case where the objective of the buffer tank is to dampen (“average out”) the fast (i.e., highfrequency) disturbances, which cannot be handled by the feedback control system. We consider
separately design procedures for (I) mixing tanks to dampen quality disturbances and (II) surge
tanks with averaging level control to handle flow-rate disturbances.
also affiliated with Norsk Hydro ASA, Corporate Research Centre, N-3907 Porsgrunn, Norway, E-mail: [email protected], Tel.: +47 35 92 40 21, Fax.: +47 35 92 32 63
Author to whom all correspondence should be addressed. E-mail: [email protected],
Tel.: +47 73 59 41 54, Fax.: +47 73 59 40 80
Extensions compared to published paper: Section “Further discussion” and Appendices A-C
Chapter 3. Buffer Tank Design for Acceptable Control Performance
40
3.1 Introduction
Buffer tanks are common in industry, under many different names, such as intermediate storage vessels, holdup tanks, surge drums, accumulators, inventories,
mixing tanks, continuous stirred tank reactors (CSTRs), and neutralization vessels. We start with a definition:
A buffer tank is a unit where the holdup (volume) is exploited to provide smoother operation.
We here focus on buffer tanks for liquids, although most of the results may
be easily extended to gas- or solid-phase systems. Buffer tanks may be divided
into two categories, namely, for (A) disturbance attenuation and (B) independent
operation:
A. Buffer tanks are installed between units to avoid propagation of disturbances
for continuous processes.
B. Buffer tanks are installed between units to allow independent operation, for
example during a temporary shutdown and between continuous and batch
process units.
In this category there is a continuous delivery or outdraw on one side and a
discontinuous delivery or outdraw on the other side. The design of the tank
size for these types of buffer tanks is often fairly straightforward (typically
equal to the batch volume) and is not covered further in this paper.
LC
Quality
Flow rate
(I) Averaging by mixing (mixing tank)
(II) Averaging level control (surge tank)
Figure 3.1: Two types of buffer tanks
In this paper we focus on category A. There are two fundamentally different
disturbances, namely, in quality and flow rate, and two approaches to dampen
them (see Figure 3.1):
3.1 Introduction
41
(I) Quality disturbances, e.g., in concentration or temperature, where we dampen by mixing. Such buffer tanks are often called mixing tanks or neutralization vessels for pH processes.
(II) Flow-rate disturbances, e.g., in the feed rate, where we dampen by temporarily changing the volume (level variation). Such buffer tanks are often
called surge tanks, intermediate storage vessels, holdup tanks, surge drums,
accumulators, or inventories.
In both cases the tank volume is exploited, and a larger volume gives better
dampening: In the first case, mixing of a larger volume means that the in-flow
entering during a longer period is mixed together, and in the second case, larger
level variations are allowed.
Often, in the design of buffer tanks, the residence or hold-up time is used as a
, where
measure instead of the volume. The residence time is defined as ? [
[
.
is the volume
and the nominal flow rate
Even if the buffer tanks are designed and implemented for control purposes,
control theory is rarely used when sizing and designing the tanks. Instead, rules
of thumb are used. For example, textbooks on chemical process design seem to
agree that a half-full residence time of 5-10 minutes is appropriate for distillation
reflux drums and that this also applies for many other buffer (surge) tanks. For
tanks between distillation columns, a half-full residence time of 10-20 minutes
is recommended (Lieberman, 1983; Sandler and Luckiewicz, 1987; Ulrich, 1984;
Walas, 1987; Wells, 1986).
Sigales (1975) sets the total residence time as the sum of the surge time and
a possible settling time. The following surge times are recommended: distillation
reflux, 5 minutes; product to storage, 2 minutes; product to heat exchanger or
other process streams, 5 minutes; product to heater, 10 minutes. The settling time
applies when there is an extra liquid phase. For water in hydrocarbons, a settling
time of 5 minutes is proposed.
None of the above references provide any justifications for their rules.
The most complete design procedure for reflux drum volumes is presented by
Watkins (1967), who proposes a half-full volume given by
[
V
(3.1)
Here (typical range 0.5-2) and (typical range 1-2) are instrumentation and
labor factors, respectively, related to buffer tanks of category B mentioned above.
For example, the value of may be based on how much time it takes for the operator to replace a disabled pump. and are reflux and product rates, and the
factor [ (typical range 1.25-4) is dependent on how well external units are operated (e.g., 1.25 for product to storage). V (typical range 1-2) is a level indicator
factor. The method gives half-full hold-up times from >7 to ; .
42
Chapter 3. Buffer Tank Design for Acceptable Control Performance
In addition to the volumes proposed above, one normally adds about 2 of
the volume to prevent overfilling (Wells, 1986). For reflux drums, 25-50% extra
volume for the vapor is recommended (Sandler and Luckiewicz, 1987).
A basic guide to the design of mixing tanks is given by (Ludwig, 1977).
The process control literature refers to the level control of buffer tanks for
flow-rate dampening (surge tanks) as averaging level control. Harriott (1964),
Hiester et al. (1987), and Marlin (1995) propose controller and tank size designs
that are based on specifying the maximum allowed change in the flow rate out of
the buffer (surge) tank because this flow acts as a disturbance for the downstream
process. However, no guidelines are given for the critical step of specifying the
outlet flow-rate change. Otherwise, these methods have similarities with the one
proposed in the present paper.
To reduce the effect of the material balance control on the quality control loop,
Buckley (1964) recommends designing the buffer tank such that the material balance control can be made 10 times slower than the quality loop. In practice, this
means that the effect of the disturbance on the quality at the worst-case frequency
is reduced by a factor of 10. This applies to both surge and mixing tanks.
There have also been proposals for optimal averaging level control, e.g.,
(McDonald et al., 1986), where the objective is to find the controller that essentially gives the best disturbance dampening for a given surge tank. To reduce
the required surge tank volume, provided one is willing to accept rare and short
large changes in the outlet flow, one may use a nonlinear controller that works as
an averaging controller when the flow changes are small but where the nonlinear
part prevents the tank from being completely empty or full, e.g., (McDonald et
al., 1986; Shunta and Fehervari, 1976; Shinskey, 1996).
Another related class of process equipment is neutralization tanks. Neutralization is a mixing process of two or more liquids of different pH. Normally this
takes place in one or more buffer (mixing) tanks in order to dampen variations in
the final product. The process design for neutralization is discussed by Shinskey
(1973) and McMillan (1984). Another design method and a critical review on the
design and control of neutralization processes with emphasis on chemical wastewater treatment is found in Walsh (1993). In Chapter 2 tank size selection for neutralization processes is discussed.
Zheng and Mahajanam (1999) propose the use of the necessary buffer tank
volume as a controllability measure.
The objective of this paper is to answer the following questions: When should
a buffer tank be installed to avoid propagation of disturbances, and how large
should the tank be? The preferred way of dealing with disturbances is feedback
control. Typically, with integral feedback control, perfect compensation may be
achieved at steady state. However, because of inherent limitations such as time
delays, the control system is generally not effective at higher frequencies, and the
3.2 Introductory example
43
process itself (including any possible buffer tanks) must dampen high-frequency
disturbances. We have the following:
The buffer tank (with transfer function ) should modify the disturbance, , such that the modified disturbance
(3.2)
can be handled by the control system. The buffer tank design problem
can be solved in two steps:
Step 1. Find the required transfer function . (Typically A? , and the task is to find the order and the time
constant ? .)
Step 2. Find a physical realization of level control tuning).
(tank volume
and possibly
In this paper we present design methods for buffer tanks based on this fundamental insight.
3.2 Introductory example
The following example illustrates how we may use (1) the control system and
(2) a buffer (mixing) tank to keep the output within its specified limits despite
disturbances.
Example 3.1 Consider the mixing of two process streams, and , with different
components (also denoted and ), as illustrated in Figure 3.2.
The objective is to mix equal amounts of and such that the excess concentration of the outlet flow is close to zero. More specifically,
we require to stay within 2 [ . The combined component and total
material balance gives the following model:
> @
(3.3)
For the case with no control and no buffer tank, the time response in the outlet
concentration, , to a step disturbance in the feed concentration, ,
is shown by the solid line (“Original”) in Figure 3.3. The value of approaches \ [ , which is 10 times larger than the accepted value.
44
Chapter 3. Buffer Tank Design for Acceptable Control Performance
qA
cA,f
qB
cB,f
Mixer
c0
CIC
Buffer tank
c
Figure 3.2: Mixing process. The concentration is controlled by manipulating the flow rate
of stream . Variations are further dampened by an extra buffer tank.
(1) We first design a feedback control system, based on measuring ,
and manipulating to counteract the disturbance. We choose a
proportional-integral (PI) composition controller,
7 . Note that the speed of the control system is limited by an effective delay . , mainly due to the concentration measurement. The
resulting response with control is shown by the dashed line. Because the
controller has integral action, the outlet concentration returns to its desired
value of \[ . However, because of the delay, the initial deviation is
still unacceptable.
(2) To deal with this, we install, in addition, a buffer tank with volume [
(residence time ) (drawn with dashed lines in Figure 3.2). We are
now able to keep the outlet concentration within its limit of . [
at all times as shown by the dash-dotted line in Figure 3.3.
Instead of the buffer tank, we could have installed a feedforward controller,
but this requires a fast (and accurate) measurement of the disturbance, ,
and a good process model. In practice, it would be very difficult to make this work
for this example.
Comment on notation: Throughout the paper, the main feedback controller for
the process is denoted , whereas the buffer tank level controller is denoted
.
In the following sections we will show how to design buffer tanks for quality
disturbances, like in the above example, as well as for flow-rate disturbances.
3.3 Step 2: Physical realization of with a buffer tank
45
10
Original
Concentration
8
6
With control
4
With control and buffer tank
2
}
Allowed
range
0
0
10
20
30
40
Time [min]
50
60
70
80
Figure 3.3: Response in the excess outlet concentration to a step in inlet quality (from
[ to [ at ) for the system in Figure 3.2. A composition controller handles the long term (“slow”) disturbance, but a buffer tank is required
to handle the short term deviations. Nominal data: , ,
, !#" , $
%!&(')" . Residence time mixer:
* . Delay in control loop * . The levels in the mixer and the buffer tank are controlled
+0!,21 3 . +0!,3. .
by adjusting the outflow with PI controllers, #+-,. /
3.3 Step 2: Physical realization of 4
tank
5768
with a buffer
Consider the effect of a disturbance, , on the controlled variable . Without any
buffer tank, the linearized model in terms of deviation variables may be written as
:9 (3.4)
where :9 is the original disturbance transfer function (without a buffer tank). To
illustrate the effect of the buffer tank, we let denote the transfer function for
the buffer tank. The disturbance passes through the buffer tank. With a buffer
tank, the model becomes (see Figure 3.4)
; :9 >< = @ ,
? (3.5)
where is the resulting modified disturbance transfer function. A typical
buffer tank transfer function is
A? (3.6)
46
Chapter 3. Buffer Tank Design for Acceptable Control Performance
Quality/Flow
disturbance
Buffer
tank
d(s)
Process
Gd0(s)
h(s)
Gd(s)
Figure 3.4: Use of buffer tank to dampen the disturbance
Note that so that the buffer tank has no steady-state effect.
We will now consider separately how transfer functions of the form (3.6)
arise for (I) quality and (II) flow-rate disturbances. In both cases, we consider a
([ , inlet flow rate [ , and outlet flow
buffer tank with liquid volume
rate \[ .
I Mixing tank for quality disturbance (
)
Let denote the inlet quality and the outlet quality (for example, concentration
or temperature). For quality disturbances, the objective of the buffer tank is to
smoothen the quality response
(3.7)
so that the variations in are smaller than those in . A component or simplified
energy balance for a single perfectly mixed tank yields >@ .
By combining this with the total material balance >@ (assuming
constant density), we obtain 2 @ , which upon linearization
and taking the Laplace transform yields
! "
"
(3.8)
where an asterisk denotes the nominal (steady-state) values and the Laplace variables , , , and now denote deviations from the nominal
values. We note that flow-rate disturbances (in ) may result in quality disturbances if we mix streams of different compositions (so that ). From (3.8),
we find that the transfer function for the tank is
? (3.9)
3.3 Step 2: Physical realization of with a buffer tank
47
where ? is the nominal residence time. We note that the buffer
(mixing) tank works as a first-order filter. Similarly, for tanks in series, we have
where
from
?
A? (3.10)
residence time in tank . We find the required volume of each tank
? , where is the nominal flow rate through tank .
)
II Surge tank for flow-rate disturbance (
For flow-rate disturbances, the objective is to use the buffer volume to smoothen
the flow-rate response
(3.11)
The total mass balance assuming constant density yields
@ (3.12)
We want to use an “averaging level control” with a “slow” level controller, because
tight level control yields @
P and P . Let denote the transfer
function for the level controller including measurement and actuator dynamics
and also the possible dynamics of an inner flow control loop. Then
(3.13)
where is the set-point for the volume. Combining this with (3.12) and taking
Laplace transforms yields
or from (3.13):
The buffer (surge) tank transfer function is thus given by
(3.14)
(3.15)
(3.16)
With a proportional controller, , we get that is a first-order filter
with ? . Alternatively, for a given , the resulting controller is
(3.17)
Chapter 3. Buffer Tank Design for Acceptable Control Performance
48
Table 3.1: Averaging level
control: Design procedure II for flow-rate disturbances for
alternative choices of .
Step
2.1. Desired (from Step 1)
2.2. Level controller, from (3.17)
2.3. from (3.18)
2.4. G5G
1st order
K
?
Y?
? 01.354
2nd order
K K Y; ?
;Y? 0 1.354
nth order
K K ? 1
?'0 1.354
Compared to the quality disturbance case, we have more freedom in selecting
, because we can quite freely select the controller . However, the liquid
level will vary, so the size of the tank must be chosen so that the level remains between its limits. The volume variation is given by (3.14), which upon combination
with (3.17) yields
(3.18)
Note that represents the deviation from the nominal volume. The maximum
value of this transfer function occurs for all of our cases at low frequencies ( ).
In Table 3.1 we have found the level controller and computed the required
total volume for ? . For example, for a first-order filter, A? , the required controller is a P controller with gain Y? and the required
volume of the tank is G G ? 01.354 .
Note that the resulting level controllers, , do not have integral action. A
level controller without integral action was also recommended and further discussed by Buckley (1964, page 167) and Shinskey (1996, page 25).
For flow-rate disturbances, a high-order can alternatively be realized using multiple tanks with a P level controller, , in each tank. However, the
required total volume is the same as that found above with a single tank and a
more complex , so the latter is most likely preferable from an economic point
of view.
3.4 Step 1: Desired buffer transfer function 4
5768
What is a desirable transfer function, ? We here present a frequency-domain
approach for answering this question. Figure 3.5 shows the frequency plot of
O? Y for to < , where ? in most cases is the total
residence time in the tanks. With a given value of ? , we see that
is
1
See Appendix C.
3.4 Step 1: Desired buffer transfer function 49
“best” if we want to reduce the effect of the disturbance at a given frequency by
a factor ( 7 ) or less; ; is “best” if the factor is between 3 and
about ( 7J<>< ), and is “best” if the factor is between about 7 and ( 7 N < ). Thus, we find that a larger order is desired when we want a large
disturbance reduction. We now derive more exactly the desired .
0
10
Gain 0.33
(min.vol. n=1)
Gain 0.144 (min.vol. n=2)
−1
10
Magnitude
Gain 0.064 (min.vol. n=3)
n=1
−2
10
n=2
n=3
n=4
−3
10
−1
0
10
1
10
2
10
10
Frequency × τh
Figure 3.5: Frequency responses for
K$ .
Let us start with an uncontrolled plant without a buffer tank. The effect of the
disturbance on the output is then
:9 (3.19)
To counteract the effect of the disturbances, we apply feedback control (
) (see Figure 3.6). The resulting closed-loop response becomes
9 (3.20)
With integral action in the controller, the sensitivity function approaches
zero at low frequencies. However, at higher frequencies, the disturbance response,
:9 , may still be too large, and this is the reason for installing a
buffer tank. The closed-loop response with a buffer tank is
; 9 <>= ? @
, (3.21)
50
Chapter 3. Buffer Tank Design for Acceptable Control Performance
d
h
Gd
Gd0
+
yr
-
u
K
+
G
y
+
Figure 3.6: Feedback control system
which is acceptable if 9 is sufficiently small at all frequencies. We need to
quantify the term “sufficiently small”, and we define it as “smaller than 1”. More
precisely, we assume that the variables and thus the model ( 9 ) has been scaled
such that
The expected disturbance is less than 1 (
The allowed output variation is less than 1 (
From (3.21) we see that to keep
we must require
when
,
)
)
(worst-case disturbance),
(3.22)
from which we can obtain the required . We illustrate the idea with an exam-
ple.
, Example 3.1 (continued) (Mixing process). Let
Linearizing and scaling the model (3.3) then yields
:9 - E .
(3.23)
We here used for the scaling the following: expected variations in ,
; 2 \[ ; range for , 7 \[ ; allowed range for : 2 ([ .
In Figure 3.7 we plot the disturbance effects , :9 , and :9
as
functions of frequency. Originally (without any buffer tank or control), we have
:9 at lower frequencies. The introduction of feedback makes :9 $ at low frequencies, whereas adding the buffer tank brings 9 ,$ also at
intermediate frequencies.
, and 3.4 Step 1: Desired buffer transfer function 51
1
10
Magnitude
Gd0
SGd0
0
10
SGd0 h
−1
10
−2
10
−1
0
10
1
10
Frequency [rad/min]
10
2
10
Figure 3.7: Original disturbance effect ( :9 ), with feedback control ( :9 ) and with
feedback control and a buffertank
( : 9 ). A buffer tank with
a residence time of
is required to bring :9
for all In the following we will present methods for finding based on the controllability requirement (3.22). There are two main cases:
S. Existing plant with an existing controller: The “counteracting” controller,
, is already designed, so is known. The “ideal” is then
simply the inverse of 9 .
N. New plant: The “counteracting” controller, , is not known so is
not known. This is the typical situation during the design stage when most
buffer tanks are designed.
In most cases we will choose
3.4.1
to be of the form
? .
given (existing plant)
We consider an existing plant where controller is known. The task is to find
such that H$ 9 . Several approaches may be suggested.
S1. Graphical approach with ? : This is done by selecting
? and adjusting ? until touches 9 at one
frequency. As a starting point we choose the following:
(a)
is the slope of
9 .
9
in a log-log plot in the frequency area where
Chapter 3. Buffer Tank Design for Acceptable Control Performance
52
9 crosses one from below.
Numerical approach with ? : With a given we find ?
such that
just touches 9 by solving the following problem:
(b)
S2
?
is the inverse of the frequency where
?
where
?
!
? !
9 I (3.24)
9 otherwise
(3.25)
Because it is not practical to calculate ? ! for all frequencies, we replace
with , where , which is a finite set of frequencies from
the range of interest. The calculation is explicit and fast, so a large number
of frequencies can be used. (This approach was used to obtain in Figure 3.7.)
As illustrated in Example 3.2 (below), for one may save some volume
with the following approach, which is more involved since it includes nonconvex
optimization.
S3. Numerical approach with “free” : We formulate a constrained optimization problem that minimizes the (total) volume of the buffer tank(s)
subject to (3.22). As in the previous method, we formulate the optimization
for a finite set of frequencies, , from the frequency range of interest.
(I) Quality disturbances: For mixing tanks
O? Q QQ A?
(3.26)
when the tanks are not necessarily equal. Because the flow rate is in ), we may minimize the total residependent of the volumes (? dence time (instead of minimizing the total volume) subject to (3.22):
?
K K QQQ
subject to
O? Q QQ A ? ?
9 (3.27)
where is a set of frequencies. This is a single-input, single-output
variant of a method proposed by Zheng and Mahajanam (1999).
3.4 Step 1: Desired buffer transfer function 53
(II) Flow-rate disturbances:
5 5 (3.28)
where we have parametrized the level controller with the parameter
vector . We minimize subject to (3.22) the required tank volume
(3.14):
9 subject to
5 (3.29)
Many controller formulations are possible, for example, the familiar
PI(D) (D=derivative) controller or a state-space formulation. We here
express the controller by a steady-state gain, , real zeros, and real poles:
5 and thus With O ? 777
and O? ? 7277 ? (3.30)
.
in (3.30) we get
? $ Q QQ QQQ A? Y; ?
(3.31)
does not give real time constants as the previous approaches.
For a first-order filter (with and ? ), there
is no extra degree of freedom in the optimization, and we get the same
result as that with (3.24).
Example 3.2 (Temperature control with flow-rate disturbance).
:9 8
:
7 ;> : > ; >
E
(3.32)
(3.33)
This may represent the process in Figure 3.8, where two streams and are
mixed, and we want to control the temperature ( ) after the mixing point. Stream
is heated in a heat exchanger, and the manipulated input, , is the secondary
54
Chapter 3. Buffer Tank Design for Acceptable Control Performance
A
B
u
d = Flow in
Buffer
tank with
"slow" level
control
LIC
y=Temperature
TIC
Figure 3.8: Temperature control with flow-rate disturbance
flow rate in this exchanger. The disturbance, , is variation from the nominal flow
rate of . , , and are scaled as outlined above.
First consider the case without the buffer tank. Because :9 > , the disturbance has a large impact on the output, and a temperature controller is certainly
required. However, this is not sufficient because, as seen in Figure 3.9, 9
exceeds 1 at higher frequencies and it approaches 100 at high frequencies.
We thus need to install a buffer tank with averaging level control to dampen
the flow-rate disturbance at higher frequencies. The slope of :9 is 2 after it
has crossed 1, so one would expect that a second order is the best.
1
10
Magnitude
SGd0(V=0)
SGdfree
(V= 56)
2
0
10
SGd2(V=72)
SGd1(V=242)
−1
10
−3
10
−2
10
−1
0
10
Frequency [rad/s]
10
Figure 3.9: A buffer tank is needed for the temperature control problem: :9 for
frequencies above % . Comparison of :9 for designs 1, 2 and 3
in Table 3.2.
For the graphical approach S1, we use ? . crosses 1
at about frequency 0.024 rad/s, corresponding to ? P 7 ; < < ; , and because
this is a flow-rate disturbance (II), we have from Table 3.1 that G5G B;Y? 0 P
:8< 0 . The required level controller is 7 ;> 5; .
For the more exact numerical approaches (S2 and S3), we consider three de-
3.4 Step 1: Desired buffer transfer function 55
Table 3.2: Buffer (surge) tank design procedure II (flow-rate disturbance) applied to the
temperature control example
Step
Design 1
1. Numerical approach to
S2: obtain 1st order
2.1. Desired (from Step 1) V 7 > < 2.2. Level controller, 2.3. ;<;
2.4. G5G
; < ; 0 1.354
Design 2
S2: 2nd order
[ & V
!Q Design 3
S3(II): 2nd order
; > N B>;
>;8
0 1.354
- V! [ [ N
N>01.354
signs, and the results are given in Table 3.2. Design 1 (with A? )
only requires a P level controller, but as expected, the required volume is large
because is first-order. Design 2 (with ? ) gives a considerably smaller required volume. From design 3 (with in (3.31)), the required
volume is even smaller than with design 2, as expected. Little is gained by increasing the order of above 2.
In Figure 3.9 we plot the resulting for the three designs, which confirms
that they stay below 1 in magnitude at all frequencies. These results are further
confirmed by the time responses to a unit step disturbance shown in Figure 3.10.
Buckley’s method (Buckley, 1964) gives a residence time of , which is
much less than the minimum required residence time of about N (see Table 3.2). The reason is that the disturbance needs to be reduced by a factor of
> , and not as Buckley implicitly assumes.
3.4.2
not given
The requirement is that (3.22) must be fulfilled; that is, the buffer tank with trans at all frequencies.
fer function must be designed such that :9
However, at the design stage the controller and thus is not known. Three approaches are suggested:
N1. Shortcut approach: The requirement (3.22) must, in particular, be satisfied at the bandwidth frequency where , and this gives the
(minimum) requirement
;
:9 <>= ? (3.34)
In Skogestad and Postlethwaite (1996, p. 173-4) it is suggested that ' , where is the effective delay around the feedback loop. However, to
Chapter 3. Buffer Tank Design for Acceptable Control Performance
56
1
0.8
Scaled output
0.6
Design 2 (V=72)
0.4
Design 3 (V=56)
0.2
0
Design 1 (V=242)
−0.2
0
20
40
60
80
100
Time[s]
120
140
160
180
200
Figure 3.10: Temperature control with flow-rate disturbance: Response in the scaled output to a unit step in the disturbance (flow rate) with different tank sizes and level controllers (Table 3.2).
3.4 Step 1: Desired buffer transfer function 57
get acceptable robustness, we here suggest to use a somewhat lower value
P
;
(3.35)
Skogestad (2003) proposes the following simple rule for estimating
?
?
;
?
B;
for PI-control
for PID-control
:
(3.36)
where is the delay, ? is the inverse of a right half-plane zero , and
? is the time lag (time constant) number ordered by size so that ? is the
largest time constant.
We now assume to get
A? , use 5;
I ; , and solve (3.34)
?
(3.37)
:9 ' . Alternatively, Figure 3.5 may be used for a
where given to read off the normalized frequency ,? where ,
and the required ? for each tank is then ? .
N2. Numerical approach based on preliminary controller design: The above
shortcut method only considers the frequency . To get a more exact design, we must consider all frequencies, and a preliminary controller design
is needed. This approach consists of two steps:
N2a. Find a preliminary controller for the process, and from this, obtain
.
N2b. Use one of the approaches S1, S2, or S3 from section 3.4.1.
For step N2a, we have used the method of Schei (1994), where we maxi
Y? , subject to a robustness
mize the low-frequency controller gain
restriction (maximum value on the peak of ):
? 2
subject to
"$
and stable
(3.38)
? ? . Compared to the
where for a PI controller
optimization problem that Schei uses, we have added the constraint that is stable. This is implemented by requiring the eigenvalues of to be in
the left half-plane, where is obtained from by replacing the delay with
58
Chapter 3. Buffer Tank Design for Acceptable Control Performance
a Padé approximation. To obtain a robust design,
should be chosen
>
7
low, typically N
; . With this controller design, we then use one of the
methods S1-S3 to design the buffer tank.
N3. Numerical approach with a simultaneous controller and buffer tank
design. A more exact approach is to combine the controller tuning and
the buffer tank design optimization into one problem. For (I) quality disturbances, the optimization problem may be formulated as an extension of
(3.27):
?
K K ? Q QQ A ? QQQ
?
subject to
5 5 H$
stable
: 9 5 :9 subject to
5 5 H$ stable
(3.39)
Likewise for (II)
5 .
where is the controller parameter vector for
flow-rate disturbances, we get from (3.29):
(3.40)
where is the controller parameter vector for the level controller ,
which enters in , and is the controller parameter vector for the feed
back controller , which enters in . To ensure effective integral
action in , these optimization problems must be extended by a constraint;
for example, if is a PI controller, a maximum value must be put on
the integral time.
Example 3.2 (continued) (Temperature control with flow-rate disturbance (II))
:9 >
8 ; >
E
(3.41)
The available information of the process is given by (3.41), and we assume that
the controller is not known. The delay is . We get the following results:
3.5 Before or after?
59
N1. The shortcut approach yields ( 7 and
all ) from (3.37) (or Figure 3.5) the following:
9 >
for
): G 5G B; > 0 1.354 .
Second-order filter ( B
; ): G 5G < 0 1.354 .
First-order filter (
N2. The Schei tuning in (3.38) followed by the optimal design (3.29) yields for
a second order ( and ) the following:
8 7 N : G 5G B
>; 0 1.354 .
B; : G G >01.354 .
N3. Simultaneous controller tuning and optimal design (3.40) yields with secondorder ( and ) the following:
8 7 N : G 5G B
>; 0 1.354 (as for method N2)
B; : G G >01.354 (as for method N2)
Note that
>7 N gives more robust (and “slow”) controller tunings
than
; and therefore requires a larger tank volume. The smallest
achievable tank volume with a second-order filter is G5G ;>80 1.354 (found
with method N3 with
free). Methods N2 and N3 yield almost identical
results for this example. The shortcut method N1 also gives a tank volume
B; .
very similar to that found with
3.5 Before or after?
If the buffer tank is placed upstream of the process, the disturbance itself is dampened before entering the process. If it is placed downstream of the process, the resulting variations in the product are dampened. The control properties are mainly
determined by the effect of input on output (as given by the transfer function
). An upstream buffer tank has no effect on , and also a downstream buffer
tank has no effect on provided we keep the original measurement. On the other
hand, placement “inside” the process normally affects . In the following we list
some points that may be considered when choosing the placement. We assume
that we prefer to have as few and small buffer tanks as possible (sometimes other
issues come into consideration, like differences in cost due to different pressure
or risk of corrosion, but this is not covered).
60
Chapter 3. Buffer Tank Design for Acceptable Control Performance
(1) In a “splitting process”, the feed flow is split into two or more flows (Figure 3.11(a)). One common example is a distillation column. To reduce the
number of tanks, it will then be best to place the buffer tank at the feed (upstream placement). An exception is if only one of the product streams needs
to be dampened, in which case a smaller product tank can be used because
each of the product streams are smaller than the feed stream.
(2) In a “mixing” process, two or more streams are mixed into one stream (Figure 3.11(b)). To reduce the number of tanks, it is here best with a downstream placement. An exception is if we only have disturbances in one
of the feed streams because the feed streams are smaller than the product
stream, leading to a smaller required size.
(a) A splitting process
(b) A mixing process
Figure 3.11: Two types of processes
(3) An advantage of a downstream placement is that a downstream buffer tank
dampens all disturbances, including disturbances in the control inputs. This
is not the case with upstream tanks, which only dampen disturbances entering upstream of the tank.
(4) An advantage of an upstream placement is that the process stays closer to
its nominal operation point and thus simplifies controller tuning and makes
the response more linear and predictable (see Example 3.3).
(5) An advantage of the “inside” placement is that it may be possible to avoid
installation of a new tank by making use of an already planned or existing
unit, for example, by increasing the size of a chemical reactor.
(6) A disadvantage with placing the buffer tank inside or downstream of the
process is that the buffer tank then may be within the control loop, and
3.5 Before or after?
61
the control performance will generally be poorer. Also, its size will effect the tuning, and the simultaneous approach (N3) is recommended. For
the downstream placement, these problems may be avoided if we keep the
measurement before the buffer tank, but then we may need an extra measurement in the buffer tank to get a more representative value for the final
product.
Example 3.3 (Distillation column). We apply the methods from section 3.4.1 to
a distillation column and compare the use of a single feed tank with the use of two
product tanks (Figure 3.12). We consider a distillation column with 40 stages (the
linearized model has 82 states; see column A from (Skogestad and Postlethwaite,
1996, p.425)). The disturbances to the column are feed flow rate and composition
( and ), and the outputs are the mole fractions of the component in
top and bottom products, respectively ( and ). The manipulated variables are
). The variables have been scaled
the reflux and the boilup ( and 2
so that a variation of in the feed flow rate corresponds to and a
variation of 2 in the feed composition corresponds to . A change in
the top and bottom product composition of 7 mole fraction units corresponds
to a change in and . Decentralized PI controllers are used to control
the compositions. In the top, N 7 :8< ; ; , and in the bottom,
7 < N ; ; . There is a delay of in each loop, which we
represent with fifth-order Padé approximations in the linear model. Nominally,
the feed flow rate is .\[ , and the top and bottom concentrations are 0.99
and 0.01, respectively.
The holdup in the reflux and the boiler are controlled with controllers (with
gain ) by the top and bottom product streams, respectively.
We consider the effect of the flow-rate disturbance, . The closed-loop gains
from to and without any buffer tank, 9 and 9 are shown with
solid lines in Figure 3.13. The gains are both above at intermediate frequencies,
so our purity requirements will not be fulfilled, unless we install a buffer tank.
Upstream placement (feed surge tank). is known, and with , (3.24) in
method S2 yields ? ><, . The resulting and are shown with
dashed lines, and we see that just hits (as expected). is also plotted
(dash-dotted) to indicate the limiting frequency, which is not at the maximum of
9 , but at a lower frequency “shoulder”. Following design procedure II, we
now get the following:
2.1
> < 2.2 The required level controller for the buffer tank is 2.3
? >2<
>2< 7 >>:>:
Chapter 3. Buffer Tank Design for Acceptable Control Performance
62
y1=xD
u1=L
V=49
LIC
V=68
d1=F
d2=zF
V=94
u2=V
y2=xB
Figure 3.12: Distillation column with either one feed surge tank or two product mixing
tanks to dampen disturbances.
2.4
G G
? 01.354 8<,
Q ; MQ 7 \[ N : [ .
Comment: Since the slope of 9 is less that 1 around the limiting frequency, higher order filters will increase the volume demand. For example, with
; , (3.24) gives ? B8N 7 N , and G G B;Y? 01.354 >7 \[ .
Downstream placement (product mixing tank). Because both 9 and 9 at some frequencies, we must apply one mixing tank for each of
the two products. When we designed the feed tank, we had to consider the worst
of 9 and 9 , but now we may consider 9 for the top product and
9 for the bottom product. With , (3.24) yields ; for the top buffer
tank and as before ><, for the bottom tank. The corresponding volumes are
; Q 7 >>7 [ (top) and >2< Q 7 > \[ (bottom), which gives a total
volume of N [ , which is the same as that for the feed tank. However, the feed
tank placement is preferred because we then need only one tank.
Nonlinear simulations. The above design is based on a linearized model, and
(as expected) the feed tank placement is further justified if we consider a nonlinear
model because the column is then less perturbed from its nominal state. This is
illustrated by the simulations in Figures 3.14, 3.15 and 3.16. If the buffer tanks
are placed downstream, the nonlinear response deviates considerably from the
linear response, and the tanks designed by linear analysis are too small. By trial
and error with disturbance step simulations on the nonlinear model, we find that
3.6 Further discussion
63
1
10
1/h
Magnitude
Limiting ω
SGd20
↓
0
10
SGd2
SGd10
SGd1
−1
10
−3
10
−2
−1
10
0
10
10
Frequency [rad/s]
9 and Figure 3.13: Feed flow disturbance for the distillation column: 9 (for
top and bottom) are both above 1 (solid line). A feed tank with averaging level con
, brings the disturbance gain to both top and bottom below 1
trol, (dashed). Note that is just touching 9 .
and ? :>: are needed for the top and bottom product tanks.
This gives a total volume of 2< [ , considerably larger than the required feed
tank of N : [ .
In conclusion, an upstream feed tank with a P controller (averaging level control) proves best for this example. The example also illustrates that for nonlinear
processes the buffer tank design methods that we have proposed are most reliable for the design of upstream buffer tanks. For (highly) nonlinear processes, the
results should, if possible, be checked with simulations on a nonlinear model.
?G
:,
3.6 Further discussion
In this paper we have assumed that the surge tank outlet flow rate is controlled,
which for example is the case when an inner flow-control loop is installed. When
such a flow loop is missing, the flow rate is level dependent (this was what Harriott
(1964) assumed). In Appendix A, we find that essentially the same results are
obtained in this case.
When comparing one large with several smaller (mixing) tanks, the actual
investment cost related to the tanks must be considered. A short discussion on
Chapter 3. Buffer Tank Design for Acceptable Control Performance
64
1
8
0
6
−1
4
−2
2
−3
0
−4
0
−2
0
50
100
150
200
250
300
Time [min.]
350
400
450
500
(a) Output . Nonlinear simulation
(solid) and linear simulation (dashed).
50
100
150
200
250
300
Time [min.]
350
400
450
500
(b) Output . Nonlinear simulation
(solid) and linear simulation (dashed).
Figure 3.14: Distillation example with no buffer tanks installed. The control system is not
able to handle the disturbance. There is a large deviation between nonlinear and linear
simulation.
0
4
−1
3
−2
2
−3
1
−4
0
50
100
150
200
250
300
Time [min.]
350
400
450
500
0
0
(a) Output . Nonlinear simulation
(solid) and linear simulation (dashed).
50
100
0
4
−1
3
−2
2
−3
1
50
100
150
200
250
300
Time [min.]
350
400
450
(a) Output . Nonlinear simulation
(solid) and linear simulation (dashed).
200
250
300
Time [min.]
350
400
450
500
(b) Output . Nonlinear simulation
(solid) and linear simulation (dashed).
Figure 3.15: Distillation example with a feed tank of and the nonlinear simulation is close to the linear one.
−4
0
150
500
0
0
50
100
[ . Both outputs stay within 150
200
250
300
Time [min.]
350
400
450
,
500
(b) Output . Nonlinear simulation
(solid) and linear simulation (dashed).
Figure 3.16: Distillation example with product tanks at the top ( ( [ ). The outputs deviate from in the nonlinear simulations.
[ ) and at the bottom
3.7 Conclusions
65
this is given in Appendix B. A small number is favoured, even if this means a
larger total volume.
The use multiple of buffer tanks in series is of interest for processes with large
disturbances, e.g., for neutralization processes. With multiple tanks one may ask
whether it is best with equal or unequal sized buffer tanks. Equal tanks are easiest
to handle, but for neutralization it has been argued (Shinskey, 1973) that unequal
tanks reduce resonance peaks. The conclusion from Chapter 2 is that there may be
a reduction in total volume with tanks of different size, but this most likely does
not compensate for the added cost of different units.
The shortcut formula (3.37) in method N1 is easy to use and convenient at
an early stage of the process design. It is especially convenient for mixing processes like neutralization Chapter 2. However, it is a necessary but not sufficient
requirement for (3.22). Two possible errors may occur:
E1. The estimate for may be wrong.
E2.
is not the “worst” frequency. We only consider
fulfilled here, :9 "$ may be violated at
(a) lower frequencies than .
(b) higher frequencies than due to peaks in
9
at . Even if it is
.
Errors E1 and E2(b) are not really a problem with the choice for used in
this paper, which allows for a robust controller tuning where is “flat” over
a frequency range and with a low peak for . Error E2(a) may be an important
issue if is of high order, and how to overcome it is discussed in the Thesis’
Appendix B (B3.1 and 4).
3.7 Conclusions
The controlled variables ( ) must be kept within certain limits despite disturbances
( ) entering the process. High-frequency components of disturbances are dampened by the process itself, while low-frequency components, e.g., the long-term
effect of a step, are handled by the control system. There are, however, always
limitations in how quickly a control system can react, for example, as a result of
delays. Thus, for some processes there is a frequency range where the original
process and the controller do not dampen the disturbance sufficiently. In this paper we introduce methods for designing buffer tanks based on this insight. The
methods consist of two steps:
66
Chapter 3. Buffer Tank Design for Acceptable Control Performance
Step 1. Find the required transfer function such that (with scaled variables)
9 $ . The methods for this have been divided
into two groups depending on whether the control system for the process
is already designed (methods S1-S3) or not (methods N1-N3). The shortcut methods (S1/S2 or N1), supplemented with nonlinear simulations, are
recommended for most practical designs.
Step 2. Design a buffer tank that realizes this transfer function . For a first-order
transfer function, O? , we have the following:
I. Quality disturbances Install a mixing tank with volume
is the nominal flow rate.
M?
, where
II. Flow-rate disturbances Install a tank with averaging level control with
gain Y? and volume
?'01.354 where 01.354 is the expected range (from minimum to maximum) in the flow-rate variation.
Sometimes a higher-order is preferable, in which case we need (I)
for quality disturbances more than one mixing tanks and (II) for flow-rate
disturbances a more complicated level controller (with lags) (see Table 3.1).
References
Buckley, P. S. (1964). Techniques of Process Control. John Wiley & Sons, Inc.. New York.
Harriott, P. (1964). Process Control. McGraw-Hill. New York.
Hiester, A. C., S. S. Melsheimer and E. F. Vogel (1987). Optimum size and location of
surge capacity in continuous chemical processes. AIChE Annual Meet, Nov. 15-20,
1987, paper86c.
Lieberman, N. P. (1983). Process Design for Reliable Operations. Gulf Publishing Company. Houston.
Ludwig, E. E. (1977). Applied Process Design for Chemical and Petrochemical Plants.
Vol. 1. Gulf Publishing Company. Houston.
Marlin, T. E. (1995). Process Control. Designing Processes and Control Systems for Dynamic Performance. McGraw-Hill, Inc.. New York.
McDonald, K. A., T. J. McAvoy and A. Tits (1986). Optimal averaging level control.
AIChE J. 32(1), 75–86.
REFERENCES
67
McMillan, G. K. (1984). pH Control. Instrument Society of America. Research Triangle
Park, NC, USA.
Peters, M. S. and K. D. Timmerhaus (1991). Plant Design and Economics for Chemical
Engineers. 4 ed.. McGraw-Hill. New York.
Sandler, H. J. and E. T. Luckiewicz (1987). Practical Process Engineering. McGraw-Hill
Book Company. New York.
Schei, T. S. (1994). Automatic tuning of PID controllers based on transfer function estimation. Automatica 30(12), 1983–1989.
Shinskey, F. G. (1973). pH and pIon Control in Process and Waste Streams. John Wiley
& Sons. New York.
Shinskey, F. G. (1996). Process Control Systems - Application, Design, and Tuning, 4th
Ed.. McGraw-Hill Inc., New York.
Shunta, J. P. and W. Fehervari (1976). Nonlinear control of liquid level. Instrum. Technol.
pp. 43–48.
Sigales, B. (1975). How to design reflux drums. Chem. Eng. 82(5), 157–160.
Skogestad, S. (2003). Simple analytic rules for model reduction and PID controller tuning.
J. Proc. Contr. 13(4), 291–309.
Skogestad, S. and I. Postlethwaite (1996). Multivariable Feedback Control. John Wiley &
Sons. Chichester, New York.
Ulrich, G. D. (1984). A Guide to Chemical Engingeering Process Design and Economics.
John Wiley & Sons. New York.
Walas, S. M. (1987). Rules of thumb, selecting and designing equipment. Chem. Eng.
94(4), 75–81.
Walsh, S. (1993). Integrated Design of Chemical Waste Water Treatment Systems. PhD
thesis. Imperial College, UK.
Watkins, R. N. (1967). Sizing separators and accumulators. Hydrocarbon Processing
46(11), 253–256.
Wells, G. L. (1986). The Art of Chemical Process Design. Elsevier. Amsterdam.
Zheng, A. and R. V. Mahajanam (1999). A quantitative controllability index. Ind. Eng.
Chem. Res. 38, 999–1006.
Chapter 3. Buffer Tank Design for Acceptable Control Performance
68
Appendix A Surge tank with level dependent flow
In section 3.3, II we assume that we control the outlet flow rate from the buffer
(surge) tank. If we instead let the controller determine the valve position, which
is the case if the cascade flow loop is omitted to save cost, the actual flow rate
depends on both the valve position and the given tank level (or volume) as well
as external pressure variations. The flow rate through a control valve is a function
of the valve position, , and the differential pressure across the valve, 0 . We
assume that the differential pressure is given by the hydrostatic pressure at the
outlet (neglecting the other pressure variations), and that the tank area of the tank
is not varying with the level. We then find a linearized model (Harriott, 1964):
0DP
0
0 or by Laplace transform
(3.42)
(3.43)
where , and now represent deviations from nominal values. A
controller acting on the valve position , is given by the following equation:
(3.44)
We insert (3.43) and (3.44) into the the buffer tank mass balance, and Laplace
transform yields:
(3.45)
(3.46)
where .
(3.14) and (3.15), and see that the effect of inlet flow
We compare this with
rate changes on and is unchanged provided , that is
!
!
(3.47)
Here > is the scaling from flow rate to valve position, while 8
represents the effect that the outlet flow rate is increasing with increasing level (“selfregulation”). The time contant, ? , is then
?
!
!
(3.48)
Appendix B Capital investments
69
From (3.48) we can see that for first order the largest possible ? is now
, so for high 8
(i.e., for low pressure drop over the control valve) a
flow cascade loop is recommended. This is in agreement with normal practice.
>
Appendix B Capital investments
We will here consider the capital investment in connection with the installation of
one buffer tank. It consists of two terms, namely a constant independent of the
tank size, and a term which relates with the tank size (Peters and Timmerhaus,
1991):
The constant term, , includes the cost of instruments (level measurements), valves
(whose size only depends on the flow rates), controllers (normally only programming and testing cost), piping (increased tank size may both increase and decrease
the amount of piping), wiring for signals and electrical power, engineering and
, includes the price of the purchased equipstart-up. The size dependent term,
ment and its installation. A common approximation is that it is proportional to the
tank weight, which (assuming that wall thickness and materials are independent
of the size) yields the typical exponent, P 7 . For equal tanks with total
volume G5G , we then have
C
G 5G S L
(3.49)
Often the constant term is large, which favors few (one) tanks. Since normally < (or even ; ), theory on the cost optimal is not interesting since
it is easy to calculate the cost for different and compare.
; >> and ; >> [ for neuExample 3.4 Walsh (1993) found tralization tanks. The resulting cost for to < tanks is shown in Figure 3.17.
Now we can combine Figures 3.5 and 3.17. We want to reduce the effect of a
quality disturbance by a factor 8 , and read from Figure 3.5 the value of
,? that corresponds to magnitude . We find that the volume with one tank is
about times larger than the total volume of two tanks, and about times larger
than the total volume of three tanks, and about : times larger than the total volume
of four tanks. In Figure 3.17 we have marked the cost of one tank of [ , two
tanks of > [ , three tanks of > \[ and four tanks of :\[ . With 8
the cost of one tank is the lowest, even though the total volume is much larger.
Chapter 3. Buffer Tank Design for Acceptable Control Performance
70
4
16
x 10
14
n=4
12
Cost [£]
n=3
10
n=2
8
6
n=1
4
2
0
10
20
30
40
50
60
Total volume [m3]
70
80
90
100
Figure
3.17: Capital investment as a function of total volume. The cost of each tank is
G5G S .
Appendix C Surge tank: Required volume with n-th order 71
Appendix C Surge tank: Required volume with n-th
order 4 5 68
In this appendix we derive the required tank volume for a desired buffer tank
transfer function ? .
For the -th order filter, ? , the resulting controller from
(3.17)
O? is of order ? . Furthermore, from (3.18):
QQQ
(3.50)
O? ? ? ? (3.51)
where is a polynomial with terms 777 with positive coefficients.
The maximum occurs at and the volume requirement is ? .
Chapter 4
Control Design for Serial Processes
Audun Faanes and Sigurd Skogestad
Department of Chemical Engineering
Norwegian University of Science and Technology
N–7491 Trondheim, Norway
Submitted to Journal of Process Control
Extended version of a paper presented at European Control Conferance, ECC’99,
Aug.31-Sept.3, 1999, Karlsruhe, Germany.
Abstract
Conceptually, a multivariable controller uses the two basic principles of “Feedforward” action, based mainly on the model (for example the off-diagonal decoupling elements of the controllers), and feedback correction, based mainly on the measurements. The basic differences
between feedback and feedforward control are well-known, and these differences also manifest
themselves in the multivariable controller.
Feedforward control may improve the performace significantly, but is sensitive to uncertainty,
especially at low frequencies. Feedback control is very effective at lower frequencies where high
feedback gains are allowed.
In this paper we aim at obtaining insight into how a multivariable feedback controller works,
with special attention to serial processes. Serial processes are important in the process industry,
and the structure of this process makes it simple to classify the different elements of the multivariable controller.
An example of neutralization of an acid in a series of three tanks is used to illustrate some of
the ideas.
Keywords: Control structure, Serial process, Multivariable control, Feedforward, Feedback
also affiliated with Statoil ASA, TEK, Process Control, N-7005 Trondheim, Norway
Author to whom all correspondence should be addressed. E-mail: [email protected]no
74
Chapter 4. Control Design for Serial Processes
4.1 Introduction
Before designing and implementing a multivariable controller, there are some
questions that are important to answer: What will the multivariable controller
really attempt to do? Will a multivariable controller significantly improve the response as compared to a simpler scheme? What must the multivariable controller
take into account to succeed? How accurate a model is needed?
One key issue with multivariable control is uncertainty. There is a fundamental difference between feedforward and feedback controllers with respect to their
sensitivity to uncertainty. Feedforward control is sensitive to static uncertainty,
while feedback is not. On the other hand, aggressively tuned feedback controllers
are very sensitive to uncertainty in the crossover frequency region. Similar differences with respect to uncertainty can be found for multivariable controllers.
Traditional single loop controllers are predominantly based on feedback, whereas
model based multivariable controllers often combine feedback and feedforward
control, and usually the component of feedforward action is significant (for example the off-diagonal “decoupling” elements of the controllers).
In this paper we discuss these issues for the important class of serial processes.
A serial process consists of a series of one-way interacting units. The states in one
unit influence the states in the downstream unit, but not the other way round. This
is very common in the process industry, where the outlet flow of one process enters
into the next. One example, which will be studied in Section 4.4, is neutralization
performed in several tanks in series. Examples of processes that are not serial
are processes with some kind of recycle of material or energy. Even for such
processes, however, parts of the process may be modelled as a serial process,
if the outlet variations of the last unit is dampened through other process units
before it is recycled, so that no significant correlation can be found between the
outlet variations and the variations in the disturbances to the first unit.
A multivariable controller often yields significant nominal improvements compared to local single-loop control. This is largely because of to the “feedforward”
action, and with model error, the feedforward effect may in fact lead to worse performance. On the other hand, use of feedback from downstream measurements
is much less dependent on the model, as use of high feedback gains at low frequencies removes the steady-state error. However, one must be careful about high
feedback gains at higher frequencies due to potential stability problems, and it is
at these higher frequencies one may have the largest benefit of the model-based
“feedforward” action of the multivariable controller.
Buckley (1964) discusses control structure design for serial processes and distinguishes between material balance control (control of inventory or pressure by
flow rate adjustments) and product quality control (control of quality parameters
such as concentration).
4.2 Model structure of serial processes
75
Shinskey (1973) and McMillan (1982) present methods for design of pH neutralization processes. Mixing tanks are used to dampen disturbances, and they find
that the total volume may be reduced by use of multiple stages with one control
loop for each tank. Another advantage with multiple stages is that one may use
successively smaller and smaller control valves, leading to a more precise manipulated variable in the last stage. McMillan and Shinskey both recommend different
sized tanks to avoid equal resonance frequencies in the tanks, but this has later
been questioned (Walsh, 1993), Chapter 2.
A discussion on the open loop response of serial process is found in Marlin
(1995, p. 156f). Morud and Skogestad (1996) note that the poles and zeros of the
transfer function of a serial process are the poles and zeros of the transfer functions of the individual units. Thus, the overall response may be predicted directly
from the individual units, in contrast to e.g. processes with recycle. Many series
connections of processing units are not really serial processes, as the response
of each unit also depends on the downstream unit (for example if the outlet flow
rate from a unit depends on the pressure in the subsequent unit) (Marlin, 1995),
(Morud, 1995, Chapter 4), (Morud and Skogestad, 1995). Morud et al. denote the
latter process structure cascades, whereas Marlin uses the terms noninteracting
and interacting series, respectively, for the two structures.
The characteristics of serial processes can be utilized when analyzing multivariable controllers for such processes. The multivariable controller can be divided into three types of controller blocks: Local feedback, feedback from downstream units and “feedforward” from upstream units. Thus, depending on the
location, the control input will be a sum of these three terms.
This division of the controller blocks has two purposes. First, it gives insight
into the behaviour of the control system. Second, it allows simple implementation.
In some cases the multivariable controller can be implemented as combinations of
conventional single loop controllers.
In Section 4.2 we develop the model structure for serial processes and discuss
some of its properties. In Section 4.3 control of serial processes is discussed. One
popular multivariable controller is MPC, and to be able to use theory for linear
systems, we summarize in Appendix A how to express an unconstrained MPC
combined with a state estimator on state space and transfer function form. This
was not available for the controller we have used, so that a detailed description is
given in Chapter 5. The ideas of the paper are illustrated through an example with
pH neutralization in three stages (section 4.4). The paper is concluded by a short
discussion (section 4.5) and the conclusions in section 4.6.
Chapter 4. Control Design for Serial Processes
76
ui-1, di-1
ui, di
ui+1, di+1
yi-2
Unit no. i-1
yi-1
Unit no. i
yi
Unit no. i+1
yi+1
(manipulated) and
(disturFigure 4.1: Serial process with exogenous variables
bances) into unit . The vector represents the outflow of unit , which continues into
unit number .
4.2 Model structure of serial processes
In this section we look closer at serial processes and develop a general transfer
function model. An example of a serial process is a process where mass and/or
energy flows from one process unit to another, and there is no recycling of mass
or energy. We define a serial process by the following (also see Figure 4.1):
A serial process can be divided into a series of sub-processes or units, where
the states in each unit depend on the states in the unit itself ( ), the states in the
upstream unit ( ), and the exogenous variables ( , ) to the unit.
The model for unit no. can then be expressed as
>@
(4.1)
where and are the state vectors for unit and unit respectively, and the
external input is divided into a vector of manipulated inputs, , and disturbances,
. We further define the outputs from a unit as a function of the states and the
external inputs for this unit
(4.2)
It is easy to also inlude direct througput terms, i.e., define ,
but is makes the expressions below slightly more complex.
around a working point, introduce We linearize (4.1) and (4.2)
, O , O , and O and let the variables
be the deviation from their working point. Applying Laplace transformation, and
recursively inserting for variables from previous tank, we obtain:
We have defined the total output vector, , as all the outputs, manipulated inputs, as all the disturbances. Defining
(4.3)
as all the
(4.4)
4.2 Model structure of serial processes
77
we get
1 ..
.
..
.
..
.
QQQ
QQQ
..
..
.
..
QQQ
QQQ
.
QQQ QQQ
..
.
.
QQQ QQQ
..
.
(4.5)
and
..
.
..
.
..
..
.
.
QQQ
QQQ
QQQ QQQ
.
..
..
QQQ
QQQ
.
QQQ QQQ
..
.
(4.6)
is replaced
where is the number of units. and are identical except in by (the disturbances to each unit are assumed independent).
We see that and are both lower block triangular. From (4.5) and
(4.6), we can deduce the following properties:
The state vector of a process unit is not influenced by control inputs and
disturbances to downstream units.
The influence from a control input or a disturbance which enters an upstream
unit, , is dampened by the transfer function
!
before it reaches the output of unit .
..
.
Chapter 4. Control Design for Serial Processes
78
The open loop stability of the total process is given by the stability of each
unit since the elements in and consists of products of ’s.
and
are block diagonal at infinite frequency ( ).
Note that the nominal model of unit can be expressed as
(4.7)
where is the transfer function from “disturbances” due to variations in the
upstream unit, to output :
(4.8)
This is illustrated in Figure 4.2.
Figure 4.2: Model structure for serial processes
4.3 Control structures for serial processes
In the previous section we introduced the concept of serial processes and Equations (4.3)-(4.6) summarize the linearized model. If a full, multivariable controller
is used to control this process, the characteristics of each blocks of this controller
4.3 Control structures for serial processes
79
can be identified. If we for simplicity assume that the set-points are zero, and we
want to control all the outputs, the control inputs are given by:
where is the controller.
We divide the controller
in :
into #
..
.
0
(4.9)
blocks of the same size as the blocks
777
777
..
.
..
.
777
..
.
(4.10)
These controller blocks can be divided into three groups:
Blocks on the diagonal ( ) These blocks use local control, where inputs to the
unit are used to control outputs of the same unit.
Blocks above the diagonal ( , ,$ ) These blocks represents feedback from
the outputs of downstream units. Intuitively, when the effective delay through
the units is large, these blocks seem ineffective since the local feedback always will be quicker. There are, however, several cases when it may prove
useful:
(1) We have no relevant control inputs downstream so local control is impossible.
(2) The downstream actuators are slow, so that it actually is more efficient
to manipulate the upstream control inputs.
(3) There are not enough degrees of freedom in the downstream units.
(4) The control inputs downstream are constrained, and insufficient to
compensate for the disturbances.
(5) The downstream actuators are expensive to use.
In the latter two cases the upstream manipulated variable can be used to
(slowly) drive the downstream ones to zero or to some other ideal resting
value. This is called input resetting and is normally used for systems where
we have more control variables than outputs (e.g., (Skogestad and Postlethwaite, 1996, page 418)).
Blocks below the diagonal ( , ) Through these blocks an output from an
upstream unit directly affects the input in a downstream unit. Since upstream units act as disturbances to downstream units (see (4.7)), these controller blocks may be viewed as “feedforward” elements.
Chapter 4. Control Design for Serial Processes
80
In analyzing the controller it is useful to plot the gain of the controller elements
as a function of frequency, see Figures 4.6, 4.8(a), 4.10(a), and 4.12(b) presented
below. A key point is to find out whether there is integral action in the feedback
part of the controller or not. Integral action requires high gain at low frequencies,
but it is not always straight-forward to interpret the gain plot of the controller
elements. For example, in Figure 4.8(a) all the elements have large gains at low
frequencies. In such cases the steady-state effect is better illustrated by plotting
the individual gains of the sensitivity function, where
is the loop transfer function. The usefulness of is seen
from the following expression
E
(4.11)
), is the reference, is the disturbance
where E is the control error ( and is the (open loop) transfer function matrix from the disturbance to the
output. To have no steady-state offset in an output we need that all elements in
the corresponding row of to be small at low frequencies. Also note that system
stability is determined by the poles of .
4.3.1 Local control (diagonal control)
Local control is by far the most common control element,
[
1#
[
#
(4.12)
#
[[
), the loop transfer function becomes
With only local control and three units (
1#
Local control:
[[ [[
[[
(4.13)
From this it follows that the stability of the closed-loop system is determined
only by the blocks on the diagonal. That is, we have closed-loop stability if and
are stable.
only if each of the individual loops 4.3.2 Pure feedforward from upstream units
The use of measurements in upstream units in the control of a unit is denoted
feedforward control:
Feedforward (
)
(4.14)
4.3 Control structures for serial processes
81
With “pure” feedforward control (only feedforward elements), the controller does
not influence stability.
From (4.7) and (4.8) we find that perfect nominal control is obtained by selecting
Q QQ (4.15)
(4.16)
The reason for the zero in (4.16) is that the disturbance is already eliminated
by (4.15). If (4.15) cannot be realised, for example if it is not casusal, (4.15) must
be modified:
(4.17)
where subcript minus indicates that negative delays and other non-causal elements
of the (total) controller has been removed (this is a simplification of the optimal
feedforward controller given by Lewin and Scali (1988) and Scali et al. (1989)).
As an example, let
,
E E (4.18)
?
?
, E (4.19)
Remark 1 It is not necessary to make causal itself. For example if has a delay of and a delay of N the delay of the “ideal” feedforward
Then
controller would have been < , which is not implementable. (4.17) states that
the controller delay shall be truncated to 0, which means that the effect of the
controller of the controller occurs < too late. But, requiring to be causal
would have given a N delay in the controller (zero in delay in plus N in ),
and the effect of the feedforward controller would have occurred < N too
late.
When (4.15) cannot be realised, feedforward from units ; 7727 can be
useful. For example, if it is causal, the following feedforward controller from unit
; eliminates the control error that “rests” after
:
C L C L (4.20)
See Appendix B for a derivation of (4.20).
Feedforward control is generally sensitive to uncertainty, and we will now
consider its effect. The nominal model is given by (4.7), and the actual model
(with uncertainty) is
(4.21)
Chapter 4. Control Design for Serial Processes
82
A pure feedforward controller from upstream units then yields the following actual
control error:
E (4.22)
With “ideal” feedforward control based on the nominal model, as given by
(4.15) and (4.16), the actual control error becomes
E ;
<> = ? ,% where denotes generalized inverse (Zhou et al., 1996, page 67), and
relative model error in . In particular, for scalar blocks
(4.23)
is a
O (4.24)
Thus model errors at any frequency, directly influences the actual control error.
Upon comparing the response with control in (4.23) with the response without
control ( in (4.21)) we see that “feedforward” (decoupling) control has a
positive (dampening) effect on disturbances from upstream units at frequencies where
$ (4.25)
or in words, as long as the relative error in is less than 1 in magnitude.
Here, an appropriate norm dependent on the definition of performance is used.
External disturbances entering directly into the process at unit , , are (of
course) not dampened by feedforward control from upstream units, but if is
measured, then separate feedforward controllers may be designed for . Feedforward control from the reference, , is also necessary to avoid control error if
and no feedback is applied.
4.3.3 Lower block triangular controller
A lower (block) triangular controller will result if we combine local feedback and
feedforward from upstream units,
Local control (
Feedforward (
)
)
4.3 Control structures for serial processes
83
):
The loop transfer function now becomes (
#
[
[
[
[[
1#
# [
# [
#
#
[
[[ [
[
[[
[[ [
[[ [[
(4.26)
The diagonal elements are feedback elements, where most of the control benefits are achieved simply by using sufficiently high gains, and an accurate process
model is not needed. The main problem is that too high gain may give closed-loop
instability.
As for the local feedback (diagonal) control structure the stability of the closedloop system is determined only by the blocks on the diagonal, that is we have
closed-loop stability if and only if each of the local loops are
stable.
Note that we also obtain this control structure if an inverse-based (decoupling)
design method ( ) is used. An example of an inverse based
controller is IMC decoupling (Morari and Zafiriou, 1989), where and are (block) diagonal matrices (with blocks corresponding to
the blocks in ). For this controller we obtain the following diagonal and subdiagonal blocks:
(4.27)
+ +
(4.28)
where denotes block of weight matrix
(this is the integrator). (4.27)
and (4.28) can be verified by calculating that
. Since the stability is
determined by the diagonal blocks, and these are the scaled inverse of the blocks
of , the weights can be selected independently for each unit, e.g. using (Rivera
et al., 1986) (for scalar blocks). If is not invertible, e.g., due to right half plane
zeros and delays, the not invertible part of is essentially factored out before the
inversion (for details, see (Morari and Zafiriou, 1989)).
Using (4.8), we note that the sub-diagonal part of the IMC controller, (4.28), is
identical to the ideal feedforward controller (4.15), except for the weights. Integral
action in the feedback part of the controller ( ) requires an integrator in
either or . For example, we may choose K where ? is the
desired closed loop time constant, (Rivera et al., 1986). Thus we see from (4.28)
that also the “feedforward” gain will be amplified at low frequencies.
Let us now consider the effect of model uncertainty. The nominal model is
given by (4.7) and the actual model by (4.21). A lower triangular controller yields
Chapter 4. Control Design for Serial Processes
84
the following actual control error:
E (4.29)
where (Skogestad and Postlethwaite, 1996)
C
L (4.30)
where and are nominal sensitivity and complementary sensitivity functions,
respectively, and relative error in (note that we in Section 4.2 let denote
something else).
Upon comparing the closed-loop response in (4.29) with the open loop response in (4.21) we see the following:
) dampens disturbances
(1) Effective local feedback control ( from the preceding tank ( ), external disturbances entering the process at
unit , and also the effect of the model error ( ) and errors in the feedforward control.
(2) For frequencies where the feedback control is not effective, i.e., , the results from Section 4.3.2, (4.15)-(4.25) can be applied except that
(4.20) must be modified due to the feedback control in unit :
C L C L (4.31)
As for the pure feedforward case, external disturbances entering the process at
unit , , are not dampened by the feedforward control from upstream units, but
are handled by the feedback control.
For serial processes with a lower block triangular controller it is particularly
simple to identify feedforward and feedback controller elements, but similar differences between the elements occur for most multivariable controllers. Such insights are important, e.g. when evaluating how the controller is affected by model
error.
A more general analysis of feedforward control under the presence of uncertainty is given in Chapter 6.
4.3 Control structures for serial processes
85
4.3.4 Full controller
), the loop transfer
With a full controller, as in (4.10), and three units (
function becomes
[
1#
# [
# #
[[ [
[
QQQ
#
[
[
[
[
#
[
[
[[ [
QQQ
[
[
[[ [[
(4.32)
In this case the stability of the closed-loop system is affected by all elements in
the controller (and in ).
As illustrated in the case study in Section 4.4, even in this case the controller
block below the diagonal may be similar to feedforward control.
4.3.5 Final control only in last unit (input resetting)
In many serial processes, the output from the last unit is by far the most important
for the overall plant economics, and the outputs in upstream units are mainly
controlled to improve control performance in the final unit. The extra degrees
of freedom are used for local disturbance rejection, but are otherwise typically
reset to some ideal resting value by adjusting set-points in upstream units.
ru2
+
IR
K12
-
r1 +
-
K11
u1
Unit y1
1
Unit
2
y2
Unit y3
3
FF
K21
ru3 +
-
IR
K23
r2 +
-
K22
+
+
u2
FF
K31
r3
+
-
K33
FF
K32
+
+
+
+
u3
Figure 4.3: Serial units controlled with a combination of local control, feedforward control and input resetting
Chapter 4. Control Design for Serial Processes
86
We may then use the following control elements:
Local control ( )
Feedforward ( )
Input resetting ( )
as illustrated in Figure 4.3. Note that we here have restricted input resetting to
operate between neighbouring units, but this is not strictly required. With local
control in the three units, feedforward from unit 1 to unit 2 and 3 and from unit
2 to unit 3, and input resetting from unit 3 to unit 2 and from unit 2 to unit 1, the
resulting full multivariable controller is:
C #
[
[
[
[
L
[
#
#
#
[
[
#
C
QQQ
C #
[
[[
[
L
[
[ L [
[ [
[
[[
[
[[
[[
[[
QQQ
[[
(4.33)
(4.34)
, where is the set point for the
with controlled output in unit 3, whereas and are the ideal resting values for the
inputs in tank 2 and 3.
The final controller in (4.33) and (4.34) may seem very complicated, but it can
usually be tuned in a rather simple cascaded manner. The feedforward elements
are normally the fastest acting and should normally be designed first. The local
feedback controllers can be tuned almost independently. Finally, the slow input
resetting is added, which will not affect closed-loop stability if it is sufficiently
slow.
4.4 Case study: pH neutralization
4.4.1 Introduction
Neutralization of strong acids or bases is often performed in several steps (tanks).
The reason for this is mainly that with a single tank the pH control is not quick
enough to compensate for disturbances (Skogestad, 1996). In (McMillan, 1984),
an analogy from golf is used: the difficulty of controlling the pH in one tank is
4.4 Case study: pH neutralization
87
compared to getting a hole in one. Using several tanks, and smaller valves for
addition of reagent for each tank, is similar to reaching the hole with a series of
shorter and shorter strokes. This is further discussed in Chapter 2.
In the present example we want to compare different control structures for
neutralization of a strong acid in three tanks (see Figure 4.4). This is clearly a
serial process. The aim of the control is to keep the outlet pH from the last tank
constant despite changes in inlet pH and inlet flow rate. For each tank the pH
can be measured, and the reagent (here base) can be added. Figure 4.4 shows the
process with only local control in each tank ( diagonal).
∆cin,max= Acid
±5mol/l q=0.005m3/s
Base
pH - 1
pHC
Base
pHI
pHC
Base
V1
pHI
pHC
pHI
V2
V3
kd ~ 106
pH 7 ±1
∆cout,max=±10-6mol/l
Figure 4.4: Neutralization of an acid in three tanks in series with local control in each
tank. Data: Outlet requirement: , set-points tank 2 and 3: ! and .
. Reactant (base): Inlet acid flow ' ( %! ) and flow rate >
. .
( ), nominal flow: >
4.4.2 Model
To study this process we use the models derived in Chapter 2. In each tank we
consider the excess
concentration, defined as ) *#+ . This gives
a bilinear model which is linearized around a steady-state working point, so that
the methods from linear control theory can be used. We get two states in each
process unit (tank), namely the concentration, , and the level. The disturbances
(feed changes mainly) enter in tank 1. We here assume that there is a delay of for the effect of a change in inlet acid or base flow rate or inlet acid concentration
to reach the outflow of the tank, e.g. due to incomplete mixing, and a further delay
of until the change can be measured. In the discrete linear state space model
these transportation delays are represented as extra states (poles in the origin). We
assume no further delay in the pipes between the tanks. The levels are assumed to
Chapter 4. Control Design for Serial Processes
88
be controlled by the outflows using a P controller such that the time constant for
the level is about 1/10 of the residence time ( 7 , where is the
volume set-point).
The volumes of the tanks are chosen to 7 N ([ , which are the smallest possible volumes according to the discussion in Skogestad (1996). The concentrations
are scaled so that a variation of corresponds to a scaled value of . The
control inputs and the disturbances are also scaled appropriately. The linear model
is used for multivariable controller design, while the simulations are performed on
the nonlinear model.
4.4.3 Model uncertainty
The model presented in the previous section was the nominal model, which will
be used in the controller design. If the model gives an exact representation of the
actual process, we say it is perfect. Due to simplifications in the modelling or
process variations, there is often a discrepancy between the model and the actual
process. Often the model is idealized, i.e., simplified, to ease the modelling work,
the identification of parameters, and the controller design.
In this example we use linearized models in the MPC design. In the design
of (SISO) feedforward controllers a further simplification is that outlet flow variations are neglected. This gives a steady-state model error, but dynamically the
error is small due to slow level control. What we here consider as the “actual
plant”, is the full nonlinear model, possibly with the following errors:
Offset of 0.2 (in scaled value) in control input measurement error of
R
[
(last tank).
in second tank.
4.4.4 Local PID-control (diagonal control)
The conventional way of controlling this process is to use local PID-control of the
pH in each tank. Starting from the tunings obtained with the method of Ziegler
and Nichols (1942), and employing some manual fine tuning (by trial and error),
we obtained
#
67 67 ; < ;
[[ 67 ; >:
;
;
;
;
; ; < 7: 7=< : ;
>7 ; < >7=< (4.35)
(4.36)
(4.37)
4.4 Case study: pH neutralization
89
Figure 4.5(a) shows the pH-response in each tank when the acid concentration
in the inflow is decreased from 2 to 2 . As expected (Skogestad,
1996), this control system is barely able to give acceptable control, in last tank. However, the nominal response can be significantly improved with
feedforward or multivariable control as shown in the following.
4.4.5 Feedforward control (control elements below the diagonal)
We now want to study the use of feedforward control from upstream units. As
before, we let the pH in the first tank be controlled with local PID control (the
same tuning as before), since we do not measure inlet disturbances to tank 1, and
feedback is therefore the only possibility. We let the pH in the second and third
tanks be controlled with feedforward control only, namely with feedforward from
to and from to [ . With “ideal” feedforward control based on the nominal
model we then get
[ [ [ (4.38)
[
(4.39)
where and [ are given by (4.8) and subscript minus indicates that the
net delay is increased to obtain a causal controller with zero or positive delay in
the controller. The two feedforward controllers will react too late due to the
measurement delays in and , and thereby introduce a transient output error.
To avoid this, the last feedforward controller, [ , from to [ , can be used to
eliminate this error by choosing [ from (4.31):
[ [ [ C [ [ [ [ L C L
[
(4.40)
Figure 4.5(b) shows a simulation on the same model as used for the feedforward controller design, and we can see that perfect control is acheived in tank 3
(solid line). However, when applied to a more realistic nonlinear model (incorporating flow rate changes), the feedforward controller fails (dotted lines).
4.4.6 Combined local PID and feedforward control (lower block
triangular control)
We now combine local PID-control in all the tanks, (4.35)-(4.37) with feedforward
control of tanks 2 and 3 (controllers , [ and [ ). In [ it is now
necessary to take into account the feedback loop of tank 2 and use Equation (4.31):
[ [ [ C X
[ [ [ [ L C L
[
(4.41)
Chapter 4. Control Design for Serial Processes
90
3
2
1
0
3
2
1
0
pH in tank 1
pH in tank 2
4
pH in tank 1
pH in tank 2
4
2
2
8
8
pH in tank 3
pH in tank 3
6
6
1
1
Control inputs u, scaled
Control inputs u, scaled
0
0
−1
0
50
100
150
time [s]
200
250
(a) Local feedback control in all three
tanks: The PID controllers must be aggressively tuned to keep the pH in the
last tank within .
−1
0
50
100
150
time [s]
200
250
(b) Feedback control in tank 1 only, and
feedforward control of tanks 2 and 3:
With a perfect model (i.e. simulation on
idealistic model) the disturbance is cancelled (solid line). With model error
(i.e.,simulation on a “realistic” nonlinear model), the response is very poor
and drifts away (dotted line). is only
given for the nominal case.
3
2
1
0
pH in tank 1
pH in tank 2
4
2
8
pH in tank 3
6
1
Control inputs u, scaled
0
−1
0
50
100
150
time [s]
200
250
(c) Local feedback control in all three
tanks combined with feedforward control of tanks 2 and 3: Even with model
error, the response in the outlet pH is
good (solid line).
Figure 4.5: Simple control structures applied to the neutralization process in Figure 4.4
(tank (dash-dotted), tank (dashed) and tank (solid)). Disturbance in inlet concentration occurs at .
4.4 Case study: pH neutralization
91
where is the PID controller of tank 2.
Again, with perfect model (i.e. simulated on the simplified model with constant flow rates) the effect of the disturbance is eliminated (same result as in Figure 4.5(b)). Simulation on the more realistic model reveals an improvement compared to the pure feedback and pure feedforward structures, as expected. The
feedforward controllers reduce the transient errors, whereas the PID controllers
remove the steady-state errors, as illustrated in Figure 4.5(c).
In Figure 4.6 the controller gains are plotted (lower left corner). The integral
actions are recognized from the high gains at low frequencies in the diagonal
elements. The sub-diagonal control elements are constant, whereas [ only has
an effect at high frequencies. This is where [ is no longer effective (error in
gives feedforward control error control for frequencies above
delay, 0
80 7 ; , see Chapter 6).
Note that with a larger model error, the positive effect of the feedforward controller may be reduced, and the feedforward action may even amplify the disturbances.
4
10
u
1
K11
−3
10
−6
0
10
10
4
10
4
10
K
K
22
u
2
21
−3
10
−3
−6
0
10
10
4
10
10
−6
0
10
10
4
10
4
10
K32
K33
u
3
K31
−3
10
−3
−6
0
10
10
y
1
10
−3
−6
0
10
10
y
2
10
−6
0
10
10
y
3
Figure 4.6: The controller gains of the lower block-diagonal control structure resulting
from combination of feedback (PID) and feedforward control (Section 4.4.6)
4.4.7 Multivariable control
Original MPC control (full multivariable controller) Figure 4.7(a) shows the
response with a 0 MPC controller ((Muske and Rawlings, 1993); see also Appendix A). To obtain the current state at each time step for the controller, a state
estimator is used. The estimated states in this “original” MPC-controller also
includes the two (unmeasured) disturbances: Inlet flow rate and inlet excess concentration, modelled as integrated white noise (we will discuss this choice later).
92
Chapter 4. Control Design for Serial Processes
The controller design is based on a discretized model, whereas in the simulation
only the controller is discrete. Even if this is a feedback controller, we see that
the disturbance response is similar to that of combined local feedback and feedforward control, and the main reason for the large improvement compared to the
local feedback case (Figure 4.5(a)) is in fact the “feedforward” effect. From the
lower plots in Figure 4.5 and Figure 4.7(a) we can see that the control input in
tanks 2 and 3 acts both earlier and with a steeper slope for MPC control than for
local control. Note that with MPC the control inputs for tanks 2 and 3 react before
the disturbance can be measured in the two tanks. The MPC also has a higher
order controller, which may explain why it reacts even faster than the combined
feedback/feedforward controller (Figure 4.5(c)).
“Feedforward” part of MPC-controller To study the “feedforward” effect separately, we design a MPC-controller that uses the pH measurement in the first tank
only, but adjust the reactant flow rates to all three tanks as shown in Figure 4.7(b).
The response for the nominal case is similar to the simulation with the full MPCcontroller shown in Figure 4.7(a). If, however, a model error is introduced, e.g. by
simulation on the nonlinear model instead, a steady-state error occurs for outlet
pH. The reason for this is the lack of feedback control in the last two tanks.
The individual gains of the 0 MPC-controller are shown as a function of
frequency in Figure 4.8(a) (solid lines). The diagonal control elements are the
local controllers in each tank, whereas the elements below the diagonal represent
the “feedforward” elements. From these plots we get an idea of how the multivariable controller works. For example, we see that the control input to tank 1
(row 1) is primarily determined by local feedback, while in tanks 2 and 3 (rows 2
and 3) it seems that “feedforward” from previous tank is more important for the
control input. In tanks 2 and 3 the control actions are smaller, which is also confirmed in the simulation (Figure 4.7(a)). The local feedback control elements on
the diagonal compare well with the PID controllers (dashed lines), except that the
gain is reduced for tanks 2 and 3, but this depends on the tuning of the MPC. At
high frequencies the “feedforward” elements are similar to the manually designed
feedforward controllers.
As discussed in Section 4.3, it is not straight-forward to interpret the steady
state behaviour from the gain plots of the controller elements when all the elements have large gains at low frequencies as in Figure 4.8(a). In Figure 4.8(b) we
.
therefore show the individual gains of the sensitivity function, To have no steady-state offset in an output we need that all elements in the corresponding row of to be small at low frequencies. From Figure 4.8(b) we then
see that we do have integral action for output 1, but not for outputs 2 and 3. We
should therefore expect steady-state offset in tank 3. However, the simulations
in Figure 4.7(a) show no offset. The reason is that the integral effect in the first
4.4 Case study: pH neutralization
3
2
1
0
93
3
2
1
0
pH in tank 1
pH in tank 2
4
pH in tank 1
pH in tank 2
4
2
2
8
pH in tank 3
8
6
pH in tank 3
6
1
1
Control inputs u, scaled
Control inputs u, scaled
0
−1
0
0
50
100
150
time [s]
200
250
) and reduced (
100
150
200
250
(b) MPC with measurement in first tank
only: With a perfect model the response
is as for the “full” MPC controller. With
model error, i.e., simulated on the “realistic” nonlinear model, the response is
poor and drifts away (dotted line for pH
in tank 3).
50
time [s]
(a) Full multivariable control: A large
improvement in nominal
performance is
possible with a
MPC-controller
compared to pure local feedback.
Figure 4.7: Full (
occurs at .)
−1
0
) MPC (Disturbance in inlet concentration
Chapter 4. Control Design for Serial Processes
94
1
10
3
10
K12
K13
u1
e
1
K11
−4
−3
10
10
1
3
10
10
K22
K23
e
u2
2
K21
−4
−3
10
10
3
1
10
K32
10
K33
e
3
u3
K31
−3
10
−4
−5
0
10
10
y1
−5
0
10
10
y2
−5
0
10
10
y3
(a)
of the control elements of the
Gain
MPC (solid). Also shown: Local
PID controllers and manually designed
feedforward elements (dashed).
10
−4
0
10
10
yr1
−4
0
10
10
yr2
−4
0
10
10
yr3
(b)
The gains
of the sensitivity
function
( +7. ) with the
MPC: Steadystate offset can be expected since some
of the elements related to control errors
and have high gain at low frequencies.
Figure 4.8: The original multivariable MPC controller: Frequency domain analysis.
tank removes the concentration effect, and the “feedforward” control gives the
correct compensation for the flow rate disturbance. However, if some unmodelled
disturbance or model error is introduced (e.g. a constant offset in [ or a measurement error in tank 2), then we do indeed get steady-state offset. This is shown in
Figure 4.9. The local PID controllers give no such steady-state offset.
Modified MPC-controller with integral action In the “original” estimator used
above we only estimated the inlet disturbances. We now redesign the controller
by estimating one disturbance in each tank: The concentration disturbance to the
first tank and disturbances in the manipulated variables in tanks 2 and 3 ( and
[ ). The resulting controller gains are shown in Figure 4.10(a). With this design
the gain in is low at low frequencies for all tanks (Figure 4.10(b)), and
the simulations in this case give no steady-state offset (Figure 4.11). This agrees
with the result from Chapter 5 that the number of disturbance estimates in the
controller must equal the number of measurements.
This illustrates one of the problems of the “feedforward” control block, namely
its sensitivity to static uncertainty. Simulations using the perfect model may lead
the designer to believe that there is integral effect in the controller even if it is not.
4.4 Case study: pH neutralization
3
2
1
0
95
3
2
1
0
pH in tank 1
pH in tank 2
4
pH in tank 1
pH in tank 2
4
2
2
8
8
pH in tank 3
6
pH in tank 3
6
1
1
Control inputs u, scaled
Control inputs u, scaled
0
0
−1
0
50
100
150
time [s]
200
250
−1
0
100
150
time [s]
200
250
(b) Measurement error: At time a
pH measurement error of ' is introduced in tank 2, and the steady-state pH
is 5.9 instead of 7 in the last tank.
(a) Unmodelled disturbance: Control
input has got an offset of 0.2 (at
time 0) compared to the model, and the
steady-state pH is in stead of in
last tank (disturbance in inlet concentration occurs at ).
Figure 4.9: The original
50
MPC has insufficient integral action
3
10
K12
1
10
K13
e
1
u1
K11
−3
10
−3
10
3
10
1
K22
10
K23
e2
u2
K21
−3
10
−3
10
3
10
1
K
K
K
32
33
e3
u3
31
10
−3
10
−3
−5
0
10
10
y
1
−5
0
10
10
−5
0
10
10
y
y
2
−4
0
10
10
−4
0
10
−4
10
r
3
10
r
2
0
10
r
1
(a) Gain of the control elements of the
modified
MPC (solid). Also
shown: Local PID controllers and manually designed feedforward elements
(dashed).
Figure 4.10: Modified
10
3
(b) The elements
+ 7. show that
there is no steady-state offset in output
3 (last tank).
MPC: Frequency domain analysis.
Chapter 4. Control Design for Serial Processes
96
3
2
1
0
3
2
1
0
pH in tank 1
pH in tank 2
4
pH in tank 1
pH in tank 2
4
2
2
8
8
pH in tank 3
6
pH in tank 3
6
1
1
Control inputs u, scaled
Control inputs u, scaled
0
−1
0
0
50
100
150
time [s]
200
250
(a) Unmodelled disturbance: Control
input has got an offset of 0.2 (at time
0) compared to the model, and with the
modified MPC we get no steady-state
offset. (disturbance in inlet concentration occurs at ). Step disturbance at time .
Figure 4.11: Modified
model errors.
−1
0
50
100
150
time [s]
200
250
(b) Measurement error: At time a
pH measurement error of ' is introduced in the tank 2, and now we get no
steady-state offset.
MPC with integral action: Closed loop simulations with
4.4 Case study: pH neutralization
97
4.4.8 MPC with input resetting
In the simulations above we gave set points for the pH in each tank. Actually
we are only interested in the pH in the last tank, so that giving set points for
the other two is not necessary. Since we have three control inputs, this leaves
two extra degrees of freedom as described in section 4.3.5, which may be used
for input resetting. The MPC controller is easily modified to accommodate this.
Figure 4.12 illustrates how this works after a unit step in the disturbance: At
steady-state all the required change in base addition is done in the first tank. Since
we do not measure the actual base addition, offsets in the control input are not
compensated for.
3
10
3
2
1
0
K11
K12
K13
K21
K22
K23
K31
K32
K33
u1
pH in tank 1
−3
10
3
pH in tank 2
4
10
u2
2
8
pH in tank 3
−3
10
6
3
10
u3
1
Control inputs u, scaled
0
−1
0
−3
10
50
100
150
time [s]
200
250
(a) and are brought back to
0 (but
it takes slightly more than
). Step
!
disturbance at time .
−5
0
10
10
−5
0
10
y
1
10
y
2
−5
0
10
10
y
3
(b) Control element gains: At steadystate the dominant control element is
+ . : is used to control at low
frequencies.
Figure 4.12: MPC with input resetting.
4.4.9 Conclusion case study
The case study shows a large improvement that is obtained by the introduction
of a multivariable controller instead of single loop control (Figure 4.5(a)). The
improvement is caused by “feedforward” effects (Figure 4.5(c)), and with model
errors, the “feedforward” may in fact lead to worse performance.
Integral action or strong gain in the local controllers at low frequencies is
required, even if the “feedforward” effect itself nominally give no steady-state.
Feedback to upstream tanks may be used to bring the inputs to their ideal resting
98
Chapter 4. Control Design for Serial Processes
positions. The example indicates that it is possible to get a good performance
with careful use of a multivariable controller or a combination of local control,
“feedforward” from tank 1 and 2 and possibly input resetting.
4.5 Discussion
There are several ways to avoid steady-state offsets with MPC controllers. The
most common method is to estimate the bias in the outputs, i.e. the difference
between the predicted and the measured outputs, and compensate for this bias.
However, performance is often improved by estimating input biases, or disturbances (Muske and Rawlings, 1993; Lee et al., 1994; Lundström et al., 1995).
In this paper we have followed this approach. We ended up with estimating the
concentration disturbance into first tank and input biases for tanks 2 and 3 (three
input biases gives similar results). Our controller handles well both input disturbances (see Figure 4.7(a)) and output disturbances or measurement errors (see
Figure 4.11(b)).
We have also tried to estimate output biases, but this gave a very slow settling
in response to inlet disturbances. The reason is the long time constants in our process, which give the output bias estimates a ramp form (Lundström et al., 1995).
The controller then faces a problem similar to following a ramp trajectory.
In Chapter 2 we found that the minimum volume in each tank is limited by the
delays in each tank. In the present paper we found that with a multivariable controller for simultaneous control of all three tanks, these limitations are no longer
valid provided a sufficiently accurate process model is used. The reason for this
is that the multivariable controller does not have to wait for the measurement in
last tank before it takes action (due to the “feedforward” effect). To be able to
achieve a nominally perfect “feedforward” control effect, the delay from at least
one control input to the output must be shorter or equal to the delay from a measurement in the disturbance to the output. The effect of model uncertainty on the
feedforward control improvements must be evaluated for the process. If there is
an improvement, one may design smaller tanks compared to the sizes given in
Chapter 2, or reduce the instrumentation.
4.6 Conclusions
An example of neutralization of a strong acid with base in a series of three tanks is
used to illustrate some of the ideas in the paper. This is obviously a serial process.
The example illustrates that a multivariable controller yields significant nominal
improvements compared to single loop PID control (compare Figure 4.7(a) with
4.7 Acknowledgements
99
Figure 4.5(a)). This is mainly due to “feedforward” elements (see Figure 4.5(c)).
Due to imperfections in the process model, including unmodelled disturbances,
an efficient feedback effect must also be included. To obtain this one must:
include measurements late in the process.
include integral action if offset free steady-state is important. For MPC control, the use of input error estimates is one efficient method, which requires
that the disturbance vector is chosen with some care.
Testing of the controller on a too idealistic process model may give the impression
that the feedback is better than it actually is. Simulations with the multivariable
controller active must include all possible disturbances, model offsets (for example one may apply the controller on a more realistic (nonlinear) process model)
and also offsets in the measurement signals.
Assuming no active constraints, a linear analysis may be used to analyze the
controller. The frequency dependent gain in each channel may give insight into
how the controller utilize each measurement and the magnitude of the control
actions for each input. The steady-state behaviour can be seen from the low frequency gains. But often more than one channel in a row have high gain at low
frequencies, for example when inversed based methods like IMC is used, and
then it is difficult to interpret the result. It is then better to consider the elements
of the sensitivity function matrix. An offset-free steady-state control for a specific
output requires that all the elements in the corresponding row have low gain at
low frequencies.
When designing the controller one must also consider which of the outputs
that is really important. If the number of inputs exceed the number of (important)
outputs, one may either give set-points to other (less important) outputs, or one
may let the controller bring some of the inputs back to ideal resting positions.
In this study we used multivariable MPC, but very similar results have also
been found for a multivariable
-controller (Faanes and Skogestad (1999), i.e.,
Thesis’ Appendix A).
4.7 Acknowledgements
Financial support from the Research Council of Norway (NFR) and the first author’s previous employer Norsk Hydro ASA is gratefully acknowledged.
References
Buckley, P. S. (1964). Techniques of Process Control. John Wiley & Sons, Inc.. New York.
100
Chapter 4. Control Design for Serial Processes
Faanes, A. and S. Skogestad (1999). Control structure selection for serial processes with
application to pH-neutralization. Proc. European Control Conferance, ECC 99, Aug.
31-Sept. 3, 1999, Karlsruhe, Germany.
Lee, J. H., M. Morari and C. E. Garcia (1994). State-space interpretation of model predictive control. Automatica 30(4), 707–717.
Lewin, D. R. and C. Scali (1988). Feedforward control in presence of uncertainty. Ind.
Eng. Chem. Res. 27, 2323–2331.
Lundström, P., J. H. Lee, M. Morari and S. Skogestad (1995). Limitations of dynamic
matrix control. Comp. Chem. Engng. 19(4), 409–421.
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Nov. 12-17 1995, Paper 189b.
Morud, J. and S. Skogestad (1996). Dynamic behaviour of integrated plants. J. Proc. Cont.
6(2/3), 145–156.
Muske, K. R. and J. B. Rawlings (1993). Model predictive control with linear models.
AIChE Journal 39(2), 262–287.
Rivera, D. E., M. Morari and S. Skogestad (1986). Internal model control. 4. PID controller design. Chem. Engn. 25, 252–265.
Scali, C., M. Hvala and D. R. Lewin (1989). Robustness issues in feedforward control..
ACC-89 pp. 577–581.
Shinskey, F. G. (1973). pH and pIon Control in Process and Waste Streams. John Wiley
& Sons. New York.
Skogestad, S. (1996). A procedure for SISO controllability analysis - with application to
design of pH neutralization processes. Computers Chem. Engng. 20(4), 373–386.
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Skogestad, S. and I. Postlethwaite (1996). Multivariable Feedback Control. John Wiley &
Sons. Chichester, New York.
Walsh, S. (1993). Integrated Design of Chemical Waste Water Treatment Systems. PhD
thesis. Imperial College, UK.
Zhou, K., J. C. Doyle and K. Glover (1996). Robust and Optimal Control. Prentice Hall.
New Jersey, US.
Ziegler, J. G. and N. B. Nichols (1942). Optimum settings for automatic controllers. Trans.
ASME 64, 759–768.
Chapter 4. Control Design for Serial Processes
102
Appendix A State space MPC used in case study
Here we briefly describe the MPC controller of Muske and Rawlings (1993) under
the assumption that the constraints are not active. For details we refer to Chapter 5.
The MPC controller uses an estimate of the current states of the process and a
state space model to predict future responses to control input movements. By letting the control input change each time step over a certain horizon, and thereafter
held constant, the optimal sequence of control inputs is calculated. The criterion
for the optimization is
C
0
,0 L
(4.42)
where is the vector of future control inputs, the first at sample number , is the output vector at time , is the control input at time , 0 is the change
in since last time step and , and are weight matrices. Note that in the
crierion we assume that the set-point for the output, . Non-zero set-points
are handled by a steady-state solver. Only the first control input is applied, since at
next time step the whole sequence is recalculated, starting from the states actually
obtained at that moment.
Without constraints the MPC can be represented as state feedback control, i.e.
the control input at time step no. can be expressed by
(4.43)
where is the state vector at time and and are constant matrices, independent of time provided the model is assumed time invariant. The dependence
of the control input at the previous step, , comes from the weight on change
in in the optimization criterion.
Since all the states are not measured, we estimate them for example with a
Kalman filter. For the MPC algorithm we use a discretized model with time step
1 second and use a zero order hold method for the discretization since the inputs
are held constant between the time steps. In the discretized model time delays are
represented exactly, as long as they are multiples of the time step.
In Chapter 5 we derive a state space formulation for the controller and the
estimator:
(4.44)
(4.45)
where is the control input at sample number , is the controller/estimator
state vector, is the measurement vector and the reference, which may be seen
as a disturbance to the controller. , , , , and are constant matrices.
Appendix A State space MPC used in case study
103
For frequency analysis of the controller we may convert this discrete controller
into a continuous one using d2c in Matlab (Tustin method), and Laplace transform
yields:
(4.46)
We have chosen weights in the MPC optimization criterion as > , and in the MPC optimization criterion (4.42). For the estima
tor the co-variance matrices are
(process noise) and (measurement noise).
Chapter 4. Control Design for Serial Processes
104
Appendix B Derivation of equations (4.20) and (4.31)
With pure feedforward control we get the following control error
E
L C (4.47)
where we have inserted feedforward from unit from (4.17). With a combination of feedback and feedforward control we get (with (4.17))
L (4.48)
In both cases “ideal” feedforward requires E for all and :
C L (4.49)
We consider first pure feedforward, , and find the transfer
function from to :
(4.50)
E
C yields
C L (4.51)
and upon inserting (4.51) into (4.49) we obtain
C L C leading to (4.20).
Second, we find the transfer function from
local feedback and feedforward,
L to
for a combination of
where . Then
C L (4.52)
(4.53)
and by inserting this into (4.49) it follows
C which gives (4.31).
L X C L (4.54)
Chapter 5
On MPC without active constraints
Audun Faanes and Sigurd Skogestad
Department of Chemical Engineering
Norwegian University of Science and Technology
N–7491 Trondheim, Norway
Submitted to Modeling, Identification and Control, MIC.
also affiliated with Statoil ASA, TEK, Process Control, N-7005 Trondheim, Norway
Author to whom all correspondence should be addressed. E-mail: [email protected],
Tel.: +47 73 59 41 54, Fax.: +47 73 59 40 80
106
Chapter 5. On MPC without active constraints
Abstract
In order to be able to use traditional tools when analysing a multivariable controller as MPC,
we develop a state space formulation of the resulting controller for MPC without constraints or
assuming that the constraints are not active. Such a derivation was not found in the literature. The
state space formulation is used in Chapters 4 and 7. The formulation includes the state estimator.
The MPC algorithm used is a receding horizon controller with infinite horizon based on a state
space process model. When no constraints are active, we obtain a state feedback controller, which
is modified to achieve either output tracking, or a combination of input and output tracking.
When the states are not available, they need to be estimated from the measurements. It is often
recommended to achieve integral action in a MPC by estimating input disturbances and include
their effect in the model. We show that to obtain offset free steady state the number of estimated
disturbances must equal the number of measurements. The estimator is included in the controller
equation to obtain the overall controller with the set-points and measurements as inputs, and which
give the manipulated variables.
One use of the state space formulation is to combine it with the process model to obtain a
closed loop model. This can for example be used to check the steady-state solution and see if
integral action is obtained.
5.1 Introduction
107
5.1 Introduction
In this paper, we develop a state-space formulation for a MPC without constraints
or assuming that the constraints are not active. This state-space formulation of
the controller enables the use of traditional tools to get insight into how the controller behaves (see Chapters 4 and 7). Maciejowski (2002) (independently) use
a linear formulation for a MPC controller to analyze its controller tuning for a
paper machine headbox. He combines the linear controller formulation with the
process model, and calculates the singular values of the sensitivity function and
the complementary sensitivity function.
The main idea behind MPC is that a model of the process is used to predict
the response of future moves of the control inputs (the inputs that the controller
can manipulate to control the process). This prediction is used to find an optimal
sequence of the control inputs. Optimal means that a certain criterion containing
an output vector and the vector of the control inputs is minimized.
In most MPC implementations the control inputs are assumed to be held constant within a given number of time intervals. At a given time, the first value in the
sequence of control inputs is implemented in the process. The prediction depends
on the current state of the process, and this will also the optimal sequence do. At
the next time step, the state being reached is therefore used in the calculation of a
new optimal control input sequence. This sequence will not necessarily be what
was computed at the previous time step, due to the effects of model errors and
unmodelled disturbances. So, at each time step we only implement the first step
in the control input sequence, and discard the rest.
Normally we include constraints in the optimization problem. These are constraints that naturally occur in a process, like the range of control valves and pump
speeds (on control inputs), and safety-related constraints on the outputs. One may
also restrict the rate of change of the control inputs.
For a review of industrial MPCs we refer to (Qin and Badgwell, 1996; Badgwell and Qin, 2002).
In this chapter, we consider the MPC formulation proposed by Muske and
Rawlings (1993). This MPC is based on a state-space model. Our assumption is
that no constraints are active, and this also covers the case when the same constraints are active all the time and the degree of freedom is reduced. Bemporad
et al. (2002) (first appeared in (Bemporad et al., 1999)) have shown that the controller also for the case with dynamic constraints is piecewise linear.
Since the models are not perfect, and there always are unmodelled disturbances, the MPC needs some correction from measurements. The most common
approach is to estimate some output bias in the measurements, and correct for
this bias. However, for integrating processes or processes with long time constants, this method has proved unsatisfactory (Muske and Rawlings, 1993; Lee
Chapter 5. On MPC without active constraints
108
et al., 1994; Lundström et al., 1995). We therefore estimate input disturbances,
which is straight forward using a state-space representation of MPC.
As known, MPC without constraints is a special case of optimal control, and
in Sections 5.2, 5.3 and 5.4 we will demonstrate how the control input can be
expressed by the current state and the previous control input. The first of these
sections, Section 5.2, covers the simple case when the reference for the output
vector is zero, while Section 5.3 handles non-zero references. When the number
of control inputs exceeds the number of outputs, the extra degree of freedom may
also be used to give references to the control inputs (Section 5.4). Since the full
state vector normally is not measured, we include a state estimator, which also
estimates input disturbances, in Section 5.5. The total controller formulation, i.e.,
the control inputs, given by the measurements, is given in Section 5.6. In Section 5.7 we find the number of estimated disturbances needed to obtain effective
integral action. We develop the closed loop model of the system in Section 5.8.
5.2 Derivation of equivalent controller from receding horizon controller without active constraints
Muske and Rawlings (1993) present a model predictive control algorithm based
on the following state-space model:
; 777
(5.1)
(5.2)
Here is the state vector, the control input vector, the vector of (unmeasured) disturbances and the output vector, all at time . The model is assumed
to be time invariant so , and are constant matrices. The optimal control
input minimizes the following infinite horizon criterion:
C
0
,0 L
(5.3)
777 Here is a vector of future moves of the
control input, of which only the first is actually implemented. The control input,
, is assumed zero for all . In the criterion it is assumed that the reference for is zero. We assume that the process is stable, and Muske and Rawlings
(1993) show how this formulation can be transformed into the following finite
optimization problem:
C L ;RC L
(5.4)
5.3 The steady-state solution
109
where , and are time independent matrices expressed by the model ma
trices, , and , and the weight matrices, , and . Since is unknown
in the future, the term from (5.1) is omitted in the derivation of (5.4). For
normal use of this MPC algorithm, the control input is found by optimizing (5.4)
subject to given constraints on the outputs, the control inputs and changes in the
control inputs. Assuming no active constraints, however, the optimum of (5.4) can
be found by setting the gradient equal to zero (Halvorsen, 1998):
C L ;
which implies
Only the first vector from ;
(5.5)
(5.6)
is applied:
(5.7)
and , respectively,
where
and consist of the first rows in and is the number of control inputs.
Since , and are constant, also
and are constant matrices. The
first term can therefore be recognized as state feedback. The second term comes
from the weight on the change in control input from the original criterion. The
matrix only contains and zeros, so when no weight is put on the change in
the control input, is zero, and .
5.3 The steady-state solution
Here, we consider tracking of outputs. If the output reference vector, , is
nonzero, (5.7) must be shifted to the steady-state values for the states and the
control inputs:
(5.8)
or
and can be found from the steady-state solver:
subject to
(5.9)
(5.10)
(5.11)
Chapter 5. On MPC without active constraints
110
'1
'1.354
(5.12)
where and are the references for the output and the control input, respectively. Again, we assume that the limitations are never active, and that we have no
extra freedom for the control inputs (number of control inputs equals number of
outputs), in which case the problem reduces to solving the equation set (5.11).
Assuming square systems (i.e., equal number of control inputs and references),
invertible) and that is
no poles in the origin (which makes invertible (it is at least quadratic from the first assumption), we get the following
solution:
(5.13)
where
C L C C C
L
L
L
(5.14)
(5.15)
Since we have no knowledge of future disturbances, we assume that it will keep
as desired, and that
its current value, that is . We note that if we assume that the disturbance enters via the control inputs, i.e., , the
expression for simplifies to
where
and
are defined in Section 5.2 and
i.e., and .
Now (5.9) can be expressed with and :
(5.16)
(5.17)
(5.18)
5.4 Generalization with tracking of inputs
In this section, we generalize the steady-state solution to include tracking of both
inputs and outputs. The total number of references that it is possible to track is
limited by the number of (independent) control inputs.
5.4 Generalization with tracking of inputs
111
We collect the inputs that we want to give a reference into the vector , and
likewise the outputs we want to give a reference into . The rest of the inputs
and outputs are assembled into and , respectively. The model may now be
formulated as
(5.19)
two matrices 1 and where we have distributed the columns of into the
corresponding to the division of , and the rows of is divided into and corresponding to the division of . At steady state and . Now
and can be expressed by , , and ( ):
where
C
C
(5.20)
(5.21)
L
L
L
(5.22)
(5.23)
are invertible. For we obtain
and L
L
C
L
where
Introduction of
and
C
C
C
provided that
(5.26)
(5.27)
(5.24)
(5.25)
yields
(5.28)
Chapter 5. On MPC without active constraints
112
5.5 State and disturbance estimator
To calculate from (5.16) or (5.28) one must know the state, , and if it is not
measured, it must be estimated from the measurements. The same applies also to
the disturbance vector . If we assume that neither the states nor the disturbances
are measured, we extend the state variable with the disturbance vector
(5.29)
As basis for a state estimator the following model based on (5.1) and (5.2) is
introduced:
(5.30)
(5.31)
where and are zero-mean, uncorrelated, normally distributed white stochas
tic noise with covariance matrices of
and respectively, and
not necessarily the same as the output vector
is the measured output vector,
that shall track a reference, and
is the corresponding matrix in the estimator
model, mapping from the states to the measured output vector. We have modelled
the disturbance as constant except for the noise.
The augmented state estimator is then formulated as
(5.32)
(5.33)
where is the estimator gain matrix, for example the Kalman filter gain. is called the a priori estimate (since it is prior to the measurement), and the a
posteriori estimate (after the measurement is available). For a Kalman filter, is
given by the solution of a Ricatti equation:
(5.34)
(5.35)
We want to express the estimator in a single expression, and this can be done in
two ways, depending on which of the two estimates one prefers to use. Alternative
1: A posteriori estimate, :
(5.36)
5.6 State-space representation of the overall controller
Alternative 2: A priori estimate, :
113
(5.37)
Remark 1 Muske and Rawlings (1993) refer to Åström (1970) who used a priori
estimate (Alternative 2), (noting that their corresponds to our ). However,
according to (Rawlings, 1999) they actually used Alternative 1 (a posteriori) in
their work. Normally the control input is implemented directly after a new measurement has been sampled, in which case the a posteriori estimate is preferred
since it utilizes this new measurement. Thus, in this paper we will use Alternative
1, the a posteriori estimate.
5.6 State-space representation of the overall controller
In this section, we will form the overall controller, containing the state feedback,
the steady-state solution and the estimator on state-space form.
With the extended state vector from (5.29) and
(5.38)
the controller equations (5.16) and (5.28) can both be expressed by
(5.39)
For (5.16) (without input resetting) and . Since generally
is not available, we use the estimate . Combination of the controller equation
(5.39) with the estimator difference equation (5.36) yields
where
(5.40)
(5.41)
. This is not an ordinary discrete state-space
formulation. First, and do not have the same index on the right side of
(5.40). To overcome this we introduce the artificial state variable :
(5.42)
(5.43)
Chapter 5. On MPC without active constraints
114
Next, the term is a problem. We first assume that in the optimization
criterion (5.3) . Then , and we get an ordinary discrete state-space
system with as the states, as the input and as the output. The reference,
, can be seen as a ”disturbance” to the controller. We may express the controller
as
where
, ,
,
,
(5.44)
and
.
For we have not yet obtained the controller on ordinary state-space
form. We first express the controller as
insert for in the expression for and
.
(5.45)
(5.46)
and re-arrange:
where in addition to the definitions above
We repeat shifted one time step,
(5.47)
We now introduce the state vector
and obtain
(5.48)
(5.49)
5.7 On the number of estimated disturbances
Again, we have which yields
For we obtain
in the expression for , and introduce
115
(5.50)
(5.51)
(5.52)
which yields the following expression for the total controller:
where
(5.53)
In summary, we have shown that with no active constraints, the MPC controller with augmented state estimator can be expressed on discrete state-space
form.
If we instead use the a priori estimate (Alternative 2), we get a different controller with other poles.
5.7 On the number of estimated disturbances
In this section, we will discuss the number of estimated disturbances (the dimension of ) necessary to avoid steady-state offset. According to Muske and Rawlings (1993), the number of elements in can not exceed the number of measurements if observability of the estimator shall be achieved. But what is the smallest
number required?
Chapter 5. On MPC without active constraints
116
We first have to specify clearer what ”no steady-state offset” means. If the
process is perturbed by measurement noise and disturbances that change their
value from time step to time step, the control will never be offset free, and no
steady state will be obtained. Thus, we will consider the response when the noise,
the model error and the disturbances are constant. (Alternatively, one may model
noise, model error and disturbances as stochastic processes and consider a large
number of experiments.)
Using as before the a posteriori estimate, the estimate of the measurement is
(5.54)
In order to obtain a offset free steady state, the estimator must provide a correct
state estimate for the MPC despite model errors, constant measurement errors
or noise and a constant input disturbance at steady state. More precisely, the
prediction of the measured output must equal the actual one:
(5.55)
We let index to denote steady state.
We want to see what this condition means for our MPC and estimator, and first
we extract the expression for from the estimator equation (5.36):
(5.56)
is the upper part of , corresponding to the dimension of . At steady
where state
which yields
(5.57)
(5.58)
To find we cannot use (5.13) or (5.20) since these include the actual state and
disturbance vectors and not their estimates. Instead we apply (5.39) which yields
for the steady-state control input
(5.59)
We insert this into (5.58) and obtain
C C
LL C L (5.60)
5.7 On the number of estimated disturbances
117
To simplify the notation we introduce the matrices
[ [
Thus (5.54)
and (5.55) yields
[
[ [
[ which
leads to the following
matrix equation
(5.61)
(5.62)
(5.63)
and obtain for the a posteriori state estimate
[
[ (5.64)
(5.65)
(5.66)
In (5.66) the number of scalar equations equals the number of measurements
). The only free variables are the elements of . To
(the number of rows in
obtain an offset free steady-state solution of the control problem there must exist
a solution of (5.66), which implies that the number of elements in must be
equal or greater than the number of measurements (independent of the size of the
reference, , and the number of control inputs, ). Thus, since the number of
estimated disturbances cannot exceed the number of measurements (see above),
we may conclude that:
If offset free steady state shall be obtained, the number of estimated
disturbances must be equal to the number of measurements.
This was, independently, also found by Muske and Badwell (2002), except that
they do not distinguish between outputs to be controlled by the MPC and the
measurements. Such a distinction proves to be useful in Chapter 7, where an
experimental illustration is given.
Remark 2 In the general case (5.66) cannot be used to determine given and
. It will often be many that fulfills (5.66), and the value of will depend on
the disturbance, measurement or model error that is present.
Example 5.1 For the neutralization example in Chapter 4 we use three measurements, and thus estimation of three disturbances is required. For the ”original”
MPC we only estimate two input disturbances, and the result is insufficient integral action, as expected. The modified MPC with three disturbance estimates gets
full integral action.
Chapter 5. On MPC without active constraints
118
5.8 Closed loop model
The combination of the process model with the controller yields the closed loop
model of the system. The process is expressed by the discrete model (5.1) and
(5.2) which we repeat for the actual process, marked with a prime:
The vector of measurements,
; 777
(5.67)
(5.68)
, is expressed by
(5.69)
where
is the matrix mapping from the states to the measured output vector
and is the measurement error. The controller is expressed by (5.44) or (5.39).
and
are then eliminated from the equations by combining the controller with
(5.67), (5.68) and (5.69). We then get the following closed loop model (where we
have omitted the tilde in the controller matrices from (5.39)):
(5.70)
(5.71)
(5.72)
We combine the process states, , and the controller states, into
and obtain the following model
where
0
0
(5.73)
(5.74)
(5.75)
(5.76)
(5.77)
(5.78)
(5.79)
5.8 Closed loop model
119
One possible use of the closed loop model is to study the steady state of input
steps. Introducing the time-shift operator E where is the time step, gives
0 The z-transform of a unit step is inputs at a time. This may be formulated as
(5.80)
. We apply a unit step on one of the
(5.81)
where , and are vectors with zeros except one element equal . From see e.g.,
(Phillips and Harbor, 1991, p. 452), we have
(5.82)
and thus
0 0 Thus the matrices 0 , and 9 (5.83)
reveal the
steady-state effect of a unit step in each of the inputs on each of the outputs. For
gives the steady-state effect of a
example, element 5; in matrix unit step in disturbance no. on output no. ; (when the controller is applied).
Example 5.2 For the neutralization example in Chapter 4 we get for the ”original” MPC with estimation of disturbances into first tank only (resulting in insufficient integral action):
Q "!
0 <DQ ""!
N Q "!
: Q S
Q "&
Q HV
; Q 67 : Q ; Q "& 6
Q ""& Q "
: Q ; Q "& 6
(5.84)
(5.85)
; Q 7 R
RQ H V
; Q [
: Q [
(5.86)
We see that we get significant deviations from set point when measurement errors
are present. For example, a measurement error of 1 in measurement no.1 gives a
Chapter 5. On MPC without active constraints
120
deviation from set-point of 1 in output 3 (element in the matrix in (5.86)).
With disturbances in all outputs (and full integral action), we obtain
RQ "&
0 Q S
Q "!
Q "&
P
: Q S
; Q "HV
<RQ "!
P ; Q " S
Q Q " & Q 6<DQ " & Q : Q S Q "&
(5.87)
(5.88)
: Q N Q> : Q ""! N >Q " Q - 6:DQ "!
(5.89)
and there are no significant steady-state errors.
5.9 Conclusions
In this paper, we have developed a state-space formulation for a MPC (for stable
processes) without constraints or assuming that the constraints are not active. This
state-space formulation of the controller makes it possible to use traditional tools
to get insight into how the controller behave (see Chapters 4 and 7). The controller
can be extended with tracking of inputs, and also include the state estimator necessary if not all the states are measured. To obtain offset-free tracking, estimates
of the input disturbances are included in the estimator and in the calculation of
steady state. We show that the length of this estimated disturbance vector must
equal the number of measurements available to the estimator.
Finally, a closed loop state-space formulation is derived, assuming a statespace formulation of the process model.
References
Åström, K. J. (1970). Introduction to Stochastic Control Theory. Mathematics in Science
and Engineering. Academic Press. New York San Francisco London.
Badgwell, T. A. and S. J. Qin (2002). Industrial model predictive control - an updated
overview. Process Control Consortium, University of California (UCSB), March 8 9, 2002.
Bemporad, A., M. Morari, V. Dua and E. N. Pistikopoulus (1999). The explicit linear quadratic regulator for constrained systems. Technical Report AUT99-16. ETH,
Zürich.
REFERENCES
121
Bemporad, A., M. Morari, V. Dua and E. N. Pistikopoulus (2002). The explicit linear
quadratic regulator for constrained systems. Automatica 38, 3–20.
Halvorsen, I. (1998). Private communication.
Lee, J. H., M. Morari and C. E. Garcia (1994). State-space interpretation of model predictive control. Automatica 30(4), 707–717.
Lundström, P., J. H. Lee, M. Morari and S. Skogestad (1995). Limitations of dynamic
matrix control. Comp. Chem. Engng. 19(4), 409–421.
Maciejowski, J. M. (2002). Predictive Control with Contraints. Prentice Hall. Harlow,
England.
Muske, K. R. and J. B. Rawlings (1993). Model predictive control with linear models.
AIChE Journal 39(2), 262–287.
Muske, K. R. and T. A. Badwell (2002). Disturbance modeling for offset-free linear model
predictive control. J. Proc. Cont. 12, 617–632.
Phillips, C. L. and R. D. Harbor (1991). Feedback Control Systems. Prentice-Hall International, Inc.. London.
Qin, S. J. and T. A. Badgwell (1996). An overview of industrial model predictive control
technology.. Presented at Chemical Process Control-V, Jan 7-12, 1996, Tahoe, CA.
Proceedings AIChE Symposium Series 316 93, 232 – 256.
Rawlings, J. B. (1999). Private communication. European Control Conference, ECC’99,
Karlsruhe, September 2, 1999.
Chapter 6
Feedforward Control under the
Presence of Uncertainty
Audun Faanes and Sigurd Skogestad
Department of Chemical Engineering
Norwegian University of Science and Technology
N–7491 Trondheim, Norway
Preliminary accepted for publication in European Journal of Control
Abstract
In this paper we study the effect of model errors on the performance of feedforward controllers. In accordance
with the sensitivity function for feedback
control, we define the feedfor
ward sensitivities,
(feedforward from disturbance) and
(feedforward from set-point), as
measures for the reduction in the output error obtained by the feedforward control. For “ideal”
feedforward controllers based on the inverted nominal model, the feedforward sensitivities equal
the relative model errors, which must thus remain less than for feedforward control to have a
positive (dampening) effect.
For some common model error classes we provide rules for when the feedforward controller
is effective, and we also design -optimal feedforward controllers.
Keywords: Process control, Linear systems, Feedforward control, Uncertainty
also affiliated with Statoil ASA, TEK, Process Control, N-7005 Trondheim, Norway
Author to whom all correspondence should be addressed. E-mail: [email protected]
also to be presented at European Control Conferance, ECC’03, September 1-4, 2003, Cambridge, UK
124
Chapter 6. Feedforward Control under the Presence of Uncertainty
6.1 Introduction
There is a fundamental difference between feedforward and feedback controllers
with respect to their sensitivity to uncertainty. Feedforward control is sensitive
to uncertainty in general (including steady state), whereas feedback control is
insensitive to uncertainty at frequencies within the system bandwidth. With no
model error a feedforward controller may remove the effect of disturbances, but
due to its dependence of the process model, it may actually amplify the effect of
a disturbance if the model is faulty.
Textbooks on control and process control focus mainly on feedback controllers.
This reflects the difference in importance and popularity of the two controllers,
but also that feedback theory is more complicated. Most of the articles on feedforward control refer to industrial applications. However, some control textbooks, e.g., Buckley (1964), Stephanopoulos (1984), Doebelin (1985), Seborg et
al. (1989), Middleton and Goodwin (1990), Coughanowr (1991), Marlin (1995),
Ogata (1996), Shinskey (1996), describe feedforward controllers and their design, and the advantages and disadvantages compared to feedback is discussed.
It is concluded that a feedforward controller may improve the performance, and
is valuable when feedback control is not sufficient, but that in practice it must be
combined with a feedback controller. It is agreed that the feedforward controller
is most efficient if good disturbance measurements and accurate models are available, but no quantitative analysis is given (with some exceptions as given in the
following). Harriott (1964) claims that in a “typical system” the disturbance effect
is reduced to ; 2 . Middleton and Goodwin (1990) demonstrate that the variation
in the gain from the inputs to the outputs (the process uncertainty) is amplified
with feedforward control. Shinskey (1996) states that the integrated error of the
output signal can be reduced by a factor of 10 even if the feedforward calculation
has 2 error, and that mass- and energy balance based feedforward controllers
typically has less than ; 2 error, leading to a reduction in integrated output error
with a factor of 50. Shinskey also provides an interesting figure showing nine
different responses to disturbance steps for a process with a pure gain (static)
feedforward controller. The nine cases are the combinations of neglected time
constants and delays in the transfer functions from the disturbance and the manipulated variable to the output (Shinskey, 1996, Figure 7.12). The figure may also
be used for dynamic feedforward controllers as a qualitative illustration of the effect of errors in delays or time constants on disturbance step responses. (Note
that Shinskey assumes that the disturbance has a negative effect on the output, in
contrast to what we assume in the present paper.)
In the context of IMC (Internal Model Control), Morari and Zafiriou (1989)
recommend a structure for the combined feedback-feedforward scheme that decouples the two functions such that the feedforward controller handles disturbance
6.1 Introduction
125
dampening and the feedback controller handles reference tracking. This is exploited in the controller tuning (assuming perfect models) since the two controllers
can be tuned independently. The traditional controllers can then be derived from
these controllers and the process models. It is shown that assuming perfect models optimal feedforward can only be better than optimal feedback if there are non
minimum phase components (such as delays and inverse responses) in the process.
Scali and co-workers (Lewin and Scali, 1988; Scali et al., 1989), also work in
the IMC context and compare the control error of optimal feedback controllers
with an optimal combination of feedback and feedforward controllers under
the presence of uncertainty. The motive is to make a fair comparison, and to give
methods for identifying when feedforward is worth the effort, and to quantify the
benefits from accurate models. Uncertainty representations, similar to the ones we
will discuss, are used. Numerical results for parametric uncertainty in first-order
processes with delay are presented for different nominal values and uncertainties.
Even for this simple case the picture gets rather complicated, as there are many
parameters that must be varied to cover all cases, both nominal parameters as
well as the parameters representing the uncertainty, so it is difficult to present
the results and give general quantitative answers. The overall conclusion is that
feedforward may make the performance poorer if the response to the manipulated
input is considerably faster than the disturbance response and the uncertainty is
large for the model of the disturbance effect.
Marlin (1995) studies the effect of model errors (one at a time) by comparing
combined feedforward and feedback control with the response when pure feedback is applied. The response to a disturbance step for a first order process with
delay is the criterion for the comparison. From his example the feedforward reduces the control error with more than 2 for parametric errors up to 2 .
A general quantitative frequency domain analysis of feedforward control under model uncertainty is proposed by Balchen (1968) (and referred in (Balchen
and Mummé, 1988)).
The aim of this article is to study feedforward control under the presence of
uncertainty and answer the following basic questions:
(1) How much does the feedforward controller reduce the control error?
(2) When is the feedforward controller amplifying the effect of disturbances on
the outputs?
(3) If combined with feedback control, when is feedforward control necessary
(and useful)?
(4) How can uncertainty be taken into account when the feedforward controller
is designed?
126
Chapter 6. Feedforward Control under the Presence of Uncertainty
The outline of the paper is as follows. We first recapitulate the characteristics of feedforward control (Section 6.2), and then define feedforward sensitivities
(Section 6.3). We then discuss the effect of model errors under feedback and
feedforward control, i.e., answer questions 1 and 2 (Section 6.4) and study some
classes of model uncertainty in Section 6.5. We illustrate some of the ideas with
an example (Section 6.6). Question 3 is discussed in Section 6.7. Proposals to
answers of Question 4 are given in Section 6.8. The article is concluded by Section 6.9.
6.2 The characteristics of feedforward control
A block diagram where feedforward from a disturbance and the reference is combined with feedback, is shown in Figure 6.1. To analyze the effect of a given
feedforward controller, we denote the feedback controller and the feedforward
action from the disturbance and the reference . With perfect measurements we then have (see Figure 6.1)
;
<> = ? Feedback
;
<>= ?
(6.1)
Feedforward
Some important characteristics of the “traditional” feedforward controller are:
d
Gdm
dm
Kff
yr
+
Gd
r
Kff,r
ff
-e
-
K
+ + + u
ym
+
G
y
+
Gm
Figure 6.1: Block scheme for feedforward control combined with a feedback controller.
We assume ideal measurements:
and .
(1) The basic task of a feedforward controller ( and ) is to use the process input, , to reduce the effect of measured disturbances and improve
set-point tracking.
6.2 The characteristics of feedforward control
127
(2) Feedforward control is “open loop” since the disturbance measurement, ,
and the reference (which are used by the feedforward controller) are independent of .
(3) For linear systems, the feedforward controller does not influence the stability of the system.
(4) The feedforward controller uses a model of the process ( and ). If the
model is faulty, then the controller based on this faulty model will not yield
the desired performance, and the controller may even amplify the effect of
the disturbance.
(5) Normally the effect of the disturbance is observed earlier in the disturbance
measurement than in the other process measurements.
(6) Referring to Figure 6.1, the closed loop response for the combination of
feedforward and feedback control is
(6.2)
where is the feedback sensitivity function.
E
Ideal feedforward control
An “ideal” feedforward controller, which is based on inverting the nominal model
(e.g., (Balchen, 1968; Balchen and Mummé, 1988) and (Morari and Zafiriou,
1989)), removes completely the effect of the disturbance and reference changes
such that E . We denote the “ideal” controller with an asterisk, and get from
(6.2)
(6.3)
Designs of robustly optimized ( -optimal) feedforward controllers presented later
in this paper, confirm that this is a good controller as to use in some practical
cases. However, there are three reasons why ideal feedforward control (E )
may not be achieved in many cases:
(a) The ideal feedforward controller in (6.3) may not be realizable. First, if
is non-minimum phase, it cannot be inverted. Second, if
has more
? , the inverse is improper and repoles than zeros, e.g.,
quires differentiation. Because of measurement noise higher-order derivatives are normally avoided (Harriott, 1964). Thus we divide into a (prac, such that
tically) invertible part, , and a not invertible allpass part,
Chapter 6. Feedforward Control under the Presence of Uncertainty
128
(Holt and Morari, 1985a; Holt and Morari, 1985b). Morari and
Zafiriou (1989) derive the -optimal feedforward controller (in the context
of IMC). A simpler alternative that we will use here is
(6.4)
(6.4) has a optimal -norm ( -optimal for impulse disturbances on the
output, , and impulses in the reference).
(b) The ideal feedforward controller in (6.3) is also not realizable if the number of outputs exceeds the number of manipulated inputs (the length of
exceeds the length of ). One must then control the (most) important outputs (reducing the length of till it equals the length of ), or find some
compromise between the outputs, for example use the pseudo-inverse of .
(c) The model used in the design of the feedforward controller differs from the
actual plant. This is the main topic of this paper.
6.3 Feedforward sensitivity functions
The closed loop response for combined feedforward and feedback control in (6.2)
may be rewritten as follows
E
(6.5)
where we define the feedforward sensitivities as
(6.6)
(6.7)
These express the effect of feedforward action on the control error. denotes
the generalized inverse of (Zhou et al., 1996, page 67). Feedback control is
effective and improves performance as long as the gain of the sensitivity function
$ . Similarly feedforward control improves the performance if
$ and
$ (6.8)
Here, an appropriate norm dependent on the definition of performance is used.
With no feedforward control , and with “ideal” feedforward control .
In the literature and are also denoted control ratio and feedforward control ratio, respectively (Balchen and Mummé, 1988). More precisely, in (Balchen
6.4 The effect of model error with feedforward control
129
and Mummé, 1988), the feedforward control ratio is defined for single-input/singleoutput (SISO) controllers as
(6.9)
where is the actual feedforward controller and is the “ideal” controller
for the actual process. For SISO controllers this is identical to the definition in
Equation (6.6).
Balchen uses a Nichols chart to determine requirements on the gain and phase
error in relative to for a given disturbance dampening (e.g. 0.1) in . The
Nichols chart used to be convenient for the study of given a transfer
function . With tools like Matlab, it is now easy to study any transfer function by defining a finite number of frequencies and calculate the gain or phase
shift over this set of frequencies. We follow this direct approach.
6.4 The effect of model error with feedforward control
In this section we restrict ourselves to single-input/single-output (SISO) processes,
i.e., with one control input, , one disturbance, , and one output . With
a nom , and an actual plant model ,
inal process model, the actual control error is:
E
C
where
L
(6.10)
(6.11)
(6.12)
(6.13)
expresses the ratio between the output when
controller is applied
a feedback
and when it is not (open loop). Similarly, and express the ratio of the
output when feedforward is applied and the output when it is not. This follows by
comparing the output error using control in (6.10) with the output error when no
control is applied ( ):
E
Note
case
with no control (
that
for the
, , .
,
(6.14)
, ), we have
Chapter 6. Feedforward Control under the Presence of Uncertainty
130
The actual sensitivity can be expressed in terms of the nominal sensitivity and
the relative error as following
(6.15)
and Here, are the nominal sensitivity and complementary
sensitivity functions, respectively, and the relative error in , i.e.,
(see also (Skogestad and Postlethwaite, 1996, Section 5.13)).
The “ideal” feedforward controller (6.3) gives with no model error
With model error we get the result
(6.16)
(6.17)
(6.18)
Here, is the relative
and the relative error in . Thus for
error in
“ideal” controllers, and are equal to (except for the sign) the relative
model errors in and , respectively, and we have that the “ideal” feedforward action reduces the control error for a frequency , as long as the relative
modelling errors are less than one, i.e.,
$ $ (6.19)
(6.20)
In Section
6.8 we discuss how to modify the ideal feedforward controller such
that '$ . However, the nominal performance becomes worse. If
is not invertible, we obtain for the feedforward controller, in (6.4)
(6.21)
(6.22)
For a given process and the knowledge of its uncertainty we can use (6.19)
and (6.20) to see whether an “ideal” feedforward controller will be effective. This
can be used to consider whether the extra controller shall be implemented, and if
6.4 The effect of model error with feedforward control
131
other control configurations or even process modifications are necessary to obtain
the desired response (e.g., introduction of buffer tanks, see Chapter 3).
If the model error (uncertainty) is sufficiently large, such that
the relative error in is larger than , then we see from (6.17) that is larger than and feedforward control makes control worse. This may quite easily happen in
practice. For example, if the gain in
by 33% and the gain in is
@ @ is increased
[ [
I
@ , @ , reduced by 33%, then & S ; R . In words, the
I
feedforward controller overcompensates for the disturbances, such that its negative counteractive effect is twice that of the original effect.
Another important insight from (6.10) and (6.17) is the following: To achieve
E #$ for with feedforward control only ( ) we must require that
the relative model error in is less than . This requirement is unlikely
to be satisfied at frequencies where is much larger than (see the following
example) and motivates the need for feedback control in such cases.
Example 6.1 Consider a plant with
>
>
(6.23)
The objective is to keep '$ for , but note that the disturbance gain
gives
at steady state is 8 . Nominally, the feedforward controller perfect control, . Now we apply this controller to the actual process where
the gains have changed by 2
(6.24)
From (6.10) the disturbance response in this case is
6 7 >; ; ; (6.25)
Thus, for a step disturbance of magnitude , the output will approach ; (much
larger than ). This means that we need to use feedback control, which is hardly
affected by the above model error. There is some benefit in using feedforward
control, though. The feedback control is required to be effective at all frequencies
where the gain from the disturbance to the output is larger than 1. Without feedforward control the feedback loop must be effective up to P Y? > . The feedforward controller brings this limit down to about ; ; . In
other words, the feedforward controller reduces the bandwidth requirement for
the feedback controller from to ; .
132
Chapter 6. Feedforward Control under the Presence of Uncertainty
6.5 Some classes of model uncertainty
In the following we will consider some examples of model uncertainties for ideal
feedforward controllers, and use (6.19) and (6.20) to analyse when
feedforward
control should be used. To simplify notation we write: and .
where and are conand Static gain uncertainty. Let
stants. (Nominally, and and a 82 gain error corresponds
to ; and ; .) In this case we have from (6.19) that ideal feedforward control reduces the error from the disturbance, , as long as
$ $ ;
$
(6.26)
and from (6.20) for the reference as long as
H$ $
$ ;
(6.27)
See Figure 6.2(a). In other words, if the effect of the input changes sign
(which is not very common), or is increased by more than 100% (which
may easily happen), feedforward actually makes the response worse. This
will also happen, as we saw above, if the gain in is increased by more
than 2 and the gain in at the same time is reduced with more than
33%, since 87 > 7 N B; 7 .
In the following we will only consider feedforward from the disturbance, .
Delay uncertainty.
We let , , and denote the delays for , , , and
, respectively. We assume so that ideal feedforward control is
feasible, and perfect models except for the delay. Now the feedforward
sensitivity becomes
E E , E E , JFE (6.28)
where 0
is the error in the difference between the
delays in and . The ideal feedforward control reduces the error at a
frequency as long as
FE ; ; 0 $ (6.29)
We note that since 0 0 , the relative delay uncertainty
is independent of the sign of 0
.
6.5 Some classes of model uncertainty
133
In Figure 6.2(b) we plot in (6.29) as a function of normalized frequency.
At low frequencies feedforward control is perfect, but for frequencies above
, feedforward has a negative effect, and in the worst 87 8 0
case (at frequency )1.354 0 ) the feedforward effect doubles the error. To avoid that the feedforward controller amplifies the control error, the
feedforward control signal may be low-pass filtered with a break frequency
or less.
at about 0
We may find the frequency, , where
analytically:
>7 0 > ; 0 P 0
(6.30)
To find the frequency 1.354 for the first maximum value of 2 we differentiate
the expression for with respect to the frequency
; ; 0
to obtain
1.354
0 O0 ; ; 0
0
(6.31)
(6.32)
Uncertainty in time constants. In the general case this is more complicated to
analyze than the gain and delay errors. We consider the situation where the
:9 A? error is in only and is restricted to one time constant: and 9 ? where is the relative error in the time constant. We then obtain the following limit for effective feedforward
? $ (6.33)
? If ; , then ? A? is always less than or
equal to one. For ; the feedforward is effective as long as
,? 6
$
; (6.34)
The maximum value of is , see Figure 6.2(c). Again this can
be used to find the frequency for which the feedforward controller shall be
active.
The situation if there is an error in only
in only .
is similar to the case with error
134
Chapter 6. Feedforward Control under the Presence of Uncertainty
Combined uncertainty in both gain and time constant (“pole uncertainty”).
Some physical parameter changes affect both the gain and the time constant ? , such that their ratio Y? remains constant. As an example, consider the following physical state-space model with a single state
@
(6.35)
where is the state, is the control input (manipulated variable),
output, and and are constants. Laplace transform yields
C L An error in
constant (? (6.36)
is the
(6.37)
will then influence both the gain ( ) and the time
R ), whereas M? remains unchanged.
:9 A? and
The
model in (6.37) can be written on the form :9 ? , where is the relative error in the gain and the
time constant
(which is equal to the relative error in ).
contains no
). We then obtain the following requirement for effective
errors (
feedforward
? $
(6.38)
The effect of model error is largest at low frequencies (below Y? )
. Feedforward has a positive effect at all
where P
frequencies when 8; . For $ >; , feedforward is effective at high
frequencies
,?
as shown in Figure 6.2(d).
In other cases and consider the physical model
@
and we get
In this case control and gives ,?
.
;
(6.39)
share the same dynamics. For example,
and an error in
(6.40)
(6.41)
does not affect feedforward
6.5 Some classes of model uncertainty
135
2
1.8
1.6
|Sff|
α
|1−___|
αd
1.4
1.2
1
π
0.8
|Sff|
0.6
0.4
0.2
0 −2
10
ω
'
corresponding to
and
.
(a) Effect of gain uncertainty
−1
10
0
1
10
2
10
ω|∆θ| [rad/s]
10
(b) Effect of time delay uncertainty
' where +
' . ' + ' !. , and , , and
are the delays in , , and ,
respectively. At low frequencies the effect is zero, but for high frequencies, it
doubles the worst-case error.
1
|1−___|
αd
αd−1
|Sff|
|Sff|
1
1
1
__________
αd(αd−2)
1_
__
αd

√
0
0
ωτd (logarithmic scale)
ωτd (logarithmic scale)
+! " 3, 1 3 . + 3, 1 3. (c) Effect of time constant uncertainty
'
,
1− 2α
√
d
corresponding to
,
.
Figure 6.2: Effect of uncertainty on #
(d) Effect of combined uncertainty in gain and time constant
correspond
'
%$ & & +' 3,21.
( +) * 3,71.
ing to
,
.
for SISO feedforward control
,
136
Chapter 6. Feedforward Control under the Presence of Uncertainty
Frequency domain representation of uncertainties. In (Lewin and Scali, 1988;
Scali et al., 1989) combinations of the above uncertainties were examined.
The analytical method we have used above is not suitable for this case,
and another approach is proposed. We want to find 1.354 , i.e., the
worst-case feedforward sensitivity for each frequency given the parametric uncertainty. Since it is impractical to find an analytical expression for
1.354 , we calculate its value for some where is a set of
frequencies in the relevant range:
'
1.354 ,
(6.42)
where and are vectors of the parameters in and , respectively. For
each parameter we have . The optimization is in general
non-convex, so that precautions must be taken to find the global optimum at
each frequency.
Example 6.2 We consider the following process (Skogestad and Postlethwaite, 1996, Example 7.3):
?
E ;
?
E , ;
? ?
(6.43)
(6.44)
i.e., nominally and are equal, but their parameters may vary independently between ; and . Nominally
7
; 7 ; (6.45)
We find that the ideal feedforward controller from the disturbance measure
ment is . Solving the optimization problem (6.42)1 gives 1.354
as shown in Figure 6.3. We can see that the ideal feedforward controller
dampens the disturbance for frequencies below 7 for all combinations of the parameters.
1
The optimization problem is non-convex so we first
make a uniform
grid in the space spanned
by the parameters and take the maximum value of
+ 3. for all points. The result of
this is used as initial
value for the routine fmincon in Matlab. AMonte-Carlo-simulation
results in
lower values of
.
+ . up to a frequency higher than 6.6 Example: Two tank process
137
2
1.8
1.6
|Sff|max
Magnitude
1.4
1.2
1
0.8
0.6
0.4
0.2
0
−2
10
−1
10
0
10
Frequency [rad/s]
Figure 6.3: 1.354 when frequency domain uncertainty is used to represent the
gain, delay and time constant uncertainties, see (6.43) and (6.44).
6.6 Example: Two tank process
Example 6.3 In this example we consider feedforward control of the process illustrated in Figure 6.4(a). A hot flow with flow rate and temperature passes
through tank 1 and into tank 2 where it is cooled by mixing with a cold flow with
flow rate and temperature . is measured before the first tank. The outlet
temperature, , shall be kept constant despite temperature variations in the hot
flow. To obtain this the measurement of is used by a feedforward controller to
adjust to compensate for the variations.
In Appendix A we derive the model on transfer function form
?
O ? A? (6.46)
E where is the disturbance, is the control input and
output that shall be kept constant. The parameters are defined by
? , ' and .
(6.47)
?
is the
,
Feedforward controller design
The ”ideal” feedforward controller is given by (6.3):
? 2 E (6.48)
Chapter 6. Feedforward Control under the Presence of Uncertainty
138
TT
d=Tin
FF
qin
Tc
d
u=qc
Tank T1
1
q1
V1
V2
Tank y=T2
2
Kff =
k'd
e-θ's
(τ'1s+1)(τ'2s+1)
kd/k
e-θs u G' = k'
(τ1s+1)
(τ'2s+1)
+y
+
(b) Block diagram. Parameters derived from the
nominal
' , , ,
data:
. In addition there is a delay, . (a) Illustration of the process.
! ,
Nominal data: , ,
, ' ' -
G'd =
Figure 6.4: The process in Example 6.3
In Figure 6.4(b) we have illustrated the process and the feedforward controller in
a block diagram. The variables of the actual plant are marked with a prime.
Sinusoidal disturbances
We will now see how a feedforward controller dampens the effect of sinusoidal
disturbances. The disturbance has amplitude and three frequencies are considered: 7J , and ; . (These three frequencies have been chosen to illustrate
$ , %P , and .) We will study six cases (the
results are summarized in Figure 6.5):
(a) No control. see Figure 6.5(a).
In the remaining cases we use the feedforward control in (6.48).
(b) Nominal case (perfect model) As seen in Figure 6.5(b), the disturbance is
perfectly cancelled by the feedforward controller.
(c) Gain error 7 , and no error in . Figure 6.5(c) illustrates that
the feedforward controller does not help, i.e., the feedforward controller
overcompensates such that the variation in has the same amplitude as
without control, as expected from (6.26). This applies to all frequencies. If
Actually in Example 6.3 we consider errors in the nominal model (
), and thereby in the
controller
, while the actual plant (
) is kept constant. This has the advantage that the
response without control remains constant, so that it is easier to identify the effect on performance
of using an incorrect model in the controller.
2
6.6 Example: Two tank process
139
the gain error is reduced, the feedforward controller has a positive effect on
the dampening compared to no control, whereas if the gain error increases
further above ; , the feedforward controller has a negative effect.
0 , which is 2 of the delay (see
(d) Delay error Figure 6.5(d)). From (6.30) the feedforward controller has a dampening ef , as confirmed by the simulation
fect up to the frequency 0
results. Even this relative small error gives a low frequency limit for where
the feedforward controller is effective.
(e) Error in time constant ? ? . This may be the result of operating tank
1 with a higher level than expected in the model. In Section 6.4 we found
that for all frequencies, the feedforward controller has a positive effect on
the dampening as long as ? $ ;Y? . When the error is larger than this than
this, as it is here, feedforward control is effective (by (6.34)) for frequencies
$ N 7 ;8> ; 7 ;8> . As illustrated in Figure 6.5(e)
at 79 , the controller has some dampening effect, while above this
frequency the controller makes the situation worse.
(f) Error in gain and time constant 7 and ? 7 Y? , see Figure 6.5(f).
At low frequencies the response is similar to a pure gain error, but this error
gives no problems for high frequency disturbances.
Step disturbances
Using the same controller, the output response ( ) to a unit step in the disturbance
( ) is shown in Figure 6.6.
(a) Gain errors give problems at low frequencies, and therefore we get an offset from set-point after a step disturbance (see Figure 6.6(a)). With pure
feedforward this is clearly the worst error for “step like” disturbances.
(b) Delay errors give problems only at high frequencies (Figure 6.6(b)), so that
the deviation from set-point has a limited duration. The performance is
improved compared to no control.
(c) Time constant errors give only transient deviations from the set-point (see
Figure 6.6(c)).
(d) Combined gain and time constant errors (Figure 6.6(d)) give the same steadystate response as the gain error. But the error is smaller in the beginning,
which makes it easier for feedback control.
Chapter 6. Feedforward Control under the Presence of Uncertainty
140
1
1
0
0
−1
−1
0.1
0.1
0
0
−0.1
−0.1
0.02
0.02
0
0
−0.02
−0.02
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
Time [s]
(a) No control
1
0
0
−1
−1
0.1
0.1
0
0
−0.1
−0.1
0.02
0.02
0
0
−0.02
−0.02
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
Time [s]
1
0
−1
−1
0.1
0.1
0
0
−0.1
−0.1
0.02
0.02
0
0
−0.02
−0.02
30
40
50
60
70
80
90
100
0
10
Time [s]
90
100
60
70
80
90
100
20
30
40
50
60
70
80
90
100
Time [s]
(e)
Error in time constant ( ): Improved performance only
for the lowest frequency.
50
0
20
80
(d) Delay error ( ):
Improved
performance
for
, no effect for ,
and larger amplitude for .
1
10
70
Time [s]
(c) Gain error ( ( ): Same
amplitude as with no control (for
all frequencies)
0
60
(b) Nominal case: Perfect control
1
0
50
Time [s]
(f) Error in gain and time constant
( and ):
Improved performance for the two
highest frequencies.
) to sinusoidal
Figure 6.5: Feedforward control
of two tank process: Response ( disturbances ( : with frequencies , and % (upper, middle and
lower plot, respectively)
6.6 Example: Two tank process
141
0.8
0.8
No control
No control
0.6
0.6
0.4
θd′=θ
θd+∆
∆θ
0.4
kd′=2kd
∆ θ = 1, 0, −1, −2 [s]
0.2
0.2
0
0
Nominal case
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
kd′=0.5kd
−0.8
−0.8
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
30
35
40
45
50
Time [s]
Time [s]
(a) Gain errors
(b) Delay errors
0.8
0.8
No control
No control
0.6
0.6
τ 1′ = α d τ 1
0.4
αd = 3, 2, 1, 1/2
0.2
kd′=2kd; τd′=2ττd
0.4
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
Nominal case
−0.8
0
kd′=0.5kd; τd′=0.5ττd
−0.8
5
10
15
20
25
30
35
40
Time [s]
(c) Errors in time constant
45
50
0
5
10
15
20
25
35
40
45
50
(d) Errors in gain and time constant
Figure 6.6: Feedforward control of two tank process: Response (
disturbances ( )
30
Time [s]
) to unit step
Chapter 6. Feedforward Control under the Presence of Uncertainty
142
6.7 When is feedforward control needed and when
is it useful?
We will now shortly discuss when a feedforward controller is needed and useful
in the combination with a feedback controller. We consider a scalar system and
assume that the variables
are scaled, so that the disturbance is within , and the
, shall stay within . We consider two cases (similar
control error, E to the buffer tank design, Chapter 3:
Given feedback controller (known ) Given the sensitivity function and
a transfer function from the disturbance
to the output of . Then
feedforward is needed (with H$ ) at all frequencies where
Unknown
(6.49)
(shortcut method) (1) Let denote the frequency up to which
, such that control is needed to achieve acceptable disturbance rejection.
(2) Let denote the frequency up to which feedback control is effective,
i.e., .$ for all $ . Approximations of the achievable
for a given process are discussed in (Skogestad and Postlethwaite,
1996, p. 173-4) and Chapter 3.
It then follows that feedforward control is needed (with H$ )
in the frequency range from to .
A similar rule is given by Middleton and Goodwin (1990), although
they denote the desired bandwidth with no reference to how to
determine this.
Feedforward control may also be needed outside the range between
. But at least we know that if
and , namely when $ , then feedforward control (or some process or instrumentation modification) is needed.
Knowing where feedforward control is needed, we may use to identify where a given feedforward controller is useful. This is illustrated in Figure 6.7.
In Figure 6.7(a), the model error is so large that feedforward control has a negative
effect on the performance for frequencies between and . In Figure 6.7(b)
feedforward control reduces the control error for some frequencies, while at others it makes the performance worse ( ). In Figure 6.7(c) feedforward
control is effective in the whole range between and .
6.7 When is feedforward control needed and when is it useful?
1
143
1
10
10
S′ff
0
10
ωd
Magnitude
Magnitude
S′ff
ωB
−1
10
ωd
FF useful
0
10
ωB
FF neg.
effect
FF not
necessary
and may
have negative
effect
−1
−4
−2
10
10
0
2
10
10
4
10
10
−4
−2
10
0
10
10
2
10
4
10
Frequency
Frequency
(a) Feedforward has a negative effect
(b) Feedforward is useful at low frequencies, but has a negative effect at
high frequencies
1
10
Magnitude
S′ff
ωd
0
10
ωB
−1
10
−4
10
−2
10
0
10
2
4
10
10
Frequency
(c) Feedforward is useful for all frequencies between and Figure
6.7: Examples
of (a) large, (b) intermediate and (c)
small relative model error,
. is the bandwidth for feedback control, and
is the required disturbance
bandwidth. More generally, feedforward control is required at frequencies where .
144
Chapter 6. Feedforward Control under the Presence of Uncertainty
Example 6.3 (continued from Section 6.6) Is the feedforward controller needed
and useful?
Figure 6.7 demonstrates that the feedforward controller must be effective for
the frequencies where the feedback loop fails to dampen disturbances. We will
here check if our feedforward controller is useful when there is a delay error in
.
the feedforward loop of 0
We apply feedback control using a measurement of . Because of the delay
and the higher-order dynamics in tank 2, the bandwidth of this control loop is
limited. We consider two different effective delays in the feedback loop: Case a)
7 N ; and Case b) .
The process model is scaled assuming that the outlet temperature is allowed to
vary 7 around the nominal value, and obtain a modified 7 N 7 . A PI controller with 7 Y? and ? O? : ? (SIMC
tuning, see (Skogestad, 2003)) is used.
Now is known, and thereby . For both cases a) and b) there is
a frequency range where (see Figure 6.8). For both cases, $
$ , so feedforward control is clearly useful.
For case a) the combination of feedforward and feedback gives acceptable
performance with "$ . However, for case b) this is
not the case, and we have an intermediate frequency range where .
We note from Figure 6.8(a) that feedforward control is needed even though
. The reason is that has slope ; whereas has slope in the
logarithmic scale.
in the feedforward
In conclusion, we see that for a delay error of 0
loop, the addition of feedforward control is useful both with the short ( 7 N ; )
and long delay ( ) in the feedback loop. For the longest delay ( ),
additional improvements (design changes) are necessary in order to achieve the
performance requirements.
6.8 Design of feedforward controllers under uncertainty
Knowledge of the model uncertainty may be utilized in the feedforward controller
design. optimal combined feedforward/feedback control under the presence
of uncertainty is derived in (Lewin and Scali, 1988; Scali et al., 1989). Here, we
discuss two other methods:
Two step procedure: 1) Choose a nominal model and design the ideal feedforward controller. 2) Modify this by introducing a low-pass filter or by
6.8 Design of feedforward controllers under uncertainty
Gd
1
10
Gd
1
10
Magnitude
Magnitude
SGd
Sff
S
0
10
SffSGd
SGd
ω B=ω
ωd
SffSGd
Sff
0
10
ωB
ωd
S
−1
10
145
−1
−2
10
−1
0
10
1
10
10
10
−2
−1
10
(a) Delay of in tank 2:
0
10
Frequency [rad/s]
1
10
10
Frequency [rad/s]
(b) Delay of in tank 2:
Figure 6.8: Example 6.3: Combination of feedback and feedforward control illustrated in
.
the frequency domain. Delay error, reducing the gain to achieve
H$ -optimal feedforward controller
Modification of ideal feedforward controller
Errors in time constants or time delays lead to reduced performance at high frequencies, and one may attempt to avoid this by adding a low-pass filter in series
with the feedforward controller. The break frequency can be chosen as the frequency where crosses . For delay error 0 the break frequency is about
80 , and
for a relative error in the time constant in the break frequency is
about ; (see Section 6.4 for details).
Low-pass filters are also often used to remove noise from the measurement to
avoid excessive wear in the actuators (e.g., (Buckley, 1964)).
Gain errors reduce the performance at all frequencies, so a low-pass filter does
not help. The only way to avoid the feedforward controller from making the situation worse, is to reduce the gain of the feedforward controller so that "$ for the whole range of the process gains. This will, however, reduce the effect of
the feedforward controller in the nominal case. If we choose (where
is the ideal controller obtained with the nominal model), we obtain
(6.50)
where and are the gain errors in and , respectively. To assure .$
, we take the smallest possible , and the largest possible and choose the
Chapter 6. Feedforward Control under the Presence of Uncertainty
146
following reduction factor, :
;
We have here assumed make use of it as long as .
(6.51)
will always be less than 1 since we only
; .
-optimal feedforward design
Normally, -design is used for feedback controllers (Doyle, 1982; Doyle, 1983;
Skogestad and Postlethwaite, 1996), but may also be applied to feedforward controllers. In this case, the whole design is taken in one step (and not by modifications on a nominal design). Figure 6.9 illustrates how the problem may be
formulated for the feedforward case. The -design algorithm finds the controller
(between the disturbance, , and ) that minimizes the weighted output, i.e., the
output of . The uncertainty block 0 may be structured so that the uncertainty
in and may be independent.
WI
∆
+
d
u
+
+
+
Gd
G
+
+
WP
Figure 6.9: Problem formulation for the design of a -optimal feedforward controller
With the presently available software we cannot handle delays in the -design.
If one knows that nominally the feedforward controller should include a delay, this
may be included manually after the -design. The nominal delays in and are then omitted in the models used for the -design.
We will now apply the two methods to the example in Section 6.6.
Example 6.3 (continued from Section
6.6)
, and add to the ideal feedforward
Low-pass filter. We consider .
controller a first-order low-pass filter with break frequency 0
From Figure 6.10(a), we see that the filtered feedforward controller makes the
nominal performance worse, especially at high frequencies where it approaches
6.8 Design of feedforward controllers under uncertainty
147
no control (compare with Figure 6.5(b)). On the other hand, with delay error (Figure 6.10(b)) the performance is slightly improved (compare with Figure 6.5(d)) at
the highest (worst) frequency, but at lower frequencies the performance remains
poorer with the filter. These results are confirmed in Figure 6.11, which shows the
magnitude at all frequencies.
The filter introduces a phase shift, and therefore a delay error of no longer
gives the same effect as R , and in the opposite direction the effect of the filter
is better.
1
1
0
0
−1
−1
0.1
0.1
0
0
−0.1
−0.1
0.02
0.02
0
0
−0.02
−0.02
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
Time [s]
50
60
70
80
90
100
Time [s]
(b) Delay error ( ' ): With the
filter the feedforward controller do not
make the performance worse for any of
these three frequencies.
(a) Nominal case: The feedforward effect is reduced or removed by the filter.
Figure 6.10: Feedforward controller with low-pass filter (response of sinusoidal disturbances with amplitude and frequencies , and on the process of Example 6.3).
-design. We consider combined gain and delay error in , and design a optimal feedforward controller using the setup in Figure 6.9. We let the uncer
, be diagonal with elements
tainty weight,
HV
879 7 ; C [ [ L
&
Q
7 C L
[ & [
; QM7 : : Q [ & [
; QM7 N : Q [ & [
(6.52)
(6.53)
represents the uncertainty in (approximately zero) and
Here
repre
2
sents the uncertainty in corresponding to ;
gain uncertainty and delay
uncertainty (Skogestad and Postlethwaite, 1996, eq. (7.27)). The performance
Chapter 6. Feedforward Control under the Presence of Uncertainty
148
Sff for ideal FF with filter
0
0
10
10
Magnitude
Magnitude
Sff for ideal FF with filter
Sff for ideal FF
Ideal FF: Sff = 0
−1
−1
10
−1
0
10
1
10
10
Frequency [rad/s]
(a) Nominal case
Figure 6.11: 10
−1
0
10
1
10
10
Frequency [rad/s]
(b) Delay error ( ' )
with and without low-pass filter (Example 6.3)
weight, , is chosen as a constant independent of frequency, and several values
for is considered (from HV to >> ). A large value of corresponds to
requiring tight control. The -controller is designed with D-K iterations using the
-toolbox in Matlab (with scaling matrices of order ; ). The delay difference between and is removed from the models used for the design, and the nominal
delay of is included manually in the feedforward controller.
The resulting is seen in Figure 6.12. From the peak value in Figure 6.12(b)
we see that with large the -optimal feedforward control is close to the
“ideal” controller in (6.48). “Detuning” ( $ ) gives little improvement
when there is a delay error, except when a large detuning ( ) is used.
However, nominal performance is then poor. This is confirmed by Figure 6.13,
which shows the response with gain and delay errors (only errors in the direction
that gives benefit are shown).
In summary, with a low weight on performance (small ), the -optimal
feedforward controller approaches no control ( ). Interestingly, with
a large weight on performance (large ) we obtain a feedforward controller
close to the ideal.
6.9 Conclusions
In this paper we have discussed and illuminated some important characteristics of
feedforward
controllers. We have defined the feedforward “sensitivity functions”,
and for the disturbance
and the reference, respectively. For
ideal feedfor
ward controllers, we find that is equal to the
and relative error in , and is equal to the relative error in (except for the
6.10 Acknowledgements
149
Sff for µoptimal FF
Sff for µoptimal FF
WP = 100,5,1,10−4
0
WP = 1000,100,5,1,10−4
0
Magnitude
10
Magnitude
10
Sff for ideal FF
Ideal FF: Sff = 0
−1
10
−1
−1
10
0
10
1
10
10
−1
0
10
(a) Nominal case (no uncertainty)
1
10
Frequency [rad/s]
10
Frequency [rad/s]
(b) Delay error ( ' )
Figure 6.12: Effect of detuned feedforward control: for -optimal feedforward controllers with performance weight, HV . ( for the ideal controller (6.48) is dashed.)
signs). A simple frequency domain analysis of and shows for which
frequencies feedforward control has a positive (dampening) effect when certain
uncertainties are present (in gain, delay, dominant time constant and a common
combination of gain and time constant). The results are summarized in Figure 6.2.
We also discuss how to analyze the effect of more complex uncertainties.
Feedforward is needed when the bandwidth, , of the feedback controller is
below the frequency for which becomes less than one (with appropriate
scaling). We must then require W$ in the frequency region between
and , or if it is known, for all frequencies where the closed loop frequency
response, , is above . See Figure 6.7 for a summary.
The ideas are illustrated with a process example.
6.10 Acknowledgements
Financial support from the Research Council of Norway (NFR) and the first author’s previous employer Norsk Hydro ASA is gratefully acknowledged. The
authors also wish to thank Ass. Prof. Torsten Wik, Chalmers University of Techology, Sweden, for useful comments.
References
Balchen, J. G. (1968). Reguleringsteknikk Bind 1 (In Norweigan) 1. Ed.. Tapir. Trondheim,
Norway.
Chapter 6. Feedforward Control under the Presence of Uncertainty
150
0.8
0.8
0.7
0.7
No control
0.6
(overlapping with WP = 10−4)
0.5
0.4
0.4
0.3
0.3
WP = 1
0.2
0.2
WP = 5
0.1
0
−0.1
No control
0.6
(overlapping with WP = 10−4)
0.5
−0.1
(and WP=1000, 100)
5
10
15
20
25
30
WP = 1000,100,5,1,10−4
0
WP = 1000,100,5,1,10−4
Ideal FF
−0.2
0
WP = 1
WP = 5
0.1
35
40
45
50
Ideal FF
−0.2
0
(and WP=1000, 100)
5
10
15
20
25
30
35
40
45
50
Time [s]
Time [s]
(b) (a) No delay error
gain error
0.8
0.7
No control
0.6
(overlapping with WP = 10−4)
0.5
0.4
0.3
WP = 1
0.2
WP = 5
0.1
WP = 1000,100,5,1,10−4
0
−0.1
Ideal FF
−0.2
0
(and WP=1000, 100)
5
10
15
20
25
30
35
40
45
50
Time [s]
(c) Delay error: 1 Figure 6.13: Effect of detuned feedforward control: Step responses for -optimal feedforward controllers with performance weight, HV .
REFERENCES
151
Balchen, J. G. and K. I. Mummé (1988). Process Control. Structures and Applications.
Van Nostrand Reinhold. New York.
Buckley, P. S. (1964). Techniques of Process Control. John Wiley & Sons, Inc.. New York.
Coughanowr, D. R. (1991). Process Systems Analysis and Control. McGraw-Hill Inc..
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Doebelin, E. O. (1985). Control System Principles and Design. John Wiley & Sons. New
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Doyle, J. (1982). Analysis of feedback systems with structured uncertainties. IEEE Proceedings, Part D 129(6), 242–250.
Doyle, J.C. (1983). Synthesis of robust controllers and filters. Proc. IEEE Conf. on Decision and Control, San Antonio, Texas pp. 109–114.
Harriott, P. (1964). Process Control. McGraw-Hill. New York.
Holt, B.R and M. Morari (1985a). Design of resilient processing plants - V. The effect of
deadtime on dynamic resilience. Chem. Eng. Sci. 40(1), 1229–1237.
Holt, B.R and M. Morari (1985b). Design of resilient processing plants - VI. The effect
of right-half-plane zeroes on dynamic resilience. Chem. Eng. Sci. 40(1), 59–74.
Lewin, D. R. and C. Scali (1988). Feedforward control in presence of uncertainty. Ind.
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Marlin, T. E. (1995). Process Control. Designing Processes and Control Systems for Dynamic Performance. McGraw-Hill, Inc.. New York.
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Scali, C., M. Hvala and D. R. Lewin (1989). Robustness issues in feedforward control..
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Chapter 6. Feedforward Control under the Presence of Uncertainty
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Skogestad, S. (2003). Simple analytic rules for model reduction and PID controller tuning.
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Skogestad, S. and I. Postlethwaite (1996). Multivariable Feedback Control. John Wiley &
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Zhou, K., J. C. Doyle and K. Glover (1996). Robust and Optimal Control. Prentice Hall.
New Jersey, US.
Appendix A Modelling of the two tank process
We here develop a model of the two tank process of Example 6.3. Energy balances
for tanks 1 and 2 can be expressed by
>@ >@
(6.54)
(6.55)
where and are the temperatures in the two tanks, is the heat capacity, the
density,( and are both assumed constant and temperature independent), and
are the volumes of tank 1 and 2, respectively, and and are the outlet flow
rates from the two tanks. By use of the mass balance for both tanks, the energy
balance simplifies to
>@
>@
(6.56)
(6.57)
Linearization around a steady-state nominal point (marked with ) under the assumption that , and are constant, yields
H0
>@
H0
>@
0 0
0 0
(6.58)
0
where . The terms with 0 and 0 are cancelled since and in tank 2 steady-state energy balance yields (6.59)
.
Appendix A Modelling of the two tank process
153
Laplace transform yields for the outlet temperature
' (6.60)
In (6.60), some delay and higher order dynamics in tank 1, i.e., between the
measurement of and the inlet of tank 2, is ignored. This is represented by a
delay, . Delay and higher order dynamics in tank 2 can be ignored since they can
be assumed equal for the disturbance and the control input. We obtain the model
(6.46) and (6.47).
Chapter 7
Offset free tracking with MPC
under uncertainty: Experimental
verification
Audun Faanes and Sigurd Skogestad
Department of Chemical Engineering
Norwegian University of Science and Technology
N–7491 Trondheim, Norway
Accepted for presentation (poster session) at AIChE Annual Meeting, November
16-21, 2003, San Francisco.
also affiliated with Statoil ASA, TEK, Process Control, N-7005 Trondheim, Norway
Author to whom all correspondence should be addressed. E-mail: [email protected],
Tel.: +47 73 59 41 54, Fax.: +47 73 59 40 80
156
Chapter 7. Offset free tracking with MPC under uncertainty
777
Abstract
In this paper, a laboratorial experiment has been used to investigate some aspects related to
integral action in MPC. We have used MPC for temperature control of a process with two tanks
in series. Since this often improves performance, we used the temperature measurements of both
tanks in the controller, even if we only are interested in the outlet temperature, and we have only
one control input. To avoid outlet temperature steady-state offset, estimates of input disturbances
have been used in the calculation of the steady-state control input. This method has been reported
in the literature as the generally most efficient.
Simulations may indicate that integral action is present and that disturbances are handled well,
but in practice unmodelled phenomena may give a poor result in the actual plant, also at steadystate. If should be verified that integral action (feedback) is actually present and not an apparent
effect of perfect feedforward control.
The experiments verify that output feedback through input disturbance estimation is efficient,
provided that it is correctly done. To obtain integral action, care must be taken when choosing
which input disturbance estimates to include. It is not sufficient to estimate a disturbance or
bias in the control input(s), even if the control input(s) are sufficient to control the process. The
present work verifies the result that the number of independent disturbance estimates must equal
the number of measurements. In our experiment the use of estimates of input disturbances to both
tanks gave satisfactory performance with no steady-state error.
Keywords: MPC, Uncertainty, Integral action, Feedforward control, Experiment
7.1 Experimental set-up
157
7.1 Experimental set-up
7.1.1 Equipment
The experimental set-up is illustrated in Figure 7.1. The aim of the process is to
keep the temperature in the circulation loop (as measured by TI2) constant by adjusting the cold-water flow-rate (marked with in the Figure) despite disturbances
(marked and ). A more detailed description of the equipment follows.
TI
H
Hot water
reservoir
Cold water
reservoir
d1
TI
1
u
Heat loss
d2
Mixing
tank
Cold water
TI
2
Main tank
y
FI
1
Circulation
Figure 7.1: The experimental set-up
Hot and cold water from two reservoirs are mixed together into a mixing
tank. The water flow rates are controlled with peristaltic pumps (Watson Marlow 505Du/RL). At a certain level in the mixing tank there is a spout acting as an
overflow drain, and the mixed water flows through this spout and through a flexible tube to the main tank, which is situated at a lower altitude. The outlet provides
a constant level in the mixing tank.
The main tank has a circulation loop with a pump (Johnson Pump F4B-8)
and a flow-rate measurement (tecfluid SC-250). The main tank temperature measurement is placed in the circulation loop, which gives an adjustable delay in the
measurement. In addition, the circulation serves for mixing.
In the circulation loop, below the main tank, there is a drainage. The drainage
flow is controlled with an on-off valve (Asco SCE030A017). The drainage keeps
158
Chapter 7. Offset free tracking with MPC under uncertainty
777
the level in the main tank approximately constant despite the inflow from the
mixing tank.
The reservoirs and the tanks are all modified beakers. The pipes of the circulation loop are made of glass.
The experiments are taking place in room temperature (about ; ). Since
), the hotthe hot-water temperature deviates considerable from this (< : water reservoir is placed on a hot-plate with thermostat to keep the hot-water
temperature approximately constant. Since the two reservoirs do not contain a
sufficient amount for the whole experiment, refill is necessary. The cold-water is
, which is considered fairly close to room temperature.
about Magnetic stirrers are placed in the hot-water reservoir and in the mixing tank.
7.1.2 Instrumentation and logging
Pt-100 elements (class B, 3 wire, single, diameter 3mm, length 150mm) are placed
in the hot-water reservoir, the mixing tank and in the circulation loop of the main
tank. The main tank level is measured with a capacitance probe (Endress+Hauser
Multicap DC11 TEN). The instruments are connected to National Instruments
Fieldpoint modules, which are further connected to a PC via the serial port. In
the PC, Bridgeview (National Instruments) is used for data display and basic control. Bridgeview also provides an OPC server interface, such that an OPC client
may read off measured data, and give values to the actuators. The temperature
controller is implemented in Matlab. The temperature measurements are read into
Matlab, and the flow rate for the peristaltic pumps are determined in Matlab, and
provided to Bridgeview via the OPC interface. Matlab is also used to plot the
results.
7.1.3 Basic control
The following basic control is implemented in Bridgeview on the connected PC:
(1) The level in the main tank is controlled by opening the drainage valve when
the main tank level reaches above ; 7 , and closing it when it is below >7 .
A manually adjustable valve is installed on the drainage tube to reduce the
drainage flow (otherwise the main tank empties too quickly compared to the
response time of the level control loop).
(2) The rotational speed of the circulation pump is set to a constant value, which
in this set-up gives a constant circulation flow-rate.
(3) The speed of the peristaltic pumps is determined from the desired flow rate
by a linear relation. A two-point calibration is used.
7.2 Process model
159
7.2 Process model
We assume perfect mixing in both tanks, and model the main tank with circulation
loop as one mixing tank. Combination of mass and energy balance for the mixing
tank (numbered ) and the main tank (numbered ; ) yields
>@
>@
(7.1)
(7.2)
where the variables are explained in Table 7.1. Here we have assumed that the
outlet flow from the mixing tank is identical to the inflow (i.e. constant level in
the tank). In addition there is a delay in tank 1 of and a delay in tank 2 of .
These represent transportation delays and neglected dynamics.
Table 7.1: The model variables of nonlinear model given by (7.1) and (7.2)
Name
Explanation
Temperature mixing tank
Temperature main tank
Volume mixing tank
Volume main tank
Temperature cold-water into mixing tank
Temperature hot-water into mixing tank
Temperature cold-water into main tank
Flow rate cold-water into mixing tank
Flow rate hot-water into mixing tank
Flow rate cold-water into main tank
Unit
Linearization around a nominal point, denoted with an asterisk, yields:
>@
O @ [email protected] ! "
"
! "
"
" "
"
!+"
"
!#" "
" "
A @ A @ A @ " "
"
(7.3)
(7.4)
(7.5)
Chapter 7. Offset free tracking with MPC under uncertainty
160
777
where the model variables are given in Table 7.2 and the model parameters are
given in Table 7.3. Here we have incorporated the delays. The linear model
is discretized with the Matlab Control Toolbox routine c2d. sample time is
chosen. The delays are implemented as extra poles in the origin in the model (by
delay2z in Matlab Control Toolbox). The linear discrete model has 27 states. Note
that in this way the delays are implemented exactly. The linear discrete model may
be formulated as
where the subscript (7.6)
(7.7)
7727 denotes the time step number.
Table 7.2: The model variables of the linear model given by (7.3) - (7.5)
Name
( )
Explanation
Unit
Variation in temperature mixing tank ( ) Variation in temperature main tank ( )
Measurement vector
The output that we want to control
Variation in cold-water flow rate into mixing
)
tank ( Variation in hot-water flow rate into mixing
tank ( )
Variation in cold-water flow rate into main
tank ( )
In this work we have used the linear model (7.6) and (7.7) for the controller,
whereas the nonlinear model (7.1) and (7.2) is used instead of the process in the
simulations referred in section 7.6.
7.3 Identification of process parameters
Most of the process parameters can be determined directly by inspection or measurements. The delays and and the nominal volume of the main tank, ,
are more difficult to quantify in this way, since they represent more than one phenomena. The main tank volume includes the recirculation loop, and the delays
represent both transportation of water and other neglected dynamics.
Therefore, three open loop experiments have been performed to determine
these three parameters. The MPC with a preliminary tuning was used to drive the
process towards a steady state, after which the controller was turned off. Three
7.3 Identification of process parameters
161
Table 7.3: The model parameters
Name
Explanation
Nominal temperature mixing tank
Nominal temperature main tank (=set-point)
Nominal liquid volume of mixing tank
(tank no.1)
Nominal liquid volume of main tank,
including circulation loop (tank no. 2)
, Cold-water temperatures (assumed constant)
Hot-water temperature
Nominal total flow from mixing tank
( )
Nominal flow rate from hot reservoir
Nominal flow rate from cold reservoir
into mixing tank
Nominal flow rate from cold reservoir
into main tank
Transportation and measurement delay in Transportation and measurement delay in 1
For experiment 1 and 2, repectively.
Value
>7 > , >7 > , >>
>8
7
<: >>
>
> 8 7 >:
>7 >:
Unit
1
1
Chapter 7. Offset free tracking with MPC under uncertainty
162
777
steps tests were performed, and in each test the process was run to the new steady
state. The results are shown in Figure 7.2.
The linear model (7.3) - (7.5) was simulated with the actual and as inputs.
The nominal volumes and the delays and were determined by trial and
error. Simulation results with the final model are compared with the experiments
in Figure 7.2. The resulting parameter values are given in Table 7.3.
7.4 Controller
The MPC used for temperature control is based on the controller proposed by
Muske and Rawlings (1993). A discrete linear model, as expressed by
(7.8)
(7.9)
is used. This model is the same as (7.6) and (7.7), except that the disturbance term
is omitted. The control input, , is found from an infinite horizon criterion:
C
L
(7.10)
where is the deviation in the main tank temperature at sample number ,
is a vector of future moves of the
7277 and control input, of which only the first is actually implemented. The control input,
, is assumed zero for all . Weight may also be put on change in the
control input, but this is omitted here.
Muske and Rawlings (1993) demonstrated how to formulate (7.10) as a finite optimisation problem. By assuming that the constraints never are active, at
optimum the control law can be formulated as state feedback:
(7.11)
is assumed constant from to .
If we have a nonzero reference for or external disturbances, however, the
control law (7.11) has no integral action, and will give steady-state offset. There
are many ways to obtain integral action, and one is to modify the control law
(7.12)
) and is the
where is the state corresponding to desired value of ( control input that is necessary to obtain the state . and are both functions
7.4 Controller
163
6
6
Simulation
2
Experiment
0
−2
2
4
6
8
10
12
14
16
18
y2
Experiment
0
−2
6
8
10
12
14
16
Simulation
18
20
2
y2
Experiment
0
−2
−4
0
2
4
6
8
10
12
14
16
18
−4
0
20
100
d1
[ml/min]
[ml/min]
4
4
[gr.C]
[gr.C]
2
0
2
6
Simulation
4
−100
y1
Experiment
0
−4
0
20
6
0
2
−2
−4
0
100
Simulation
4
y1
[gr.C]
[gr.C]
4
u
2
4
6
8
2
4
6
8
10
12
14
16
18
20
10
12
Time [min]
14
16
18
20
u
0
d1
−100
2
4
6
8
10
12
Time [min]
14
16
18
20
0
(b) Step in hot flow rate: from to
, corresponding to
a change in from to ! % . Cold flow
rate , corresponding to ! % .
(a) Step in cold flow rate: from ' 3 to , corresponding
to a change in
from
%
% . Hot flow
to
rate , corresponding to .
6
[gr.C]
4
2
Simulation
−2
−4
0
y1
Experiment
0
2
4
6
8
10
12
14
16
18
20
6
[gr.C]
4
y2
2
Experiment
0
−2
−4
0
Simulation
2
4
6
8
10
12
14
16
[ml/min]
18
20
d1
100
u
0
−100
0
2
4
6
8
10
12
Time [min]
14
16
18
20
(c) Step in hot flow rate: from ' !
to , corresponding to a change in
from ! to % . Cold
, corresponding to
flow rate
! .
Figure 7.2: The resulting linear model: Open loop simulations compared with the open
loop experiments
Chapter 7. Offset free tracking with MPC under uncertainty
164
777
of the reference and disturbances. is known, and is held constant during the
experiments. Disturbances, however, are here assumed unknown, and must therefore be estimated from the temperature measurements. For processes with large
time constants better performance is obtained if we estimate input disturbances
(Lee et al., 1994; Lundström et al., 1995). The states, , must also be estimated,
so we define an extended state vector including a disturbance estimate vector, ,
of length :
(7.13)
In the experiments we will investigate the use of two different vectors . First we
let be the input disturbance to the mixing tank ( ). Second we let be
the input disturbance to both the mixing and the main tank ( ; ). We assume
that the disturbances are integrated white noise, and introduce the extended model
;
<>= ?
; <>= ?
<> =
;
?
where
(7.14)
(7.15)
,
..
.
(7.16)
and and are zero-mean, not correlated,
normally distributed white stochastic
noise with covariance matrices of
and , respectively. We design a Kalman
filter:
(7.17)
(7.18)
where and are a priori and a posteriori estimates of , respectively, and is the estimator gain matrix given by the solution of a Ricatti equation:
(7.19)
(7.20)
7.5 Experimental procedure
165
The steady-state solutions and can be expressed by the disturbance estimate
and the reference. This yields for the control law:
(7.21)
Values for the weight and covariance matrices:
8N Q 7 ,
-
>>
(7.22)
(7.23)
is number of states and is number of estimated disturbances.
The large difference in magnitude between and is a result of not having
scaled the model. For a variation in between 7 and 7 and between >
and > , the two terms are in the same order of magnitude for the limiting values:
7 >Q 7 8 Q 8N Q -
7=< ;
(7.24)
(7.25)
7.5 Experimental procedure
In each experiment, the process was run to a steady-state working point. The
following sequence of disturbances was then introduced:
To introduce disturbance :
> Increase hot flow rate back from < > to > (1) Reduce hot flow rate from
(2)
>
to <
To introduce disturbance :
3. Start addition of cold-water to main tank
4. Stop addition of cold-water to main tank
Two such sequences (1.- 4.) was performed with the MPC for the temperature
control active.
Prior to the experiments, we performed a simulation with the nonlinear model
of the process (7.1) and (7.2), which was implemented in Simulink (a Matlab
toolbox). In the simulation we only introduced disturbance (steps 1. and 2.).
In the experiments we wanted to investigate the effect of different disturbance
vectors, , to be estimated and used in the calculation of steady-state control and
state vector:
166
Chapter 7. Offset free tracking with MPC under uncertainty
777
The simulation and experiment no.1 Estimate of the disturbance into the mixing tank only.
Experiment no.2 Estimate of the disturbances into the mixing tank and the main
tank.
The change in hot flow was done by adjusting the speed of the peristaltic pump
via the Matlab user interface.
The addition of cold-water to the main tank was done by pouring from a jug.
During 7 minutes < (experiment 1) and < (experiment 2) cold-water
was added. This gives a mean flow rate of N >7=<, 2 and N <7 , respectively for the two experiments.
During the two experiments the hot-water temperature varied between <>: and
, whereas during the simulations the temperature was held constant.
7.6 Results
First the disturbance was applied (as described in section 7.5) to the nonlinear
model of the process (7.1) and (7.2), implemented in Simulink. In Figure 7.3 we
see the response when no temperature control is applied.
[gr.C]
2
−2
0
200
[ml/min]
y2
0
y1
5
10
15
20
10
15
20
Time [min]
25
30
35
25
30
35
0
d1
−200
0
5
Figure 7.3: Open loop simulation with the same disturbances as the experiment
In Figure 7.4 the closed loop simulation is shown. Note that (solid line)
is the important output (temperature) which we want to return to its set-point as
quickly as possible. Disturbance is estimated and used in the calculation of
steady state. We see that the disturbance is well handled by the controller.
In Figures 7.5 and 7.6 we see the results of the experiments. In contrast to
the simulation, the controller with estimation of only one disturbance failed to
achieve the desired steady state, both before and after the disturbances was added
(experiment 1, Figure 7.5). We also see that is above . The reason for this
7.7 Discussion
167
[gr.C]
2
0
−2
[ml/min]
0
200
y2
y1
5
10
20
25
30
35
20
25
30
35
25
30
35
d1
0
u
−200
0
15
5
0.02
10
15
Estimated d1
0
−0.02
0
5
10
15
20
Time [min]
Figure 7.4: Simulation: MPC with estimate of
is mainly heat loss, and there was also a small difference in the calibration of the
temperature elements. The model does not cover these effects.
In the experiments we also introduced disturbance . In experiment 1 where
this disturbance was not modelled, the controller failed to bring back to steady
state.
In Figure 7.6 we can see that in experiment 2 we reached the desired steady
state for the temperature in the main tank, . To compensate for the heat loss,
the controller increased the temperature in tank 1 ( ). Both disturbances were
handled well.
7.7 Discussion
In the experiments the estimator exploited two measurements: The measurement
in addition to which is the output of real interest. With estimation of two
input disturbances an offset free steady state was obtained, whereas with only one
input estimate insufficient integral action was obtained. This is in accordance with
the theoretical results in Chapter 5. We there found that the number of estimated
input disturbances must equal the number of measurements if steady-state offset
shall be avoided.
We have also simulated the case when is omitted, i.e. only is used by the
MPC. Then it is sufficient to only estimate one disturbance in the second tank ( ).
Normally this controller will give a poorer performance, since the early information of disturbances to the first tank from is not exploited, but for the controller
Chapter 7. Offset free tracking with MPC under uncertainty
[gr.C]
2
0
y1
y2
−2
[ml/min]
0
200
5
10
15
20
25
30
35
25
30
35
25
30
35
d2
d1
0
u
−200
0
5
10
0.02
15
20
Estimated d1
0
−0.02
0
5
10
15
20
Time [min]
Figure 7.5: Experiment 1: MPC with estimate of
[gr.C]
2
0
0
200
5
10
15
20
25
30
35
25
30
35
25
30
35
d2
d1
−200
u
5
0.02
10
15
20
Estimated d2
0
Estimated d1
−0.02
0
y2
0
0
y1
−2
[ml/min]
168
5
10
15
20
Time [min]
Figure 7.6: Experiment 2: MPC with estimate of
and
777
7.7 Discussion
169
tunings we have chosen, the performance was actually slightly improved for the
controller without .
We will compare our MPC controllers (with estimation of and with estimation of and ) in the frequency domain. This is possible since the constraints
in the control input, , is never active. In Chapter 5 we derive a state-space formulation for the combination of the controller and the estimator for this case. The
controller may further be expressed by an approximated continuous state-space
formulation (by d2c in Control Toolbox in Matlab), which is easily converted to a
transfer function formulation:
(7.26)
We will study the magnitude of , but first it is convenient to introduce scaled
variables. The maximum possible variation in in each direction is 1.354 > 2 , and 1.354 7 is the maximum desired variation in . We there
fore
introduce
the
scaled
variables
and
'
.
1
5
3
4
1.354 such that both
and stay within . The corresponding controller equation for the scaled
system is
(7.27)
where
Y1.3542 '1.354 .
In Figure 7.7 we have illustrated the magnitude of
for the two types
of controllers. We see that the controller with only one disturbance estimation has
low gain at low frequencies and higher gain from than from (Figure 7.7(a)),
whereas for the controller with two disturbances the low frequency gain from is high (Figure 7.7(b)). (Figure 7.7(b) also reveals that the gain from is low for
all frequencies, which explains why the use of in the control did not improve
performance as expected.)
3
3
u′
10
u′
10
−3
10
−3
−6
0
10
10
−6
0
10
y′1
10
10
−6
0
10
10
y′2
−6
(a) Estimates 0
10
y′1
10
y′2
(b) Estimates and Figure 7.7: In this work, we have assumed that the constraints never are active in the
design and analysis of the controller. In this set-up, this will at least be the case
170
Chapter 7. Offset free tracking with MPC under uncertainty
777
close to steady state. This means that the result will be the same if we use an
ordinary MPC for the same example.
7.8 Conclusions
In a laboratorial experiment, we have used MPC combined with an estimator for
temperature control of a process with two tanks in series. Since this often improves performance, we used the temperature measurements of both tanks in the
controller, even if we only are interested in the last temperature, and we have
only one control input. To avoid steady-state offset, we have estimated input disturbances, and used these estimates in the calculation of the steady-state control
input.
Simulations may indicate that integral action is present and that disturbances
are handled well, but in practice unmodelled phenomena may give a poor result
in the actual plant, also at steady state. It should be verified that integral action
(feedback) is actually present and not an apparent effect of perfect “feedforward
control”.
Estimates of input disturbances have been described in the literature as efficient for a quick response back to the desired steady state. The present work
confirms this provided that it is correctly done.
To obtain integral action, care must be taken when choosing which input disturbance estimates to include. It is not enough to estimate a disturbance or bias
in the control input(s), even if the control input(s) are sufficient to control the
process. The number of disturbance estimates must equal the number of measurements. In our experiment, the use of estimates of input disturbances to both tanks
gave satisfactory performance with no steady-state error.
7.9 Acknowledgements
The experimental equipment has been set up at Norsk Hydro Research Centre,
and was originally designed by Jostein Toft, Arne Henriksen and Terje Karstang.
Norsk Hydro ASA has financed the experiments.
References
Lee, J. H., M. Morari and C. E. Garcia (1994). State-space interpretation of model predictive control. Automatica 30(4), 707–717.
REFERENCES
171
Lundström, P., J. H. Lee, M. Morari and S. Skogestad (1995). Limitations of dynamic
matrix control. Comp. Chem. Engng. 19(4), 409–421.
Muske, K. R. and J. B. Rawlings (1993). Model predictive control with linear models.
AIChE Journal 39(2), 262–287.
Chapter 8
Conclusions and directions for
future work
8.1 Conclusions
8.1.1 Buffer tank design
The first part of this thesis treats the design of buffer tanks for control purposes.
The basic idea is that the buffer tank shall handle disturbances in the frequency
range where neither the (original) process nor the basic feedback control system
dampens them sufficiently. Chapter 2 addresses control-related design for neutralization plants. One or several mixing tanks are usually installed to smoothen
disturbances that cannot be handled by the control system. Control theory has
been applied to determine the required number of mixing tanks and their volumes,
assuming strong acids and
bases. Skogestad (1996) derived a minimum required
total volume, I , where is the flow rate, is the number of
tanks, is the delay in each tank and is the scaled disturbance gain. With PI
or PID control in each tank, we compute numerically the required volume for different tunings, and based on this we recommend a total volume of G G ; .
We recommend identical tank sizes (in contrast to Shinskey (1973) and McMillan
(1984)).
Chapter 3 extends the ideas from Chapter 2 to buffer tanks for all kind of
processes. We first find the required buffer tank transfer function such that (with
scaled variables) the gain from the disturbance to the output (including the process, the feedback loop, and the buffer tank) is less than . We realize this transfer
function with either one or several mixing tanks (for quality disturbances) or a
surge tank with “slow” level control (for flow-rate disturbances).
The work is based on (Skogestad, 1994). In the present work more “accurate”
numerical and graphical methods have been included, and we have distinguished
174
Chapter 8. Conclusions and directions for future work
between the case when the feedback loop of the original plant is given (such that
the sensitivity function, , is known), and the case when it is not. Aspects regarding the buffer tank placement (before or after the process) are discussed. A
literature survey and several process examples are included.
8.1.2 Feedforward control under the presence of uncertainty
In Chapter 6 feedforward control under the presence of model uncertainty is discussed, and we define the feedforward sensitivity functions, and for the
disturbance and the reference, respectively. For “ideal” feedforward controllers,
we find that is equal to the relative error in , and is equal to the
relative error in (except for the signs). A simple frequency domain analysis of
and shows for which frequencies feedforward control has a dampening
effect when some common model errors are present ( in gain, delay, dominant
time constant, or a common combination of gain and time constant). The effect of
more complex uncertainties is also discussed.
Feedforward is needed when the bandwidth, , of the feedback controller is
below the frequency for which becomes less than one (with appropriate
scaling). We must require $ in the frequency region between and
, or if it is known, for all frequencies where the magnitude of the closed loop
disturbance response, , is above .
To make the feedforward controller more robust, two methods have been proposed: 1) Adding a low-pass filter to the nominal design and 2) -optimal feedforward controller design.
8.1.3 Multivariable control under the presence of uncertainty
Serial processes are very common in the process industry, and in Chapter 4 we
use this class of processes to illustrate that a multivariable controller may actually
use the two basic principles of “feedforward” action (based mainly on the model),
and feedback correction (based mainly on measurements) simultaneously. The
feedforward action may improve the performance significantly, but is sensitive
to uncertainty, in particular at low frequencies. Therefore it is important to include efficient feedback control by using measurements late in the process, and to
include integral action if offset-free steady-state is important.
In Chapter 4 we see that testing the process on a too idealistic process model
may give the impression that the control is better than it actually is. This is confirmed by the experiments reported in Chapter 7 (Model predictive control, MPC,
is used for temperature control of a process with two tanks in series). Simulations may indicate that integral action is present and that disturbances are handled
well, but unmodelled phenomena may give a poor result in the actual plant, also at
8.1 Conclusions
175
steady state. It should be verified that integral action (feedback) is actually present
and not an apparent effect of “ideal feedforward control”.
Estimates of input disturbances have been described in the literature as efficient for a quick response back to the desired steady state. The experiments
confirm this provided that it is correctly done. Care must be taken when choosing
which input disturbance estimates to include. It is not enough to estimate a disturbance or bias in the control input(s), even if the control input(s) are sufficient to
control the process. The number of disturbance estimates must equal the number
of measurements (as found theoretically in Chapter 5).
When designing the controller, one must also consider which of the outputs
that are really important. If the number of inputs exceed the number of (important)
outputs, one may either give set-points to other (less important) outputs, or one
may let the controller bring some of the inputs back to ideal resting positions
(Chapter 4).
As a tool to understand the model predictive controller (MPC), in Chapter 5
we derive a (linear, discrete) state-space realization of a MPC controller (Muske
and Rawlings, 1993) under the assumption that it is operated with no active constraints. A generalization to tracking of both inputs and outputs is derived. The
final controller expression also includes a state estimator that is extended with input disturbance states. We have not found such a derivation of a MPC controller
on state-space form elsewhere.
A direct result is that to obtain integral action with input bias estimation, it
is required to include the same number of input biases as measurements. Combined with the process model (also on state-space form), the closed loop model is
determined, and this can, for example, be used to check the steady-state solution.
The state-space MPC formulation has been applied (in Chapters 4 and 7) to
obtain the frequency dependent gain for each controller channel and the magnitude of each of the elements in the sensitivity function matrix. The frequency
dependent gain in each channel may give insight into how the controller utilizes
each measurement and the magnitude of the control actions for each input. The
steady-state behaviour can be seen from the low-frequency gains. But, often more
than one channel in a row have high gain at low frequencies, and then it is difficult
to interpret the result. It is then better to consider the elements of the sensitivity
function matrix. An offset-free, steady-state control for a specific output requires
that all the elements in the corresponding row have low gain at low frequencies.
Chapter 8. Conclusions and directions for future work
176
8.2 Directions for further work
8.2.1 Serial processes: Selection of manipulated inputs and measurements
A general question related to control structure design is the choice of manipulated
inputs and measurements. In Section 4.4 we study a serial process with three
units, and with one candidate measurement (pH) and one candidate manipulated
input (addition of a reactant) in each unit. To save installation and operational
costs, one may omit one or more of the instruments or actuators. From Table 8.1
we see there are 49 possible combinations. Often one would like to monitor the
final output, in which case the number of possible combinations is 28.
Table 8.1: Possible combinations of inputs and measurements for the example in Section
4.4. The last column is for the case with a measurement in the last unit.
Inputs
Measurements
No of combinations
3
3
3
2
2
2
1
1
1
Total
3
2
1
3
2
1
3
2
1
1
3
3
3
9
9
3
9
9
49
No of combinations
pH in last tank used
1
2
1
3
6
3
3
6
3
28
In general, if one may choose from to
inputs and from to
measurements, the number of combinations is given by (Nett, 1989):
possible
(8.1)
In the example
.
To illustrate the problem, we will here compare two realistic combinations
from the example:
(1) pH measurement and reactant addition in tanks and .
(2) pH measurement and reactant addition in tanks ; and .
8.2 Directions for further work
177
In both cases we keep the measurement and reactant addition in the last tank,
since normally we want to measure the product quality, and the late reactant addition minimizes the delay in the last control loop. When we omit reactant addition
to a tank, the steady-state pH will be the same as the inflow pH. From the simulations in Figure 8.1 we see that the resulting pH-response in the last tank is similar
to the full instrumentation case (compare with Figure 4.7(a)). We see that the
small deviation in the pH of last tank has a shorter duration for case 1 (with no
instrumentation in tank 2). In case 2 (Figure 8.1(b)) the control inputs have not
reached their steady state after ;> ( reaches 7 < ).
The simulations indicate that with a multivariable controller one may omit the
instrumentation in one of the three tanks.
10
5
0
4
1
0
−1
−2
−3
pH in tank 1
pH in tank 1
4
pH in tank 2
pH in tank 2
2
2
8
8
pH in tank 3
pH in tank 3
6
6
1
1
Control inputs u, scaled
Control inputs u
0
0
−1
0
50
100
150
time [s]
200
250
(a) Instrumentation is removed from
tank 2. pH set-point in tanks 1 and 2
are both set to 2.4.
−1
0
50
100
150
time [s]
200
250
(b) Instrumentation is removed from
tank 1. At steady state the pH in tank 1
is equal to the influent pH. pH set-point
is in tank 2.
Figure 8.1: pH measurement in and reactant addition to two tanks only.
as with full instrumentation)
(not
Even if the final results for the two cases are similar, one may point out some
important distinctions: In case 1, the total control loop includes all three tanks,
whereas in case 2, only the two last tanks are included. In case 1, therefore the
feedback loop from the last tank to the first is slower, but on the other hand, the
“feedforward” controller element can be made close to “ideal”, in contrast to case
2 (because of the delays).
A further analysis of the differences between different control configurations
would be useful, both as a basis for recommendations to process designers, but
also to get a deeper understanding of the process and the controller.
178
Chapter 8. Conclusions and directions for future work
8.2.2 MIMO feedforward controllers under the presence of uncertainty
MPC vendors often offer feedforward control from measured disturbances (e.g.,
Honeywell (1999) and ABB (2003)), and therefore the study of multivariable
feedforward controllers (from multiple measurements to multiple control inputs)
has become more interesting. The theory of Chapter 6 covers multiple-input,
multiple-output (MIMO) feedforward controllers, but the application of the theory
to MIMO examples is still remaining.
8.2.3 Effect of model uncertainty on the performance of multivariable controllers
In this thesis we have studied some aspects of multivariable control under the
presence of uncertainty. The basic idea is that a multivariable controller consists
of both “feedforward” and feedback control elements, and these two types of elements respond differently to model error. We believe that a closer look into some
of the following thoughts might be useful
Identify elements or blocks of a multivariable controller that may degrade
the performance, and redesign the controller to avoid this. In principle, it
should be possible to identify such elements from the process model. One
way to change (or remove) a controller element is to change the corresponding part of the model, for example, by removing the relationship in the
model between the control input and the output.
One method to investigate, is to consider feedforward
elements (either
manually or automatically detected) and compute for expected
model errors to determine the frequency range for which the controller element is effective. If there are any feedback element (e.g., ) that also
controls output , one may compute to see if this control element
remove errors introduced by the feedforward branch. If the frequency range
for which the feedforward element is effective is not overlapping with the
range where it is needed, it is better to leave this controller element out. A
simple example using this method has been presented (Faanes and Skogestad, 2003).
Automatically detect feedforward control elements. Sometimes this is not
an easy thing to do manually. One possible automatic method is (from the
process model) to determine which outputs depend on which inputs when
all the loops are closed. A control element from measurement to manipulated variable is feedforward control if 1) is (closed loop) independent
REFERENCES
179
of , 2) there is another output which depends on , and 3) there is
another input that both and depend on. An output is (closed loop) dependent of an input if a change in the input leads to a change in the output
(when all the loops are closed).
Due to other feedback loops or weak dependencies in the process, a control element may fail to fulfil the criteria for a feedforward controller, even
though it has many similarities with feedforward control. This is seen in the
case study in Chapter 4. For such cases it may be better to find an appropriate definition for the “degree of feedforward action” for a (total) controller
or its control elements. This may for example be a number between 0 and 1
where 1 corresponds to pure feedforward control and 0 corresponds to pure
feedback control.
8.2.4 MPC with integral action
There are many ways of obtaining integral action with mode predictive controllers
(MPC). Output bias estimation is the most popular. Another is input disturbance
or bias estimation (which we have used). Alternatively, integration may be introduced in the process model itself (for example by integrating the control input)
with the disadvantage that the MPC optimization problem has grown, and also
that the “new process” includes poles at the imaginary axis. For example, this
means that the state-space formulation we derived in Chapter 5 must be modified
since it only applies to stable processes.
We believe that a comparison of the different methods would be useful. The
recent paper by Muske and Badwell (2002) is a good starting point. It is also
interesting to consider the methods proposed for integral action for linear qudratic
(LQ) controllers, since a criterion for obtaining offset-free steady state is that none
of the constraints are active (Muske and Badwell, 2002).
References
ABB (2003). IndustrialIT Solutions for advanced process control and optimization.
Brochure.
Faanes, A. and S. Skogestad (2003). Feedforward control under the presence of uncertainty. Nordic Process Control Workshop 11, January 9-11, 2003, Trondheim.
Honeywell, Hi-Spec Solutions (1999). RMPCT implementation course.
McMillan, G. K. (1984). pH Control. Instrument Society of America. Research Triangle
Park, NC, USA.
180
Chapter 8. Conclusions and directions for future work
Muske, K. R. and J. B. Rawlings (1993). Model predictive control with linear models.
AIChE Journal 39(2), 262–287.
Muske, K. R. and T. A. Badwell (2002). Disturbance modeling for offset-free linear model
predictive control. J. Proc. Cont. 12, 617–632.
Nett, C. N. (1989). A quantitative approach to the selection and partitioning of measurements and manipulations for the control of complex systems. Presentation at Caltech Control Workshop, Pasadena, USA.
Shinskey, F. G. (1973). pH and pIon Control in Process and Waste Streams. John Wiley
& Sons. New York.
Skogestad, S. (1994). Design modifications for improved controllability - with application
to design of buffer tanks. AIChE Annual Meeting, San Francisco, Nov. 1994.
Skogestad, S. (1996). A procedure for SISO controllability analysis - with application to
design of pH neutralization processes. Computers Chem. Engng. 20(4), 373–386.
Appendix A
Control Structure Selection for
Serial Processes with Application to
pH-Neutralization
Audun Faanes and Sigurd Skogestad
Extract from paper presented at European Control Conferance, ECC’99,
Aug.31-Sept.3, 1999, Karlsruhe, Germany.
Abstract
In this paper we aim at obtaining insight into how a multivariable feedback controller works,
with special attention to serial processes.
Keywords: Control structure, Serial process, Multivariable control, Feedforward, Feedback
182
Appendix A. Control Structure Selection for Serial Processes
7277
A.1 Example: pH neutralization
Neutralization of strong acids or bases is often performed in several steps. The
reason for this is mainly that the pH control in one tank cannot be quick enough to
compensate for disturbances (Skogestad, 1996). In (McMillan, 1984), an analogy
from golf is used: the difficulty of controlling the pH in one tank is compared
to getting a hole in one. Using several tanks, and smaller valves for addition of
reagent for each tank, is compared to the easier task of reaching the hole with a
series of shorter and shorter strokes.
In this example, control structures for neutralization of a strong acid by use
of three tanks in series are discussed. The aim of the control is to keep the outlet
pH from last tank constant despite changes in inlet pH or flow. This is obviously
a serial process, since the flow goes from one tank to another. For each tank, the
pH can be measured, and the reagent can also be added to each tank. Referring to
Figure 4.1, the three units (i-1, i and i+1) correspond to the three tanks (1, 2 and
3).
To study this process we model each tank as described in (Skogestad, 1996).
concentrations, that is *#,+ .
In each tank we model the excess
This gives bilinear models, which are further linearized around a stationary working point so that methods from linear control theory can be used. We get two
states in each process unit (tank), namely the concentration, , and the level. The
disturbances enter tank 1 only. We here assume that there is a delay of 5 seconds
for the effect of a change in inlet acid or base flow or inlet concentration to reach
the outflow of the tank, e.g. due to incomplete mixing, and a further delay of 5
seconds until the change can be measured. In the linear state space model these
transportation delays are modelled by Padé-approximations of 4th order. There is
assumed no further delay in the pipes between the tanks. We assume that the levels are controlled by the outflows using a P controller such that the time constant
for the level is about 1/10 the time constants for the concentrations.
The volumes of the tanks were chosen to 7 N [ , the smallest possible volumes according to the discussion in (Skogestad, 1996). The acid inflow (distur R . The pH of the final product in tank 3 should be ,
bance) has and we selected the set-points in tank 1 as 1.65 and in tank 2 as 3.8. The concentrations are scaled so that a variation of around these set-points corresponds
to a scaled value of . The control inputs and the disturbances are also scaled appropriately. The linear model was used for multivariable controller design, while
the simulations are performed on the nonlinear model.
A conventional way of controlling this process is to use local control of the
pH in each tank using PID-controllers. Figure A.1 shows the response of pH in
each tank when the acid concentration in the inflow is decreased from 10mol/l to
5mol/l. As expected from (Skogestad, 1996), this control system is barely able
pH 1
3
2
1
0
pH 2
A.1 Example: pH neutralization
4
183
pH 3
2
8
6
1
u
0
−1
0
50
100
150
time [s]
200
250
Figure A.1: With only local control, PID controllers must be agressively tuned to keep
the pH in the last tank within . (Disturbance in inlet concentration occurs at .)
pH 1
3
2
1
0
pH 2
to give acceptable control. However, the nominal response can be significantly
improved with multivariable control.
4
pH 3
2
8
6
u
1
0
−1
0
50
100
150
time [s]
200
250
Figure A.2: A large improvement in nominal performance is possible with multivariable
control. (Disturbance in inlet concentration occurs at )
Figure A.2 shows the response with a 0 multivariable
controller designed with performance weights on the outputs and on the control inputs in all
tanks, and with composition into tank 1 as a disturbance. The main reason for the
large improvement is the feedforward effect discussed in section 4.3.
The gain of the elements in the multivariable controller as a function of fre-
184
Appendix A. Control Structure Selection for Serial Processes
7277
4
u1
10
−4
10
4
u2
10
−4
10
4
u3
10
−4
10
−4
4
10
10
y1
−4
4
10
10
−4
4
10
10
y2
Figure A.3: Gain of the control elements of the original
controllers are dashed.)
y3
controller. (Local PID
quency are shown in Figure A.3. The diagonal control elements are the local
controllers in each tank, whereas the elements below the diagonal represent the
”feedforward” elements. From such plots we get an idea of how the multivariable
controller works. For example, we see that the control input to tank 1 (row 1)
is primarily determined by local feedback, while in tank 2 it seems that ”feedforward” from tank 1 is most decisive for the control input. In tank 3 the control
actions are smaller. This is also seen from the simulation in Figure A.2 (the solid
line in the plot of ).
We observe that none of the control elements have any integrators, even though
the simulation in Figure A.2 show no steady-state offset. However, if some model
error is introduced ( ; 2 reduced gain in tank 2 and 3), we do get a steady-state
offset. Figure A.4 shows the start of the response, it finally ends up slightly above
: . Local PID controllers give no such steady-state offset.
We subsequently redesigned the controller to get three integrators in the control loop shape (Figure A.5). The simulation in this case gives no steady-state
offset. This illustrates one of the problems of the ”feedforward” control block,
namely the sensitivity to static uncertainty. Simulations on the perfect model may
lead the designer to believe that no integrator is necessary.
controller was designed
To study the feed forward effect separately, a
using the measurement in tank 1, and control inputs in all tanks. The result is
local control in tank 1 and feed forward from tank 1 to tanks 2 and 3. Simulation
on the linear model gives the same result as for the 0 controller (Figure A.2),
whereas nonlinear simulation gives steady-state offset due to static model error
and no feedback in tanks 2 and 3.
pH 1
3
2
1
0
pH 2
A.1 Example: pH neutralization
4
185
pH 3
2
8
6
1
u
0
−1
0
50
100
150
time [s]
200
250
Figure A.4: Model error gives steady-state offset with original
controller.
4
u1
10
−4
10
4
u2
10
−4
10
4
u3
10
−4
10
−4
4
10
10
y1
−4
4
10
10
−4
4
10
10
y2
Figure A.5: Gain of the control elements of the redesigned
PID controllers are dashed).
y3
controller. (Local
186
Appendix A. Control Structure Selection for Serial Processes
7277
pH 1
3
2
1
0
pH 2
The effect of feedback from downstream tanks, i.e. the blocks above the diagonal from the discussion in section 4.3, is illustrated through the following simulations. We introduce a static measurement noise in tank 2 of 1 unit. In
Figure A.6 we see the response for the process with local control with PID. We
can see that the pH in tank 3 relatively quickly returns to a pH of 7. The problem
is the control input in tank 3, which stabilizes at a level away from the point in the
middle of the range (0), which we consider as the ideal resting position. Since we
really are interested in the pH in only the last tank, we get two extra degrees of
freedom, which can be used for resetting the control inputs of the last two tanks.
Figure A.7 shows the simulation for the multivariable controller. Here we see that
both the pH and the control input in tank 3 go to their desired values. The actual
pH in tank 2 is increased to the correct value to obtain this. This illustrates that the
elements above the diagonal in the multivariable controller give input resetting.
4
pH 3
2
8
6
u
1
0
−1
0
50
100
150
time [s]
200
250
Figure A.6: Steady-state measurement noise in tank 2: Local control with PID do not
bring the control input for tank 3, [ , back to the ideal resting position. (u-plot: solid
line.)
To summarize the example we can say that the multivariable controller gives
significant improvements compared to local control based on PID. This is especially due to the feedforward effect, and with large model errors, the feedforward
may lead to worse performance. Integral action is important in the controllers,
even if the feedforward effect may give no stationary deviation for the nominal
case. The inputs in the last two tanks are reset to their ideal resting position with
the multivariable controller, because of the feedback from downstream tanks.
A.2 Conclusion
pH 1
3
2
1
0
pH 2
187
4
pH 3
2
8
6
u
1
0
−1
0
2000
4000
6000
time [s]
8000
10000
Figure A.7: Steady-state measurement noise in tank 2: The multivariable controller has
built in input resetting, and brings [ back to the ideal resting position (u-plot: solid).
Note that the timescale differs from the other plots.
A.2 Conclusion
An example of neutralization of a strong acid with base in a series of three tanks is
used to illustrate some of the ideas in the paper. This process is obviously serial.
The example illustrates that the multivariable controller yields significant nominal
improvements compared to local control based on PID. But this is especially due
to feedforward, and with model errors, the feedforward may in fact lead to worse
performance. Integral action or strong gain in the local controllers at low frequencies is important to obtain no steady-state offset, even if the feedforward effect
itself may nominally give no steady-state. Feedback to upstream tanks brings the
inputs to their ideal resting positions, also when a wrong pH measurement give
problems in an upstream tank. The example indicates that it is possible to get a
good performance with careful use of a multivariable controller or a combination
of local control, feed forward from tank 1 and input resetting.
-contoller, but similar results have also been found
In this study we used a
for a MPC controller.
References
McMillan, G. K. (1984). pH Control. Instrument Society of America. Research Triangle
Park, NC, USA.
Skogestad, S. (1996). A procedure for SISO controllability analysis - with application to
design of pH neutralization processes. Computers Chem. Engng. 20(4), 373–386.
Appendix B
A Systematic Approach to the
Design of Buffer Tanks
Audun Faanes and Sigurd Skogestad
Presented1 at PSE’2000, July 16-21, 2000, Keystone, Colorado, USA, Supplement to
Computers and Chemical Engineering, 24, pp. 1395-1401
Abstract
Buffer tanks are often designed and implemented for control purposes, yet control theory is rarely
used when sizing and designing buffer tanks and their control system. Instead, rules of thumb
such as “10 min residence time” are used. The objective of this paper is to provide a systematic
approach. We consider mainly the case where the objective of the buffer tank is to dampen (“average out”) the fast (i.e. high frequency) disturbances, e.g. in flow and concentration, which cannot
be handled by the feedback control system.
Keywords: Process control, process design, buffer tanks
1
In the present version some corrections and clarifying modifications from the original text
have been made. The most important error was step 3 in Table B.1. Some missing values have
been provided for the examples, and equation (B.26) has been modified. A concluding section that
was omitted due to spatial limitations has been included.
190
Appendix B. A Systematic Approach to the Design of Buffer Tanks
B.1 Introduction
The objective of this paper is to provide a systematic approach to the design of
buffer tanks based on control theory. The background for this approach is that
buffer tanks often are implemented for control purposes. Even so, control theory
is rarely used when sizing and designing the tanks. Instead, rules of thumb are
used.
Text books on chemical process design seem to agree that a half-full residence
time of 5-10 minutes is appropriate for reflux drums and that this also applies for
other buffer tanks. For tanks between distillation columns a half-full residence
time of 10-20 minutes is recommended. ((Lieberman, 1983), (Sandler and Luckiewicz, 1987), (Ulrich, 1984), (Walas, 1987) and (Wells, 1986)). Sigales (1975) is
more specific concerning what follows after the drum. None of these references
give any justifications for their choice. (Watkins, 1967) gives a reflux drum volume dependent on instrumentation and labor factors (both related to operational
use of the buffer tank), reflux and product rates, and a factor dependent on how
well external units are operated. The method gives half full hold-up times from
1.5 to 32 min.
Design of vessels to dampen flow variations is presented by Harriott (1964) using a specification of outlet flow rate change given a certain step in inlet flow. This
method has similarities with the one presented for flow variations in the present
paper.
Another related class of process equipment is neutralization tanks. The main
problems for this process are large and varying process gain and delays in the control loop. Design is described in (Shinskey, 1973) and (McMillan, 1984). Another
design method and a critical review is found in (Walsh, 1993).
Zheng and Mahajanam (1999) find the necessary buffer tank volume by optimization and use it as a controllability measure.
A stated above, due to limitations in the control system, there is a limitation
in frequencies above which the control system is not effective. The process itself
must dampen the disturbances in this area. If it initially does not, addition of one
or more buffer tanks is necessary. In this paper we present design methods for
buffer tanks based on this fundamental understanding.
B.2 Transfer functions for buffer tanks
Consider the effect of a disturbance, , on the controlled variable, . The linearized model in terms of deviation variables may be written as
(B.1)
B.2 Transfer functions for buffer tanks
191
To illustrate the effect of the buffer tank, we express the dynamic model of the
tank with the transfer function . The disturbances passes through the buffer
tank (e.g. see Figure B.1), so that the process with a buffer tank may be expressed
by
9 (B.2)
where 9 is the disturbance transfer function of the original plant, and is the modified disturbance transfer function. A typical buffer tank transfer function is
(B.3)
A? Note that so that the buffer tank has no steady state effect.
Quality/ Flow
disturbance
Buffer tank
h(s)
Process
Gd0(s)
Gd(s)
Figure B.1: Example of how a buffer tank dampens disturbances.
[ , inlet flow-rate [ ,
We consider a buffer tank with liquid volume
outlet flow-rate . Further we let and denote the inlet and outlet quality
(concentration or temperature), respectively. A component or simplified energy
balance for a perfectly mixed tank yields
>@ (B.4)
In addition we have the total mass balance (assuming constant density):
>@ (B.5)
B.2.1 Quality disturbance
For quality disturbances the objective of the buffer tank is to smoothen the quality
response, , so that the variations in are smaller than in .
Appendix B. A Systematic Approach to the Design of Buffer Tanks
192
Combining (B.4) and (B.5) yields
linearization yields
!
"
G
and for a single buffer tank
(B.6)
where denotes the nominal (steady state) values. Note that the dynamics of
(level control) have no effect on the linearized response of . Furthermore for the
case with a single feed stream and the dynamics of have no effect on
the response of . In any case we find that the transfer function for quality is
? (B.7)
where ?
is called the residence time (steady state). We can see that
the buffer tank works as a first order filter. Similarly for buffer tanks in series
we have
? (B.8)
where ?
is the total residence time.
B.2.2 Flow rate disturbance
For flow rate disturbances the objective of the buffer tank is to smoothen the flow
response, . Note that we need to use a “slow” level controller,
as tight level control yields P . Let denote the transfer function for the
level controller including measurement and actuator dynamics and the possible
dynamics of an inner flow control loop. Then , where
is
the
set-point
for
the
volume.
Combining
this
with
the
total
mass
balance
(B.5) yields
(B.9)
The buffer tank transfer function is thus given by
(B.10)
In this case we have more freedom in selecting since we can select the
controller . With a proportional controller , we get that is a
first order filter with ? . For a given the controller is
' (B.11)
B.3 Controllability analysis
193
B.3 Controllability analysis
We here provide a review of some controllability results which are subsequently
used for buffer tank design. We consider SISO (single input-single output) systems. Consider a linear process in terms of deviation variables
(B.12)
the manipulated input and the disturbance (includ-
Here denotes the output, ing disturbances entering at the input which are frequently referred to as “load
changes”). We assume throughout this paper that the model has been scaled such
that expected disturbances make the magnitude of the elements of lie within for all frequencies and the requirement for the scaled output vector, , is that the
magnitude of each element in shall lie between R and for all frequencies, and
is scaled so that the manipulated input range corresponds to a variation of in
.
Feedback control yields , and from this we eliminate to get
(B.13)
is the set-point, and and are the sensitivity function and the com-
plementary sensitivity function, respectively. We ignore set-point changes and get
the following expression for the effect of disturbances
(B.14)
Two different requirements must be fulfilled to get acceptable control performance. The first relates to the speed of response to reject disturbances. From
(B.14) we see that to keep
$ when , we must require
We define as the frequency where .
(B.15)
At higher frequencies we
cannot rely on feedback control for disturbance rejection, so that
(B.16)
For acceptable performance and robustness we have the following maximum
value of the bandwidth (Skogestad, 1999), (Skogestad and Postlethwaite, 1996):
(B.17)
Appendix B. A Systematic Approach to the Design of Buffer Tanks
194
where
1999):
is the effective delay. With PI or PID control we have (Skogestad,
?
?
;
B;
?
for PI
for PID
(B.18)
where is the delay, ? , where is a right half plane zero, and ? is lag
number ordered by size so that ? is the largest time constant. For more realistic
PI controllers, must be reduced compared to (B.17). Ziegler-Nichols tuning
gives 87 , while a more robust tuning (Skogestad, 1999) gives
;
(B.19)
Note that (B.16) is only a necessary requirement, as (B.15) needs to be satisfied
for $ . In particular, (B.15) may impose additional requirements if is of
high order; this is discussed later.
In words (B.16) tells us that at sufficiently high frequencies the process must
be “self-regulating”. If (B.16) is not satisfied then we need to modify the process.
One commonly used approach is to add buffer tanks as illustrated in Figure B.1,
such that the “new” disturbance response becomes as in equation (B.2).
The second limitation relates to input constraints for disturbances, but will not
be covered by this article.
B.3.1 Additional requirements due to high order
As mentioned, (B.16) is only a necessary requirement as (B.15) needs to be satisfied also for B$ . To investigate this further we make the following approximation of the sensitivity function, , with the loop transfer function, (
):
P (B.20)
Inserting this approximation into (B.15), we obtain
(B.21)
Now it may be difficult to have sufficiently high roll-off (slope) in the loop transfer
at frequencies below the bandwidth
function to get (even though we satisfy it at the bandwidth). The problem is that a high roll-off
in yields a large phase lag, and we get instability problems. For reasonable
robustness and performance we must have that the slope for is about -1 near the
bandwidth . In this case it is difficult to make general formulas for the buffer
tank design. Graphical or optimization based solutions are probably simplest. One
particular case is studied later.
B.4 Quality variations
195
We can get a steeper slope around the bandwidth, however, with multiple control loops. E.g. with a series of buffer tanks and control in each tank, the total
slope of is (even though it is -1 for each individual tank).
B.4 Quality variations
When the main source of disturbances are variations in the inflow quality (temperature or concentration) they may be smoothened by a mixing tank. With perfect
mixing and a residence time of ? ( denotes hold-up), the outflow quality is
roughly speaking the sliding mean of the input quality within a time window of
length ? . The transfer function for one buffer tank is given by (B.7). We may
also consider using a series of buffer tanks. For equal tanks in series with a total
residence time of ? , and total volume , the transfer function is given by (B.8).
0
10
Gain 0.33 (min.vol.1 tank)
Gain 0.144 (min.vol. 2 tanks)
−1
10
Magnitude
Gain 0.064 (min.vol. 3 tanks)
n=1
−2
10
n=2
n=3
n=4
−3
10
−1
0
10
1
10
10
2
10
Frequency × τh
Figure B.2: Quality disturbance: Frequency responses for n tanks in series with total
residence time , K .
In Figure B.2 we show the amplitude plot of for ; < equal tanks
in series with a given total residence time ?/ . Physically, on the x-axis is shown
the normalized frequency, ,? , of the sinusoidal varying input concentration,
[email protected] [email protected] ,@ 196
Appendix B. A Systematic Approach to the Design of Buffer Tanks
into the first tank, and on the y-axis is shown the normalized output concentration
from tank , , where and denote the magnitude of the sinusoidal
variations. Note that both axis are logarithmic.
M? , we have P , which means that slow siAt low frequencies, nusoidal variations are unaffected when they pass through the tanks. However, fast
variations (with high frequencies) are dampened by the tanks which tend to “average out” the variations. At sufficiently high frequencies, M? , we find that
(log-scale) as a function of frequency (log-scale) approaches a straight
line. This follows because the high-frequency asymptote is ? Q (in
words, “the slope is ” at high frequencies for tanks in series). Thus, at high
frequencies the use of many tanks is “better”, in terms of providing more dampening for a given total volume. On the other hand, the frequency where the asymptote crosses magnitude 1 (its “break” or “corner” frequency) is Y? Y? ,
which is at a lower frequency when is smaller, so at lower frequencies fewer
tanks is better. This is also seen from the more exact plot in Figure B.2.
The plot may be used to obtain the total required volume of the buffer tanks
if we at a given frequency specify the factor by which we want to reduce the
disturbance. The required “gain” of the buffer transfer function is then and
we can read off ,? and with a given value of obtain the total residence time
? . Typically, the given frequency is the achievable closed-loop bandwidth of the
feedback control system, , and is the value of at this frequency.
We see that one tank is “best” if we want to reduce the effect of the disturbance
at a given frequency by a factor 7 or less; two tanks is “best” if
the factor is between 3 and about 7J<>< , and three tanks is “best” if the
factor is between about 7 and 7 N < . The word “best” has been put in
quotes because we here only consider the total combined volume of the tanks. In
practice, there are several other factors that favor using as few tanks as possible;
S ), the cost of
this includes the scaling law for cost (typically, cost scales with
additional equipment like pipes, pumps, sensors, control systems, etc. as well as
other controllability considerations (slope condition on ). Therefore, one would
probably consider using only one tank also when we want to reduce the effect of
the disturbance by a factor > , even though in this case the volume of one
tank is about 5 times larger than the total volume of two tanks, and more than 7
times larger than the total volume of three tanks (this is seen from
Figure B.2 by
reading off the value of ,?/ that corresponds to magnitude
).
To satisfy the necessary condition (B.16) we need to select such that
:9 (B.22)
We introduce the factor by which the effect of the disturbance must be reduced
9 (B.23)
B.4 Quality variations
197
We must at least require . As mentioned this may be solved
graphically using Figure B.2, but alternatively we can find the analytical solution
from (B.8) and (B.17):
I (B.24)
? For one tank and we have the appropriate formula ? . For
;
the use of (B.24) assumes that the total slope of around can be . This
can be achieved with local quality control in each tank, e.g. for a neutralization
plant, it must be possible to measure the concentration and automatically add a
reactant in each tank.
To find the optimal number of tanks one must then take into account equipment, piping, control systems (each tanks may require a level controller), etc. as
mentioned above. Normally the optimal number of tanks will not be large, so that
the cost calculations has to be made for a limited number of cases.
Example B.1 Consider mixing of two process streams, and as illustrated in
Figure B.3. The concentration and flow rate of stream are denoted and ,
and for stream they are called and ( and may also be temperatures).
The two streams with total flowrate [ , are mixed in a mixing tank of
[ , and the concentration of the outlet flow is denoted . The concentrations
since stream never
represent the difference between component 1 and 2. has less of component 1, whereas is negative. The objective is to mix equal
amounts of the components such that is zero. This concentration is controlled by manipulating the flow rate of . First we check if this controller,
together with the mixing tank, is sufficient for suppressing disturbances in the
concentration of stream . Combination of component balance and total material
balance gives the following model:
> @
(B.25)
This model is linearized and scaled (as described in the controllability section).
We require a variation in less than of the variation in . The scaled
deviation variables are marked with a prime and we get the following model after
Laplace transformation
; (B.26)
where we have assumed constant . We study concentration disturbances, leading to 9 and further ; . Mainly due
to the measurement, the control loop has an effective delay of 7 With a robust
controller tuning, (B.19) gives a bandwidth of 7
" .
198
Appendix B. A Systematic Approach to the Design of Buffer Tanks
qA
cA
qB
cB
FIC
Tank 1
CIC
c0
Tank 2
c
Figure B.3: Extra buffer tank for a mixing process. Concentration is controlled by
manipulating flow rate of stream B. Nominal data: [ , [ ,
[
[
[
, , . Range, used for scaling: Expected variations in : [ . Range for : [ . Allowed range for :
[ .
B.4 Quality variations
199
and 9 are shown in Figure B.4 (dashed lines). We see that
for all frequencies, so that input constraints pose no problems in
this case. In the figure the bandwidth frequency, , is also marked. We see that
:9 at frequencies above the bandwidth, so a standard (robust) control
system is not sufficient to fulfil the requirements on the outlet concentration. To
solve this problem, we may either improve the control system (e.g. feedforward
control), increase the volume of the mixing tank, or install an extra buffer tank.
In this case we assume that the latter alternative is the best, and introduce a new
tank after the mixing tank (dashed in Figure B.3). We see from Figure B.4 that the
gain must be reduced with 10 at the bandwidth ( ), and obtain from (B.24)
( ) a required residence time of the buffer tank of ; , corresponding to a
Y? ; ,\[ . The modified disturbance transfer function gain,
volume of
, is shown with a solid line in Figure B.4. The slope is -1 or smaller below the
bandwidth, so that we need not consider the problem discussed in section B.3.1.
is plotted (dash-dotted) to illustrate this ( $
). is below
1 for all frequencies (dashed). Figure B.5 shows the response of a unit step in
concentration of stream with (solid) and without (dashed) the extra buffer tank.
We see that it is kept below 7J with the extra buffer tank present.
9
2
10
G0
1
10
Magnitude
Gd
Gd0
0
10
ωB
SGd
−1
L
10
−2
10
−2
10
−1
10
Figure B.4: With an extra buffer tank, the bandwidth.
0
10
Frequency [rad/s]
1
10
2
10
is brought below 1 for all frequencies above
If the slope of is steeper than the slope of , ?/ is too optimistic. We will
however analyze one case. We assume 9 has slope R so that has slope
200
Appendix B. A Systematic Approach to the Design of Buffer Tanks
0.8
0.7
0.6
Concentration
0.5
Concentration without extra buffer tank
0.4
0.3
0.2
Concentration with extra buffer tank
0.1
0
−0.1
0
10
20
30
40
Time [s]
50
60
70
80
Figure B.5: With an extra buffer tank the outlet concentration is kept within 0.1 from
set-point despite a unit step in disturbance. This is not the case without the extra buffer
tank.
;
above the frequency M?/ , where ? is the buffer tank residence time. Further
we assume that has slope near the bandwidth and that it increases to ;
due to an integrator in the controller below M? , where ? is the integral
time. A robust choice of ? is : (Skogestad, 1999). Using geometry it is easy
to show that in this case ? : . Compared to (B.24) for one tank we see that
the residence time for this case is increased by a factor of : .
Example B.2 Consider the process from example B.1, modified so that the mea[ and the variation resurement delay is 7J , the volume of the first tank is
quirements for the outlet concentration is 0.01. The concentration in the first tank
is controlled with a robust PI controller (Skogestad, 1999). In this case the slope
of is ; around the bandwidth, and (B.24) leads to a residence time of
7 , which is insufficient. In Figure B.6 a residence time of ? : 7 ; is applied. The method uses asymptotes, and we see that is just touching the asymptote of . itself is a distance above so the result
here is slightly conservative. By optimization one find a minimum residence time
of ; 7 < required to fulfil (B.21) for this controller tuning.
B.5 Flow variations
By exploiting the volume of the buffer tank, flow variations in the outflow may be
dampened using a slow level control. The outflow will then be dependent on the
B.5 Flow variations
201
2
10
1
Magnitude
10
Gd0
0
10
ωB
SGd
L
−1
10
Gd
1/τ1
1/τ2
1/τI
−2
10
−2
10
−1
10
0
10
Frequency [rad/s]
1
10
2
10
in the second tank, Figure B.6: With a residence time of for all frequencies, and disturbances are rejected.
[ and the inlet and outlet flowchosen controller. Denote the tank volume
rates and respectively. The transfer function for the buffer tank is then given
by (B.10). Compared to the quality disturbance case, we have more freedom in
selecting , since we can select the controller . But the level will vary, so
the size of the tank must be chosen so that the level remains between its limits.
The volume variation is given by ' , and combination
with (B.11) yields:
(B.27)
' which is used to find the required tank volume. The tank size design consists of
the following steps:
(1) Select satisfied.
such that if has the desired shape, that is such that (B.16) is
(2) Find the corresponding controller from (B.11) (is it realizable?)
(3) Find the largest effect of on
).
from (B.27) (usually at steady state,
(4) Obtain the required total volume from the expected range of
0 ).
(denoted
In table B.1 we have applied the method for first and second order filtering.
202
Appendix B. A Systematic Approach to the Design of Buffer Tanks
Table B.1: Flowrate disturbance: Procedure for buffer tank design applied to first and
second order filtering
Step
1. Desired 2. from (B.11)
3.
4.
1st order
?
from (B.27)
G5G
A?
Y? ?0
2nd order
A? K ;Y? ;Y? 0 B.5.1 First-order filtering
With K the required
controller is a P-controller with gain Y? .
From (B.27), K K . The maximum value of this transfer function
occurs at low frequencies ( ), and the required volume of the tank is G5G ? 0 . Adding a slow integral action to the controller will not affect these
results considerably. Such an integral action will reset the volume to its nominal
value. This is not always desired, however. If e.g. is at its maximum, we may
want the volume to stay at a large value to anticipate a possible large reduction in
.
B.5.2 Second-order filtering
With K we get from (B.11) that the required controller is a lag
;Y? K (B.28)
and from (B.27) the response of the volume deviation is
B;Y? O? >; A ? This has its largest value equal to ;Y?
volume is ;Y? 0 .
at low frequencies ( (B.29)
), and the required
B.6 Conclusions
The objective of the control system is to counteract disturbances. However, the
maximum achievable control bandwidth is approximately equal to the inverse of
the effective process delay, i.e. FP . For “fast” disturbances, above the
bandwidth frequency, one must rely on the process itself, including any buffer
REFERENCES
203
tanks, to dampen the disturbances. The requirement is that the effect of disturbances on the controlled variable (usually concentration), should be less than 1 (in
scaled units) at frequencies above the bandwidth. Specifically, if the magnitude
of the original disturbance transfer function is larger than 1 at frequencies
above the bandwidth, then we must add one or more buffer tanks, with overall
transfer function , such that is less than 1. In the paper we present
design methods for sizing buffer tanks based on this fundamental insight.
The two fundamentally different sources of disturbances are variations in flowrate and variations in quality (concentration, temperature). Quality variations are
dampened by mixing, and it may be adventageous to use several smaller rather
than a single large buffer tank. Figure B.2 shows how depends on the number
of tanks and total residence time ?/ . If we define as the value of at the
bandwidth frequency , then the design objective is that
should be less than
1/f at this frequency, and we derive in (B.24) the required value for ? . The volume
in each buffer tank is then Y?Y where is the total flowrate. If the resulting
slope of around the bandwidth is steeper than -1, then we need to increase the
volume or add local feedback loops. The design method is illustrated in Examples
B.1 and B.2.
Flowrate variations are dampened using a slow level controller in the
buffer tank, and there is no advantage of using several tanks as we may include
dynamics in . Table B.1 gives a design procedure for flowrate disturbances.
In conclusion, buffer tanks are designed and implemented for control purposes, yet control theory is rarely used when sizing and designing buffer tanks
and their control system. In this paper we have presented a systematic approach
for design of buffer tanks to dampen disturbances in quality and flowrate.
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