Using the Control System Toolbox

Using the Control System Toolbox
Control System
Toolbox
For Use with MATLAB
®
Computation
Visualization
Programming
Using the Control System Toolbox
Version 5
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Using the Control System Toolbox
 COPYRIGHT 2000-2002 by The MathWorks, Inc.
The software described in this document is furnished under a license agreement. The software may be used
or copied only under the terms of the license agreement. No part of this manual may be photocopied or reproduced in any form without prior written consent from The MathWorks, Inc.
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Printing History: November 2000
June 2001
July 2002
Online only
Online only
Online only
Revised for Version 5 (Release 12)
(Reorganized and a name change)
Revised for Version 5.1 (Release 12.1)
Revised for Version 5.2 (Release 13)
Contents
LTI Models
1
LTI Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Using LTI Models in the Control System Toolbox . . . . . . . . . . .
LTI Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Precedence Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Viewing LTI Systems As Matrices . . . . . . . . . . . . . . . . . . . . . . .
Command Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-2
1-3
1-3
1-5
1-5
1-6
Creating LTI Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-8
Transfer Function Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-8
Zero-Pole-Gain Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-12
State-Space Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-14
Descriptor State-Space Models . . . . . . . . . . . . . . . . . . . . . . . . . 1-16
Frequency Response Data (FRD) Models . . . . . . . . . . . . . . . . . 1-17
Discrete-Time Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-19
Data Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-23
LTI Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Generic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Model-Specific Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Setting LTI Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Accessing Property Values Using get . . . . . . . . . . . . . . . . . . . .
Direct Property Referencing . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Insight into LTI Properties . . . . . . . . . . . . . . . . . .
1-25
1-25
1-26
1-28
1-30
1-31
1-32
Model Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Explicit Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Automatic Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Caution About Model Conversions . . . . . . . . . . . . . . . . . . . . . .
1-39
1-39
1-40
1-40
Time Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supported Functionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Specifying Input/Output Delays . . . . . . . . . . . . . . . . . . . . . . . .
Specifying Delays on the Inputs or Outputs . . . . . . . . . . . . . .
Specifying Delays in Discrete-Time Models . . . . . . . . . . . . . . .
Retrieving Information About Delays . . . . . . . . . . . . . . . . . . .
1-42
1-42
1-43
1-47
1-49
1-50
i
Padé Approximation of Time Delays . . . . . . . . . . . . . . . . . . . . . 1-51
Simulink Block for LTI Systems . . . . . . . . . . . . . . . . . . . . . . . 1-53
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-55
Operations on LTI Models
2
Precedence and Property Inheritance . . . . . . . . . . . . . . . . . . 2-3
Extracting and Modifying Subsystems . . . . . . . . . . . . . . . . . .
Referencing FRD Models Through Frequencies . . . . . . . . . . . . .
Referencing Channels by Name . . . . . . . . . . . . . . . . . . . . . . . . .
Resizing LTI Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-5
2-7
2-8
2-9
Arithmetic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inversion and Related Operations . . . . . . . . . . . . . . . . . . . . . .
Transposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pertransposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-11
2-11
2-13
2-13
2-14
2-14
Model Interconnection Functions . . . . . . . . . . . . . . . . . . . . . 2-16
Concatenation of LTI Models . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16
Feedback and Other Interconnection Functions . . . . . . . . . . . 2-18
Continuous/Discrete Conversions of LTI Models . . . . . . . .
Zero-Order Hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
First-Order Hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tustin Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tustin with Frequency Prewarping . . . . . . . . . . . . . . . . . . . . .
Matched Poles and Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Discretization of Systems with Delays . . . . . . . . . . . . . . . . . . .
ii
Contents
2-20
2-20
2-22
2-22
2-23
2-23
2-23
Resampling of Discrete-Time Models . . . . . . . . . . . . . . . . . . . 2-26
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-27
Model Analysis Tools
3
General Model Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 3-3
Model Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5
State-Space Realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-8
Arrays of LTI Models
4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
When to Collect a Set of Models in an LTI Array . . . . . . . . . . .
Restrictions for LTI Models Collected in an Array . . . . . . . . . .
Where to Find Information on LTI Arrays . . . . . . . . . . . . . . . . .
4-2
4-2
4-2
4-3
The Concept of an LTI Array . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4
Higher Dimensional Arrays of LTI Models . . . . . . . . . . . . . . . . 4-6
Dimensions, Size, and Shape of an LTI Array . . . . . . . . . . . . 4-7
size and ndims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9
reshape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-11
Building LTI Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Generating LTI Arrays Using rss . . . . . . . . . . . . . . . . . . . . . . .
Building LTI Arrays Using for Loops . . . . . . . . . . . . . . . . . . . .
Building LTI Arrays Using the stack Function . . . . . . . . . . . .
Building LTI Arrays Using tf, zpk, ss, and frd . . . . . . . . . . . . .
4-12
4-12
4-12
4-15
4-17
iii
Indexing Into LTI Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Accessing Particular Models in an LTI Array . . . . . . . . . . . . .
Extracting LTI Arrays of Subsystems . . . . . . . . . . . . . . . . . . .
Reassigning Parts of an LTI Array . . . . . . . . . . . . . . . . . . . . . .
Deleting Parts of an LTI Array . . . . . . . . . . . . . . . . . . . . . . . . .
4-20
4-20
4-21
4-22
4-23
Operations on LTI Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example: Addition of Two LTI Arrays . . . . . . . . . . . . . . . . . . .
Dimension Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Special Cases for Operations on LTI Arrays . . . . . . . . . . . . . .
Other Operations on LTI Arrays . . . . . . . . . . . . . . . . . . . . . . . .
4-24
4-25
4-26
4-26
4-29
Customization
5
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2
The Property and Preferences Hierarchy . . . . . . . . . . . . . . . . . . 5-3
Setting Toolbox Preferences
6
Toolbox Preferences Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Units Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Style Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Characteristics Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SISO Tool Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-2
6-3
6-3
6-4
6-5
Setting Tool Preferences
7
iv
Contents
LTI Viewer Preferences Editor . . . . . . . . . . . . . . . . . . . . . . . . . .
Units Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Style Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Characteristics Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-2
7-3
7-3
7-4
Parameters Panpanee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5
SISO Tool Preferences Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6
Units Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7
Style Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-8
Options Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-10
Line Colors Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-12
Customizing Response Plot Properties
8
Response Plots Property Editor . . . . . . . . . . . . . . . . . . . . . . . . .
Labels Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Limits Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Units Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Style Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Characteristics Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8-3
8-4
8-4
8-5
8-7
8-8
Property Editing for Subplots . . . . . . . . . . . . . . . . . . . . . . . . . 8-10
Customizing Plots Inside the SISO Design Tool . . . . . . . . .
Root Locus Property Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Labels Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Limits Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Options Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Open-Loop Bode Property Editor . . . . . . . . . . . . . . . . . . . . . . .
Labels Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Limits Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Options Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Open-Loop Nichols Property Editor . . . . . . . . . . . . . . . . . . . . .
Labels Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Limits Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Options Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Prefilter Bode Property Editor . . . . . . . . . . . . . . . . . . . . . . . . .
8-11
8-11
8-12
8-13
8-14
8-16
8-16
8-17
8-18
8-19
8-20
8-21
8-21
8-22
v
Design Case Studies
9
Yaw Damper for a 747 Jet Transport . . . . . . . . . . . . . . . . . . . . 9-3
Computing Open-Loop Eigenvalues . . . . . . . . . . . . . . . . . . . . . . 9-4
Open-Loop Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-5
Root Locus Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-9
Washout Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-14
Hard-Disk Read/Write Head Controller . . . . . . . . . . . . . . . .
Deriving the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Model Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Adding a Compensator Gain . . . . . . . . . . . . . . . . . . . . . . . . . . .
Adding a Lead Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Design Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9-20
9-20
9-21
9-23
9-24
9-27
LQG Regulation: Rolling Mill Example . . . . . . . . . . . . . . . . .
Process and Disturbance Models . . . . . . . . . . . . . . . . . . . . . . . .
LQG Design for the x-Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
LQG Design for the y-Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cross-Coupling Between Axes . . . . . . . . . . . . . . . . . . . . . . . . . .
MIMO LQG Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9-31
9-31
9-34
9-41
9-43
9-46
Kalman Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Discrete Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Steady-State Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time-Varying Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time-Varying Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9-50
9-50
9-51
9-57
9-58
9-61
Reliable Computations
10
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-3
Conditioning and Numerical Stability . . . . . . . . . . . . . . . . . 10-5
Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-5
vi
Contents
Numerical Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-7
Choice of LTI Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-9
State Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-9
Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-9
Zero-Pole-Gain Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-14
Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-16
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-18
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-19
Tool and Viewer Quick Start
11
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-2
SISO Design Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-3
Importing and Exporting Models . . . . . . . . . . . . . . . . . . . . . . . 11-4
Configuring the Feedback Structure . . . . . . . . . . . . . . . . . . . . 11-7
Tuning Compensators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-8
Viewing Loop Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-13
Viewing System Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-14
Storing and Retrieving Designs . . . . . . . . . . . . . . . . . . . . . . . 11-15
Customizing the SISO Design Tool . . . . . . . . . . . . . . . . . . . . . 11-16
LTI Viewer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Right-Click Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
LTI Viewer Toolbar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Basic LTI Viewer Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Importing and Exporting Models . . . . . . . . . . . . . . . . . . . . . .
Selecting Response Types . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Analyzing MIMO Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Customizing the LTI Viewer . . . . . . . . . . . . . . . . . . . . . . . . . .
11-18
11-18
11-19
11-19
11-20
11-22
11-23
11-26
vii
Index
viii Contents
1
LTI Models
Creating LTI Models . . . . . . . . . . . . . . . . 1-8
LTI Properties . . . . . . . . . . . . . . . . . . . 1-25
Model Conversion
. . . . . . . . . . . . . . . . . 1-39
Time Delays . . . . . . . . . . . . . . . . . . . . 1-42
Simulink Block for LTI Systems . . . . . . . . . . . 1-53
References . . . . . . . . . . . . . . . . . . . . . 1-55
1
LTI Models
The Control System Toolbox offers extensive tools to manipulate and analyze
linear time-invariant (LTI) models. It supports both continuous- and
discrete-time systems. Systems can be single-input/single-output (SISO) or
multiple-input/multiple-output (MIMO). In addition, you can store several LTI
models in an array under a single variable name. See Chapter 4, “Arrays of LTI
Models” for information on LTI arrays.
This section introduces key concepts about the MATLAB representation of LTI
models, including LTI objects, precedence rules for operations, and an analogy
between LTI systems and matrices. In addition, it summarizes the basic
commands you can use on LTI objects.
LTI Models
You can specify LTI models as:
• Transfer functions (TF), for example,
s+2
P ( s ) = --------------------------s 2 + s + 10
• Zero-pole-gain models (ZPK), for example,
H(z ) =
2 ( z – 0.5 )
------------------------z ( z + 0.1 )
2
(z + z + 1)
--------------------------------------------( z + 0.2 ) ( z + 0.1 )
• State-space models (SS), for example,
dx
------ = Ax + Bu
dt
y = Cx + Du
where A, B, C, and D are matrices of appropriate dimensions, x is the state
vector, and u and y are the input and output vectors.
• Frequency response data (FRD) models
FRD models consist of sampled measurements of a system’s frequency
response. For example, you can store experimentally collected frequency
response data in an FRD.
1-2
Using LTI Models in the Control System Toolbox
You can manipulate TF, SS, and ZPK models using the arithmetic and model
interconnection operations described in Chapter 2, “Operations on LTI Models”
and analyze them using the model analysis functions, such as bode and step.
FRD models can be manipulated and analyzed in much the same way you
analyze the other model types, but analysis is restricted to frequency-domain
methods.
Using a variety of design techniques, you can design compensators for systems
specified with TF, ZPK, SS, and FRD models. These techniques include root
locus analysis, pole placement, LQG optimal control, and frequency domain
loop-shaping. For FRD models, you can either:
• Obtain an identified TF, SS, or ZPK model using system identification
techniques.
• Use frequency-domain analysis techniques.
Other Uses of FRD Models
FRD models are unique model types available in the Control System Toolbox
collection of LTI model types, in that they don’t have a parametric
representation. In addition to the standard operations you may perform on
FRD models, you can also use them to:
• Perform frequency-domain analysis on systems with nonlinearities using
describing functions.
• Validate identified models against experimental frequency response data.
LTI Objects
Depending on the type of model you use, the data for your model may consist
of a simple numerator/denominator pair for SISO transfer functions, four
matrices for state-space models, and multiple sets of zeros and poles for MIMO
zero-pole-gain models or frequency and response vectors for FRD models. For
convenience, the Control System Toolbox provides customized data structures
(LTI objects) for each type of model. These are called the TF, ZPK, SS, and FRD
objects. These four LTI objects encapsulate the model data and enable you to
manipulate LTI systems as single entities rather than collections of data
vectors or matrices.
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1
LTI Models
Creating an LTI Object: An Example
An LTI object of the type TF, ZPK, SS, or FRD is created whenever you invoke
the corresponding constructor function, tf, zpk, ss, or frd. For example,
P = tf([1 2],[1 1 10])
creates a TF object, P, that stores the numerator and denominator coefficients
of the transfer function
s+2
P ( s ) = --------------------------s 2 + s + 10
See “Creating LTI Models” on page 1-8 for methods for creating all of the LTI
object types.
LTI Properties and Methods
The LTI object implementation relies on MATLAB object-oriented
programming capabilities. Objects are MATLAB structures with an additional
flag indicating their class (TF, ZPK, SS, or FRD for LTI objects) and have
pre-defined fields called object properties. For LTI objects, these properties
include the model data, sample time, delay times, input or output names, and
input or output groups (see “LTI Properties” on page 1-25 for details). The
functions that operate on a particular object are called the object methods.
These may include customized versions of simple operations such as addition
or multiplication. For example,
P = tf([1 2],[1 1 10])
Q = 2 + P
performs transfer function addition.
2
2s + 3s + 22
Q ( s ) = 2 + P ( s ) = ----------------------------------2
s + s + 10
The object-specific versions of such standard operations are called overloaded
operations. For more details on objects, methods, and object-oriented
programming, see Classes and Objects”in the MATLAB documentation. For
details on operations on LTI objects, see Chapter 2, “Operations on LTI
Models.”
1-4
Precedence Rules
Operations like addition and commands like feedback operate on more than
one LTI model at a time. If these LTI models are represented as LTI objects of
different types (for example, the first operand is TF and the second operand is
SS), it is not obvious what type (for example, TF or SS) the resulting model
should be. Such type conflicts are resolved by precedence rules. Specifically, TF,
ZPK, SS, and FRD objects are ranked according to the precedence hierarchy.
FRD > SS > ZPK > TF
Thus ZPK takes precedence over TF, SS takes precedence over both TF and
ZPK, and FRD takes precedence over all three. In other words, any operation
involving two or more LTI models produces:
• An FRD object if at least one operand is an FRD object
• An SS object if no operand is an FRD object and at least one operand is an
SS object
• A ZPK object if no operand is an FRD or SS object and at least one is an ZPK
object
• A TF object only if all operands are TF objects
Operations on systems of different types work as follows: the resulting type is
determined by the precedence rules, and all operands are first converted to this
type before performing the operation.
Viewing LTI Systems As Matrices
In the frequency domain, an LTI system is represented by the linear input/
output map
y = Hu
This map is characterized by its transfer matrix H, a function of either the
Laplace or Z-transform variable. The transfer matrix H maps inputs to
outputs, so there are as many columns as inputs and as many rows as outputs.
If you think of LTI systems in terms of (transfer) matrices, certain basic
operations on LTI systems are naturally expressed with a matrix-like syntax.
For example, the parallel connection of two LTI systems sys1 and sys2 can be
expressed as
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1
LTI Models
sys = sys1 + sys2
because parallel connection amounts to adding the transfer matrices.
Similarly, subsystems of a given LTI model sys can be extracted using
matrix-like subscripting. For instance,
sys(3,1:2)
provides the I/O relation between the first two inputs (column indices) and the
third output (row index), which is consistent with
y1
H ( 1, 1 ) H ( 2, 1 )
y 2 = H ( 2, 1 ) H ( 2, 2 ) u 1 u 2
H ( 3, 1 ) H ( 3, 2 )
y3
for y = Hu.
Command Summary
The next two tables give an overview of the main commands you can apply to
LTI models.
Table 1-1: Creating LTI Models
1-6
Command
Description
drss
Generate random discrete state-space model.
dss
Create descriptor state-space model.
filt
Create discrete filter with DSP convention.
frd
Create an FRD model.
frdata
Retrieve FRD model data.
get
Query LTI model properties.
set
Set LTI model properties.
rss
Generate random continuous state-space model.
ss
Create a state-space model.
Table 1-1: Creating LTI Models (Continued)
Command
Description
ssdata, dssdata
Retrieve state-space data (respectively, descriptor
state-space data).
tf
Create a transfer function.
tfdata
Retrieve transfer function data.
zpk
Create a zero-pole-gain model.
zpkdata
Retrieve zero-pole-gain data.
Table 1-2: Converting LTI Models
Command
Description
c2d
Continuous- to discrete-time conversion.
d2c
Discrete- to continuous-time conversion.
d2d
Resampling of discrete-time models.
frd
Conversion to an FRD model.
pade
Padé approximation of input delays.
ss
Conversion to state space.
tf
Conversion to transfer function.
zpk
Conversion to zero-pole-gain.
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LTI Models
Creating LTI Models
The functions tf, zpk, ss, and frd create transfer function models,
zero-pole-gain models, state-space models, and frequency response data
models, respectively. These functions take the model data as input and produce
TF, ZPK, SS, or FRD objects that store this data in a single MATLAB variable.
This section shows how to create continuous or discrete, SISO or MIMO LTI
models with tf, zpk, ss, and frd.
Note You can only specify TF, ZPK, and SS models for systems whose
transfer matrices have real-valued coefficients.
Transfer Function Models
This section explains how to specify continuous-time SISO and MIMO transfer
function models. The specification of discrete-time transfer function models is
a simple extension of the continuous-time case (see “Discrete-Time Models” on
page 1-19). In this section you can also read about how to specify transfer
functions consisting of pure gains.
SISO Transfer Function Models
A continuous-time SISO transfer function
(s)
h(s) = n
----------d(s)
is characterized by its numerator n ( s ) and denominator d ( s ) , both
polynomials of the Laplace variable s.
There are two ways to specify SISO transfer functions:
• Using the tf command
• As rational expressions in the Laplace variable s
To specify a SISO transfer function model h ( s ) = n ( s ) ⁄ d ( s ) using the tf
command, type
h = tf(num,den)
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Creating LTI Models
where num and den are row vectors listing the coefficients of the polynomials
n ( s ) and d ( s ), respectively, when these polynomials are ordered in descending
powers of s. The resulting variable h is a TF object containing the numerator
and denominator data.
For example, you can create the transfer function h ( s ) = s ⁄ ( s 2 + 2s + 10 ) by
typing
h = tf([1 0],[1 2 10])
MATLAB responds with
Transfer function:
s
-------------s^2 + 2 s + 10
Note the customized display used for TF objects.
You can also specify transfer functions as rational expressions in the Laplace
variable s by:
1 Defining the variable s as a special TF model
s = tf('s');
2 Entering your transfer function as a rational expression in s
For example, once s is defined with tf as in 1,
H = s/(s^2 + 2*s +10);
produces the same transfer function as
h = tf([1 0],[1 2 10]);
Note You need only define the variable s as a TF model once. All of the
subsequent models you create using rational expressions of s are specified as
TF objects, unless you convert the variable s to ZPK. See “Model Conversion”
on page 1-39 for more information.
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LTI Models
MIMO Transfer Function Models
MIMO transfer functions are two-dimensional arrays of elementary SISO
transfer functions. There are several ways to specify MIMO transfer function
models, including:
• Concatenation of SISO transfer function models
• Using tf with cell array arguments
Consider the rational transfer matrix
s–1
-----------s+1
. H(s ) =
s+2
---------------------------2
s + 4s + 5
You can specify H ( s ) by concatenation of its SISO entries. For instance,
h11 = tf([1 1],[1 1]);
h21 = tf([1 2],[1 4 5]);
or, equivalently,
s = tf('s')
h11 = (s 1)/(s+1);
h21 = (s+2)/(s^2+4*s+5);
can be concatenated to form H ( s ).
H = [h11; h21]
This syntax mimics standard matrix concatenation and tends to be easier and
more readable for MIMO systems with many inputs and/or outputs. See “Model
Interconnection Functions” on page 2-16 for more details on concatenation
operations for LTI systems.
Alternatively, to define MIMO transfer functions using tf, you need two cell
arrays (say, N and D) to represent the sets of numerator and denominator
polynomials, respectively. See Structures and Cell Arrays in the MATLAB
documentation for more details on cell arrays.
For example, for the rational transfer matrix H ( s ) , the two cell arrays N and D
should contain the row-vector representations of the polynomial entries of
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Creating LTI Models
N(s) = s – 1
s+2
D( s) =
s+1
s2
+ 4s + 5
You can specify this MIMO transfer matrix H ( s ) by typing
N = {[1 1];[1 2]}; % cell array for N(s)
D = {[1 1];[1 4 5]}; % cell array for D(s)
H = tf(N,D)
MATLAB responds with
Transfer function from input to output...
s
1
#1: ----s + 1
#2:
s + 2
------------s^2 + 4 s + 5
Notice that both N and D have the same dimensions as H. For a general MIMO
transfer matrix H ( s ), the cell array entries N{i,j} and D{i,j} should be
row-vector representations of the numerator and denominator of H ij ( s ), the
ijth entry of the transfer matrix H ( s ) .
Pure Gains
You can use tf with only one argument to specify simple gains or gain matrices
as TF objects. For example,
G = tf([1 0;2 1])
produces the gain matrix
G = 1 0
2 1
while
E = tf
creates an empty transfer function.
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1
LTI Models
Zero-Pole-Gain Models
This section explains how to specify continuous-time SISO and MIMO
zero-pole-gain models. The specification for discrete-time zero-pole-gain
models is a simple extension of the continuous-time case. See “Discrete-Time
Models” on page 1-19.
SISO Zero-Pole-Gain Models
Continuous-time SISO zero-pole-gain models are of the form
( s – z 1 ) ... ( s – z m )
h ( s ) = k ------------------------------------------------( s – p 1 ) ... ( s – p n )
where k is a real-valued scalar (the gain), and z 1 ,..., z m and p 1 ,..., p n are the
real or complex conjugate pairs of zeros and poles of the transfer function h ( s ) .
This model is closely related to the transfer function representation: the zeros
are simply the numerator roots, and the poles, the denominator roots.
There are two ways to specify SISO zero-pole-gain models:
• Using the zpk command
• As rational expressions in the Laplace variable s
The syntax to specify ZPK models directly using zpk is
h = zpk(z,p,k)
where z and p are the vectors of zeros and poles, and k is the gain. This
produces a ZPK object h that encapsulates the z, p, and k data. For example,
typing
h = zpk(0, [1 i 1+i 2],
2)
produces
Zero/pole/gain:
2 s
-------------------(s 2) (s^2
2s + 2)
You can also specify zero-pole-gain models as rational expressions in the
Laplace variable s by:
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Creating LTI Models
1 Defining the variable s as a ZPK model
s = zpk('s')
2 Entering the transfer function as a rational expression in s.
For example, once s is defined with zpk,
H =
2s/((s
2)*(s^2 + 2*s + 2));
returns the same ZPK model as
h = zpk([0], [2
1 i
1+i ],
2);
Note You need only define the ZPK variable s once. All subsequent rational
expressions of s will be ZPK models, unless you convert the variable s to TF.
See “Model Conversion” on page 1-39 for more information on conversion to
other model types.
MIMO Zero-Pole-Gain Models
Just as with TF models, you can also specify a MIMO ZPK model by
concatenation of its SISO entries (see “Model Interconnection Functions” on
page 2-16).
You can also use the command zpk to specify MIMO ZPK models. The syntax
to create a p-by-m MIMO zero-pole-gain model using zpk is
H = zpk(Z,P,K)
where
• Z is the p-by-m cell array of zeros (Z{i,j} = zeros of H ij ( s ) )
• P is the p-by-m cell array of poles (P{i,j} = poles of H ij ( s ) )
• K is the p-by-m matrix of gains (K(i,j) = gain of H ij ( s ) )
For example, typing
Z = {[], 5;[1 i 1+i] []};
P = {0,[ 1
1];[1 2 3],[]};
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1
LTI Models
K = [ 1
3;2
0];
H = zpk(Z,P,K)
creates the two-input/two-output zero-pole-gain model
H(s ) =
–1
-----s
3(s + 5)
-------------------2
(s + 1)
2 ( s 2 – 2s + 2 ) -------------------------------------------------( s – 1 ) ( s – 2 )( s – 3 )
0
Notice that you use [] as a place-holder in Z (or P) when the corresponding
entry of H ( s ) has no zeros (or poles).
State-Space Models
State-space models rely on linear differential or difference equations to
describe the system dynamics. Continuous-time models are of the form
dx
------ = Ax + Bu
dt
y = Cx + Du
where x is the state vector and u and y are the input and output vectors. Such
models may arise from the equations of physics, from state-space
identification, or by state-space realization of the system transfer function.
Use the command ss to create state-space models
sys = ss(A,B,C,D)
For a model with Nx states, Ny outputs, and Nu inputs
• A is an Nx-by-Nx real-valued matrix.
• B is an Nx-by-Nu real-valued matrix.
• C is an Ny-by-Nx real-valued matrix.
• D is an Ny-by-Nu real-valued matrix.
This produces an SS object sys that stores the state-space matrices
A, B, C, and D. For models with a zero D matrix, you can use D = 0 (zero) as a
shorthand for a zero matrix of the appropriate dimensions.
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Creating LTI Models
As an illustration, consider the following simple model of an electric motor.
dθ
d2θ
---------- + 2 ------ + 5θ = 3I
dt
dt 2
where θ is the angular displacement of the rotor and I the driving current. The
relation between the input current u = I and the angular velocity y = dθ ⁄ dt
is described by the state-space equations
dx = Ax + Bu
-----dt
y = Cx
where
θ
x = dθ
-----dt
A =
0 1
– 5 –2
B = 0
3
C = 01
This model is specified by typing
sys = ss([0 1; 5
2],[0;3],[0 1],0)
to which MATLAB responds
a =
x1
x2
x1
0
5.00000
x1
x2
u1
0
3.00000
y1
x1
0
y1
u1
0
x2
1.00000
2.00000
b =
c =
x2
1.00000
d =
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1
LTI Models
In addition to the A, B, C, and D matrices, the display of state-space models
includes state names, input names, and output names. Default names (here,
x1, x2, u1, and y1) are displayed whenever you leave these unspecified. See
“LTI Properties” on page 1-25 for more information on how to specify state,
input, or output names.
Descriptor State-Space Models
Descriptor state-space (DSS) models are a generalization of the standard
state-space models discussed above. They are of the form
dx
E ------ = Ax + Bu
dt
y = Cx + Du
The Control System Toolbox supports only descriptor systems with a
nonsingular E matrix. While such models have an equivalent explicit form
dx
------ = ( E – 1 A )x + ( E –1 B )u
dt
y = Cx + Du
it is often desirable to work with the descriptor form when the E matrix is
poorly conditioned with respect to inversion.
The function dss is the counterpart of ss for descriptor state-space models.
Specifically,
sys = dss(A,B,C,D,E)
creates a continuous-time DSS model with matrix data A,B,C,D,E. For
example, consider the dynamical model
dω
J -------- + Fω = T
dt
y=ω
with vector ω of angular velocities. If the inertia matrix J is poorly conditioned
with respect to inversion, you can specify this system as a descriptor model by
sys = dss( F,eye(n),eye(n),0,J)
1-16
% n = length of vector ω
Creating LTI Models
Frequency Response Data (FRD) Models
In some instances, you may only have sampled frequency response data, rather
than a transfer function or state-space model for the system you want to
analyze or control. For information on frequency response analysis of linear
systems, see Chapter 8 of [1].
For example, suppose the frequency response function for the SISO system you
want to model is G(w). Suppose, in addition, that you perform an experiment
to evaluate G(w) at a fixed set of frequencies, w 1, w 2, …, w n. You can do this by
driving the system with a sequence of sinusoids at each of these frequencies, as
depicted below.
sin w i t
G(w) =
yi ( t )
Here w i is the input frequency of each sinusoid, i = 1 ... n, and G(w) =
G ( w ) exp ( j ∠G ( w ) ) . The steady state output response of this system satisfies
y i ( t ) = G ( wi ) sin ( w i t + ∠G ( wi ) ) ; i = 1…n
A frequency response data (FRD) object is a model form you can use to store
frequency response data (complex frequency response, along with a
corresponding vector of frequency points) that you obtain either through
simulations or experimentally. In this example, the frequency response data is
obtained from the set of response pairs: { (G ( w i ),wi) }, i = 1…n .
Once you store your data in an FRD model, you can treat it as an LTI model,
and manipulate an FRD model in most of the same ways you manipulate TF,
SS, and ZPK models.
The basic syntax for creating a SISO FRD model is
sys = frd(response,frequencies,units)
where
• frequencies is a real vector of length Nf.
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LTI Models
• response is a vector of length Nf of complex frequency response values for
these frequencies.
• units is an optional string for the units of frequency: either 'rad/s' (default)
or 'Hz'
For example, the MAT-file LTIexamples.mat contains a frequency vector freq,
and a corresponding complex frequency response data vector respG. To load
this frequency-domain data and construct an FRD model, type
load LTIexamples
sys = frd(respG,freq)
Continuous-time frequency response with 1 output and 1 input
at 5 frequency points.
From input 1 to:
Frequency(rad/s)
output 1
----------------------1
0.812505 0.000312i
2
0.092593 0.462963i
4
0.075781 0.001625i
5
0.043735 0.000390i
The syntax for creating a MIMO FRD model is the same as for the SISO case,
except that response is a p-by-m-by-Nf multidimensional array, where p is the
number of outputs, m is the number of inputs, and Nf is the number of
frequency data points (the length of frequency).
The following table summarizes the complex-valued response data format for
FRD models.
Table 1-3: Data Format for the Argument response in FRD Models
1-18
Model Form
Response Data Format
SISO model
Vector of length Nf for which response(i) is the
frequency response at the frequency frequency(i)
Creating LTI Models
Table 1-3: Data Format for the Argument response in FRD Models (Continued)
Model Form
Response Data Format
MIMO model
with Ny outputs
and Nu inputs
Ny-by-Nu-by-Nf multidimensional array for which
response(i,j,k) specifies the frequency response
from input j to output i at frequency frequency(k)
S1-by-...-by-Sn
array of models
with Ny outputs
and Nu inputs
Ny-by-Nu-by-S1-by-...-by-Sn multidimensional array,
for which response(i,j,k,:) specifies the array of
frequency response data from input j to output i at
frequency frequency(k)
Discrete-Time Models
Creating discrete-time models is very much like creating continuous-time
models, except that you must also specify a sampling period or sample time for
discrete-time models. The sample time value should be scalar and expressed in
seconds. You can also use the value –1 to leave the sample time unspecified.
To specify discrete-time LTI models using tf, zpk, ss, or frd, simply append
the desired sample time value Ts to the list of inputs.
sys1
sys2
sys3
sys4
=
=
=
=
tf(num,den,Ts)
zpk(z,p,k,Ts)
ss(a,b,c,d,Ts)
frd(response,frequency,Ts)
For example,
h = tf([1
1],[1
0.5],0.1)
creates the discrete-time transfer function h ( z ) = ( z – 1 ) ⁄ ( z – 0.5 ) with
sample time 0.1 seconds, and
sys = ss(A,B,C,D,0.5)
specifies the discrete-time state-space model
x [ n + 1 ] = Ax [ n ] + Bu [ n ]
y [ n ] = Cx [ n ] + Du [ n ]
with sampling period 0.5 second. The vectors x [ n ], u [ n ], y [ n ] denote the
values of the state, input, and output vectors at the nth sample.
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LTI Models
By convention, the sample time of continuous-time models is Ts = 0. Setting
Ts = 1 leaves the sample time of a discrete-time model unspecified. For
example,
h = tf([1
0.2],[1 0.3], 1)
produces
Transfer function:
z
0.2
------z + 0.3
Sampling time: unspecified
Note Do not simply omit Ts in this case. This would make h a
continuous-time transfer function.
If you forget to specify the sample time when creating your model, you can still
set it to the correct value by reassigning the LTI property Ts. See “Sample
Time” on page 1-33 for more information on setting this property.
Discrete-Time TF and ZPK Models
You can specify discrete-time TF and ZPK models using tf and zpk as indicated
above. Alternatively, it is often convenient to specify such models by:
1 Defining the variable z as a particular discrete-time TF or ZPK model with
the appropriate sample time
2 Entering your TF or ZPK model directly as a rational expression in z.
This approach parallels the procedure for specifying continuous-time TF or
ZPK models using rational expressions. This procedure is described in “SISO
Transfer Function Models” on page 1-8 and “SISO Zero-Pole-Gain Models” on
page 1-12.
For example,
z = tf('z', 0.1);
H = (z+2)/(z^2 + 0.6*z + 0.9);
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Creating LTI Models
creates the same TF model as
H = tf([1 2], [1 0.6 0.9], 0.1);
Similarly,
z = zpk('z', 0.1);
H = [z/(z+0.1)/(z+0.2) ; (z^2+0.2*z+0.1)/(z^2+0.2*z+0.01)]
produces the single-input, two-output ZPK model
Zero/pole/gain from input to output...
z
#1: --------------(z+0.1) (z+0.2)
#2:
(z^2 + 0.2z + 0.1)
-----------------(z+0.1)^2
Sampling time: 0.1
Note that:
• The syntax z = tf('z') is equivalent to z = tf('z', 1) and leaves the
sample time unspecified. The same applies to z = zpk('z').
• Once you have defined z as indicated above, any rational expressions in z
creates a discrete-time model of the same type and with the same sample
time as z.
Discrete Transfer Functions in DSP Format
In digital signal processing (DSP), it is customary to write discrete transfer
functions as rational expressions in z –1 and to order the numerator and
denominator coefficients in ascending powers of z – 1. For example, the
numerator and denominator of
1 + 0.5z – 1
H ( z – 1 ) = ---------------------------------------1 + 2z – 1 + 3z – 2
would be specified as the row vectors [1 0.5] and [1 2 3], respectively. When
the numerator and denominator have different degrees, this convention
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1
LTI Models
clashes with the “descending powers of z ” convention assumed by tf (see
“Transfer Function Models” on page 1-8, or tf). For example,
h = tf([1 0.5],[1 2 3])
produces the transfer function
z + 0.5
---------------------------2
z + 2z + 3
which differs from H ( z – 1 ) by a factor z .
To avoid such convention clashes, the Control System Toolbox offers a separate
function filt dedicated to the DSP-like specification of transfer functions. Its
syntax is
h = filt(num,den)
for discrete transfer functions with unspecified sample time, and
h = filt(num,den,Ts)
to further specify the sample time Ts. This function creates TF objects just like
tf, but expects num and den to list the numerator and denominator coefficients
in ascending powers of z – 1 . For example, typing
h = filt([1 0.5],[1 2 3])
produces
Transfer function:
1 + 0.5 z^ 1
------------------1 + 2 z^ 1 + 3 z^ 2
Sampling time: unspecified
You can also use filt to specify MIMO transfer functions in z – 1. Just as for tf,
the input arguments num and den are then cell arrays of row vectors of
appropriate dimensions (see “Transfer Function Models” on page 1-8 for
details). Note that each row vector should comply with the “ascending powers
of z – 1 ” convention.
1-22
Creating LTI Models
Data Retrieval
The functions tf, zpk, ss, and frd pack the model data and sample time in a
single LTI object. Conversely, the following commands provide convenient data
retrieval for any type of TF, SS, or ZPK model sys, or FRD model sysfr.
[num,den,Ts] = tfdata(sys)
% Ts = sample time
[z,p,k,Ts] = zpkdata(sys)
[a,b,c,d,Ts] = ssdata(sys)
[a,b,c,d,e,Ts] = dssdata(sys)
[response,frequency,Ts] = frdata(sysfr)
Note that:
• sys can be any type of LTI object, except an FRD model
• sysfr, the input argument to frdata, can only be an FRD model
You can use any variable names you want in the output argument list of any of
these functions. The ones listed here correspond to the model property names
described in Tables 2-2 – 2.5.
The output arguments num and den assigned to tfdata, and z and p assigned
to zpkdata, are cell arrays, even in the SISO case. These cell arrays have as
many rows as outputs, as many columns as inputs, and their ijth entry
specifies the transfer function from the jth input to the ith output. For example,
H = [tf([1
1],[1 2 10]) , tf(1,[1 0])]
creates the one-output/two-input transfer function
s–1 H ( s ) = -----------------------------2
s + 2s + 10
1
--s
Typing
[num,den] = tfdata(H);
num{1,1}, den{1,1}
displays the coefficients of the numerator and denominator of the first input
channel.
ans =
0
1
1
1-23
1
LTI Models
ans =
1
2
10
Note that the same result is obtained using
H.num{1,1}, H.den{1,1}
See “Direct Property Referencing” on page 1-31 for more information about this
syntax.
To obtain the numerator and denominator of SISO systems directly as row
vectors, use the syntax
[num,den,Ts] = tfdata(sys,'v')
For example, typing
sys = tf([1 3],[1 2 5]);
[num,den] = tfdata(sys,'v')
produces
num =
0
1
3
2
5
den =
1
Similarly,
[z,p,k,Ts] = zpkdata(sys,'v')
returns the zeros, z, and the poles, p, as vectors for SISO systems.
1-24
LTI Properties
LTI Properties
The previous section shows how to create LTI objects that encapsulate the
model data and sample time. You also have the option to attribute additional
information, such as the input names or notes on the model history, to LTI
objects. This section gives a complete overview of the LTI properties, i.e., the
various pieces of information that can be attached to the TF, ZPK, SS, and FRD
objects. Type help ltiprops for online help on available LTI properties.
From a data structure standpoint, the LTI properties are the various fields in
the TF, ZPK, SS, and FRD objects. These fields have names (the property
names) and are assigned values (the property values). We distinguish between
generic properties, common to all four types of LTI objects, and model-specific
properties that pertain only to one particular type of model.
Generic Properties
The generic properties are those shared by all four types of LTI models (TF,
ZPK, SS, and FRD objects). They are listed in the table below.
Table 1-4: Generic LTI Properties
Property Name
Description
Property Value
ioDelay
I/O delay(s)
Matrix
InputDelay
Input delay(s)
Vector
InputGroup
Input channel groups
Cell array
InputName
Input channel names
Cell vector of strings
Notes
Notes on the model history
Text
OutputDelay
Output delay(s)
Vector
OutputGroup
Output channel groups
Cell array
OutputName
Output channel names
Cell vector of strings
Ts
Sample time
Scalar
Userdata
Additional data
Arbitrary
1-25
1
LTI Models
The sample time property Ts keeps track of the sample time (in seconds) of
discrete-time systems. By convention, Ts is 0 (zero) for continuous-time
systems, and Ts is 1 for discrete-time systems with unspecified sample time.
Ts is always a scalar, even for MIMO systems.
The InputDelay, OutputDelay, and ioDelay properties allow you to specify
time delays in the input or output channels, or for each input/output pair. Their
default value is zero (no delay). See “Time Delays” on page 1-42 for details on
modeling delays.
The InputName and OutputName properties enable you to give names to the
individual input and output channels. The value of each of these properties is
a cell vector of strings with as many cells as inputs or outputs. For example,
the OutputName property is set to
{ 'temperature' ; 'pressure' }
for a system with two outputs labeled temperature and pressure. The default
value is a cell of empty strings.
Using the InputGroup and OutputGroup properties of LTI objects, you can
create different groups of input or output channels, and assign names to the
groups. For example, you may want to designate the first four inputs of a
five-input model as controls, and the last input as noise. See “Input Groups
and Output Groups” on page 1-36 for more information.
Finally, Notes and Userdata are available to store additional information on
the model. The Notes property is dedicated to any text you want to supply with
your model, while the Userdata property can accommodate arbitrary
user-supplied data. They are both empty by default.
For more detailed information on how to use LTI properties, see “Additional
Insight into LTI Properties” on page 1-32.
Model-Specific Properties
The remaining LTI properties are specific to one of the four model types (TF,
ZPK, SS, or FRD). For single LTI models, these are summarized in the
1-26
LTI Properties
following four tables. The property values differ for LTI arrays. See set for
more information on these values.
Table 1-5: TF-Specific Properties
Property Name
Description
Property Value
den
Denominator(s)
Real cell array of row vectors
num
Numerator(s)
Real cell array of row vectors
Variable
Transfer function
variable
String 's', 'p', 'z', 'q', or
'z^ 1'
Table 1-6: ZPK-Specific Properties
Property Name
Description
Property Value
k
Gains
Two-dimensional real matrix
p
Poles
Cell array of column vectors
Variable
Transfer function
variable
String 's', 'p', 'z', 'q', or
'z^ 1'
z
Zeros
Cell array of column vectors
Table 1-7: SS-Specific Properties
Property Name
Description
Property Value
a
State matrix A
2-D real matrix
b
Input-to-state matrix B
2-D real matrix
c
State-to-output matrix C
2-D real matrix
d
Feedthrough matrix D
2-D real matrix
e
Descriptor E matrix
2-D real matrix
Nx
Number of states
Scalar integer
StateName
State names
Cell vector of strings
1-27
1
LTI Models
Table 1-8:
FRD-Specific Properties
Property Name
Description
Property Value
Frequency
Frequency data points
Real-valued vector
ResponseData
Frequency response
Complex-valued
multidimensional array
Units
Units for frequency
String 'rad/s' or 'Hz'
Most of these properties are dedicated to storing the model data. Note that the
E matrix is set to [] (the empty matrix) for standard state-space models, a
storage-efficient shorthand for the true value E = I .
The Variable property is only an attribute of TF and ZPK objects. This
property defines the frequency variable of transfer functions. The default
values are 's' (Laplace variable s ) in continuous time and 'z' (Z-transform
variable z ) in discrete time. Alternative choices include 'p' (equivalent to s )
and 'q' or 'z^ 1' for the reciprocal q = z –1 of the z variable. The influence of
the variable choice is mostly limited to the display of TF or ZPK models. One
exception is the specification of discrete-time transfer functions with tf (see tf
for details).
Note that tf produces the same result as filt when the Variable property is
set to 'z^ 1' or 'q'.
Finally, the StateName property is analogous to the InputName and OutputName
properties and keeps track of the state names in state-space models.
Setting LTI Properties
There are three ways to specify LTI property values:
• You can set properties when creating LTI models with tf, zpk, ss, or frd.
• You can set or modify the properties of an existing LTI model with set.
• You can also set property values using structure-like assignments.
This section discusses the first two options. See “Direct Property Referencing”
on page 1-31 for details on the third option.
1-28
LTI Properties
The function set for LTI objects follows the same syntax as its Handle
Graphics counterpart. Specifically, each property is updated by a pair of
arguments
PropertyName,PropertyValue
where
• PropertyName is a string specifying the property name. You can type the
property name without regard for the case (upper or lower) of the letters in
the name. Actually, you need only type any abbreviation of the property
name that uniquely identifies the property. For example, 'user' is sufficient
to refer to the Userdata property.
• PropertyValue is the value to assign to the property (see set for details on
admissible property values).
As an illustration, consider the following simple SISO model for a heating
system with an input delay of 0.3 seconds, an input called “energy,” and an
output called “temperature.”
energy
e – 0.3s
1
-----------s+1
temperature
delay
Figure 1-1: A Simple Heater Model
You can use a TF object to represent this delay system, and specify the time
delay, the input and output names, and the model history by setting the
corresponding LTI properties. You can either set these properties directly
when you create the LTI model with tf, or by using the set command.
For example, you can specify the delay directly when you create the model, and
then use the set command to assign InputName, OutputName, and Notes to
sys.
sys = tf(1,[1 1],'Inputdelay',0.3);
set(sys,'inputname','energy','outputname','temperature',...
'notes','A simple heater model')
1-29
1
LTI Models
Finally, you can also use the set command to obtain a listing of all settable
properties for a given LTI model type, along with valid values for these
properties. For the transfer function sys created above
set(sys)
produces
num: Ny-by-Nu cell of row vectors (Nu = no. of inputs)
den: Ny-by-Nu cell of row vectors (Ny = no. of outputs)
Variable: [ 's' | 'p' | 'z' | 'z^-1' | 'q' ]
Ts: scalar
InputDelay: Nu-by-1 vector
OutputDelay: Ny-by-1 vector
ioDelay: Ny-by-Nu array (I/O delays)
InputName: Nu-by-1 cell array of strings
OutputName: Ny-by-1 cell array of strings
InputGroup: M-by-2 cell array if M input groups
OutputGroup: P-by-2 cell array if P output groups
Notes: array or cell array of strings
UserData: arbitrary
Accessing Property Values Using get
You access the property values of an LTI model sys with get. The syntax is
PropertyValue = get(sys,PropertyName)
where the string PropertyName is either the full property name, or any
abbreviation with enough characters to identify the property uniquely. For
example, typing
h = tf(100,[1 5 100],'inputname','voltage',...
'outputn','current',...
'notes','A simple circuit')
get(h,'notes')
produces
ans =
'A simple circuit'
1-30
LTI Properties
To display all of the properties of an LTI model sys (and their values), use the
syntax get(sys). In this example,
get(h)
produces
num = {[0 0 100]}
den = {[1 5 100]}
Variable = 's'
Ts = 0
InputDelay = 0
OutputDelay = 0
ioDelay = 0
InputName = {'voltage'}
OutputName = {'current'}
InputGroup = {0x2 cell}
OutputGroup = {0x2 cell}
Notes = {'A simple circuit'}
UserData = []
Notice that default (output) values have been assigned to any LTI properties in
this list that you have not specified.
Finally, you can also access property values using direct structure-like
referencing. This topic is explained in the next section.
Direct Property Referencing
An alternative way to query/modify property values is by structure-like
referencing. Recall that LTI objects are basic MATLAB structures except for
the additional flag that marks them as TF, ZPK, SS, or FRD objects (see “LTI
Objects” on page 1-3). The field names for LTI objects are the property names,
so you can retrieve or modify property values with the structure-like syntax.
PropertyValue = sys.PropertyName% gets property value
sys.PropertyName = PropertyValue% sets property value
These commands are respectively equivalent to
PropertyValue = get(sys,'PropertyName')
set(sys,'PropertyName',PropertyValue)
For example, type
1-31
1
LTI Models
sys = ss(1,2,3,4,'InputName','u');
sys.a
and you get the value of the property “a” for the state-space model sys.
ans =
1
Similarly,
sys.a =
1;
resets the state transition matrix for sys to –1.
Unlike standard MATLAB structures, you do not need to type the entire field
name or use upper-case characters. You only need to type the minimum
number of characters sufficient to identify the property name uniquely. Thus
either of the commands
sys.InputName
sys.inputn
produces
ans =
'u'
Any valid syntax for structures extends to LTI objects. For example, given the
TF model h ( p ) = 1 ⁄ p
h = tf(1,[1,0],'variable','p');
you can reset the numerator to p + 2 by typing
h.num{1} = [1 2];
or equivalently, with
h.num{1}(2) = 2;
Additional Insight into LTI Properties
By reading this section, you can learn more about using the Ts, InputName,
OutputName, InputGroup, and OutputGroup LTI properties through a set of
examples. For basic information on Notes and Userdata, see “Generic
1-32
LTI Properties
Properties” on page 1-25. For detailed information on the use of InputDelay,
OutputDelay, and ioDelay, see “Time Delays” on page 1-42.
Sample Time
The sample time property Ts is used to specify the sampling period (in seconds)
for either discrete-time or discretized continuous-time LTI models. Suppose
you want to specify
z
H ( z ) = --------------------------2
2z + z + 1
as a discrete-time transfer function model with a sampling period of 0.5
seconds. To do this, type
h = tf([1 0],[2 1 1],0.5);
This sets the Ts property to the value 0.5, as is confirmed by
h.Ts
ans =
0.5000
For continuous-time models, the sample time property Ts is 0 by convention.
For example, type
h = tf(1,[1 0]);
get(h,'Ts')
ans =
0
To leave the sample time of a discrete-time LTI model unspecified, set Ts to – 1.
For example,
h = tf(1,[1
1], 1)
produces
Transfer function:
1
----z
1
1-33
1
LTI Models
Sampling time: unspecified
The same result is obtained by using the Variable property.
h = tf(1,[1
1],'var','z')
In operations that combine several discrete-time models, all specified sample
times must be identical, and the resulting discrete-time model inherits this
common sample time. The sample time of the resultant model is unspecified if
all operands have unspecified sample times. With this inheritance rule for Ts,
the following two models are equivalent.
tf(0.1,[1
1],0.1) + tf(1,[1 0.5], 1)
and
tf(0.1,[1
1],0.1) + tf(1,[1 0.5],0.1)
Note that
tf(0.1,[1
1],0.1) + tf(1,[1 0.5],0.5)
returns an error message.
??? Error using ==> lti/plus
In SYS1+SYS2, both models must have the same sample time.
Caution. Resetting the sample time of a continuous-time LTI model sys from
zero to a nonzero value does not discretize the original model sys. The
command
set(sys,'Ts',0.1)
only affects the Ts property and does not alter the remaining model data. Use
c2d and d2c to perform continuous-to-discrete and discrete-to-continuous
conversions. For example, use
sysd = c2d(sys,0.1)
to discretize a continuous system sys at a 10Hz sampling rate.
Use d2d to change the sample time of a discrete-time system and resample it.
Input Names and Output Names
You can use the InputName and OutputName properties (in short, I/O names) to
assign names to any or all of the input and output channels in your LTI model.
1-34
LTI Properties
For example, you can create a SISO model with input thrust, output
velocity, and transfer function H ( p ) = 1 ⁄ ( p + 10 ) by typing
h = tf(1,[1 10]);
set(h,'inputname','thrust','outputname','velocity',...
'variable','p')
Equivalently, you can set these properties directly by typing
h = tf(1,[1 10],'inputname','thrust',...
'outputname','velocity',...
'variable','p')
This produces
Transfer function from input "thrust" to output "velocity":
1
-----p + 10
Note how the display reflects the input and output names and the variable
selection.
In the MIMO case, use cell vectors of strings to specify input or output channel
names. For example, type
num = {3 , [1 2]};
den = {[1 10] , [1 0]};
H = tf(num,den);
% H(s) has one output and two inputs
set(H,'inputname',{'temperature' ; 'pressure'})
The specified input names appear in the display of H.
Transfer function from input "temperature" to output:
3
-----s + 10
Transfer function from input "pressure" to output:
s + 2
----s
1-35
1
LTI Models
To leave certain names undefined, use the empty string '' as in
H = tf(num,den,'inputname',{ 'temperature' ; '' })
Input Groups and Output Groups
In many applications, you may want to create several (distinct or intersecting)
groups of input or output channels and name these groups. For example, you
may want to label one set of input channels as noise and another set as
controls.
To see how input and output groups (I/O groups) work:
1 Create a random state-space model with one state, three inputs, and three
outputs.
2 Assign the first two inputs to a group named controls, the first output to a
group named temperature, and the last two outputs to a group named
measurements.
To do this, type
h = rss(1,3,3);
set(h, 'InputGroup',{[1 2] 'controls'})
set(h, 'OutputGroup', {[1] 'temperature'; [2 3] 'measurements'})
h
and MATLAB returns a state-space model of the following form.
a =
x1
x1
0.64884
x1
u1
0.12533
y1
y2
x1
1.1909
1.1892
b =
c =
1-36
u2
0
u3
0
LTI Properties
y3
0
y1
y2
y3
u1
0.32729
0
0
d =
I/O Groups:
Group Name
controls
temperature
measurements
I/O
I
O
O
u2
0
0
2.1832
u3
0.1364
0
0
Channel(s)
1,2
1
2,3
Continuous-time model.
Notice that the middle column of the I/O group listing indicates whether the
group is an input group (I) or an output group (O).
In general, to specify M input groups (or output groups), you need an M-by-2
cell array organized as follows.
Group Names
Vectors of Channel Indices
{
Channels for Group 1
,
Name for Group 1;
Channels for Group 2
,
Name for Group 2;
Channels for Group M
,
Name for Group M
}
Figure 1-2: Two Column Cell Array
When you specify the cell array for input (or output) groups, keep in mind:
• Each row of this cell array designates a different input (output) group.
1-37
1
LTI Models
• You can add input (or output) groups by appending rows to the cell array.
• You can choose not to assign any of the group names when you assign the
groups, and leave off the second column of this array. In that case,
- Empty strings are assigned to the group names by default.
- If you append rows to a cell array with no group names assigned, you have
to assign empty strings ('') to the group names.
For example,
h.InputGroup = [h.InputGroup; {[3] 'disturbance'}];
adds another input group called disturbance to h.
You can use regular cell array syntax for accessing or modifying I/O group
components. For example, to delete the first output group, temperature, type
h.OutputGroup(1,:) = []
ans =
[1x2 double]
'measurements'
Similarly, you can add or delete channels from an existing input or output
group. Recalling that input group channels are stored in the first column of the
corresponding cell array, to add channel three to the input group controls,
type
h.inputgroup{1,1} = [h.inputgroup{1,1} 3]
or, equivalently,
h.inputgroup{1,1} = [1 2 3]
1-38
Model Conversion
Model Conversion
There are four LTI model types you can use with the Control System Toolbox:
TF, ZPK, SS, and FRD. This section shows how to convert models from one type
to the other.
Explicit Conversion
Model conversions are performed by tf, ss, zpk, and frd. Given any TF, SS, or
ZPK model sys, the syntax for conversion to another model type is
sys = tf(sys)
% Conversion to TF
sys = zpk(sys)
% Conversion to ZPK
sys = ss(sys)
% Conversion to SS
sys = frd(sys,frequency)
% Conversion to FRD
Notice that FRD models can’t be converted to the other model types. In
addition, you must also include a vector of frequencies (frequency) as an input
argument when converting to an FRD model.
For example, you can convert the state-space model
sys = ss( 2,1,1,3)
to a zero-pole-gain model by typing
zpk(sys)
to which MATLAB responds
Zero/pole/gain:
3 (s+2.333)
----------(s+2)
Note that the transfer function of a state-space model with data ( A, B, C, D ) is
–1
H ( s ) = D + C ( sI – A ) B
for continuous-time models, and
1-39
1
LTI Models
–1
H ( z ) = D + C ( zI – A ) B
for discrete-time models.
Automatic Conversion
Some algorithms operate only on one type of LTI model. For example, the
algorithm for zero-order-hold discretization with c2d can only be performed on
state-space models. Similarly, commands like tfdata expect one particular
type of LTI models (TF). For convenience, such commands automatically
convert LTI models to the appropriate or required model type. For example, in
sys = ss(0,1,1,0)
[num,den] = tfdata(sys)
tfdata first converts the state-space model sys to an equivalent transfer
function in order to return numerator and denominator data.
Note that conversions to state-space models are not uniquely defined. For this
reason, automatic conversions to state space are disabled when the result
depends on the choice of state coordinates, for example, in commands like
initial or kalman.
Caution About Model Conversions
When manipulating or converting LTI models, keep in mind that:
• The three LTI model types TF, ZPK, and SS, are not equally well-suited for
numerical computations. In particular, the accuracy of computations using
high-order transfer functions is often poor. Therefore, it is often preferable to
work with the state-space representation. In addition, it is often beneficial to
balance and scale state-space models using ssbal. You get this type of
balancing automatically when you convert any TF or ZPK model to state
space using ss.
• Conversions to the transfer function representation using tf may incur a
loss of accuracy. As a result, the transfer function poles may noticeably differ
from the poles of the original zero-pole-gain or state-space model.
• Conversions to state space are not uniquely defined in the SISO case, nor are
they guaranteed to produce a minimal realization in the MIMO case. For a
given state-space model sys,
1-40
Model Conversion
ss(tf(sys))
may return a model with different state-space matrices, or even a different
number of states in the MIMO case. Therefore, if possible, it is best to avoid
converting back and forth between state-space and other model types.
1-41
1
LTI Models
Time Delays
Using the ioDelay, InputDelay, and OutputDelay properties of LTI objects,
you can specify delays in both continuous- and discrete-time LTI models. With
these properties, you can, for example, represent:
• LTI models with independent delays for each input/output pair. For
example, the continuous-time model with transfer function
H(s ) =
e
– 0.1s
10
2
--s
e
e
– 0.3s
– 0.2s
s+1
--------------s + 10
s–1
-----------s+5
• State-space models with delayed inputs and/or delayed outputs. For
example,
x· ( t ) = Ax ( t ) + Bu ( t – τ )
y ( t ) = Cx ( t – θ ) + Du ( t – ( θ + τ ) )
where τ is the time delay between the input u ( t ) and the state vector x ( t ) ,
and θ is the time delay between x ( t ) and the output y ( t ) .
You can assign the delay properties ioDelay, InputDelay, and OutputDelay
either when first creating your model with the tf, zpk, ss, or frd constructors,
or later with the set command (see “LTI Properties and Methods” on page 1-4
for details).
Supported Functionality
Most analysis commands support time delays, including:
• All time and frequency response commands
• Conversions between model types
• Continuous-to-discrete conversions (c2d)
• Horizontal and vertical concatenation
• Series, parallel, and feedback interconnections of discrete-time models with
delays
1-42
Time Delays
• Interconnections of continuous-time delay systems as long as the resulting
transfer function from input j to output i is of the form exp ( – sτ ij ) h ij ( s )
where h ij ( s ) is a rational function of s
• Padé approximation of time delays (pade)
Specifying Input/Output Delays
Using the ioDelay property, you can specify frequency-domain models with
independent delays in each entry of the transfer function. In continuous time,
such models have a transfer function of the form
e
– sτ 11
H( s) =
h 11 ( s )
... e
– sτ 1m
:
e
– sτ p1
h p1 ( s ) ... e
h 1m ( s )
:
– sτ p m
= [ exp ( – sτ ij ) h ij ( s ) ]
h pm ( s )
where the h ij ’s are rational functions of s , and τ ij is the time delay between
input j and output i . See “Specifying Delays in Discrete-Time Models” on page
1-49 for details on the discrete-time counterpart. We collectively refer to the
scalars τ ij as the I/O delays.
The syntax to create H ( s ) above is
H = tf(num,den,'ioDelay',Tau)
or
H = zpk(z,p,k,'ioDelay',Tau)
where
• num, den (respectively, z, p, k) specify the rational part [ h ij ( s ) ] of the transfer
function H ( s )
• Tau is the matrix of time delays for each I/O pair. That is, Tau(i,j) specifies
the I/O delay τ ij in seconds. Note that Tau and H ( s ) should have the same
row and column dimensions.
You can also use the ioDelay property in conjunction with state-space models,
as in
sys = ss(A,B,C,D,'ioDelay',Tau)
1-43
1
LTI Models
This creates the LTI model with the following transfer function.
H(s ) =
exp ( – sτ ij ) r ij ( s )
Here r ij ( s ) is the ( i, j ) entry of
–1
R ( s ) = D + C ( sI – A ) B
Note State-space models with I/O delays have only a frequency-domain
interpretation. They cannot, in general, be described by state-space equations
with delayed inputs and outputs.
Distillation Column Example
This example is adapted from [2] and illustrates the use of I/O delays in process
modeling. The process of interest is the distillation column depicted by the
figure below. This column is used to separate a mix of methanol and water (the
feed) into bottom products (mostly water) and a methanol-saturated distillate.
1-44
Time Delays
Enriched vapor
Cooling water
Condensate
Distillate
Feed
Reflux
Reboiler
Vapor
Steam flow
Bottom liquid
Bottom products
Figure 1-3: Distillation Column
Schematically, the distillation process functions as follows:
• Steam flows into the reboiler and vaporizes the bottom liquid. This vapor is
reinjected into the column and mixes with the feed
• Methanol, being more volatile than water, tends to concentrate in the vapor
moving upward. Meanwhile, water tends to flow downward and accumulate
as the bottom liquid
• The vapor exiting at the top of the column is condensed by a flow of cooling
water. Part of this condensed vapor is extracted as the distillate, and the rest
of the condensate (the reflux) is sent back to the column.
• Part of the bottom liquid is collected from the reboiler as bottom products
(waste).
The regulated output variables are:
• Percentage X D of methanol in the distillate
• Percentage X B of methanol in the bottom products.
1-45
1
LTI Models
The goal is to maximize X D by adjusting the reflux flow rate R and the steam
flow rate S in the reboiler.
To obtain a linearized model around the steady-state operating conditions, the
transient responses to pulses in steam and reflux flow are fitted by first-order
plus delay models. The resulting transfer function model is
– 1s
XD ( s )
XB ( s )
=
– 3s
12.8e
– 18.9e
------------------------ ------------------------16.7e + 1 21.0s + 1 R ( s )
– 7s
– 3s S ( s )
6.6e
– 19.4e
------------------------ ------------------------10.9s + 1 14.4s + 1
Note the different time delays for each input/output pair.
You can specify this MIMO transfer function by typing
H = tf({12.8 18.9;6.6 19.4},...
{[16.7 1] [21 1];[10.9 1] [14.4 1]},...
'iodelay',[1 3;7 3],...
'inputname',{'R' , 'S'},...
'outputname',{'Xd' , 'Xb'})
The resulting TF model is displayed as
Transfer function from input "R" to output...
12.8
Xd: exp( 1*s) * ---------16.7 s + 1
Xb:
6.6
exp( 7*s) * ---------10.9 s + 1
Transfer function from input "S" to output...
18.9
Xd: exp( 3*s) * -------21 s + 1
Xb:
1-46
19.4
exp( 3*s) * ---------14.4 s + 1
Time Delays
Specifying Delays on the Inputs or Outputs
While ideal for frequency-domain models with I/O delays, the ioDelay property
is inadequate to capture delayed inputs or outputs in state-space models. For
example, the two models
 ·
( M 1 )  x ( t ) = – x ( t ) + u ( t – 0.1 )
 y( t) = x( t)
 ·
( M2 )  z ( t ) = –z ( t ) + u( t )
 y ( t ) = z ( t – 0.1 )
share the same transfer function
– 0.1s
e
h ( s ) = ---------------s+1
As a result, they cannot be distinguished using the ioDelay property (the I/O
delay value is 0.1 seconds in both cases). Yet, these two models have different
state trajectories since x ( t ) and z ( t ) are related by
z ( t ) = x ( t – 0.1 )
Note that the 0.1 second delay is on the input in the first model, and on the
output in the second model.
InputDelay and OutputDelay Properties
When the state trajectory is of interest, you should use the InputDelay and
OutputDelay properties to distinguish between delays on the inputs and
delays on the outputs in state-space models. For example, you can accurately
specify the two models above by
M1 = ss( 1,1,1,0,'inputdelay',0.1)
M2 = ss( 1,1,1,0,'outputdelay',0.1)
In the MIMO case, you can specify a different delay for each input (or output)
channel by assigning a vector value to InputDelay (or OutputDelay). For
example,
sys = ss(A,[B1 B2],[C1;C2],[D11 D12;D21 D22])
sys.inputdelay = [0.1 0]
sys.outputdelay = [0.2 0.3]
creates the two-input, two-output model
1-47
1
LTI Models
x· ( t ) = Ax ( t ) + B 1 u 1 ( t – 0.1 ) + B 2 u 2 ( t )
y 1 ( t + 0.2 ) = C 1 x ( t ) + D 11 u 1 ( t – 0.1 ) + D 12 u 2 ( t )
y 2 ( t + 0.3 ) = C 2 x ( t ) + D 21 u 1 ( t – 0.1 ) + D 22 u 2 ( t )
You can also use the InputDelay and OutputDelay properties to conveniently
specify input or output delays in TF, ZPK, or FRD models. For example, you
can create the transfer function
1
--s
H(s ) =
2
-----------s+1
e
– 0.1s
by typing
s = tf('s');
H = [1/s ; 2/(s+1)];
H.inputdelay = 0.1
% rational part
The resulting model is displayed as
Transfer function from input to output...
1
#1: exp( 0.1*s) * s
#2:
2
exp( 0.1*s) * ----s + 1
By comparison, to produce an equivalent transfer function using the ioDelay
property, you would need to type
H = [1/s ; 2/(s+1)];
H.iodelay = [0.1 ; 0.1];
Notice that the 0.1 second delay is repeated twice in the I/O delay matrix. More
generally, for a TF, ZPK, or FRD model with input delays [ α 1, ..., α m ] and
output delays [ β 1, ..., β p ] , the equivalent I/O delay matrix is
1-48
Time Delays
α1 + β1
α2 + β1
α1 + β2
α2 + β2
:
α1 + βp
:
α2 + βp
...
αm + β1
αm + β2
...
:
αm + βp
Specifying Delays in Discrete-Time Models
You can also use the ioDelay, InputDelay, and OutputDelay properties to
specify delays in discrete-time LTI models. You specify time delays in
discrete-time models with integer multiples of the sampling period. The integer
k you supply for the time delay of a discrete-time model specifies a time delay
–k
of k sampling periods. Such a delay contributes a factor z to the transfer
function.
For example,
h = tf(1,[1 0.5 0.2],0.1,'inputdelay',3)
produces the discrete-time transfer function
Transfer function:
1
z^( 3) * ----------------z^2 + 0.5 z + 0.2
Sampling time: 0.1
Notice the z^( 3) factor reflecting the three-sampling-period delay on the
input.
Mapping Discrete-Time Delays to Poles at the Origin
Since discrete-time delays are equivalent to additional poles at z = 0 , they can
be easily absorbed into the transfer function denominator or the state-space
equations. For example, the transfer function of the delayed integrator
y[ k + 1] = y[ k ] + u[ k – 2 ]
is
1-49
1
LTI Models
–2
z
H ( z ) = ----------z–1
You can specify this model either as the first-order transfer function 1 ⁄ ( z – 1 )
with a delay of two sampling periods on the input
Ts = 1;
% sampling period
H1 = tf(1,[1 1],Ts,'inputdelay',2)
or directly as a third-order transfer function:
H2 = tf(1,[1
1 0 0],Ts)
% 1/(z^3 z^2)
While these two models are mathematically equivalent, H1 is a more efficient
representation both in terms of storage and subsequent computations.
When necessary, you can map all discrete-time delays to poles at the origin
using the command delay2z. For example,
H2 = delay2z(H1)
absorbs the input delay in H1 into the transfer function denominator to produce
the third-order transfer function
Transfer function:
1
--------z^3
z^2
Sampling time: 1
Note that
H2.inputdelay
now returns 0 (zero).
Retrieving Information About Delays
There are several ways to retrieve time delay information from a given LTI
model sys:
• Use property display commands to inspect the values of the ioDelay,
InputDelay, and OutputDelay properties. For example,
1-50
Time Delays
sys.iodelay
get(sys,'inputdelay')
• Use the helper function hasdelay to determine if sys has any delay at all.
The syntax is
hasdelay(sys)
which returns 1 (true) if sys has any delay, and 0 (false) otherwise
• Use the function totaldelay to determine the total delay between each input
and each output (cumulative contribution of the ioDelay, InputDelay, and
OutputDelay properties). Type help totaldelay or see the Reference pages
for details.
Padé Approximation of Time Delays
The function pade computes rational approximations of time delays in
continuous-time LTI models. The syntax is
sysx = pade(sys,n)
where sys is a continuous-time model with delays, and the integer n specifies
the Padé approximation order. The resulting LTI model sysx is of the same
type as sys, but is delay free.
For models with multiple delays or a mix of input, output, and I/O delays, you
can use the syntax
sysx = pade(sys,ni,no,nio)
where the vectors ni and no, and the matrix nio specify independent
approximation orders for each input, output, and I/O delay, respectively. Set
ni=[] if there are no input delays, and similarly for no and nio.
For example, consider the “Distillation Column Example” on page 1-44. The
two-input, two-output transfer function in this example is
– 1s
H( s) =
– 3s
18.9e
12.8e - –
----------------------------------------------16.7e + 1 21.0s + 1
– 7s
– 3s
– 19.4e
6.6e
------------------------ ------------------------10.9s + 1 14.4s + 1
1-51
1
LTI Models
To compute a Padé approximation of H(s) using:
• A first-order approximation for the 1 second and 3 second delays
• A second-order approximation for the 7 second delay,
type
pade(H,[],[],[1 1;2 1])
where H is the TF representation of H ( s ) defined in the distillation column
example. This command produces a rational transfer function.
Transfer function from input "R" to output...
12.8 s + 25.6
Xd: --------------------16.7 s^2 + 34.4 s + 2
Xb:
6.6 s^2
5.657 s + 1.616
--------------------------------------10.9 s^3 + 10.34 s^2 + 3.527 s + 0.2449
Transfer function from input "S" to output...
18.9 s
12.6
Xd: ---------------------21 s^2 + 15 s + 0.6667
Xb:
1-52
19.4 s
12.93
-------------------------14.4 s^2 + 10.6 s + 0.6667
Simulink Block for LTI Systems
Simulink Block for LTI Systems
You can incorporate LTI objects into Simulink diagrams using the LTI System
block shown below.
This mask is linked to an LTI
block in a Simulink diagram.
Double-click on the block in your
Simulink diagram to display or
modify model information.
The LTI System block can be accessed either by typing
ltiblock
at the MATLAB prompt or by selecting Control System Toolbox from the
Blocksets and Toolboxes section of the main Simulink library.
The LTI System block consists of the dialog box shown on the right in the figure
above. In the editable text box labeled LTI system variable, enter either the
variable name of an LTI object located in the MATLAB workspace (for
example, sys) or a MATLAB expression that evaluates to an LTI object (for
example, tf(1,[1 1])). The LTI System block accepts both continuous and
discrete LTI objects in either transfer function, zero-pole-gain, or state-space
form. Simulink converts the model to its state-space equivalent prior to
initializing the simulation.
Use the editable text box labeled Initial states to enter an initial state vector
for state-space models. The concept of “initial state” is not well-defined for
1-53
1
LTI Models
transfer functions or zero-pole-gain models, as it depends on the choice of state
coordinates used by the realization algorithm. As a result, you cannot enter
nonzero initial states when you supply TF or ZPK models to LTI blocks in a
Simulink diagram.
Note:
• For MIMO systems, the input delays stored in the LTI object must be either
all positive or all zero.
• LTI blocks in a Simulink diagram cannot be used for FRD models or LTI
arrays.
1-54
References
References
[1] Dorf, R.C. and R.H. Bishop, Modern Control Systems, Addison-Wesley,
Menlo Park, CA, 1998.
[2] Wood, R.K. and M.W. Berry, “Terminal Composition Control of a Binary
Distillation Column,” Chemical Engineering Science, 28 (1973), pp. 1707-1717.
1-55
1
LTI Models
1-56
2
Operations on LTI Models
IPrecedence and Property Inheritance . . . . . . . . 2-3
Extracting and Modifying Subsystems . .
Referencing FRD Models Through Frequencies
Referencing Channels by Name . . . . . . .
Resizing LTI Systems . . . . . . . . . . .
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Arithmetic Operations . . .
Addition and Subtraction . . .
Multiplication . . . . . . . .
Inversion and Related Operations
Transposition . . . . . . . .
Pertransposition . . . . . . .
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2-5
2-7
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2-9
Model Interconnection Functions . . . . . . . . . . 2-16
Concatenation of LTI Models . . . . . . . . . . . . . . 2-16
Feedback and Other Interconnection Functions . . . . . . 2-18
Continuous/Discrete Conversions of LTI Models
Zero-Order Hold . . . . . . . . . . . . . . . .
First-Order Hold . . . . . . . . . . . . . . . .
Tustin Approximation . . . . . . . . . . . . . .
Tustin with Frequency Prewarping . . . . . . . .
Matched Poles and Zeros . . . . . . . . . . . .
Discretization of Systems with Delays . . . . . . .
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Resampling of Discrete-Time Models . . . . . . . . . 2-26
References . . . . . . . . . . . . . . . . . . . . . 2-27
2
Operations on LTI Models
You can perform basic matrix operations such as addition, multiplication, or
concatenation on LTI models. Such operations are “overloaded,” which means
that they use the same syntax as they do for matrices, but are adapted so as to
apply to the LTI model context. These overloaded operations and their
interpretation in this context are discussed in this chapter. You can read about
discretization methods in this chapter as well. The following topics and
operations on LTI models are covered in this chapter:
• Precedence and Property Inheritance
• Extracting and Modifying Subsystems
• Arithmetic Operations
• Model Interconnection Functions
• Continuous/Discrete-Time Conversions of LTI Models
• Resampling of Discrete-Time Models
These operations can be applied to LTI models of different types. As a result,
before discussing operations on LTI models, we discuss model type precedence
and how LTI model properties are inherited when models are combined using
these operations. To read about how you can apply these operations to arrays
of LTI models, see “Operations on LTI Arrays” on page 4-24. To read about the
available functions with which you can analyze LTI models, see Chapter 5,
“Model Analysis Tools,”
2-2
Precedence and Property Inheritance
Precedence and Property Inheritance
You can apply operations to LTI models of different types. The resulting type
is then determined by the rules discussed in “Precedence Rules” on page 2-5.
For example, if sys1 is a transfer function and sys2 is a state-space model,
then the result of their addition
sys = sys1 + sys2
is a state-space model, since state-space models have precedence over transfer
function models.
To supersede the precedence rules and force the result of an operation to be a
given type, for example, a transfer function (TF), you can either
• Convert all operands to TF before performing the operation
• Convert the result to TF after performing the operation
Suppose, in the above example, you want to compute the transfer function of
sys. You can either use a priori conversion of the second operand
sys = sys1 + tf(sys2);
or a posteriori conversion of the result
sys = tf(sys1 + sys2)
Note These alternatives are not equivalent numerically; computations are
carried out on transfer functions in the first case, and on state-space models in
the second case.
Another issue is property inheritance, that is, how the operand property values
are passed on to the result of the operation. While inheritance is partly
operation-dependent, some general rules are summarized below:
• In operations combining discrete-time LTI models, all models must have
identical or unspecified (sys.Ts = 1) sample times. Models resulting from
such operations inherit the specified sample time, if there is one.
• Most operations ignore the Notes and Userdata properties.
2-3
2
Operations on LTI Models
• In general, when two LTI models sys1 and sys2 are combined using
operations such as +, *, [,], [;], append, and feedback, the resulting model
inherits its I/O names and I/O groups from sys1 and sys2. However,
conflicting I/O names or I/O groups are not inherited. For example, the
InputName property for sys1 + sys2 is left unspecified if sys1 and sys2 have
different InputName property values.
• A model resulting from operations on TF or ZPK models inherits its
Variable property value from the operands. Conflicts are resolved according
the following rules:
- For continuous-time models, 'p' has precedence over 's'.
- For discrete-time models, 'z^ 1' has precedence over 'q' and 'z', while
'q' has precedence over 'z'.
2-4
Extracting and Modifying Subsystems
Extracting and Modifying Subsystems
Subsystems relate subsets of the inputs and outputs of a system. The transfer
matrix of a subsystem is a submatrix of the system transfer matrix. For
example, if sys is a system with two inputs, three outputs, and I/O relation
y = Hu
then H ( 3, 1 ) gives the relation between the first input and third output.
y 3 = H ( 3,1 ) u 1
Accordingly, use matrix-like subindexing to extract this subsystem.
SubSys = sys(3,1)
The resulting subsystem SubSys is an LTI model of the same type as sys, with
its sample time, time delay, I/O name, and I/O group property values inherited
from sys.
For example, if sys has an input group named controls consisting of channels
one, two, and three, then SubSys also has an input group named controls with
the first channel of SubSys assigned to it.
If sys is a state-space model with matrices a, b, c, d, the subsystem sys(3,1)
is a state-space model with data a, b(:,1), c(3,:), d(3,1). Note the following
rules when extracting subystems:
• In the expression sys(3,1), the first index selects the output channel while
the second index selects the input channel.
• When extracting a subsystem from a given state-space model, the resulting
state-space model may not be minimal. Use the command sminreal to
eliminate unnecessary states in the subsystem.
You can use similar syntax to modify the LTI model sys. For example,
sys(3,1) = NewSubSys
redefines the I/O relation between the first input and third output, provided
NewSubSys is a SISO LTI model.
2-5
2
Operations on LTI Models
The following rules apply when modifying LTI models:
• sys, the LTI model that has had a portion reassigned, retains its original
model type (TF, ZPK, SS, or FRD) regardless of the model type of NewSubSys.
• Subsystem assignment does not reassign any I/O names or I/O group names
of NewSubSys that are already assigned to NewSubSys.
• Reassigning parts of a MIMO state-space model generally increases its
order.
• If NewSubSys is an FRD model, then sys must also be an FRD model.
Furthermore, their frequencies must match.
Other standard matrix subindexing extends to LTI objects as well. For
example,
sys(3,1:2)
extracts the subsystem mapping the first two inputs to the third output.
sys(:,1)
selects the first input and all outputs, and
sys([1 3],:)
extracts a subsystem with the same inputs, but only the first and third outputs.
For example, consider the two-input/two-output transfer function
1
----------------s + 0.1
. T(s) =
s–1
---------------------------s 2 + 2s + 2
0
1
--s
To extract the transfer function T 11 ( s ) from the first input to the first output,
type
T(1,1)
Transfer function:
1
------s + 0.1
2-6
Extracting and Modifying Subsystems
Next reassign T 11 ( s ) to 1 ⁄ ( s + 0.5 ) and modify the second input channel of T
by typing
T(1,1) = tf(1,[1 0.5]);
T(:,2) = [ 1 ; tf(0.4,[1 0]) ]
Transfer function from input 1 to output...
1
#1: ------s + 0.5
#2:
s
1
------------s^2 + 2 s + 2
Transfer function from input 2 to output...
#1: 1
#2:
0.4
--s
Referencing FRD Models Through Frequencies
You can extract subsystems from FRD models, as you do with other LTI model
types, by indexing into input and output (I/O) dimensions. You can also extract
subsystems by indexing into the frequencies of an FRD model.
To index into the frequencies of an FRD model, use the string 'Frequency' (or
any abbreviation, such as, 'freq', as long as it does not conflict with existing
I/O channel or group names) as a keyword. There are two ways you can specify
FRD models using frequencies:
• Using integers to index into the frequency vector of the FRD model
• Using a Boolean (logical) expression to specify desired frequency points in an
FRD model
For example, if sys is an FRD model with five frequencies, (e.g.,
sys.Frequency=[1 1.1 1.2 1.3 1.4]), then you can create a new FRD model
sys2 by indexing into the frequencies of sys as follows.
sys2 = sys('frequency', 2:3);
2-7
2
Operations on LTI Models
sys2.Frequency
ans =
1.1000
1.2000
displays the second and third entries in the frequency vector.
Similarly, you can use logical indexing into the frequencies.
sys2 = sys('frequency',sys.Frequency >1.0 & sys.Frequency <1.15);
sys2.freq
ans =
1.1000
You can also combine model extraction through frequencies with indexing into
the I/O dimensions. For example, if sys is an FRD model with two inputs, two
outputs, and frequency vector [2.1 4.2 5.3], with sys.Units specified in rad/
s, then
sys2 = sys(1,2,'freq',1)
specifies sys2 as a SISO FRD model, with one frequency data point, 2.1 rad/s.
Referencing Channels by Name
You can also extract subsystems using I/O group or channel names. For
example, if sys has an input group named noise, consisting of channels two,
four, and five, then
sys(1,'noise')
is equivalent to
sys(1,[2 4 5])
Similarly, if pressure is the name assigned to an output channel of the LTI
model sys, then
sys('pressure',1) = tf(1, [1 1])
reassigns the subsystem from the first input of sys to the output labeled
pressure.
2-8
Extracting and Modifying Subsystems
You can reference a set of channels by input or output name by using a cell
array of strings for the names. For example, if sys has one output channel
named pressure and one named temperature, then these two output channels
can be referenced using
sys({'pressure','temperature'})
Resizing LTI Systems
Resizing a system consists of adding or deleting inputs and/or outputs. To
delete the first two inputs, simply type
sys(:,1:2) = []
In deletions, at least one of the row/column indexes should be the colon (:)
selector.
To perform input/output augmentation, you can proceed by concatenation or
subassignment. Given a system sys with a single input, you can add a second
input using
sys = [sys,h];
or, equivalently, using
sys(:,2) = h;
where h is any LTI model with one input, and the same number of outputs as
sys. There is an important difference between these two options: while
concatenation obeys the precedence rules (see page 2-5), subsystem assignment
does not alter the model type. So, if sys and h are TF and SS objects,
respectively, the first statement produces a state-space model, and the second
statement produces a transfer function.
For state-space models, both concatenation and subsystem assignment
increase the model order because they assume that sys and h have
independent states. If you intend to keep the same state matrix and only
update the input-to-state or state-to-output relations, use set instead and
modify the corresponding state-space data directly. For example,
sys = ss(a,b1,c,d1)
set(sys,'b',[b1 b2],'d',[d1 d2])
2-9
2
Operations on LTI Models
adds a second input to the state-space model sys by appending the B and D
matrices. You should simultaneously modify both matrices with a single set
command. Indeed, the statements
sys.b = [b1 b2]
and
set(sys,'b',[b1 b2])
cause an error because they create invalid intermediate models in which the B
and D matrices have inconsistent column dimensions.
2-10
Arithmetic Operations
Arithmetic Operations
You can apply almost all arithmetic operations to LTI models, including those
shown below.
Operation
Description
+
Addition
–
Subtraction
*
Multiplication
/
Right matrix divide
\
Left matrix divide
inv
Matrix inversion
'
Pertransposition
.'
Transposition
^
Powers of an LTI model (as in s^2)
Addition and Subtraction
Adding LTI models is equivalent to connecting them in parallel. Specifically,
the LTI model
sys = sys1 + sys2
2-11
2
Operations on LTI Models
represents the parallel interconnection shown below.
sys1
y1
+
u
y
+
sys2
y2
sys
If sys1 and sys2 are two state-space models with data A 1, B 1, C 1, D 1 and
A 2, B 2, C 2, D 2 , the state-space data associated with sys1 + sys2 is
A1 0
,
0 A2
B1
,
C1 C2 ,
B2
D1 + D2
Scalar addition is also supported and behaves as follows: if sys1 is MIMO and
sys2 is SISO, sys1 + sys2 produces a system with the same dimensions as
sys1 whose ijth entry is sys1(i,j) + sys2.
Similarly, the subtraction of two LTI models
sys = sys1
sys2
is depicted by the following block diagram.
sys1
y1
+
u
–
sys2
y2
sys
2-12
y
Arithmetic Operations
Multiplication
Multiplication of two LTI models connects them in series. Specifically,
sys = sys1 * sys2
returns an LTI model sys for the series interconnection shown below.
v
u
sys2
sys1
y
Notice the reverse orders of sys1 and sys2 in the multiplication and block
diagram. This is consistent with the way transfer matrices are combined in a
series connection: if sys1 and sys2 have transfer matrices H 1 and H 2 , then
y = H1 v = H 1 ( H2 u ) = ( H1 × H2 ) u
For state-space models sys1 and sys2 with data A 1, B 1, C 1, D 1 and
A 2, B 2, C 2, D 2 , the state-space data associated with sys1*sys2 is
A1 B1 C 2
0
A2
B1 D2
,
B2
,
C1 D1 C2 ,
D1 D2
Finally, if sys1 is MIMO and sys2 is SISO, then sys1*sys2 or sys2*sys1 is
interpreted as an entry-by-entry scalar multiplication and produces a system
with the same dimensions as sys1, whose ijth entry is sys1(i,j)*sys2.
Inversion and Related Operations
Inversion of LTI models amounts to inverting the following input/output
relationship.
y = Hu
→
–1
u = H y
This operation is defined only for square systems (that is, systems with as
many inputs as outputs) and is performed using
inv(sys)
The resulting inverse model is of the same type as sys. Related operations
include:
2-13
2
Operations on LTI Models
• Left division sys1\sys2, which is equivalent to inv(sys1)*sys2
• Right division sys1/sys2, which is equivalent to sys1*inv(sys2)
For a state-space model sys with data A, B, C, D , inv(sys) is defined only
when D is a square invertible matrix, in which case its state-space data is
–1
A – BD C ,
BD
–1
,
–1
–D C ,
D
–1
Transposition
You can transpose an LTI model sys using
sys.'
This is a literal operation with the following effect:
• For TF models (with input arguments, num and den), the cell arrays num and
den are transposed.
• For ZPK models (with input arguments, z, p, and k), the cell arrays, z and p,
and the matrix k are transposed.
• For SS models (with model data A, B, C, D ), transposition produces the
state-space model AT, CT, BT, DT.
• For FRD models (with complex frequency response matrix Response), the
matrix of frequency response data at each frequency is transposed.
Pertransposition
For a continuous-time system with transfer function H ( s ) , the pertransposed
system has the transfer function
G ( s ) = [ H ( –s ) ]T
The discrete-time counterpart is
–1
G(z ) = [ H( z ) ]
T
Pertransposition of an LTI model sys is performed using
sys'
You can use pertransposition to obtain the Hermitian (conjugate) transpose of
the frequency response of a given system. The frequency response of the
2-14
Arithmetic Operations
pertranspose of H ( s ), G ( s ) = [ H ( – s ) ] T , is the Hermitian transpose of the
frequency response of H ( s ): G ( jw ) = H ( jw ) H .
To obtain the Hermitian transpose of the frequency response of a system sys
over a frequency range specified by the vector w, type
freqresp(sys', w);
2-15
2
Operations on LTI Models
Model Interconnection Functions
The Control System Toolbox provides a number of functions to help with the
model building process. These include model interconnection functions to
perform I/O concatenation ([,], [;], and append), general parallel and series
connections (parallel and series), and feedback connections (feedback and
lft). These functions are useful to model open- and closed-loop systems.
Interconnection Operator
Description
[,]
Concatenates horizontally
[;]
Concatenates vertically
append
Appends models in a block diagonal
configuration
augstate
Augments the output by appending states
connect
Forms an SS model from a block diagonal
LTI object for an arbitrary interconnection
matrix
feedback
Forms the feedback interconnection of two
models
lft
Produces the LFT interconnection
(Redheffer Star product) of two models
parallel
Forms the generalized parallel connection
of two models
series
Forms the generalized series connection of
two models
Concatenation of LTI Models
LTI model concatenation is done in a manner similar to the way you
concatenate matrices in MATLAB, using
sys = [sys1 , sys2]% horizontal concatenation
sys = [sys1 ; sys2]% vertical concatenation
2-16
Model Interconnection Functions
sys = append(sys1,sys2)% block diagonal appending
In I/O terms, horizontal and vertical concatenation have the following
block-diagram interpretations (with H 1 and H 2 denoting the transfer
matrices of sys1 and sys2).
H1
u1
H1
y1
H2
y2
+
+
u2
y
u
H2
y = H1 , H 2
u1
y1
u2
y2
Horizontal Concatenation
=
H1
u
H2
Vertical Concatenation
You can use concatenation as an easy way to create MIMO transfer functions
or zero-pole-gain models. For example,
H = [ tf(1,[1 0])
1 ; 0
tf([1
1],[1 1]) ]
specifies
1
--H( s) = s
0
1
s – 1----------s+1
Use
append(sys1,sys2)
2-17
2
Operations on LTI Models
to specify the block-decoupled LTI model interconnection.
u1
y1
sys1
sys1 0
0 sys2
u2
sys2
y2
Transfer Function
Appended Models
See append for more information on this function.
Feedback and Other Interconnection Functions
The following LTI model interconnection functions are useful for specifying
closed- and open-loop model configurations:
• feedback puts two LTI models with compatible dimensions in a feedback
configuration.
• series connects two LTI models in series.
• parallel connects two LTI models in parallel.
• lft performs the Redheffer star product on two LTI models.
• connect works with append to apply an arbitrary interconnection scheme to
a set of LTI models.
For example, if sys1 has m inputs and p outputs, while sys2 has p inputs and
m outputs, then the negative feedback configuration of these two LTI models
+
u
sys1
sys2
2-18
y
Model Interconnection Functions
is realized with
feedback(sys1,sys2)
This specifies the LTI model with m inputs and p outputs whose I/O map is
( I + sys1 ⋅ sys2 ) –1 sys1
See the reference pages online for more information on feedback, series,
parallel, lft, and connect.
2-19
2
Operations on LTI Models
Continuous/Discrete Conversions of LTI Models
The function c2d discretizes continuous-time TF, SS, or ZPK models.
Conversely, d2c converts discrete-time TF, SS, or ZPK models to continuous
time. Several discretization/interpolation methods are supported, including
zero-order hold (ZOH), first-order hold (FOH), Tustin approximation with or
without frequency prewarping, and matched poles and zeros.
The syntax
sysd = c2d(sysc,Ts);
sysc = d2c(sysd);
% Ts = sampling period in seconds
performs ZOH conversions by default. To use alternative conversion schemes,
specify the desired method as an extra string input:
sysd = c2d(sysc,Ts,'foh');% use first-order hold
sysc = d2c(sysd,'tustin');% use Tustin approximation
The conversion methods and their limitations are discussed next.
Zero-Order Hold
Zero-order hold (ZOH) devices convert sampled signals to continuous-time
signals for analyzing sampled continuous-time systems. The zero-order-hold
discretization H d ( z ) of a continuous-time LTI model H ( s ) is depicted in the
following block diagram.
y( t)
u(t)
u[k]
ZOH
H( s)
y[ k ]
Ts
Hd ( z )
The ZOH device generates a continuous input signal u(t) by holding each
sample value u[k] constant over one sample period.
u(t) = u[k] ,
2-20
kT s ≤ t ≤ ( k + 1 )T s
Continuous/Discrete Conversions of LTI Models
The signal u ( t ) is then fed to the continuous system H ( s ) , and the resulting
output y ( t ) is sampled every T s seconds to produce y [ k ] .
Conversely, given a discrete system H d ( z ) , the d2c conversion produces a
continuous system H ( s ) whose ZOH discretization coincides with H d ( z ) . This
inverse operation has the following limitations:
• d2c cannot operate on LTI models with poles at z = 0 when the ZOH is used.
• Negative real poles in the z domain are mapped to pairs of complex poles in
the s domain. As a result, the d2c conversion of a discrete system with
negative real poles produces a continuous system with higher order.
The next example illustrates the behavior of d2c with real negative poles.
Consider the following discrete-time ZPK model.
hd = zpk([], 0.5,1,0.1)
Zero/pole/gain:
1
------(z+0.5)
Sampling time: 0.1
Use d2c to convert this model to continuous-time
hc = d2c(hd)
and you get a second-order model.
Zero/pole/gain:
4.621 (s+149.3)
--------------------(s^2 + 13.86s + 1035)
Discretize the model again
c2d(hc,0.1)
and you get back the original discrete-time system (up to canceling the pole/
zero pair at z=–0.5):
Zero/pole/gain:
(z+0.5)
2-21
2
Operations on LTI Models
--------(z+0.5)^2
Sampling time: 0.1
First-Order Hold
First-order hold (FOH) differs from ZOH by the underlying hold mechanism.
To turn the input samples u [ k ] into a continuous input u ( t ) , FOH uses linear
interpolation between samples.
t – kT s
u ( t ) = u [ k ] + ------------------ ( u [ k + 1 ] – u [ k ] ) ,
Ts
kTs ≤ t ≤ ( k + 1 )T s
This method is generally more accurate than ZOH for systems driven by
smooth inputs. Due to causality constraints, this option is only available for
c2d conversions, and not d2c conversions.
Note This FOH method differs from standard causal FOH and is more
appropriately called triangle approximation (see [2], p. 151). It is also known
as ramp-invariant approximation because it is distortion-free for ramp inputs.
Tustin Approximation
The Tustin or bilinear approximation uses the approximation
z = e
sT s
1 + sT s ⁄ 2
≈ -------------------------1 – sT s ⁄ 2
to relate s-domain and z-domain transfer functions. In c2d conversions, the
discretization H d ( z ) of a continuous transfer function H ( s ) is derived by
H d ( z ) = H ( s' ) , where
2 z–1
s' = ------ -----------Ts z + 1
Similarly, the d2c conversion relies on the inverse correspondence
2-22
Continuous/Discrete Conversions of LTI Models
H ( s ) = H d ( z' ), where
1 + sT s ⁄ 2
z' = ------------------------1 – sT s ⁄ 2
Tustin with Frequency Prewarping
This variation of the Tustin approximation uses the correspondence
z–1
ω
s' = --------------------------------- -----------tan ( ωT s ⁄ 2 ) z + 1
H d ( z ) = H ( s' ) ,
This change of variable ensures the matching of the continuous- and
discrete-time frequency responses at the frequency ω .
H ( jω ) = H d ( e
jωT s
)
Matched Poles and Zeros
The matched pole-zero method applies only to SISO systems. The continuous
and discretized systems have matching DC gains and their poles and zeros
correspond in the transformation
z = e
sT s
See [2], p. 147 for more details.
Discretization of Systems with Delays
You can also use c2d to discretize SISO or MIMO continuous-time models with
time delays. If Ts is the sampling period used for discretization:
• A delay of tau seconds in the continuous-time model is mapped to a delay of
k sampling periods in the discretized model, where k = fix(tau/Ts).
• The residual fractional delay tau k*Ts is absorbed into the coefficients of
the discretized model (for the zero-order-hold and first-order-hold methods
only).
For example, to discretize the transfer function
H( s) = e
– 0.25s
10
------------------------------2
s + 3s + 10
2-23
2
Operations on LTI Models
using zero-order hold on the input, and a 10 Hz sampling rate, type
h = tf(10,[1 3 10],'inputdelay',0.25);
hd = c2d(h,0.1)
This produces the discrete-time transfer function
Transfer function:
0.01187 z^2 + 0.06408 z + 0.009721
z^( 2) * ---------------------------------z^3
1.655 z^2 + 0.7408 z
Sampling time: 0.1
Here the input delay in H ( s ) amounts to 2.5 times the sampling period of 0.1
seconds. Accordingly, the discretized model hd inherits an input delay of two
sampling periods, as confirmed by the value of hd.inputdelay. The residual
half-period delay is factored into the coefficients of hd by the discretization
algorithm.
The step responses of the continuous and discretized models are compared in
the figure below. This plot was produced by the command
2-24
Continuous/Discrete Conversions of LTI Models
step(h,'--',hd,'-')
Step Response
1.4
1.2
Amplitude
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (sec.)
Note The Tustin and matched pole/zero methods are accurate only for delays
that are integer multiples of the sampling period. It is therefore preferable to
use the zoh and foh discretization methods for models with delays.
2-25
2
Operations on LTI Models
Resampling of Discrete-Time Models
You can resample a discrete-time TF, SS, or ZPK model sys1 by typing
sys2 = d2d(sys1,Ts)
The new sampling period Ts does not have to be an integer multiple of the
original sampling period. For example, typing
h1 = tf([1 0.4],[1
h2 = d2d(h1,0.25);
0.7],0.1);
resamples h1 at the sampling period of 0.25 seconds, rather than 0.1 seconds.
You can compare the step responses of h1 and h2 by typing
step(h1,'--',h2,'-')
The resulting plot is shown on the figure below (h1 is the dashed line).
2-26
References
References
[1] Åström, K.J. and B. Wittenmark, Computer-Controlled Systems: Theory
and Design, Prentice-Hall, 1990, pp. 48–52.
[2] Franklin, G.F., J.D. Powell, and M.L. Workman, Digital Control of Dynamic
Systems, Second Edition, Addison-Wesley, 1990.
2-27
2
Operations on LTI Models
2-28
3
Model Analysis Tools
General Model Characteristics
. . . . . . . . . . . 3-2
Model Dynamics . . . . . . . . . . . . . . . . . . 3-4
State-Space Realizations
. . . . . . . . . . . . . . 3-7
3
Model Analysis Tools
General Model Characteristics
General model characteristics include the model type, I/O dimensions,
and continuous or discrete nature. Related commands are listed in the
table below. These commands operate on continuous- or discrete-time
LTI models or arrays of LTI models of any type.
General Model Characteristics Commands
class
Display model type ('tf', 'zpk', 'ss', or 'frd').
hasdelay
Test true if LTI model has any type of delay.
isa
Test true if LTI model is of specified class.
isct
Test true for continuous-time models.
isdt
Test true for discrete-time models.
isempty
Test true for empty LTI models.
isproper
Test true for proper LTI models.
issiso
Test true for SISO models.
ndims
Display the number of model/array dimensions.
reshape
Change the shape of an LTI array.
size
Output/input/array dimensions. Used with special
syntax, size also returns the number of state
dimensions for state-space models, and the number
of frequencies in an FRD model.
This example illustrates the use of some of these commands. See the
related reference pages for more details.
H = tf({1 [1
1]},{[1 0.1] [1 2 10]})
Transfer function from input 1 to output:
1
------s + 0.1
3-2
General Model Characteristics
Transfer function from input 2 to output:
s
1
-------------s^2 + 2 s + 10
class(H)
ans =
tf
size(H)
Transfer function with 2 input(s) and 1 output(s).
[ny,nu] = size(H)% Note: ny = number of outputs
ny =
1
nu =
2
isct(H)% Is this system continuous?
ans =
1
isdt(H)% Is this system discrete?
ans =
0
3-3
3
Model Analysis Tools
Model Dynamics
The Control System Toolbox offers commands to determine the system
poles, zeros, DC gain, norms, etc. You can apply these commands to
single LTI models or LTI arrays. The following table gives an overview of
these commands.
Model Dynamics
covar
Covariance of response to white noise.
damp
Natural frequency and damping of system poles.
dcgain
Low-frequency (DC) gain.
dsort
Sort discrete-time poles by magnitude.
esort
Sort continuous-time poles by real part.
norm
Norms of LTI systems ( H 2 and L ∞ ).
pole, eig
System poles.
pzmap
Pole/zero map.
zero
System transmission zeros.
With the exception of L ∞ norm, these commands are not supported for
FRD models.
Here is an example of model analysis using some of these commands.
h = tf([4 8.4 30.8 60],[1 4.12 17.4 30.8 60])
Transfer function:
4 s^3 + 8.4 s^2 + 30.8 s + 60
--------------------------------------s^4 + 4.12 s^3 + 17.4 s^2 + 30.8 s + 60
pole(h)
ans =
1.7971 + 2.2137i
3-4
Model Dynamics
1.7971
2.2137i
0.2629 + 2.7039i
0.2629
2.7039i
zero(h)
ans =
0.0500 + 2.7382i
0.0500
2.7382i
2.0000
dcgain(h)
ans =
1
[ninf,fpeak] = norm(h,inf)% peak gain of freq. response
ninf =
1.3402
% peak gain
fpeak =
1.8537
% frequency where gain peaks
These functions also operate on LTI arrays and return arrays. For
example, the poles of a three dimensional LTI array sysarray are
obtained as follows.
sysarray = tf(rss(2,1,1,3))
Model sysarray(:,:,1,1)
=======================
Transfer function:
-0.6201 s - 1.905
--------------------s^2 + 5.672 s + 7.405
Model sysarray(:,:,2,1)
=======================
Transfer function:
0.4282 s^2 + 0.3706 s + 0.04264
------------------------------s^2 + 1.056 s + 0.1719
3-5
3
Model Analysis Tools
Model sysarray(:,:,3,1)
=======================
Transfer function:
0.621 s + 0.7567
--------------------s^2 + 2.942 s + 2.113
3x1 array of continuous-time transfer functions.
pole(sysarray)
ans(:,:,1) =
-3.6337
-2.0379
ans(:,:,2) =
-0.8549
-0.2011
ans(:,:,3) =
-1.6968
-1.2452
3-6
State-Space Realizations
State-Space Realizations
The following functions are useful to analyze, perform state coordinate
transformations on, and derive canonical state-space realizations for
single state-space LTI models or LTI arrays of state-space models.
State-Space Realizations
canon
Canonical state-space realizations.
ctrb
Controllability matrix.
ctrbf
Controllability staircase form.
gram
Controllability and observability gramians.
obsv
Observability matrix.
obsvf
Observability staircase form.
ss2ss
State coordinate transformation.
ssbal
Diagonal balancing of state-space realizations.
The function ssbal uses a simple diagonal similarity transformation
–1
–1
( A, B, C ) → ( T AT, T B, CT )
to balance the state-space data ( A, B, C ) . This is accomplished by
reducing the norm of the matrix.
–1
T AT
CT
–1
T B
0
Such balancing usually improves the numerical conditioning of
subsequent state-space computations. Note that conversions to
state-space using ss produce balanced realizations of transfer functions
and zero-pole-gain models.
By contrast, the canonical realizations produced by canon, ctrbf, or
obsvf are often badly scaled, sensitive to perturbations of the data, and
3-7
3
Model Analysis Tools
poorly suited for state-space computations. Consequently, it is wise to
use them only for analysis purposes and not in control design algorithms.
3-8
4
Arrays of LTI Models
Introduction . . . . . . . . . . . . . . .
When to Collect a Set of Models in an LTI Array .
Restrictions for LTI Models Collected in an Array
Where to Find Information on LTI Arrays . . .
.
.
.
.
.
.
.
.
.
.
.
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.
.
.
4-2
4-2
4-2
4-3
The Concept of an LTI Array . . . . . . . . . . . . 4-4
Higher Dimensional Arrays of LTI Models . . . . . . . . 4-6
Dimensions, Size, and Shape of an LTI Array . . . . . 4-7
size and ndims . . . . . . . . . . . . . . . . . . . . 4-9
reshape . . . . . . . . . . . . . . . . . . . . . . 4-11
Building LTI Arrays . . . . . . . . .
Generating LTI Arrays Using rss . . . . .
Building LTI Arrays Using for Loops . . .
Building LTI Arrays Using the stack Function
Building LTI Arrays Using tf, zpk, ss, and frd
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4-12
4-12
4-12
4-15
4-17
Indexing Into LTI Arrays . . . . . . .
Accessing Particular Models in an LTI Array
Extracting LTI Arrays of Subsystems . . .
Reassigning Parts of an LTI Array . . . .
Deleting Parts of an LTI Array . . . . . .
.
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4-20
4-20
4-21
4-22
4-23
Operations on LTI Arrays . . . . . .
Example: Addition of Two LTI Arrays . .
Dimension Requirements . . . . . . .
Special Cases for Operations on LTI Arrays
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. 4-24
. 4-25
. 4-26
. 4-26
Other Operations on LTI Arrays
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. . . . . . . . . . . . 4-29
4
Arrays of LTI Models
Introduction
In many applications, it is useful to consider collections of linear, time
invariant (LTI) models. For example, you may want to consider a model with a
single parameter that varies, such as
sys1 = tf(1, [1 1 1]);
sys2 = tf(1, [1 1 2]);
sys3 = tf(1, [1 1 3]);
and so on. A convenient way to store and analyze a collection like this is to use
LTI arrays. Continuing this example, you can create this LTI array and store
all three transfer functions in one variable.
sys_ltia = (sys1, sys2, sys3);
You can use the LTI array sys_ltia just like you would use, for example, sys1.
You can use LTI arrays to collect a set of LTI models into a single MATLAB
variable. You then use this variable to manipulate or analyze the entire
collection of models in a vectorized fashion. You access the individual models
in the collection through indexing rather than by individual model names.
LTI arrays extend the concept of single LTI models in a similar way to how
multidimensional arrays extend two-dimensional matrices in MATLAB (see
Multidimensional Arrays in the MATLAB documentation).
When to Collect a Set of Models in an LTI Array
You can use LTI arrays to represent
• A set of LTI models arising from the linearization of a nonlinear system at
several operating points
• A collection of transfer functions that depend on one or more parameters
• A set of LTI models arising from several system identification experiments
applied to one plant
• A set of gain-scheduled LTI controllers
• A list of LTI models you want to collect together under the same name
Restrictions for LTI Models Collected in an Array
For each model in an LTI array, the following properties must be the same:
4-2
Introduction
• The number of inputs and outputs
• The sample time, for discrete-time models
• The I/O names and I/O groups
Note You cannot specify Simulink LTI blocks with LTI arrays.
Where to Find Information on LTI Arrays
The next two sections give examples that illustrate the concept of an LTI array,
its dimensions, and size. To read about how to build an LTI array, go to
“Building LTI Arrays” on page 4-12. The remainder of the chapter is devoted to
indexing and operations on LTI Arrays. You can also apply the analysis
functions in the Control System Toolbox to LTI arrays. See Chapter 5, “Model
Analysis Tools,” for more information on these functions. You can also view
response plots of LTI arrays with the LTI Viewer.
4-3
4
Arrays of LTI Models
The Concept of an LTI Array
To visualize the concept of an LTI array, consider the set of five transfer
function models shown below. In this example, each model has two inputs and
two outputs. They differ by parameter variations in the individual model
components.
1.1
-----------s+1
0
1.3
----------------s + 1.1
0
1.11
----------------s + 1.2
0
1.15
----------------s + 1.3
0
1.09
----------------s + 1.4
0
0
1
-----------s+5
0
1
----------------s + 5.2
0
1
----------------s + 5.4
0
1
----------------s + 5.6
0
1
----------------s + 5.8
Figure 4-1: Five LTI Models to be Collected in an LTI Array
This LTI array embodies a
1.09
----------------0
1-by-5 list of models.
1.15 s + 1.4
----------------0
1
s + 1.3 0
----------------s
+
5.8
1.11
1
----------------- 00 ---------------s + 1.2
s + 5.6
1.3
1
----------------0
----------------0
1.1s + 1.1
s + 5.4
------------ 0
1
s+1 0
----------------1 s + 5.2
0 -----------Each element of the LTI array
s+5
is a model.
Figure 4-2: An LTI Array Containing These Five Models
4-4
The Concept of an LTI Array
Just as you might collect a set of two-by-two matrices in a multidimensional
array, you can collect this set of five transfer function models as a list in an LTI
array under one variable name, say, sys. Each element of the LTI array is an
LTI model.
Individual models in the LTI array sys are accessed via indexing. The general
form for the syntax you use to access data in an LTI array is
sysa(Outputs,Inputs,Models)
The first index
selects the output
channels.
The second index
selects the input
channels.
The remaining indices select particular
models in the LTI array by their array
coordinates.
For example, you can access the third model in sys with sys(:,:,3). The
following illustrates how you can use indexing to select models or their
components from sys.
sysa(2,2,3) selects
1.09
----------------0
1.15
s
+ -1.4 0
---------------s + 1.3
1.11
1
----------------- 00 ---------------s + 1.2
1s + 5.8
----------------1.3
0
----------------0
1s + 5.6
s + 1.1
1.1
---------------------------- 0 0
s + 5.4
s+1
1
----------------0
1 s + 5.2
0 -----------s+5
1.11
----------------s + 1.2
0
0
1
----------------s + 5.4
the (2,2) entry of the
third model in the array.
sysa(:,:,3) selects the third model in the array.
Figure 4-3: Using Indices to Select Models and Their Components
See “Indexing Into LTI Arrays” for more information on indexing.
4-5
4
Arrays of LTI Models
Higher Dimensional Arrays of LTI Models
You can also collect a set of models in a two-dimensional array. The following
diagram illustrates a 2-by-3 array of six, two-output, one-input models called
m2d.
m2d(:,:,1,3)
Each entry in this 2-by-3 array of
models is a two-output, one-input
transfer function.
3.42
-------------------s + 2.84
7.29
m2d(:,:,1,1)
m2d(:,:,2,1)
3.36
----------------s + 2.9
7.23
m2d(:,:,1,2)
m2d(:,:,2,2)
3.4
-------------------s + 2.86
7.27
m2d(:,:,1,3)
m2d(:,:,2,3)
m2d(:,:,1,3) extracts the model in
the (1,3) position of the array.
3.45
-------------------s + 2.81
7.32
Figure 4-4: m2d: A 2-by-3 Array of Two-Output, One-Input Models
More generally, you can organize models into a 3-D or higher-dimensional
array, in much the same way you arrange numerical data into
multidimensional arrays (see Multidimensional Arrays in the MATLAB
documentation).
4-6
Dimensions, Size, and Shape of an LTI Array
Dimensions, Size, and Shape of an LTI Array
The dimensions and size of a single LTI model are determined by the output
and input channels. An array of LTI models has additional quantities that
determine its dimensions, size, and shape.
There are two sets of dimensions associated with LTI arrays:
• The I/O dimensions—the output dimension and input dimension common to
all models in the LTI array
• The array dimensions—the dimensions of the array of models itself
The size of the LTI array is determined by:
• The lengths of the I/O dimensions—the number of outputs (or inputs)
common to all models in the LTI array
• The length of each array dimension—the number of models along that array
dimension
4-7
4
Arrays of LTI Models
.
The next figure illustrates the concepts of dimension and size for the LTI array
m2d, a 2-by-3 array of one-input, two-output transfer function models.
m2d(:,:,1,1)
m2d(:,:,1,2)
m2d(:,:,1,3)
m2d(:,:,2,1)
m2d(:,:,2,2)
m2d(:,:,2,3)
T he
le n
gt h
of
th e
fi rs
t ar
r ay
di m
ens
ion
is 2
The length of the second array dimension is 3.
3.4
-------------------s + 2.86
7.27
3.45
-------------------s + 2.81
7.32
m2d(:,:,2,3)
Output dimension (length 2)
3.36
----------------s + 2.9
7.23
3.45
-------------------s + 2.81
7.32
Input dimension (length 1)
Figure 4-5: Dimensions and Size of m2d, an LTI Array
You can load this sample LTI array into your workspace by typing
load LTIexamples
size(m2d)
2x3 array of continuous-time transfer functions
Each transfer function has 2 outputs and 1 input.
The I/O dimensions correspond to the row and column dimensions of the
transfer matrix. The two I/O dimensions are both of length 1 for SISO models.
For MIMO models the lengths of these dimensions are given by the number of
outputs and inputs of the model.
Five related quantities are pertinent to understanding the array dimensions:
4-8
Dimensions, Size, and Shape of an LTI Array
• N, the number of models in the LTI array
• K, the number of array dimensions
• S 1 S 2 …S K, the list of lengths of the array dimensions
- S i is the number of models along the i
th
dimension.
• S 1 – by – S 2 – by – … – by – S K , the configuration of the models in the array
- The configuration determines the shape of the array.
- The product of these integers S 1 × S 2 × … × S K is N.
In the example model m2d,:
• The length of the output dimension, the first I/O dimension, is 2, since there
are two output channels in each model.
• The length of the input dimension, the second I/O dimension, is 1, since there
is only one input channel in each model.
• N, the number of models in the LTI array, is 6.
• K, the number of array dimensions, is 2.
• The array dimension lengths are [2 3].
• The array configuration is 2-by-3.
size and ndims
You can access the dimensions and shape of an LTI array using
• size to determine the lengths of each of the dimensions associated with an
LTI array
• ndims to determine the total number of dimensions in an LTI array
When applied to an LTI array, size returns
[Ny Nu S1 S2 ... Sk]
where
• Ny is the number of outputs common to all models in the LTI array.
• Nu is the number of inputs common to all models in the LTI array.
• S1 S2 ... Sk are the lengths of the array dimensions of a k-dimensional
array of models. Si is the number of models along the ith array dimension.
4-9
4
Arrays of LTI Models
Note the following when using the size function:
• By convention, a single LTI model is treated as a 1-by-1 array of models.
For single LTI models, size returns only the I/O dimensions [Ny Nu].
• For LTI arrays, size always returns at least two array dimensions. For
example, the size of a 2-by-1 LTI array in [Ny Nu 2 1]
• size ignores trailing singleton dimensions beyond the second array
dimension. For example, size returns [Ny Nu 2 3] for a 2-by-3-by-1-by-1 LTI
array of models with Ny outputs and Nu inputs.
The function ndims returns the total number of dimensions in an LTI array:
• 2, for single LTI models
• 2 + p, for LTI arrays, where p (greater than 2) is the number of array
dimensions
Note that
ndims (sys) = length(size(sys))
To see how these work on the sample 2-by-3 LTI array m2d of two-output,
one-input models, type
load LTIexamples
s = size(m2d)
s =
2
1
2
3
Notice that size returns a vector whose entries correspond to the length of
each of the four dimensions of m2d: two outputs and one input in a 2-by-3 array
of models. Type
ndims(m2d)
ans =
4
to see that there are indeed four dimensions attributed to this LTI array.
4-10
Dimensions, Size, and Shape of an LTI Array
reshape
Use reshape to reorganize the arrangement (array configuration) of the models
of an existing LTI array.
For example, to arrange the models in an LTI Array sys as a w 1 × … × w p
array, type
reshape(sys,w1,...,wp)
where w1,...,wp are any set of integers whose product is N, the number of
models in sys.
You can reshape the LTI array m2d into a 3-by-2, a 6-by-1, or a 1-by-6 array
using reshape. For example, type
load LTIexamples
sys = reshape(m2d,6,1);
size(sys)
6x1 array of continuous-time transfer functions
Each transfer function has 2 outputs and 1 inputs.
s = size(sys)
s =
2
1
6
1
4-11
4
Arrays of LTI Models
Building LTI Arrays
There are several ways to build LTI arrays:
• Using a for loop to assign each model in the array
• Using stack to concatenate LTI models into an LTI array
• Using tf, zpk, ss, or frd
In addition, you can use the command rss to generate LTI arrays of random
state-space models.
Generating LTI Arrays Using rss
A convenient way to generate arrays of state-space models with the same
number of states in each model is to use rss. The syntax is
rss(N,P,M,sdim1,...,sdimk)
where
• N is the number of states of each model in the LTI array.
• P is the number of outputs of each model in the LTI array.
• M is the number of inputs of each model in the LTI array.
• sdim1,...,sdimk are the lengths of the array dimensions.
For example, to create a 4-by-2 array of random state-space models with three
states, one output, and one input, type
sys = rss(3,2,1,4,2);
size(sys)
4x2 array of continuous-time state-space models
Each model has 2 outputs, 1 input, and 3 states.
Building LTI Arrays Using for Loops
Consider the following second-order SISO transfer function that depends on
two parameters, ζ and ω
ω2
. H ( s ) = --------------------------------------2
s + 2ζωs + ω 2
4-12
Building LTI Arrays
Suppose, based on measured input and output data, you estimate confidence
intervals [ω 1,ω 2] , and [ζ 1,ζ 2] for each of the parameters, ω and ζ. All of the
possible combinations of the confidence limits for these model parameter
values give rise to a set of four SISO models.
ω1
ω2
ω 12
ω 12
ζ 1 H 11 ( s ) = -------------------------------------------- H 12 ( s ) = -------------------------------------------2
2
s + 2ζ 1 ω 1 s + ω 1
s 2 + 2ζ 2 ω 1 s + ω 12
ω 22
ω 22
ζ 2 H ( s ) = -------------------------------------------=
-------------------------------------------H
(
s
)
22
21
s 2 + 2ζ 2 ω 2 s + ω 22
s 2 + 2ζ 1 ω 2 s + ω 22
Figure 4-6: Four LTI Models Depending on Two Parameters
You can arrange these four models in a 2-by-2 array of SISO transfer functions
called H.
ω1
ω2
ζ1
H(:,:,1,1)
H(:,:,1,2)
ζ2
H(:,:,2,1)
H(:,:,2,2)
Each entry of this 2-by-2 array is
a SISO transfer function model.
Figure 4-7: The LTI Array H
Here, for i,j ∈ { 1, 2 } ,
H(:,:,i,j)
represents the transfer function
ω j2
--------------------------------------------s 2 + 2ζ ω s + ω 2
i j
j
corresponding to the parameter values ζ = ζ i and ω = ω j .
4-13
4
Arrays of LTI Models
The first two colon indices ( : ) select all I/O channels from the I/O dimensions
of H. The third index of H refers to the first array dimension ( ζ), while the fourth
index is for the second array dimension (ω).
Suppose the limits of the ranges of values for ζ and ω are [0.66,0.76] and
[1.2,1.5], respectively. Enter these at the command line.
zeta = [0.66,0.75];
w = [1.2,1.5];
Since the four models have the same parametric structure, it’s convenient to
use two nested for loops to construct the LTI array.
for i = 1:2
for j = 1:2
H(:,:,i,j) = tf(w(j)^2,[1 2*zeta(i)*w(j) w(j)^2]);
end
end
H now contains the four models in a 2-by-2 array. For example, to display the
transfer function in the (1,2) position of the array, type
H(:,:,1,2)
Transfer function:
2.25
------------------s^2 + 1.98 s + 2.25
4-14
Building LTI Arrays
For the purposes of efficient computation, you can initialize an LTI array to
zero, and then reassign the entire array to the values you want to specify. The
general syntax for zero assignment of LTI arrays is
Lengths of the output/input dimensions
Lengths of the array dimensions
sysa = tf(zeros(Ny,Nu,S1,...,SK))
sysa = zpk(zeros(Ny,Nu,S1,...,SK))
sysa = ss(zeros(Ny,Nu,S1,...,SK,Nx))
sysa = frd(zeros(Ny,Nu,Nf,S1,...,SK))
The number of frequency vectors in the FRD
The maximum number of states in any model in the LTI array
To initialize H in the above example to zero, type
H = tf(zeros(1,1,2,2));
before you implement the nested for loops.
Building LTI Arrays Using the stack Function
Another way to build LTI arrays is using the function stack. This function
operates on single LTI models as well as LTI arrays. It concatenates a list of
LTI arrays or single LTI models only along the array dimension. The general
syntax for stack is
stack(Arraydim,sys1,sys2...)
where
• Arraydim is the array dimension along which to concatenate the LTI models
or arrays.
• sys1, sys2, ... are the LTI models or LTI arrays to be concatenated.
4-15
4
Arrays of LTI Models
When you concatenate several models or LTI arrays along the jth array
dimension, such as in
stack(j,sys1,sys2,...,sysn)
• The lengths of the I/O dimensions of sys1,...,sysn must all match.
• The lengths of all but the jth array dimension of sys1,...,sysn must match.
For example, if two TF models sys1 and sys2 have the same number of inputs
and outputs,
sys = stack(1,sys1,sys2)
concatenates them into a 2-by-1 array of models.
There are two principles that you should keep in mind:
• stack only concatenates along an array dimension, not an I/O dimension.
• To concatenate LTI models or LTI arrays along an input or output
dimension, use the bracket notation ([,] [;]). See “Model Interconnection
Functions” for more information on the use of bracket notation to
concatenate models. See also “Special Cases for Operations on LTI Arrays”
for some examples of this type of concatenation of LTI arrays.
Here’s an example of how to build the LTI array H using the function stack.
% Set up the parameter vectors.
zeta = [0.66,0.75];
w = [1.2,1.5];
% Specify the four
%
H11 = tf(w(1)^2,[1
H12 = tf(w(2)^2,[1
H21 = tf(w(1)^2,[1
H22 = tf(w(2)^2,[1
individual models with those parameters.
2*zeta(1)*w(1)
2*zeta(1)*w(2)
2*zeta(2)*w(1)
2*zeta(2)*w(2)
w(1)^2]);
w(2)^2]);
w(1)^2]);
w(2)^2]);
% Set up the LTI array using stack.
COL1 = stack(1,H11,H21); % The first column of the 2-by-2 array
COL2 = stack(1,H12,H22); % The second column of the 2-by-2 array
H = stack(2, COL1, COL2); % Concatenate the two columns of models.
4-16
Building LTI Arrays
Notice that this result is very different from the single MIMO LTI model
returned by
H = [H11,H12;H21,H22];
Accessing LTI Arrays of Variable Order
For arrays of state-space models with variable order, you cannot use the dot
operator (e.g., sys.a) to access arrays. Use the syntax
[a,b,c,d] = ssdata(sys,'cell')
to extract the state-space matrices of each model as separate cells in the cell
arrays a, b, c, and d.
Building LTI Arrays Using tf, zpk, ss, and frd
You can also build LTI arrays using the tf, zpk, ss, and frd constructors. You
do this by using multidimensional arrays in the input arguments for these
functions.
Specifying Arrays of TF models tf
For TF models, use
sys = tf(num,den)
where
• Both num and den are multidimensional cell arrays the same size as sys (see
“size and ndims” on page 4-9).
• sys(i,j,n1,...,nK) is the (i, j) entry of the transfer matrix for the model
located in the ( n 1, …, n K ) position of the array.
• num(i,j,n1,...,nK) is a row vector representing the numerator polynomial
of sys(i,j,n1,...,nK).
• den(i,j,n1,...,nK) is a row vector representing denominator polynomial
of sys(i,j,n1,...,nK).
See “MIMO Transfer Function Models” on page 2-10 for related information on
the specification of single TF models.
Specifying Arrays of ZPK Models Using zpk
For ZPK models, use
4-17
4
Arrays of LTI Models
sys = zpk(zeros,poles,gains)
where
• Both zeros and poles are multidimensional cell arrays whose cell entries
contain the vectors of zeros and poles for each I/O pair of each model in the
LTI array.
• gains is a multidimensional array containing the scalar gains for each I/O
pair of each model in the array.
• The dimensions (and their lengths) of zeros, poles, and gains, determine
those of the LTI array, sys.
Specifying Arrays of SS Models Using ss
To specify arrays of SS models, use
sys = ss(a,b,c,d)
where a, b, c, and d are real-valued multidimensional arrays of appropriate
dimensions. All models in the resulting array of SS models have the same
number of states, outputs, and inputs.
Note You cannot use the ss constructor to build an array of state-space
models with different numbers of states. Use stack to build such LTI arrays.
The Size of LTI Array Data for SS Models
The size of the model data for arrays of state-space models is summarized in
the following table.
Data
4-18
Size (Data)
a
N s N s S 1S 2 …S K
b
N s N u S 1 S 2 …S K
c
Ny Ns S1 S2 … SK
d
N y N u S 1 S 2 …S K
Building LTI Arrays
where
• N s is the maximum of the number of states in each model in the array.
• N u is the number of inputs in each model.
• N y is the number of outputs in each model.
• S 1, S 2, …, S K are the lengths of the array dimensions.
Specifying Arrays of FRD Models Using frd
To specify a K-dimensional array of p-output, m-input FRD models for which
S 1, S 2, …, S K are the lengths of the array dimensions, use
sys = frd(response,frequency,units)
where
• frequency is a real vector of n frequency data points common to all FRD
models in the LTI array.
• response is a p-by-m-by-n-by- S 1 -by- …-by- S K complex-valued
multidimensional array.
• units is the optional string specifying 'rad/s' or 'Hz'.
Note that for specifying an LTI array of SISO FRD models, response can also
be a multidimensional array of 1-by-n matrices whose remaining dimensions
determine the array dimensions of the FRD.
4-19
4
Arrays of LTI Models
Indexing Into LTI Arrays
You can index into LTI arrays in much the same way as you would for
multidimensional arrays to
• Access models
• Extract subsystems
• Reassign parts of an LTI array
• Delete parts of an LTI array
When you index into an LTI array sys, the indices should be organized
according to the following format
sys(Outputs, Inputs, n 1, …, n K )
where
• Outputs are indices that select output channels.
• Inputs are indices that select input channels.
• n 1, …, n K are indices into the array dimensions that select one model or a
subset of models in the LTI array.
Note on Indexing into LTI Arrays of FRD models. For FRD models, the array indices
can be followed by the keyword 'frequency' and some expression selecting a
subset of the frequency points as in
sys (outputs, inputs, n1,...,nk, 'frequency', SelectedFreqs)
See “Referencing FRD Models Through Frequencies” for details on frequency
point selection in FRD models.
Accessing Particular Models in an LTI Array
To access any given model in an LTI array:
• Use colon arguments (:,:) for the first two indices to select all I/O channels.
• The remaining indices specify the model coordinates within the array.
For example, if sys is a 5-by-2 array of state-space models defined by
sys = rss(4,3,2,5,2);
4-20
Indexing Into LTI Arrays
you can access (and display) the model located in the (3,2) position of the array
sys by typing
sys(:,:,3,2)
If sys is a 5-by-2 array of 3-output, 2-input FRD models, with frequency vector
[1,2,3,4,5], then you can access the response data corresponding to the
middle frequency (3 rad/s), of the model in the (3,1) position by typing
sys(:,:,3,1,'frequency',3.0)
To access all frequencies of this model in the array, you can simply type
sys(:,:,3,1)
Single Index Referencing of Array Dimensions
You can also access models using single index referencing of the array
dimensions.
For example, in the 5-by-2 LTI array sys above, you can also access the model
located in the (3,2) position by typing
sys(:,:,8)
since this model is in the eighth position if you were to list the 10 models in the
array by successively scanning through its entries along each of its columns.
For more information on single index referencing, see “Advanced Indexing”
under “M-File Programming” in the MATLAB online documentation.
Extracting LTI Arrays of Subsystems
To select a particular subset of I/O channels from all the models in an LTI
array, use the syntax described in “Extracting and Modifying Subsystems” on
page 3-5. For example,
sys = rss(4,3,2,5,2);
A = sys(1, [1 2])
or equivalently,
A = sys(1,[1 2],:,:)
4-21
4
Arrays of LTI Models
selects the first two input channels, and the first output channel in each model
of the LTI array A, and returns the resulting 5-by-2 array of one-output,
two-input subsystems.
You can also combine model selection with I/O selection within an LTI array.
For example, to access both:
• The state-space model in the (3,2) array position
• Only the portion of that model relating the second input to the first output
type
sys(1,2,3,2)
To access the subsystem from all inputs to the first two output channels of this
same array entry, type
sys(1:2,:,3,2)
Reassigning Parts of an LTI Array
You can reassign entire models or portions of models in an LTI array. For
example,
sys = rss(4,3,2,5,2); % 5X2 array of state-space models
H = rss(4,1,1,5,2);
% 5X2 array of SISO models
sys(1,2) = H
reassigns the subsystem from input two to output one, for all models in the LTI
array sys. This SISO subsystem of each model in the LTI array is replaced
with the LTI array H of SISO models. This one-line assignment command is
equivalent to the following 10-step nested for loop.
for k = 1:5
for j = 1:2
sys(1,2,k,j) = H(:,:,k,j);
end
end
Notice that you don’t have to use the array dimensions with this assignment.
This is because I/O selection applies to all models in the array when the array
indices are omitted.
Similarly, the commands
4-22
Indexing Into LTI Arrays
sys(:,:,3,2) = sys(:,:,4,1);
sys(1,2,3,2) = 0;
reassign the entire model in the (3,2) position of the LTI array sys and the (1,2)
subsystem of this model, respectively.
Deleting Parts of an LTI Array
You can use indexing to delete any part of an LTI array by reassigning it to be
empty ([]). For instance,
sys = rss(4,3,2,5,2);
sys(1,:) = [];
size(sys)
5x2 array of continuous-time state-space models
Each model has 2 outputs, 2 inputs, and 4 states.
deletes the first output channel from every model of this LTI array.
Similarly,
sys(:,:,[3 4],:) = []
deletes the third and fourth rows of this two-dimensional array of models.
4-23
4
Arrays of LTI Models
Operations on LTI Arrays
Using LTI arrays, you can apply almost all of the basic model operations that
work on single LTI models to entire sets of models at once. These basic
operations include
• The arithmetic operations: +, , *, /,\,',.'
• The functions: concatenation along I/O dimensions ([,], [;]), feedback,
append, series, parallel, and lft
When you apply any of these operations to two (or more) LTI arrays (for
example, sys1 and sys2), the operation is implemented on a model-by-model
basis. Therefore, the kth model of the resulting LTI array is derived from the
application of the given operation to the kth model of sys1 and the kth model
of sys2.
For example, if sys1 and sys2 are two LTI arrays and
sysa = op(sys1,sys2)
then the kth model in the resulting LTI array sys is obtained by adding the kth
models in sys1 to the kth model in sys2
sysa(:,:,k) = sys1(:,:,k) + sys2(:,:,k)
You can also apply any of the response plotting functions such as step, bode,
and nyquist to LTI arrays. These plotting functions are also applied on a model
by model basis.
4-24
Operations on LTI Arrays
Example: Addition of Two LTI Arrays
The following diagram illustrates the addition of two 3-by-1 LTI arrays
sys1+sys2.
sys2(:,:,3)
sys1(:,:,3
1
-----------s+2
+
sys1(:,:,2)
1
----------------s + 2.5
sys1
2s + 6.5
--------------------------------2
s + 6.5s + 9
=
sys2(:,:,2)
+
sys1(:,:,1
1
----------------s + 2.9
1 ---------------s + 4.5
sys(:,:,3)
2.1
-----------s+4
sys(:,:,2)
=
3.15 + 9.25
-----------------------------------2
s + 6.5s + 10
sys(:,:,1)
sys2(:,:,1)
+
1.5
----------------s + 3.9
=
+
sys2
=
2.5s + 8.25
-------------------------------------------2
s + 6.8s + 11.31
sysa
Figure 4-8: The Addition of Two LTI Arrays
The summation of these LTI arrays
sysa = sys1+sys2
is equivalent to the following model-by-model summation:
for k = 1:3
sysa(:,:,k)=sys1(:,:,k) + sys2(:,:,k)
end
4-25
4
Arrays of LTI Models
Note that:
• Each model in sys1 and sys2 must have the same number of inputs and
outputs. This is required for the addition of two LTI arrays.
• The lengths of the array dimensions of sys1 and sys2 must match.
Dimension Requirements
In general, when you apply any of these basic operations to two or more LTI
arrays:
• The I/O dimensions of each of the LTI arrays must be compatible with the
requirements of the operation.
• The lengths of array dimensions must match.
The I/O dimensions of each model in the resulting LTI array are determined by
the operation being performed. See Chapter 3, “Operations on LTI Models,” for
requirements on the I/O dimensions for the various operations.
For example, if sys1 and sys2 are both 1-by-3 arrays of LTI models with two
inputs and two outputs, and sys3 is a 1-by-3 array of LTI models with two
outputs and 1 input, then
sys1 + sys2
is an LTI array with the same dimensions as sys1 and sys2.
sys1 * sys3
is a 1-by-3 array of LTI models with two outputs and one input, and
[sys1,sys3]
is a 1-by-3 array of LTI models with two outputs and three inputs.
Special Cases for Operations on LTI Arrays
There are some special cases in coding operations on LTI arrays.
Consider
sysa = op(sys1,sys2)
where op is a symbol for the operation being applied. sys1 is an LTI array, and
sysa (the result of the operation) is an LTI array with the same array
4-26
Operations on LTI Arrays
dimensions as sys1. You can use shortcuts for coding sysa = op(sys1,sys2)
in the following cases:
• For operations that apply to LTI arrays, sys2 does not have to be an array.
It can be a single LTI model (or a gain matrix) whose I/O dimensions satisfy
the compatibility requirements for op (with those of each of the models in
sys1). In this case, op applies sys2 to each model in sys1, and the kth model
in sys satisfies
sysa(:,:,k) = op(sys1(:,:,k),sys2)
• For arithmetic operations, such as +, *, /, and \, sys2 can be either a single
SISO model, or an LTI array of SISO models, even when sys1 is an LTI array
of MIMO models. This special case relies on the MATLAB scalar expansion
capabilities for arithmetic operations.
- When sys2 is a single SISO LTI model (or a scalar gain), op applies sys2
to sys1 on an entry-by-entry basis. The ijth entry in the kth model in sysa
satisfies
sysa(i,j,k) = op(sys1(i,j,k),sys2)
- When sys2 is an LTI array of SISO models (or a multidimensional array
of scalar gains), op applies sys2 to sys1 on an entry-by-entry basis for each
model in sysa.
sysa(i,j,k) = op(sys1(i,j,k),sys2(:,:,k))
Examples of Operations on LTI Arrays with Single LTI Models
Suppose you want to create an LTI array containing three models, where, for
τ in the set { 1.1, 1.2, 1.3 } , each model H τ ( s ) has the form
1
----------s
+τ
Hτ ( s ) =
–1
0
1
--s
You can do this efficiently by first setting up an LTI array h containing the
SISO models 1 ⁄ ( s + τ ) and then using concatenation to form the LTI array H of
MIMO LTI models H τ ( s ), τ ∈ { 1.1, 1.2, 1.3 }. To do this, type
tau = [1.1 1.2 1.3];
for i=1:3
% Form LTI array h of SISO models.
4-27
4
Arrays of LTI Models
h(:,:,i)=tf(1,[1 tau]);
end
H = [h 0; 1 tf(1,[1 0])]; %Concatenation: array h & single models
size(H)
3x1 array of continuous-time transfer functions
Each transfer function has 2 output(s) and 2 input(s).
Similarly, you can use append to perform the diagonal appending of each model
in the SISO LTI array h with a fixed single (SISO or MIMO) LTI model.
S = append(h,tf(1,[1 3])); % Append a single model to h.
specifies an LTI array S in which each model has the form
1
----------s
+τ
Sτ ( s )=
0
0
1
-----------s+3
You can also combine an LTI array of MIMO models and a single MIMO LTI
model using arithmetic operations. For example, if h is the LTI array of three
SISO models defined above,
[h,h] + [tf(1,[1 0]);tf(1,[1 5])]
adds the single one-output, two-input LTI model [1/s 1/(s + 5)] to every
model in the 3-by-1 LTI array of one-output, two-input models [h,h]. The
result is a new 3-by-2 array of models.
Examples: Arithmetic Operations on LTI Arrays and SISO Models
Using the LTI array of one-output, two-input state-space models [h,h],
defined in the previous example,
tf(1,[1 3]) + [h,h]
adds a single SISO transfer function model to each entry in each model of the
LTI array of MIMO models [h,h].
Finally,
G = rand(1,1,3,1);
sysa = G + [h,h]
4-28
Operations on LTI Arrays
adds the array of scalars to each entry of each MIMO model in the LTI array
[h,h] on a model-by-model basis. This last command is equivalent to the
following for loop.
hh = [h,h];
for k = 1:3
sysa(:,:,k) = G(1,1,k) + hh(:,:,k);
end
Other Operations on LTI Arrays
You can also apply the analysis functions, such as bode, nyquist, and step, to
LTI arrays.
4-29
4
Arrays of LTI Models
4-30
5
Customization
Introduction . . . . . . . . . . . . . . . . . . . . 5-2
The Property and Preferences Hierarchy . . . . . . . . . 5-3
5
Customization
Introduction
The Control System Toolbox provides editors that allow you to set properties
and preferences in the SISO Design Tool, the LTI Viewer, and in any response
plots that you create from the MATLAB prompt.
Properties refer to settings that are specific to an individual response plot. This
includes the following:
• Axes labels, and limits
• Data units and scales
• Plot styles, such as grids, fonts, and axes foreground colors
• Plot characteristics, such as rise time, peak response, and gain and phase
margins.
Preferences refers to properties that persist either:
• Within a single session for a specific instance of an LTI Viewer or a SISO
Design Tool
• Across Control System Toolbox sessions
The former are called tool preferences, the latter toolbox preferences.
5-2
Introduction
The Property and Preferences Hierarchy
This diagram explains the hierarchy from properties, which are local, to
toolbox preferences, which are global and persist from session to session.
Toolbox Preferences
Persist across sessions
User Preferences
Inheritance
Saved
to disk
Not saved
Tool Preferences
Specific to an instance
of a tool
to disk
SISO Design
DesignTool
Tool
SISO
Inheritance
LTI Viewer
Not saved
Plot Properties
Specific to an instance
of a plot
to disk
Response
Response Plot
Plot
Response
Response Plot
Plot
Response
Response Plot
Plot
5-3
5
Customization
5-4
6
Setting Toolbox
Preferences
Toolbox Preferences Editor
Units Pane . . . . . .
Style Pane . . . . . .
Characteristics Pane . .
SISO Tool Pane
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6-2
6-3
6-3
6-4
. . . . . . . . . . . . . . . . . . . 6-5
6
Setting Toolbox Preferences
The Toolbox Preferences editor allows you to set plot preferences that will
persist from session to session. This is the highest level shown in “The Property
and Preferences Hierarchy” on page 5-3.
Toolbox Preferences Editor
To open the Toolbox Preferences Editor, select Toolbox Preferences under the
File menu of the LTI Viewer or the SISO Design Tool. Alternatively, you can
type
ctrlpref
at the MATLAB prompt.
Note To get help on panes in the Control System Toolbox Preferences editor,
you can click on the tabs.
.
The Control System Toolbox Preferences Editor
6-2
Units Pane
Note To get help on panes in the Control System Toolbox Preferences editor,
click on the tabs.
Use the Units pane to set preferences for the following:
• Frequency — Radians per second (rad/s) or Hertz (Hz)
• Magnitude — Decibels (dB) or absolute value (abs)
• Phase — Degrees or radians
For frequency and magnitude axes, you can select logarithmic or linear scales.
Style Pane
Note Click on the tabs to get help on panes in the Control System Toolbox
Preferences editor.
6-3
6
Setting Toolbox Preferences
Use the Style pane to toggle grid visibility and set font preferences and axes
foreground colors for all plots you create using the Control System Toolbox.
This figure shows the Style pane.
You have the following choices:
• Grid — Activate grids by default in new plots
• Font preferences — Set the font size, weight (bold), and angle (italic)
• Colors — Specify the color vector to use for the axes foreground, which
includes the X-Y axes, grid lines, and tick labels. Use a three-element vector
to represent red, green, and blue (RGB) values. Vector element values can
range from 0 to 1.
If you do not want to specify RGB values numerically, click the Select button
to open the Select Colors window. See “Select colors” on page 7-9 for more
information.
Characteristics Pane
Note Click on the tabs to get help on panes in the Control System Toolbox
Preferences editor.
6-4
The Characteristics pane has selections for response characteristics and phase
wrapping. This figure shows the Characteristics pane with default settings.
The following are the available options for the Characteristics pane:
• Response Characteristics:
- Specify settling time tolerance — You can set the threshold of the settling
time calculation to any percentage from 0 to 100%. The default is 2%.
- Specify rise time boundaries — The standard definition of rise time is the
time it takes the signal to go from 10% to 90% of the final value. You can
choose any percentages you like (from 0% to 100%), provided that the first
value is smaller than the second.
• Phase Wrapping — By default, the phase is not wrapped. Wrap the phrase
by clearing this box. If the phase is wrapped, all phase values are shifted
such that their equivalent value displays in the range [-180°, 180°).
SISO Tool Pane
Note Click on the tabs below to get help on panes in the Control System
Toolbox Preferences editor.
6-5
6
Setting Toolbox Preferences
The SISO Tool pane has settings for the SISO Design Tool. This figure shows
the SISO Tool pane with default settings.
You can make the following selections:
• Compensator Format — You can select either the time-constant format or
the zero/pole/gain format. The time-constant format is
( 1 + Tz 1 s )
dcgain × ---------------------------- …
( 1 + Tp 1 s )
where Tz1, Tz2, ..., are the zero time constants, and Tp1, Tp2, ..., are the pole
time constants.
The zero/pole/gain format is a variation on the time-constant format.
( s + z1 )
K × -------------------( s + p1 )
In this case, the gain is compensator gain; z1, z2, ... and p1, p2, ..., are the zero
and pole locations, respectively.
• Bode Options — By default, the SISO Design Tool shows the plant and
sensor poles and zeros as blue x’s and o’s, respectively. Clear this box to
eliminate the plant’s poles and zeros from the Bode plot. Note that the
compensator poles and zeros (in red) will still appear.
6-6
7
Setting Tool Preferences
LTI Viewer Preferences Editor
Units Pane . . . . . . . .
Style Pane . . . . . . . .
Characteristics Pane . . . .
Parameters Panpanee . . . .
SISO Tool Preferences Editor .
Units Pane . . . . . . . .
Style Pane . . . . . . . .
Options Pane . . . . . . .
Line Colors Pane . . . . . .
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. 7-2
. 7-3
. 7-3
. 7-4
. 7-5
. 7-6
. 7-7
. 7-8
. 7-10
. 7-12
7
Setting Tool Preferences
Both the LTI Viewer and the SISO Design Tool have Tool Preferences Editors.
These editors comprise the middle layer of “The Property and Preferences
Hierarchy” on page 5-3.
Both editors allow you to set default characteristics for specific instances of LTI
Viewers and SISO Design Tools. If you open a new instance of either, each
defaults to the characteristics specified in the Toolbox Preferences editor.
LTI Viewer Preferences Editor
Select LTI Viewer Preferences under the Edit menu of the LTI Viewer to
open the LTI Viewer Preferences editor, which is a tool for customizing
various LTI Viewer properties, including units, fonts, and various other viewer
characteristics. This figure shows the editor open to its first pane.
Note Click on the tabs to get help on LTI Viewer Preference editor.
The LTI Viewer Preferences Editor
7-2
Units Pane
Note Click on the pane tabs below to get help on LTI Viewer Preference
editor panes.
You can select the following on the Units pane:
• Frequency — Radians per second (rad/sec) or Hertz (Hz)
• Magnitude — Decibels (dB) or absolute value (abs)
• Phase — Degrees or radians
For frequency and magnitude axes, you can select logarithmic or linear scales.
Style Pane
Note Click on the tabs to get help on LTI Viewer Preference editor.
7-3
7
Setting Tool Preferences
Use the Style pane to toggle grid visibility and set font preferences and axes
foreground colors for all plots in the LTI Viewer. This figure shows the Style
pane.
You have the following choices:
• Grid — Activate grids for all plots in the LTI Viewer
• Fonts — Set the font size, weight (bold), and angle (italic)
• Colors — Specify the color vector to use for the axes foreground, which
includes the X-Y axes, grid lines, and tick labels. Use a three-element vector
to represent red, green, and blue (RGB) values. Vector element values can
range from 0 to 1.
• If you do not want to specify the RGB values numerically, press the Select
button to open the Select Colors window. See “Select colors” on page 7-9 for
more information.
Characteristics Pane
Note Click on the tabs to get help on LTI Viewer Preference editor.
7-4
The Characteristics pane, shown below, has selections for response
characteristics and phase wrapping.
The following choices are available:
• Response Characteristics:
- Specify settling time tolerance — You can set the threshold of the settling
time calculation to any percentage from 0 to 100%. The default is 2%.
- Specify rise time boundaries — The standard definition of rise time is the
time it takes the signal to go from 10% to 90% of the final value. You can
choose any percentages you like (from 0% to 100%), provided that the first
value is smaller than the second.
• Phase Wrapping — By default, the phase is not wrapped. Wrap the phrase
by clearing this box. If the phase is wrapped, all phase values are shifted
such that their equivalent value displays in the range [-180°, 180°).
Parameters Panpanee
Note Click on the tabs to get help on LTI Viewer Preference editor.
7-5
7
Setting Tool Preferences
Use the Parameters pane, shown below, to specify input vectors for time and
frequency simulation.
The defaults are to generate time and frequency vectors for your plots
automatically. You can, however, override the defaults as follows:
• Time Vector:
- Define stop time — Specify the final time value for your simulation
- Define vector — Specify the time vector manually using equal-sized time
steps
• Frequency Vector:
- Define range — Specify the bandwidth of your response. Whether it’s in
rad/sec or Hz depends on the selection you made in the Units pane.
- Define vector — Specify the vector for your frequency values. Any real,
positive, strictly monotonically increasing vector is valid.
SISO Tool Preferences Editor
Note Click on the tabs to get help on SISO Tool Preference editor.
7-6
To open the SISO Tool Preferences editor, select SISO Tool Preferences
from the Edit menu of the SISO Design Tool. This window opens.
The SISO Tool Preferences Editor
Units Pane
Note Click on the pane tabs below to get help on SISO Tool Preference editor
panes.
7-7
7
Setting Tool Preferences
The Units pane has settings for the following units:
• Frequency — Radians per second (rad/sec) or Hertz (Hz)
• Magnitude — Decibels (dB) or absolute value (abs)
• Phase — Degrees or radians
For frequency and magnitude axes, you can select logarithmic or linear scales.
Style Pane
Note Click on the tabs to get help on SISO Tool Preference editor.
Use the Style pane to toggle grid visibility and set font preferences and axes
foreground colors for all plots in the SISO Design Tool. This figure shows the
Style pane.
Click on the Grids, Fonts, and Colors
panels for help contents.
Grids Panel
Select the box to activate grids for all plots in the SISO Design Tool
Fonts Panel
Set the font size, weight (bold), and angle (italic) by using the menus and check
boxes.
7-8
Colors Panel
Specify the color vector to use for the axes foreground, which includes the X-Y
axes, grid lines, and tick labels. Use a three-element vector to represent red,
green, and blue (RGB) values. Vector element values can range from 0 to 1.
Select colors. Click the Select button to open the Select Color window for the
axes foreground.
You can use this window to choose axes foreground colors without having to set
RGB (red-green-blue) values numerically. To make your selections, click on the
colored rectangles and press OK. If you want a broader range of colors, click the
Define Custom Colors button. This extends the Select Color window, as
shown in this figure.
7-9
7
Setting Tool Preferences
You can pick colors from the color spectrum located in the upper right corner of
the window. To select a custom color, follow these steps:
1 Place your cursor at a point in the color spectrum that has a color you want
to define.
2 Left-click. Notice that the hue, saturation, luminescence (lum.), red, green,
and blue fields specify the numerical values for the selected color.
3 Press Add to Custom Colors. This adds the selected color to the row of boxes
labeled Custom Color. You can now use this color just like the basic colors.
Options Pane
Note Click on the tabs to get help on SISO Tool Preference editor.
7-10
The Options pane, shown below, has selections for compensator format and
Bode diagrams.
You can make the following selections:
• Compensator Format — Select the time constant, natural frequency, or
zero/pole/gain format. The time constant format is a factorization of the
compenator transfer function of the form
( 1 + Tz 1 s )
dcgain × ---------------------------- …
( 1 + Tp 1 s )
where Tz1, Tz2, ..., are the zero time constants, and Tp1, Tp2, ..., are the pole
time constants.
The natural frequency format is
( 1 + s ⁄ ωz1 )
dcgain × ------------------------------------ …
( 1 + s ⁄ ( ω p1 ) )
where ωz1,ωz2, ... and ωp1, ωp2, ..., are the natural frequencies of the zeros and
poles, respectively.
The zero/pole/gain format is
( s + z1 )
K × -------------------( s + p1 )
where z1, z2, ... and p1, p2, ..., are the zero and pole locations, respectively.
7-11
7
Setting Tool Preferences
• Bode Options — By default, the SISO Design Tool shows the plant and
sensor poles and zeros as blue x’s and o’s, respectively. Clear this check box
to eliminate the plant’s poles and zeros from the Bode plot. Note that the
compensator poles and zeros (in red) will still appear.
Line Colors Pane
Note Click on the pane tabs below to get help on SISO Tool Preference editor.
The Line Colors pane, shown below, has selections for specify the colors of the
lines in the response plots of the SISO Design Tool.
Click on the Select
button for help on
choosing colors.
To change the colors of plot lines associated with parts of your model, specify a
three-element vector to represent red, green, and blue (RGB) values. Vector
element values can range from 0 to 1.
If you do not want to specify the RGB values numerically, click the Select
button to open the Select Color window. See “Select colors” on page 7-9 for more
information.
7-12
8
Customizing Response
Plot Properties
Response Plots Property Editor
Labels Pane . . . . . . . .
Limits Pane . . . . . . . .
Units Pane . . . . . . . .
Style Pane . . . . . . . .
Characteristics Pane . . . .
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8-3
8-4
8-4
8-5
8-7
8-8
Property Editing for Subplots . . . . . . . . . . . . 8-10
Customizing Plots Inside the SISO Design Tool
Root Locus Property Editor . . . . . . . . . .
Labels Pane . . . . . . . . . . . . . . . . .
Limits Pane . . . . . . . . . . . . . . . . .
Options Pane . . . . . . . . . . . . . . . .
Open-Loop Bode Property Editor . . . . . . . .
Labels Pane . . . . . . . . . . . . . . . . .
Limits Pane . . . . . . . . . . . . . . . . .
Options Pane . . . . . . . . . . . . . . . .
Open-Loop Nichols Property Editor . . . . . . .
Labels Pane . . . . . . . . . . . . . . . . .
Limits Pane . . . . . . . . . . . . . . . . .
Options Pane . . . . . . . . . . . . . . . .
Prefilter Bode Property Editor . . . . . . . . .
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. 8-11
. 8-11
. 8-12
. 8-13
. 8-14
. 8-16
. 8-16
. 8-17
. 8-18
. 8-19
. 8-20
. 8-21
. 8-21
. 8-22
8
Customizing Response Plot Properties
The lowest level of “The Property and Preferences Hierarchy” on page 5-3 is
setting response plot properties. If you have created a response plot, there are
two ways to open the Property Editor:
• Double-click in the plot region
• Select Properties from the right-click menu
Before looking at the Property Editor, open a step response plot using this
commands.
load ltiexamples
step(sys_dc)
This creates a step plot. Select Properties from the right-click window. Note
that when you open the Property Editor, a black dashed box appears around
the step response, as this figure shows.
The dashed line
indicates that the
Property Editor is
active for this plot.
A SISO System Step Response
8-2
Response Plots Property Editor
Note Click on the tabs to get help on panes in the Property Editor.
This figure shows the Property Editor dialog box for this step response.
The Property Editor for the Step Response
In general, you can change the following properties of response plots:
• Labels — Titles and X- and Y-labels
• Limits — Numerical ranges of the X and Y axes
• Units — Where applicable (e.g., rad/s to Hertz). If you cannot customize
units, as is the case with step responses, the Property Editor will display that
no units ar available for the selected plot.
• Style — Show a grid and adjust font properties, such as font size, bold and
italics
• Characteristics — Where applicable, these include peak response, settling
time, phase and gain margins, etc. Plot characteristics change with each plot
response type. The Property Editor displays only the characteristics that
make sense for the selected response plot. For example, phase and gain
margins are not available for step responses.
As you make changes in the Property Editor, they display immediately in the
response plot. Conversely, if you make changes in a plot using right-click
8-3
8
Customizing Response Plot Properties
menus, the Property Editor for that plot automatically updates. The Property
Editor and its associated plot are dynamically linked.
Labels Pane
Note Click on the tabs below to get help on the Property Editor.
To specify new text for plot titles and axis labels, type the new string in the field
next to the label you want to change. Note that the label changes immediately
as you type, so you can see how the new text looks as you are typing.
Limits Pane
Note Click on the tabs to get help on the Property Editor.
8-4
The Control System Toolbox selects default values for the axes limits to make
sure that the maximum and minimum x and y values are displayed. If you want
to override the default settings, change the values in the Limits fields. The
Auto-Scale box automatically clears if you click on a different field. The new
limits appear immediately in the response plot.
To reestablish the default values, select the Auto-Scale box again.
Units Pane
Note Click on the tabs to get help on the Property Editor.
8-5
8
Customizing Response Plot Properties
You can use the Units pane to change units in your response plot. The contents
of this pane depend on the response plot associated with the editor.
Note that for step and impulse responses, there are no alternate units available
(only time and amplitude are possible in the toolbox). This table lists the
options available for the other response objects. Use the menus to toggle
between units.
Table 8-1: Optional Unit Conversions for Response Plots
8-6
Response Plot
Unit Conversions
Bode and
Bode Magnitude
Frequency in rad/s or Hertz (Hz) using logarithmic or
linear scale
Magnitude in decibels (dB) or the absolute value
Phase in degrees or radians
Impulse
None
Nichols Chart
and Nyquist
Diagram
Frequency in rad/s or Hertz
Magnitude in decibels or the absolute value
Phase in degrees or radians
Pole/Zero Map
Frequency in rad/s or Hertz
Table 8-1: Optional Unit Conversions for Response Plots
Response Plot
Unit Conversions
Singular Values
Frequency in rad/s or Hertz using logarithmic or linear
scale
Magnitude in decibels or the absolute value
Step
None
Style Pane
Note Click on the tabs to get help on the Property Editor.
Use the Style pane to toggle grid visibility and set font preferences and axes
foreground colors for response plots.
You have the following choices:
• Grid — Activate grids by default in new plots
• Fonts — Set the font size, weight (bold), and angle (italic)
• Colors — Specify the color vector to use for the axes foreground, which
includes the X-Y axes, grid lines, and tick labels. Use a three-element vector
8-7
8
Customizing Response Plot Properties
to represent red, green, and blue (RGB) values. Vector element values can
range from 0 to 1.
If you do not want to specify RGB values numerically, click the Select button
to open the Select Color window. See “Select colors” on page 7-9 for more
information.
Characteristics Pane
Note Click on the tabs to get help on the Property Editor.
The Characteristics pane allows you to customize response characteristics for
plots. Each response plot has its own set of characteristics; the table below lists
8-8
them. Use the check boxes to activate the feature and the fields to specify rise
or settling time percentages.
Table 8-2: Response Characteristic Options for Response Plots
Plot
Customizable Feature
Bode Diagram
Show peak response
Show minimum stability margins
Show all stability margins
Unwrap phase (default is wrapped)
Bode Magnitude
Show peak response
Impulse
Show peak response
Show settling time within xx% (specify the percentage)
Nichols Chart
Show peak response
Show minimum stability margins
Show all stability margins
Unwrap phase (default is wrapped)
Nyquist
Diagram
Show peak response
Show minimum stability margins
Show all stability margins
Pole/Zero Map
None
Sigma
Show peak response
Step
Show peak response
Show settling time within xx% (specify the percentage)
Show rise time from xx to yy% (specify the percentages)
Show steady state
8-9
8
Customizing Response Plot Properties
Property Editing for Subplots
If you create more than one plot in a single figure window, you can edit each
plot individually. For example, the following code creates a figure with two
plots, a step and an impulse response with two randomly selected systems.
subplot(2,1,1)
step(rss(2,1))
subplot(2,1,2)
impulse(rss(1,1))
After the figure window appears, double-click in the upper (step response) plot
to activate the Property Editor. You will see a dashed line appear around the
step response, indicating that it is the active plot for the editor. To switch to the
lower (impulse response) plot, just click once in the impulse response plot
region. The dashed box switches to the impulse response, and the Property
Editor updates as well.
8-10
Customizing Plots Inside the SISO Design Tool
Customizing Plots Inside the SISO Design Tool
Customizing plots inside the SISO Design Tool is similar to how you customize
any response plot. The Control System Toolbox provides the following property
editors specific to the SISO Design Tool:
• “Root Locus Property Editor”
• “Open-Loop Bode Property Editor”
• “Open-Loop Nichols Property Editor”
• “Prefilter Bode Property Editor”
You can use each of these property editors to create the customized plots within
the SISO Design tool.
Root Locus Property Editor
There are three ways to open the Property Editor for root locus plots:
• Double-click in the root locus away from the curve
• Select Properties from the right-click menu
• Select Root Locus and then Properties from Edit in the menu bar
Note Click on the tabs to get help on the Root Locus Property Editor.
8-11
8
Customizing Response Plot Properties
This figure shows the Property Editor: Root Locus window.
Labels Pane
Note Click on the tabs to get help on the Root Locus Property Editor.
8-12
Customizing Plots Inside the SISO Design Tool
You can use the Label pane to specify plot titles and axis labels. To specify a
new label, type the string in the appropriate field. The root locus plot
automatically updates.
Limits Pane
Note Click on the pane tabs below to get help on panes in the Root Locus
Property Editor.
The SISO Design Tool specifies default values for the real and imaginary axes
ranges to make sure that all the poles and zeros in your model appear in the
root locus plot. Use the Limits pane, shown below, to override the default
settings.
To change the limits, specify the new limits in the real and imaginary axes
Limits fields. The Auto-Scale check box automatically clears once you click in
a different field. Your root locus diagram updates immediately. If you want to
reapply the default limits, select the Auto-Scale check boxes again.
The Limit Stack panel provides support for storing and retrieving custom limit
specifications. There are four buttons available:
8-13
8
Customizing Response Plot Properties
— Add the current limits to the stack
— Retrieve the previous stack entry
— Retrieve the next stack entry
— Remove the current limits from the stack
Using these buttons, you can store and retrieve any number of saved custom
axes limits.
Options Pane
Note Click on the tabs to get help on the Root Locus Property Editor.
The Options pane contains settings for adding a grid and changing the plot’s
aspect ratio. This figure shows the Options pane.
8-14
Customizing Plots Inside the SISO Design Tool
Select Show grid to display a grid on the root locus. If you have damping ratio
constraints on your root locus, selecting Display damping ratios as % peak
overshoot displays the damping ratio values along the grid lines. This figure
shows both options activated for an imported model, Gservo. If you want to
verify these settings, type
load ltiexamples
at the MATLAB prompt and import Gservo from the workspace into your SISO
Design Tool.
Displaying Damping Ratio Values
The numbers displayed on the root locus gridlines are the damping ratios as a
percentage of the overshoot values.
If you select the Equal check box in the Aspect Ratio panel, the x and y-axes
are set to equal limit values.
8-15
8
Customizing Response Plot Properties
Open-Loop Bode Property Editor
The Property Editor for open-loop Bode diagrams is identical to the one for root
locus, with one exception, the Options pane. Also, note the prefilter and
open-loop Bode diagram property editors are identical.
As is the case with the root locus Property Editor, there are three ways to open
the Bode diagram property editor:
• Double-click in the Bode magnitude or phase plot away from the curve
• Select Properties from the right-click menu
• Select Open-Loop Bode and then Properties from Edit in the menu bar
Note Click on the tabs to get help on the Open-Loop Bode Property editor.
This figure shows the Property Editor: Open-Loop Bode editor.
Labels Pane
Note Click on the tabs to get help on the Open-Loop Bode Property editor.
8-16
Customizing Plots Inside the SISO Design Tool
You can use the Label pane to specify plot titles and axis labels. To specify a
new label, type the string in the appropriate field. The Bode diagram
automatically updates.
Limits Pane
Note Click on the tabs to get help on the Open-Loop Bode Property editor.
8-17
8
Customizing Response Plot Properties
The Control System Toolbox sets default limits for the frequency, magnitude,
and phase scales for your plots. Use the Limits pane to override the default
values.
To change the limits, specify the new values in the Limits fields for frequency,
magnitude, and phase. The Auto-Scale check box automatically deactivates
once you click in a different field. The Bode diagram updates immediately.
To restore the default settings, select the Auto-Scale boxes again.
Options Pane
Note Click on the tabs to get help on the Open-Loop Bode Property editor.
8-18
Customizing Plots Inside the SISO Design Tool
This figure shows the Options pane for Bode diagrams.
The following options are available from this pane:
• Grid — Select Show grid to display grid lines.
• Magnitude/Phase — There are three radio buttons; you can toggle between
the following displays:
- Show magnitude & phase
- Show magnitude only
- Show phase only
• Response Characteristics — Select Show stability margins to display the
phase and gain margins on your Bode diagram. The margins appear as
brown stems, and the Bode diagram displays the numerical values of the
margins in one of the bottom corners of the gain and phase plots.
The Bode diagram in “Displaying Damping Ratio Values” on page 8-15, has
the stability margins displayed.
Open-Loop Nichols Property Editor
As is the case with the root locus Property Editor, there are three ways to open
the Nichols plot property editor:
• Double-click in the Nichols plot away from the curve
• Select Properties from the right-click menu
8-19
8
Customizing Response Plot Properties
• Select Open-Loop Nichols and then Properties from Edit in the menu bar
Note Click on the tabs to get help on the Open-Loop Nichols Property editor.
This figure shows the Property Editor: Open-Loop Nichols editor.
Labels Pane
Note Click on the tabs to get help on the Open-Loop Nichols Property editor.
8-20
Customizing Plots Inside the SISO Design Tool
You can use the Label pane to specify plot titles and axis labels. To specify a
new label, type the string in the appropriate field. The Nichols plot
automatically updates.
Limits Pane
Note Click on the tabs to get help on the Open-Loop Nichols Property editor.
The Control System Toolbox sets default limits for the frequency, magnitude,
and phase scales for your plots. Use the Limits pane to override the default
values.
To change the limits, specify the new values in the Limits fields for open-loop
phase and/or gain. The Auto-Scale check box automatically deactivates once
you click in a different field. The Nichols plot updates immediately.
To restore the default settings, select the Auto-Scale boxes again.
Options Pane
Note Click on the tabs to get help on the Open-Loop Nichols Property editor.
8-21
8
Customizing Response Plot Properties
This figure shows the Options pane for Bode diagrams.
The following options are available from this pane:
• Grid — Select Show grid to display grid lines.
• Response Characteristics — Select Show stability margins to display the
phase and gain margins on your Nichols plot.
Prefilter Bode Property Editor
The Prefilter Bode Property editor is identical to the Open-Loop Bode
diagram property editor. There are three ways to open the prefilter editor:
• Double-click in the prefilter Bode magnitude or phase plot away from the
curve
• Select Properties from the right-click menu
• Select Prefilter Bode and then Properties from Edit in the menu bar
See “Open-Loop Bode Property Editor” on page 8-16 for a description of the
features of this editor.
8-22
9
Design Case Studies
Yaw Damper for a 747 Jet Transport
Computing Open-Loop Eigenvalues . .
Open-Loop Analysis . . . . . . . .
Root Locus Design . . . . . . . . .
Washout Filter Design . . . . . . .
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Hard-Disk Read/Write Head Controller
Deriving the Model . . . . . . . . . .
Model Discretization . . . . . . . . .
Adding a Compensator Gain . . . . . .
Adding a Lead Network . . . . . . . .
Design Analysis . . . . . . . . . . .
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. 9-20
. 9-20
. 9-21
. 9-23
. 9-24
. 9-27
LQG Regulation: Rolling Mill Example
Process and Disturbance Models . . . .
LQG Design for the x-Axis . . . . . . .
LQG Design for the y-Axis . . . . . . .
Cross-Coupling Between Axes . . . . .
MIMO LQG Design . . . . . . . . . .
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. 9-31
. 9-31
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. 9-46
Kalman Filtering . . . .
Discrete Kalman Filter . .
Steady-State Design . . .
Time-Varying Kalman Filter
Time-Varying Design . . .
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References
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9-3
9-4
9-5
9-9
9-14
. . . . . . . . . . . . . . . . . . . . . 9-61
9
Design Case Studies
This chapter contains four detailed case studies of control system design and
analysis using the Control System Toolbox:
• “Yaw Damper for a 747 Jet Transport” — Illustrating the classical design
process
• “Hard-Disk Read/Write Head Controller” — Illustrating classical digital
controller design
• “LQG Regulation: Rolling Mill Example” — Using linear quadratic Gaussian
techniques to regulate the beam thickness in a steel rolling mill
• “Kalman Filtering”— Kalman filtering that illustrates both steady-state and
time-varying Kalman filter design and simulation
Demonstration files for these case studies are available as jetdemo.m,
diskdemo.m, milldemo.m, and kalmdemo.m. To run any of these demonstrations,
type the corresponding name at the command line, for example,
jetdemo
9-2
Yaw Damper for a 747 Jet Transport
Yaw Damper for a 747 Jet Transport
This case study demonstrates the tools for classical control design by stepping
through the design of a yaw damper for a 747 jet transport aircraft.
The jet model during cruise flight at MACH = 0.8 and H = 40,000 ft. is
A = [-0.0558
0.5980
-3.0500
0
B = [ 0.0729
-4.7500
.15300
0
C = [0
0
1
0
D = [0
0
0
0];
-0.9968
-0.1150
0.3880
0.0805
0.0802
-0.0318
-0.4650
1.0000
0.0415
0
0
0];
0.0000
0.00775
0.1430
0];
0
0
0
1];
The following commands specify this state-space model as an LTI object and
attach names to the states, inputs, and outputs.
states = {'beta' 'yaw' 'roll' 'phi'};
inputs = {'rudder' 'aileron'};
outputs = {'yaw' 'bank angle'};
sys = ss(A,B,C,D,'statename',states,...
'inputname',inputs,...
'outputname',outputs);
You can display the LTI model sys by typing sys. MATLAB responds with
a =
beta
yaw
roll
phi
beta
-0.0558
0.598
-3.05
0
yaw
-0.9968
-0.115
0.388
0.0805
roll
0.0802
-0.0318
-0.465
1
phi
0.0415
0
0
0
9-3
9
Design Case Studies
b =
beta
yaw
roll
phi
rudder
0.0729
-4.75
0.153
0
yaw
bank angle
beta
0
0
aileron
0
0.00775
0.143
0
c =
yaw
1
0
roll
0
0
phi
0
1
d =
yaw
bank angle
rudder
0
0
aileron
0
0
Continuous-time model.
The model has two inputs and two outputs. The units are radians for beta
(sideslip angle) and phi (bank angle) and radians/sec for yaw (yaw rate) and
roll (roll rate). The rudder and aileron deflections are in radians as well.
Computing Open-Loop Eigenvalues
Compute the open-loop eigenvalues and plot them in the s -plane.
damp(sys)
Eigenvalue
-7.28e-003
-5.63e-001
-3.29e-002 + 9.47e-001i
-3.29e-002 - 9.47e-001i
9-4
Damping
Freq. (rad/s)
1.00e+000
1.00e+000
3.48e-002
3.48e-002
7.28e-003
5.63e-001
9.47e-001
9.47e-001
Yaw Damper for a 747 Jet Transport
pzmap(sys)
This model has one pair of lightly damped poles. They correspond to what is
called the “Dutch roll mode.”
Suppose you want to design a compensator that increases the damping of these
poles, so that the resulting complex poles have a damping ratio ζ > 0.35 with
natural frequency ω n < 1 rad/sec. You can do this using the Control System
toolbox analysis tools.
Open-Loop Analysis
First, perform some open-loop analysis to determine possible control
strategies. Start with the time response (you could use step or impulse here).
9-5
9
Design Case Studies
impulse(sys)
The impulse response confirms that the system is lightly damped. But the time
frame is much too long because the passengers and the pilot are more
concerned about the behavior during the first few seconds rather than the first
few minutes. Next look at the response over a smaller time frame of 20 seconds.
9-6
Yaw Damper for a 747 Jet Transport
impulse(sys,20)
Look at the plot from aileron (input 2) to bank angle (output 2). To show only
this plot, right-click and choose I/O Selector, then click on the (2,2) entry. The
I/O Selector should look like this.
9-7
9
Design Case Studies
The new figure is shown below.
The aircraft is oscillating around a nonzero bank angle. Thus, the aircraft is
turning in response to an aileron impulse. This behavior will prove important
later in this case study.
Typically, yaw dampers are designed using the yaw rate as sensed output and
the rudder as control input. Look at the corresponding frequency response.
sys11=sys('yaw','rudder') % Select I/O pair.
9-8
Yaw Damper for a 747 Jet Transport
bode(sys11)
From this Bode diagram, you can see that the rudder has significant effect
around the lightly damped Dutch roll mode (that is, near ω = 1 rad/sec).
Root Locus Design
A reasonable design objective is to provide a damping ration ζ > 0.35 with a
natural frequency ω n < 1.0 rad/sec. Since the simplest compensator is a static
gain, first try to determine appropriate gain values using the root locus
technique.
% Plot the root locus for the rudder to yaw channel
9-9
9
Design Case Studies
rlocus(sys11)
This is the root locus for negative feedback and shows that the system goes
unstable almost immediately. If, instead, you use positive feedback, you may
be able to keep the system stable.
rlocus(-sys11)
9-10
Yaw Damper for a 747 Jet Transport
sgrid
This looks better. By using simple feedback, you can achieve a damping ratio
of ζ = 0.45 . Click on the blue curve and move the data marker to track the
9-11
9
Design Case Studies
gain and damping values. To achieve a 0.45 damping ratio, the gain should be
about 2.85. This figure shows the data marker with similar values.
Next, close the SISO feedback loop.
K = 2.85;
cl11 = feedback(sys11,-K);
% Note: feedback assumes negative
% feedback by default
Plot the closed-loop impulse response for a duration of 20 seconds, and compare
it to the open-loop impulse response.
9-12
Yaw Damper for a 747 Jet Transport
impulse(sys11,'b--',cl11,'r',20)
The closed-loop response settles quickly and does not oscillate much,
particularly when compared to the open-loop response.
Now close the loop on the full MIMO model and see how the response from the
aileron looks. The feedback loop involves input 1 and output 1 of the plant (use
feedback with index vectors selecting this input/output pair). At the MATLAB
prompt, type
cloop = feedback(sys,-K,1,1);
damp(cloop)
% closed-loop poles
Eigenvalue
Damping
-3.42e-001
-2.97e-001 + 6.06e-001i
-2.97e-001 - 6.06e-001i
-1.05e+000
Freq. (rad/s)
1.00e+000
4.40e-001
4.40e-001
1.00e+000
3.42e-001
6.75e-001
6.75e-001
1.05e+000
9-13
9
Design Case Studies
Plot the MIMO impulse response.
impulse(sys,'b--',cloop,'r',20)
The yaw rate response is now well damped, but look at the plot from aileron
(input 2) to bank angle (output 2). When you move the aileron, the system no
longer continues to bank like a normal aircraft. You have over-stabilized the
spiral mode. The spiral mode is typically a very slow mode and allows the
aircraft to bank and turn without constant aileron input. Pilots are used to this
behavior and will not like your design if it does not allow them to fly normally.
This design has moved the spiral mode so that it has a faster frequency.
Washout Filter Design
What you need to do is make sure the spiral mode does not move further into
the left-half plane when you close the loop. One way flight control designers
have addressed this problem is to use a washout filter kH ( s ) where
s
H ( s ) = ----------s+a
9-14
Yaw Damper for a 747 Jet Transport
The washout filter places a zero at the origin, which constrains the spiral mode
pole to remain near the origin. We choose a = 0.2 for a time constant of five
seconds and use the root locus technique to select the filter gain H. First specify
the fixed part s ⁄ ( s + a ) of the washout by
H = zpk(0,-0.2,1);
Connect the washout in series with the design model sys11 (relation between
input 1 and output 1) to obtain the open-loop model
oloop = H * sys11;
and draw another root locus for this open-loop model.
rlocus(-oloop)
sgrid
Create and drag a data marker around the upper curve to locate the maximum
damping, which is about ζ = 0.3 .
9-15
9
Design Case Studies
This figure shows a data marker at the maximum damping ratio; the gain is
approximately 2.07.
Look at the closed-loop response from rudder to yaw rate.
K = 2.07;
cl11 = feedback(oloop,-K);
9-16
Yaw Damper for a 747 Jet Transport
impulse(cl11,20)
The response settles nicely but has less damping than your previous design.
Finally, you can verify that the washout filter has fixed the spiral mode
problem. First form the complete washout filter kH ( s ) (washout + gain).
WOF = -K * H;
Then close the loop around the first I/O pair of the MIMO model sys and
simulate the impulse response.
cloop = feedback(sys,WOF,1,1);
% Final closed-loop impulse response
9-17
9
Design Case Studies
impulse(sys,'b--',cloop,'r',20)
The bank angle response (output 2) due to an aileron impulse (input 2) now has
the desired nearly constant behavior over this short time frame. To inspect the
9-18
Yaw Damper for a 747 Jet Transport
response more closely, use the I/O Selector in the right-click menu to select the
(2,2) I/O pair.
Although you did not quite meet the damping specification, your design has
increased the damping of the system substantially and now allows the pilot to
fly the aircraft normally.
9-19
9
Design Case Studies
Hard-Disk Read/Write Head Controller
Hard Disk Drive
Disk Platen
Ω
Disk Drive Motor
Read/Write
Head
l
θ
Solenoid
This case study demonstrates the ability to perform classical digital control
design by going through the design of a computer hard-disk read/write head
position controller.
Deriving the Model
Using Newton’s law, a simple model for the read/write head is the differential
equation
2
d θ
dθ
J ---------- + C ------- + Kθ = K i i
2
dt
dt
where J is the inertia of the head assembly, C is the viscous damping
coefficient of the bearings, K is the return spring constant, Ki is the motor
torque constant, θ is the angular position of the head, and i is the input
current.
Taking the Laplace transform, the transfer function from i to θ is
Ki
H ( s ) = --------------------------------2
Js + Cs + K
2
Using the values J = 0.01 kg m , C = 0.004 Nm/(rad/sec), K = 10 Nm/rad,
and K i = 0.05 Nm/rad, form the transfer function description of this system.
At the MATLAB prompt, type
9-20
Hard-Disk Read/Write Head Controller
J =
num
den
H =
.01; C = 0.004; K = 10; Ki = .05;
= Ki;
= [J C K];
tf(num,den)
MATLAB responds with
Transfer function:
0.05
----------------------0.01 s^2 + 0.004 s + 10
Model Discretization
The task here is to design a digital controller that provides accurate positioning
of the read/write head. The design is performed in the digital domain. First,
discretize the continuous plant. Because our plant will be equipped with a
digital-to-analog converter (with a zero-order hold) connected to its input, use
c2d with the 'zoh' discretization method. Type
Ts = 0.005;
% sampling period = 0.005 second
Hd = c2d(H,Ts,'zoh')
Transfer function:
6.233e-05 z + 6.229e-05
----------------------z^2 - 1.973 z + 0.998
Sampling time: 0.005
You can compare the Bode plots of the continuous and discretized models with
9-21
9
Design Case Studies
bode(H,'-',Hd,'--')
To analyze the discrete system, plot its step response, type
9-22
Hard-Disk Read/Write Head Controller
step(Hd)
The system oscillates quite a bit. This is probably due to very light damping.
You can check this by computing the open-loop poles. Type
% Open-loop poles of discrete model
damp(Hd)
Eigenvalue
9.87e-01 + 1.57e-01i
9.87e-01 - 1.57e-01i
Magnitude
9.99e-01
9.99e-01
Equiv. Damping
6.32e-03
6.32e-03
Equiv. Freq.
3.16e+01
3.16e+01
The poles have very light equivalent damping and are near the unit circle. You
need to design a compensator that increases the damping of these poles.
Adding a Compensator Gain
The simplest compensator is just a gain, so try the root locus technique to select
an appropriate feedback gain.
9-23
9
Design Case Studies
rlocus(Hd)
As shown in the root locus, the poles quickly leave the unit circle and go
unstable. You need to introduce some lead or a compensator with some zeros.
Adding a Lead Network
Try the compensator
z+a
D ( z ) = ----------z+b
with a = – 0.85 and b = 0 .
The corresponding open-loop model
u
9-24
D(z)
Hd ( z )
Compensator
Plant
y
Hard-Disk Read/Write Head Controller
is obtained by the series connection
D = zpk(0.85,0,1,Ts)
oloop = Hd * D
Now see how this compensator modifies the open-loop frequency response.
bode(Hd,'--',oloop,'-')
The plant response is the dashed line and the open-loop response with the
compensator is the solid line.
The plot above shows that the compensator has shifted up the phase plot
(added lead) in the frequency range ω > 10 rad/sec.
Now try the root locus again with the plant and compensator as open loop.
rlocus(oloop)
zgrid
9-25
9
Design Case Studies
Open the Property Editor by right-clicking in the plot away from the curve.
On the Limits page, set the x- and y-axis limits from -1 to 1.01. This figure
shows the result.
This time, the poles stay within the unit circle for some time (the lines drawn
by zgrid show the damping ratios from ζ = 0 to 1 in steps of 0.1). Use a data
9-26
Hard-Disk Read/Write Head Controller
marker to find the point on the curve where the gain equals 4.111e+03. This
figure shows the data marker at the correct location.
Design Analysis
To analyze this design, form the closed-loop system and plot the closed-loop
step response.
K = 4.11e+03;
cloop = feedback(oloop,K);
9-27
9
Design Case Studies
step(cloop)
This response depends on your closed loop set point. The one shown here is
relatively fast and settles in about 0.07 seconds. Therefore, this closed loop disk
drive system has a seek time of about 0.07 seconds. This is slow by today's
standards, but you also started with a very lightly damped system.
Now look at the robustness of your design. The most common classical
robustness criteria are the gain and phase margins. Use the function margin to
determine these margins. With output arguments, margin returns the gain and
phase margins as well as the corresponding crossover frequencies. Without
output argument, margin plots the Bode response and displays the margins
graphically.
To compute the margins, first form the unity-feedback open loop by connecting
the compensator D ( z ) , plant model, and feedback gain k in series.
olk = K * oloop;
9-28
Hard-Disk Read/Write Head Controller
oloop
u
+
D(z)
–
Plant
y
k
Next apply margin to this open-loop model. Type
[Gm,Pm,Wcg,Wcp] = margin(olk);
Margins = [Gm Wcg Pm Wcp]
Margins =
3.7987
296.7978
43.2031
106.2462
To obtain the gain margin in dB, type
20*log10(Gm)
ans =
11.5926
You can also display the margins graphically by typing
margin(olk)
9-29
9
Design Case Studies
The command produces the plot shown below.
This design is robust and can tolerate a 11 dB gain increase or a 40 degree
phase lag in the open-loop system without going unstable. By continuing this
design process, you may be able to find a compensator that stabilizes the
open-loop system and allows you to reduce the seek time.
9-30
LQG Regulation: Rolling Mill Example
LQG Regulation: Rolling Mill Example
This case study demonstrates the use of the LQG design tools in a process
control application. The goal is to regulate the horizontal and vertical thickness
of the beam produced by a hot steel rolling mill. This example is adapted from
[1]. The full plant model is MIMO and the example shows the advantage of
direct MIMO LQG design over separate SISO designs for each axis. Type
milldemo
at the command line to run this demonstration interactively.
Process and Disturbance Models
The rolling mill is used to shape rectangular beams of hot metal. The desired
outgoing shape is sketched below.
rolling cylinders
y
x
shaped beam
9-31
9
Design Case Studies
This shape is impressed by two pairs of rolling cylinders (one per axis)
positioned by hydraulic actuators. The gap between the two cylinders is called
the roll gap.
rolling mill stand
incoming beam
shaped beam
x-axis
rolling cylinders
The objective is to maintain the beam thickness along the x- and y-axes within
the quality assurance tolerances. Variations in output thickness can arise from
the following:
• Variations in the thickness/hardness of the incoming beam
• Eccentricity in the rolling cylinders
Feedback control is necessary to reduce the effect of these disturbances.
Because the roll gap cannot be measured close to the mill stand, the rolling
force is used instead for feedback.
The input thickness disturbance is modeled as a low pass filter driven by white
noise. The eccentricity disturbance is approximately periodic and its frequency
is a function of the rolling speed. A reasonable model for this disturbance is a
second-order bandpass filter driven by white noise.
9-32
LQG Regulation: Rolling Mill Example
This leads to the following generic model for each axis of the rolling process.
Open-loop Model for x- or y-axis
H( s)
u
+
hydraulic actuator
f1
–
+
+
δ
force-to-gap gain
Fe ( s )
we
gx
eccentricity model
Fi ( s )
wi
f2
+
f
+
input disturbance model
u
δ
f
w i, w e
command
thickness gap (in mm)
incremental rolling force
driving white noise for disturbance models
The measured rolling force variation f is a combination of the incremental
force delivered by the hydraulic actuator and of the disturbance forces due to
eccentricity and input thickness variation. Note that:
• The outputs of H ( s ), F e ( s ) , and F i ( s ) are the incremental forces delivered
by each component.
• An increase in hydraulic or eccentricity force reduces the output thickness
gap δ .
• An increase in input thickness increases this gap.
The model data for each axis is summarized below.
9-33
9
Design Case Studies
Model Data for the x-Axis
8
2.4 × 10
H x ( s ) = -----------------------------------2
2
s + 72s + 90
4
10
F ix ( s ) = ------------------s + 0.05
4
3 × 10 s
F ex ( s ) = ----------------------------------------2
2
s + 0.125s + 6
g x = 10
–6
Model Data for the y-Axis
8
7.8 × 10
H y ( s ) = ------------------------------------2
2
s + 71s + 88
4
2 × 10
F iy ( s ) = -------------------s + 0.05
5
10 s
F ey ( s ) = -------------------------------------------2
2
s + 0.19s + 9.4
g y = 0.5 × 10
–6
LQG Design for the x-Axis
As a first approximation, ignore the cross-coupling between the x- and y-axes
and treat each axis independently. That is, design one SISO LQG regulator for
each axis. The design objective is to reduce the thickness variations δ x and δ y
due to eccentricity and input thickness disturbances.
Start with the x -axis. First specify the model components as transfer function
objects.
% Hydraulic actuator (with input "u-x")
Hx = tf(2.4e8,[1 72 90^2],'inputname','u-x')
9-34
LQG Regulation: Rolling Mill Example
% Input thickness/hardness disturbance model
Fix = tf(1e4,[1 0.05],'inputn','w-ix')
% Rolling eccentricity model
Fex = tf([3e4 0],[1 0.125 6^2],'inputn','w-ex')
% Gain from force to thickness gap
gx = 1e-6;
Next build the open-loop model shown in “Open-loop Model for x- or y-axis”
above. You could use the function connect for this purpose, but it is easier to
build this model by elementary append and series connections.
% I/O map from inputs to forces f1 and f2
Px = append([ss(Hx) Fex],Fix)
% Add static gain from f1,f2 to outputs
Px = [-gx gx;1 1] * Px
x-gap
and
x-force
% Give names to the outputs:
set(Px,'outputn',{'x-gap' 'x-force'})
Note To obtain minimal state-space realizations, always convert transfer
function models to state space before connecting them. Combining transfer
functions and then converting to state space may produce nonminimal
state-space models.
The variable Px now contains an open-loop state-space model complete with
input and output names.
Px.inputname
ans =
'u-x'
'w-ex'
'w-ix'
Px.outputname
9-35
9
Design Case Studies
ans =
'x-gap'
'x-force'
The second output 'x-force' is the rolling force measurement. The LQG
regulator will use this measurement to drive the hydraulic actuator and reduce
disturbance-induced thickness variations δ x .
The LQG design involves two steps:
1 Design a full-state-feedback gain that minimizes an LQ performance
measure of the form
J ( ux ) =
∞
2
∫0  qδx + rux dt
2
2 Design a Kalman filter that estimates the state vector given the force
measurements 'x-force'.
The performance criterion J ( u x ) penalizes low and high frequencies equally.
Because low-frequency variations are of primary concern, eliminate the
high-frequency content of δ x with the low-pass filter 30 ⁄ ( s + 30 ) and use the
filtered value in the LQ performance criterion.
lpf = tf(30,[1 30])
% Connect low-pass filter to first output of Px
Pxdes = append(lpf,1) * Px
set(Pxdes,'outputn',{'x-gap*' 'x-force'})
% Design the state-feedback gain using LQRY and q=1, r=1e-4
kx = lqry(Pxdes(1,1),1,1e-4)
9-36
LQG Regulation: Rolling Mill Example
Note lqry expects all inputs to be commands and all outputs to be
measurements. Here the command 'u-x' and the measurement 'x-gap*'
(filtered gap) are the first input and first output of Pxdes. Hence, use the
syntax Pxdes(1,1) to specify just the I/O relation between 'u-x' and
'x-gap*'.
Next, design the Kalman estimator with the function kalman. The process noise
wx =
w ex
w ix
has unit covariance by construction. Set the measurement noise covariance to
1000 to limit the high frequency gain, and keep only the measured output
'x-force' for estimator design.
estx = kalman(Pxdes(2,:),eye(2),1000)
Finally, connect the state-feedback gain kx and state estimator estx to form
the LQG regulator.
Regx = lqgreg(estx,kx)
This completes the LQG design for the x -axis.
Let’s look at the regulator Bode response between 0.1 and 1000 rad/sec.
9-37
9
Design Case Studies
bode(Regx,{0.1 1000})
The phase response has an interesting physical interpretation. First, consider
an increase in input thickness. This low-frequency disturbance boosts both
output thickness and rolling force. Because the regulator phase is
approximately 0o at low frequencies, the feedback loop then adequately reacts
by increasing the hydraulic force to offset the thickness increase. Now consider
the effect of eccentricity. Eccentricity causes fluctuations in the roll gap (gap
between the rolling cylinders). When the roll gap is minimal, the rolling force
increases and the beam thickness diminishes. The hydraulic force must then
be reduced (negative force feedback) to restore the desired thickness. This is
exactly what the LQG regulator does as its phase drops to -180o near the
natural frequency of the eccentricity disturbance (6 rad/sec).
Next, compare the open- and closed-loop responses from disturbance to
thickness gap. Use feedback to close the loop. To help specify the feedback
connection, look at the I/O names of the plant Px and regulator Regx.
Px.inputname
ans =
9-38
LQG Regulation: Rolling Mill Example
'u-x'
'w-ex'
'w-ix'
Regx.outputname
ans =
'u-x'
Px.outputname
ans =
'x-gap'
'x-force'
Regx.inputname
ans =
'x-force'
This indicates that you must connect the first input and second output of Px to
the regulator.
clx = feedback(Px,Regx,1,2,+1)
% Note: +1 for positive feedback
You are now ready to compare the open- and closed-loop Bode responses from
disturbance to thickness gap.
9-39
9
Design Case Studies
bode(Px(1,2:3),'--',clx(1,2:3),'-',{0.1 100})
The dashed lines show the open-loop response. Note that the peak gain of the
eccentricity-to-gap response and the low-frequency gain of the
input-thickness-to-gap response have been reduced by about 20 dB.
Finally, use lsim to simulate the open- and closed-loop time responses to the
white noise inputs w ex and w ix . Choose dt=0.01 as sampling period for the
simulation, and derive equivalent discrete white noise inputs for this sampling
rate.
dt = 0.01
t = 0:dt:50
% time samples
% Generate unit-covariance driving noise wx = [w-ex;w-ix].
% Equivalent discrete covariance is 1/dt
wx = sqrt(1/dt) * randn(2,length(t))
9-40
LQG Regulation: Rolling Mill Example
lsim(Px(1,2:3),':',clx(1,2:3),'-',wx,t)
The dotted lines correspond to the open-loop response. In this simulation, the
LQG regulation reduces the peak thickness variation by a factor 4.
LQG Design for the y-Axis
The LQG design for the y -axis (regulation of the y thickness) follows the exact
same steps as for the x -axis.
% Specify model components
Hy = tf(7.8e8,[1 71 88^2],'inputn','u-y')
Fiy = tf(2e4,[1 0.05],'inputn','w-iy')
Fey = tf([1e5 0],[1 0.19 9.4^2],'inputn','w-ey')
gy = 0.5e-6
% force-to-gap gain
% Build open-loop model
Py = append([ss(Hy) Fey],Fiy)
Py = [-gy gy;1 1] * Py
set(Py,'outputn',{'y-gap' 'y-force'})
9-41
9
Design Case Studies
% State-feedback gain design
Pydes = append(lpf,1) * Py
% Add low-freq. weigthing
set(Pydes,'outputn',{'y-gap*' 'y-force'})
ky = lqry(Pydes(1,1),1,1e-4)
% Kalman estimator design
esty = kalman(Pydes(2,:),eye(2),1e3)
% Form SISO LQG regulator for y-axis and close the loop
Regy = lqgreg(esty,ky)
cly = feedback(Py,Regy,1,2,+1)
Compare the open- and closed-loop response to the white noise input
disturbances.
dt = 0.01
t = 0:dt:50
wy = sqrt(1/dt) * randn(2,length(t))
9-42
LQG Regulation: Rolling Mill Example
lsim(Py(1,2:3),':',cly(1,2:3),'-',wy,t)
The dotted lines correspond to the open-loop response. The simulation results
are comparable to those for the x -axis.
Cross-Coupling Between Axes
The x / y thickness regulation, is a MIMO problem. So far you have treated
each axis separately and closed one SISO loop at a time. This design is valid as
long as the two axes are fairly decoupled. Unfortunately, this rolling mill
process exhibits some degree of cross-coupling between axes. Physically, an
increase in hydraulic force along the x -axis compresses the material, which in
turn boosts the repelling force on the y -axis cylinders. The result is an increase
in y -thickness and an equivalent (relative) decrease in hydraulic force along
the y-axis.
9-43
9
Design Case Studies
The figure below shows the coupling.
Coupling Between the x- and y-axes
+
δx
+
ux
x-axis
w ex
model
gx
w ix
–
+
g xy
g yx
+
–
uy
fx
fy
y-axis
w ey
w iy
model
gy
+
+
δy
g xy = 0.1
g yx = 0.4
Accordingly, the thickness gaps and rolling forces are related to the outputs
δ x, f x, ... of the x- and y-axis models by
9-44
LQG Regulation: Rolling Mill Example
δx
δy
fx
0
0
gyx g x
δx
0
1
g xy g y
0
δy
0
0
1
– g yx
fx
0
0
– g xy
1
fy













fy
=
1
cross-coupling matrix
Let’s see how the previous “decoupled” LQG design fares when cross-coupling
is taken into account. To build the two-axes model, shown in “Coupling
Between the x- and y-axes” above, append the models Px and Py for the x - and
y-axes.
P = append(Px,Py)
For convenience, reorder the inputs and outputs so that the commands and
thickness gaps appear first.
P = P([1 3 2 4],[1 4 2 3 5 6])
P.outputname
ans =
'x-gap'
'y-gap'
'x-force'
'y-force'
Finally, place the cross-coupling matrix in series with the outputs.
gxy = 0.1; gyx = 0.4;
CCmat = [eye(2) [0 gyx*gx;gxy*gy 0] ; zeros(2) [1 -gyx;-gxy 1]]
Pc = CCmat * P
Pc.outputname = P.outputname
To simulate the closed-loop response, also form the closed-loop model by
feedin = 1:2
% first two inputs of Pc are the commands
feedout = 3:4
% last two outputs of Pc are the measurements
cl = feedback(Pc,append(Regx,Regy),feedin,feedout,+1)
You are now ready to simulate the open- and closed-loop responses to the
driving white noises wx (for the x-axis) and wy (for the y-axis).
9-45
9
Design Case Studies
wxy = [wx ; wy]
lsim(Pc(1:2,3:6),':',cl(1:2,3:6),'-',wxy,t)
The response reveals a severe deterioration in regulation performance along
the x-axis (the peak thickness variation is about four times larger than in the
simulation without cross-coupling). Hence, designing for one loop at a time is
inadequate for this level of cross-coupling, and you must perform a joint-axis
MIMO design to correctly handle coupling effects.
MIMO LQG Design
Start with the complete two-axis state-space model Pc derived above. The
model inputs and outputs are
Pc.inputname
ans =
'u-x'
'u-y'
'w-ex'
9-46
LQG Regulation: Rolling Mill Example
'w-ix'
'w_ey'
'w_iy'
P.outputname
ans =
'x-gap'
'y-gap'
'x-force'
'y-force'
As earlier, add low-pass filters in series with the 'x-gap' and 'y-gap' outputs
to penalize only low-frequency thickness variations.
Pdes = append(lpf,lpf,eye(2)) * Pc
Pdes.outputn = Pc.outputn
Next, design the LQ gain and state estimator as before (there are now two
commands and two measurements).
k = lqry(Pdes(1:2,1:2),eye(2),1e-4*eye(2))
est = kalman(Pdes(3:4,:),eye(4),1e3*eye(2))
RegMIMO = lqgreg(est,k)
% LQ gain
% Kalman estimator
% form MIMO LQG regulator
The resulting LQG regulator RegMIMO has two inputs and two outputs.
RegMIMO.inputname
ans =
'x-force'
'y-force'
RegMIMO.outputname
ans =
'u-x'
'u-y'
Plot its singular value response (principal gains).
9-47
9
Design Case Studies
sigma(RegMIMO)
Next, plot the open- and closed-loop time responses to the white noise inputs
(using the MIMO LQG regulator for feedback).
% Form the closed-loop model
cl = feedback(Pc,RegMIMO,1:2,3:4,+1);
% Simulate with lsim using same noise inputs
9-48
LQG Regulation: Rolling Mill Example
lsim(Pc(1:2,3:6),':',cl(1:2,3:6),'-',wxy,t)
The MIMO design is a clear improvement over the separate SISO designs for
each axis. In particular, the level of x / y thickness variation is now comparable
to that obtained in the decoupled case. This example illustrates the benefits of
direct MIMO design for multivariable systems.
9-49
9
Design Case Studies
Kalman Filtering
This final case study illustrates the use of the Control System Toolbox for
Kalman filter design and simulation. Both steady-state and time-varying
Kalman filters are considered.
Consider the discrete plant
x [ n + 1 ] = Ax [ n ] + B ( u [ n ] + w [ n ] )
y [ n ] = Cx [ n ]
with additive Gaussian noise w [ n ] on the input u [ n ] and data
A = [1.1269
1.0000
0
-0.4940
0
1.0000
0.1129
0
0];
B = [-0.3832
0.5919
0.5191];
C = [1 0 0];
Our goal is to design a Kalman filter that estimates the output y [ n ] given the
inputs u [ n ] and the noisy output measurements
y v [ n ] = Cx [ n ] + v [ n ]
where v [ n ] is some Gaussian white noise.
Discrete Kalman Filter
The equations of the steady-state Kalman filter for this problem are given as
follows.
Measurement update
xˆ [ n n ] = xˆ [ n n – 1 ] + M ( y v [ n ] – Cxˆ [ n n – 1 ] )
Time update
xˆ [ n + 1 n ] = Axˆ [ n n ] + Bu [ n ]
9-50
Kalman Filtering
In these equations:
• xˆ [ n n – 1 ] is the estimate of x [ n ] given past measurements up to y v [ n – 1 ]
• xˆ [ n n ] is the updated estimate based on the last measurement y v [ n ]
Given the current estimate xˆ [ n n ] , the time update predicts the state value at
the next sample n + 1 (one-step-ahead predictor). The measurement update
then adjusts this prediction based on the new measurement y v [ n + 1 ] . The
correction term is a function of the innovation, that is, the discrepancy.
y v [ n + 1 ] – Cxˆ [ n + 1 n ] = C ( x [ n + 1 ] – xˆ [ n + 1 n ] )
between the measured and predicted values of y [ n + 1 ] . The innovation gain
M is chosen to minimize the steady-state covariance of the estimation error
given the noise covariances
T
E ( w [ n ]w [ n ] ) = Q ,
T
E ( v [ n ]v [ n ] ) = R
You can combine the time and measurement update equations into one
state-space model (the Kalman filter).
xˆ [ n + 1 n ] = A ( I – MC ) xˆ [ n n – 1 ] + B AM
u[n]
yv [ n ]
yˆ [ n n ] = C ( I – MC ) xˆ [ n n – 1 ] + CM y v [ n ]
This filter generates an optimal estimate yˆ [ n n ] of y [ n ] . Note that the filter
state is xˆ [ n n – 1 ] .
Steady-State Design
You can design the steady-state Kalman filter described above with the
function kalman. First specify the plant model with the process noise.
x [ n + 1 ] = Ax [ n ] + Bu [ n ] + Bw [ n ]
y [ n ] = Cx [ n ]
(state equation)
(measurement equation)
This is done by
% Note: set sample time to -1 to mark model as discrete
Plant = ss(A,[B B],C,0,-1,'inputname',{'u' 'w'},...
9-51
9
Design Case Studies
'outputname','y');
Assuming that Q = R = 1 , you can now design the discrete Kalman filter by
Q = 1; R = 1;
[kalmf,L,P,M] = kalman(Plant,Q,R);
This returns a state-space model kalmf of the filter as well as the innovation
gain
M
M =
3.7980e-01
8.1732e-02
-2.5704e-01
The inputs of kalmf are u and y v , and its outputs are the plant output and
state estimates y e = yˆ [ n n ] and xˆ [ n n ] .
u
ye
kalmf
yv
xˆ [ n n ]
Kalman filter
Because you are interested in the output estimate y e , keep only the first output
of kalmf. Type
kalmf = kalmf(1,:);
kalmf
a =
x1_e
x2_e
x3_e
x1_e
0.7683
0.6202
-0.081732
x2_e
-0.494
0
1
x1_e
u
-0.3832
y
0.3586
b =
9-52
x3_e
0.1129
0
0
Kalman Filtering
x2_e
x3_e
0.5919
0.5191
0.3798
0.081732
y_e
x1_e
0.6202
x2_e
0
y_e
u
0
y
0.3798
c =
x3_e
0
d =
I/O groups:
Group name
KnownInput
Measurement
OutputEstimate
I/O
I
I
O
Channel(s)
1
2
1
Sampling time: unspecified
Discrete-time model.
To see how the filter works, generate some input data and random noise and
compare the filtered response y e with the true response y . You can either
generate each response separately, or generate both together. To simulate each
response separately, use lsim with the plant alone first, and then with the
plant and filter hooked up together. The joint simulation alternative is detailed
next.
The block diagram below shows how to generate both true and filtered outputs.
u
Plant
Process noise
y
yv
Kalman
filter
ye
Sensor noise
y
9-53
9
Design Case Studies
You can construct a state-space model of this block diagram with the functions
parallel and feedback. First build a complete plant model with u, w, v as
inputs and y and y v (measurements) as outputs.
a
b
c
d
P
=
=
=
=
=
A;
[B B 0*B];
[C;C];
[0 0 0;0 0 1];
ss(a,b,c,d,-1,'inputname',{'u' 'w' 'v'},...
'outputname',{'y' 'yv'});
Then use parallel to form the following parallel connection.
w
y
v
P
yv
u
Kalman
yv
filter
ye
sys = parallel(P,kalmf,1,1,[],[])
Finally, close the sensor loop by connecting the plant output y v to the filter
input y v with positive feedback.
% Close loop around input #4 and output #2
SimModel = feedback(sys,1,4,2,1)
% Delete yv from I/O list
SimModel = SimModel([1 3],[1 2 3])
The resulting simulation model has w, v, u as inputs and y, y e as outputs.
SimModel.inputname
ans =
'w'
9-54
Kalman Filtering
'v'
'u'
SimModel.outputname
ans =
'y'
'y_e'
You are now ready to simulate the filter behavior. Generate a sinusoidal input
u and process and measurement noise vectors w and v .
t = [0:100]';
u = sin(t/5);
n = length(t)
randn('seed',0)
w = sqrt(Q)*randn(n,1);
v = sqrt(R)*randn(n,1);
Now simulate with lsim.
[out,x] = lsim(SimModel,[w,v,u]);
y = out(:,1);
ye = out(:,2);
yv = y + v;
% true response
% filtered response
% measured response
and compare the true and filtered responses graphically.
subplot(211), plot(t,y,'--',t,ye,'-'),
xlabel('No. of samples'), ylabel('Output')
title('Kalman filter response')
subplot(212), plot(t,y-yv,'-.',t,y-ye,'-'),
9-55
9
Design Case Studies
xlabel('No. of samples'), ylabel('Error')
The first plot shows the true response y (dashed line) and the filtered output
y e (solid line). The second plot compares the measurement error (dash-dot)
with the estimation error (solid). This plot shows that the noise level has been
significantly reduced. This is confirmed by the following error covariance
computations.
MeasErr = y-yv;
MeasErrCov = sum(MeasErr.*MeasErr)/length(MeasErr);
EstErr = y-ye;
EstErrCov = sum(EstErr.*EstErr)/length(EstErr);
The error covariance before filtering (measurement error) is
MeasErrCov
MeasErrCov =
1.1138
while the error covariance after filtering (estimation error) is only
9-56
Kalman Filtering
EstErrCov
EstErrCov =
0.2722
Time-Varying Kalman Filter
The time-varying Kalman filter is a generalization of the steady-state filter for
time-varying systems or LTI systems with nonstationary noise covariance.
Given the plant state and measurement equations
x [ n + 1 ] = Ax [ n ] + Bu [ n ] + Gw [ n ]
y v [ n ] = Cx [ n ] + v [ n ]
the time-varying Kalman filter is given by the recursions
Measurement update
xˆ [ n n ] = xˆ [ n n – 1 ] + M [ n ] ( y v [ n ] – Cxˆ [ n n – 1 ] )
T –1
T
M [ n ] = P [ n n – 1 ]C ( R [ n ] + CP [ n n – 1 ]C )
P [ n n ] = ( I – M [ n ]C ) P [ n n – 1 ]
Time update
xˆ [ n + 1 n ] = Axˆ [ n n ] + Bu [ n ]
T
P [ n + 1 n ] = AP [ n n ]A + GQ [ n ]G
T
with xˆ [ n n – 1 ] and xˆ [ n n ] as defined in “Discrete Kalman Filter”, and in the
following.
T
Q [ n ] = E ( w [ n ]w [ n ] )
T
R [ n ] = E ( v [ n ]v [ n ] )
T
P[ n n] = E({ x [ n] – x [ n n ] }{ x [ n] – x [ n n ]} )
T
P[ n n – 1] = E( { x[ n] – x[ n n – 1] }{ x [n ] – x [n n – 1 ]} )
9-57
9
Design Case Studies
For simplicity, we have dropped the subscripts indicating the time dependence
of the state-space matrices.
Given initial conditions x [ 1 0 ] and P [ 1 0 ] , you can iterate these equations to
perform the filtering. Note that you must update both the state estimates
x [ n . ] and error covariance matrices P [ n . ] at each time sample.
Time-Varying Design
Although the Control System Toolbox does not offer specific commands to
perform time-varying Kalman filtering, it is easy to implement the filter
recursions in MATLAB. This section shows how to do this for the stationary
plant considered above.
First generate noisy output measurements
% Use process noise w and measurement noise v generated above
sys = ss(A,B,C,0,-1);
y = lsim(sys,u+w);
% w = process noise
yv = y + v;
% v = measurement noise
Given the initial conditions
x[ 1 0] = 0 ,
P [ 1 0 ] = BQB
T
you can implement the time-varying filter with the following for loop.
P = B*Q*B';
% Initial error covariance
x = zeros(3,1);
% Initial condition on the state
ye = zeros(length(t),1);
ycov = zeros(length(t),1);
for i=1:length(t)
% Measurement update
Mn = P*C'/(C*P*C'+R);
x = x + Mn*(yv(i)-C*x);
P = (eye(3)-Mn*C)*P;
ye(i) = C*x;
errcov(i) = C*P*C';
% Time update
9-58
% x[n|n]
% P[n|n]
Kalman Filtering
x = A*x + B*u(i);
P = A*P*A' + B*Q*B';
end
% x[n+1|n]
% P[n+1|n]
You can now compare the true and estimated output graphically.
subplot(211), plot(t,y,'--',t,ye,'-')
title('Time-varying Kalman filter response')
xlabel('No. of samples'), ylabel('Output')
subplot(212), plot(t,y-yv,'-.',t,y-ye,'-')
xlabel('No. of samples'), ylabel('Output')
The first plot shows the true response y (dashed line) and the filtered response
y e (solid line). The second plot compares the measurement error (dash-dot)
with the estimation error (solid).
The time-varying filter also estimates the covariance errcov of the estimation
error y – y e at each sample. Plot it to see if your filter reached steady state (as
you expect with stationary input noise).
subplot(211)
9-59
9
Design Case Studies
plot(t,errcov), ylabel('Error covar')
From this covariance plot, you can see that the output covariance did indeed
reach a steady state in about five samples. From then on, your time-varying
filter has the same performance as the steady-state version.
Compare with the estimation error covariance derived from the experimental
data. Type
EstErr = y-ye;
EstErrCov = sum(EstErr.*EstErr)/length(EstErr)
EstErrCov =
0.2718
This value is smaller than the theoretical value errcov and close to the value
obtained for the steady-state design.
Finally, note that the final value M [ n ] and the steady-state value M of the
innovation gain matrix coincide.
Mn, M
9-60
Kalman Filtering
Mn =
0.3798
0.0817
-0.2570
M =
0.3798
0.0817
-0.2570
References
[1] Grimble, M.J., Robust Industrial Control: Optimal Design Approach for
Polynomial Systems, Prentice Hall, 1994, p. 261 and pp. 443-456.
9-61
9
Design Case Studies
9-62
10
Reliable Computations
Introduction . . . . . . . . . . . . . . . . . . . . 10-2
Conditioning and Numerical Stability . . . . . . . . 10-4
Conditioning . . . . . . . . . . . . . . . . . . . . 10-4
Numerical Stability . . . . . . . . . . . . . . . . . . 10-6
Choice of LTI Model
State Space . . . . .
Transfer Function . .
Zero-Pole-Gain Models
Scaling
Summary
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10-13
. . . . . . . . . . . . . . . . . . . . . 10-15
. . . . . . . . . . . . . . . . . . . . 10-17
References . . . . . . . . . . . . . . . . . . . . 10-18
10
Reliable Computations
Introduction
When working with low-order SISO models (less than five states), computers
are usually quite forgiving and insensitive to numerical problems. You
generally won't encounter any numerical difficulties and MATLAB will give
you accurate answers regardless of the model or conversion method you choose.
For high order SISO models and MIMO models, however, the finite-precision
arithmetic of a computer is not so forgiving and you must exercise caution.
In general, to get a numerically accurate answer from a computer, you need
• A well-conditioned problem
• An algorithm that is numerically stable in finite-precision arithmetic
• A good software implementation of the algorithm
A problem is said to be well-conditioned if small changes in the data cause only
small corresponding changes in the solution. If small changes in the data have
the potential to induce large changes in the solution, the problem is said to be
ill-conditioned. An algorithm is numerically stable if it does not introduce any
more sensitivity to perturbation than is already inherent in the problem. Many
numerical linear algebra algorithms can be shown to be backward stable; i.e.,
the computed solution can be shown to be (near) the exact solution of a slightly
perturbed original problem. The solution of a slightly perturbed original
problem will be close to the true solution if the problem is well-conditioned.
Thus, a stable algorithm cannot be expected to solve an ill-conditioned problem
any more accurately than the data warrant, but an unstable algorithm can
produce poor solutions even to well-conditioned problems. For further details
and references to the literature see [5].
While most of the tools in the Control System Toolbox use reliable algorithms,
some of the tools do not use stable algorithms and some solve ill-conditioned
problems. These unreliable tools work quite well on some problems (low-order
systems) but can encounter numerical difficulties, often severe, when pushed
on higher-order problems. These tools are provided because
• They are quite useful for low-order systems, which form the bulk of
real-world engineering problems.
• Many control engineers think in terms of these tools.
• A more reliable alternative tool is usually available in this toolbox.
10-2
Introduction
• They are convenient for pedagogical purposes.
At the same time, it is important to appreciate the limitations of computer
analyses. By following a few guidelines, you can avoid certain tools and models
when they are likely to get you into trouble. The following sections try to
illustrate, through examples, some of the numerical pitfalls to be avoided. We
also encourage you to get the most out of the good algorithms by ensuring, if
possible, that your models give rise to problems that are well-conditioned.
10-3
10
Reliable Computations
Conditioning and Numerical Stability
Two of the key concepts in numerical analysis are the conditioning of problems
and the stability of algorithms.
Conditioning
Consider the linear system Ax = b given by
A =
0.7800
0.9130
0.5630
0.6590
b =
0.2170
0.2540
The true solution is x = [1, -1]' and you can calculate it approximately using
MATLAB.
x = A\b
x =
1.0000
-1.0000
format long, x
x =
0.99999999991008
-0.99999999987542
Of course, in real problems you almost never have the luxury of knowing the
true solution. This problem is very ill-conditioned. To see this, add a small
perturbation to A
E =
0.0010
-0.0020
0.0010
-0.0010
and solve the perturbed system ( A + E )x = b
xe = (A+E)\b
xe =
-5.0000
7.3085
10-4
Conditioning and Numerical Stability
Notice how much the small change in the data is magnified in the solution.
One way to measure the magnification factor is by means of the quantity
A
A
–1
called the condition number of A with respect to inversion. The condition
number determines the loss in precision due to roundoff errors in Gaussian
elimination and can be used to estimate the accuracy of results obtained from
matrix inversion and linear equation solution. It arises naturally in
–1
perturbation theories that compare the perturbed solution ( A + E ) b with the
–1
true solution A b .
In MATLAB, the function cond calculates the condition number in 2-norm.
cond(A) is the ratio of the largest singular value of A to the smallest. Try it for
the example above. The usual rule is that the exponent log10(cond(A)) on the
condition number indicates the number of decimal places that the computer
can lose to roundoff errors.
IEEE standard double precision numbers have about 16 decimal digits of
accuracy, so if a matrix has a condition number of 1010, you can expect only six
digits to be accurate in the answer. If the condition number is much greater
than 1/sqrt(eps), caution is advised for subsequent computations. For IEEE
arithmetic, the machine precision, eps, is about 2.2 × 10 -16, and 1/sqrt(eps)
= 6.7 × 10 8.
Another important aspect of conditioning is that, in general, residuals are
reliable indicators of accuracy only if the problem is well-conditioned. To
illustrate, try computing the residual vector r = Ax – b for the two candidate
solutions x = [0.999 -1.001]' and x = [0.341 -0.087]'. Notice that the
second, while clearly a much less accurate solution, gives a far smaller
residual. The conclusion is that residuals are unreliable indicators of relative
solution accuracy for ill-conditioned problems. This is a good reason to be
concerned with computing or estimating accurately the condition of your
problem.
Another simple example of an ill-conditioned problem is the n -by- n matrix
with ones on the first upper-diagonal.
A = diag(ones(1,n-1),1);
This matrix has n eigenvalues at 0. Now consider a small perturbation of the
–n
data consisting of adding the number 2 to the first element in the last ( n th)
10-5
10
Reliable Computations
row of A. This perturbed matrix has n distinct eigenvalues λ 1, ..., λ n with
λ k = 1 ⁄ 2 exp ( j2πk ⁄ n ) . Thus, you can see that this small perturbation in the
n
data has been magnified by a factor on the order of 2 to result in a rather
large perturbation in the solution (the eigenvalues of A). Further details and
related examples are to be found in [7].
It is important to realize that a matrix can be ill-conditioned with respect to
inversion but have a well-conditioned eigenproblem, and vice versa. For
example, consider an upper triangular matrix of ones (zeros below the
diagonal) given by
A = triu(ones(n));
This matrix is ill-conditioned with respect to its eigenproblem (try small
perturbations in A(n,1) for, say, n=20), but is well-conditioned with respect to
inversion (check its condition number). On the other hand, the matrix
A = 1 1
1 1+δ
has a well-conditioned eigenproblem, but is ill-conditioned with respect to
inversion for small δ .
Numerical Stability
Numerical stability is somewhat more difficult to illustrate meaningfully.
Consult the references in [5], [6], and [7] for further details. Here is one small
example to illustrate the difference between stability and conditioning.
Gaussian elimination with no pivoting for solving the linear system Ax = b is
known to be numerically unstable. Consider
A = 0.001 1.000
1.000 – 1.000
b = 1.000
0.000
All computations are carried out in three-significant-figure decimal arithmetic.
–1
The true answer x = A b is approximately
x = 0.999
0.999
10-6
Conditioning and Numerical Stability
Using row 1 as the pivot row (i.e., subtracting 1000 times row 1 from row 2) you
arrive at the equivalent triangular system.
0.001 1.000 x 1
0 – 1000 x 2
=
1.000
– 1000
Note that the coefficient multiplying x 2 in the second equation should be
– 1001, but because of roundoff, becomes – 1000 . As a result, the second
equation yields x 2 = 1.000 , a good approximation, but now back-substitution
in the first equation
0.001x 1 = 1.000 – ( 1.000 ) ( 1.000 )
yields x 1 = 0.000 . This extremely bad approximation of x 1 is the result of
numerical instability. The problem itself can be shown to be quite
well-conditioned. Of course, MATLAB implements Gaussian elimination with
pivoting.
10-7
10
Reliable Computations
Choice of LTI Model
Now turn to the implications of the results in the last section on the linear
modeling techniques used for control engineering. The Control System Toolbox
includes the following types of LTI models that are applicable to discussions of
computational reliability:
• State space
• Transfer function, polynomial form
• Transfer function, factored zero-pole-gain form
The following subsections show that state space is most preferable for
numerical computations.
State Space
The state-space representation is the most reliable LTI model to use for
computer analysis. This is one of the reasons for the popularity of “modern”
state-space control theory. Stable computer algorithms for eigenvalues,
frequency response, time response, and other properties of the ( A, B, C, D )
quadruple are known [5] and implemented in this toolbox. The state-space
model is also the most natural model in the MATLAB matrix environment.
Even with state-space models, however, accurate results are not guaranteed,
because of the problems of finite-word-length computer arithmetic discussed in
the last section. A well-conditioned problem is usually a prerequisite for
obtaining accurate results and makes it important to have reasonable scaling
of the data. Scaling is discussed further in the “Scaling” section later in this
chapter.
Transfer Function
Transfer function models, when expressed in terms of expanded polynomials,
tend to be inherently ill-conditioned representations of LTI systems. For
systems of order greater than 10, or with very large/small polynomial
coefficients, difficulties can be encountered with functions like roots, conv,
bode, step, or conversion functions like ss or zpk.
10-8
Choice of LTI Model
A major difficulty is the extreme sensitivity of the roots of a polynomial to its
coefficients. This example is adapted from Wilkinson, [6] as an illustration.
Consider the transfer function
1
1
H ( s ) = ------------------------------------------------------------- = ----------------------------------------------------------20
19
( s + 1 ) ( s + 2 )... ( s + 20 )
s + 210s + ... + 20!
The A matrix of the companion realization of H ( s ) is
A =
0
0
:
0
– 20!
1
0
:
0
.
0
1
.
...
...
...
...
.
.
.
0
0
:
1
– 210
Despite the benign looking poles of the system (at -1,-2,..., -20) you are faced
18
with a rather large range in the elements of A , from 1 to 20! ≈ 2.4 × 10 . But
19
the difficulties don’t stop here. Suppose the coefficient of s in the transfer
– 23
– 23
–7
function (or A ( n, n ) ) is perturbed from 210 to 210 + 2
(2
≈ 1.2 × 10 ).
Then, computed on a VAX (IEEE arithmetic has enough mantissa for only
n = 17 ), the poles of the perturbed transfer function (equivalently, the
eigenvalues of A ) are
eig(A)'
ans =
Columns 1 through 7
-19.9998 -19.0019 -17.9916 -17.0217 -15.9594 -15.0516 -13.9504
Columns 8 through 14
-13.0369 -11.9805 -11.0081
-9.9976
-9.0005
-7.9999
Columns 15 through 20
-6.0000 -5.0000 -4.0000
-3.0000
-2.0000
-1.0000
-7.0000
The problem here is not roundoff. Rather, high-order polynomials are simply
intrinsically very sensitive, even when the zeros are well separated. In this
–9
case, a relative perturbation of the order of 10 induced relative
–2
perturbations of the order of 10 in some roots. But some of the roots changed
10-9
10
Reliable Computations
very little. This is true in general. Different roots have different sensitivities to
different perturbations. Computed roots may then be quite meaningless for a
polynomial, particularly high-order, with imprecisely known coefficients.
Finding all the roots of a polynomial (equivalently, the poles of a transfer
function or the eigenvalues of a matrix in controllable or observable canonical
form) is often an intrinsically sensitive problem. For a clear and detailed
treatment of the subject, including the tricky numerical problem of deflation,
consult [6].
It is therefore preferable to work with the factored form of polynomials when
available. To compute a state-space model of the transfer function H ( s )
defined above, for example, you could expand the denominator of H , convert
the transfer function model to state space, and extract the state-space data by
H1 = tf(1,poly(1:20))
H1ss = ss(H1)
[a1,b1,c1] = ssdata(H1)
However, you should rather keep the denominator in factored form and work
with the zero-pole-gain representation of H ( s ) .
H2 = zpk([],1:20,1)
H2ss = ss(H2)
[a2,b2,c2] = ssdata(H2)
Indeed, the resulting state matrix a2 is better conditioned.
[cond(a1)
cond(a2)]
ans =
2.7681e+03
8.8753e+01
and the conversion from zero-pole-gain to state space incurs no loss of accuracy
in the poles.
format long e
[sort(eig(a1))
sort(eig(a2))]
ans =
9.999999999998792e-01
2.000000000001984e+00
3.000000000475623e+00
3.999999981263996e+00
10-10
1.000000000000000e+00
2.000000000000000e+00
3.000000000000000e+00
4.000000000000000e+00
Choice of LTI Model
5.000000270433721e+00
5.999998194359617e+00
7.000004542844700e+00
8.000013753274901e+00
8.999848908317270e+00
1.000059459550623e+01
1.099854678336595e+01
1.200255822210095e+01
1.299647702454549e+01
1.400406940833612e+01
1.499604787386921e+01
1.600304396718421e+01
1.699828695210055e+01
1.800062935148728e+01
1.899986934359322e+01
2.000001082693916e+01
5.000000000000000e+00
6.000000000000000e+00
7.000000000000000e+00
8.000000000000000e+00
9.000000000000000e+00
1.000000000000000e+01
1.100000000000000e+01
1.200000000000000e+01
1.300000000000000e+01
1.400000000000000e+01
1.500000000000000e+01
1.600000000000000e+01
1.700000000000000e+01
1.800000000000000e+01
1.900000000000000e+01
2.000000000000000e+01
There is another difficulty with transfer function models when realized in
state-space form with ss. They may give rise to badly conditioned eigenvector
matrices, even if the eigenvalues are well separated. For example, consider the
normal matrix
A = [5
4
1
1
4
5
1
1
1
1
4
2
1
1
2
4]
Its eigenvectors and eigenvalues are given as follows.
[v,d] = eig(A)
v =
0.7071
-0.7071
0.0000
-0.0000
-0.0000
0.0000
0.7071
-0.7071
-0.3162
-0.3162
0.6325
0.6325
0.6325
0.6325
0.3162
0.3162
1.0000
0
0
0
2.0000
0
0
0
5.0000
0
0
0
d =
10-11
10
Reliable Computations
0
0
0
10.0000
The condition number (with respect to inversion) of the eigenvector matrix is
cond(v)
ans =
1.000
Now convert a state-space model with the above A matrix to transfer function
form, and back again to state-space form.
b = [1 ; 1 ; 0 ; -1];
c = [0 0 2 1];
H = tf(ss(A,b,c,0));
[Ac,bc,cc] = ssdata(H)
% Transfer function
% Convert back to state space
The new A matrix is
Ac =
18.0000
16.0000
0
0
-6.0625
0
4.0000
0
2.8125
0
0
1.0000
-1.5625
0
0
0
Note that Ac is not a standard companion matrix and has already been
balanced as part of the ss conversion (see ssbal for details).
Note also that the eigenvectors have changed.
[vc,dc] = eig(Ac)
10-12
vc =
-0.5017
-0.8026
-0.3211
-0.0321
0.2353
0.7531
0.6025
0.1205
0.0510
0.4077
0.8154
0.4077
0.0109
0.1741
0.6963
0.6963
dc =
10.0000
0
0
0
0
5.0000
0
0
0
0
2.0000
0
0
0
0
1.0000
Choice of LTI Model
The condition number of the new eigenvector matrix
cond(vc)
ans =
34.5825
is thirty times larger.
The phenomenon illustrated above is not unusual. Matrices in companion form
or controllable/observable canonical form (like Ac) typically have
worse-conditioned eigensystems than matrices in general state-space form
(like A). This means that their eigenvalues and eigenvectors are more sensitive
to perturbation. The problem generally gets far worse for higher-order systems.
Working with high-order transfer function models and converting them back
and forth to state space is numerically risky.
In summary, the main numerical problems to be aware of in dealing with
transfer function models (and hence, calculations involving polynomials) are
• The potentially large range of numbers leads to ill-conditioned problems,
especially when such models are linked together giving high-order
polynomials.
• The pole locations are very sensitive to the coefficients of the denominator
polynomial.
• The balanced companion form produced by ss, while better than the
standard companion form, often results in ill-conditioned eigenproblems,
especially with higher-order systems.
The above statements hold even for systems with distinct poles, but are
particularly relevant when poles are multiple.
Zero-Pole-Gain Models
The third major representation used for LTI models in MATLAB is the
factored, or zero-pole-gain (ZPK) representation. It is sometimes very
convenient to describe a model in this way although most major design
methodologies tend to be oriented towards either transfer functions or
state-space.
In contrast to polynomials, the ZPK representation of systems can be more
reliable. At the very least, the ZPK representation tends to avoid the
10-13
10
Reliable Computations
extraordinary arithmetic range difficulties of polynomial coefficients, as
illustrated in the “Transfer Function” section. The transformation from state
space to zero-pole-gain is stable, although the handling of infinite zeros can
sometimes be tricky, and repeated roots can cause problems.
If possible, avoid repeated switching between different model representations.
As discussed in the previous sections, when transformations between models
are not numerically stable, roundoff errors are amplified.
10-14
Scaling
Scaling
State space is the preferred model for LTI systems, especially with higher order
models. Even with state-space models, however, accurate results are not
guaranteed, because of the finite-word-length arithmetic of the computer. A
well-conditioned problem is usually a prerequisite for obtaining accurate
results.
You should generally normalize or scale the ( A, B, C, D ) matrices of a system
to improve their conditioning. An example of a poorly scaled problem might be
a dynamic system where two states in the state vector have units of light years
and millimeters. You would expect the A matrix to contain both very large and
very small numbers. Matrices containing numbers widely spread in value are
often poorly conditioned both with respect to inversion and with respect to their
eigenproblems, and inaccurate results can ensue.
Normalization also allows meaningful statements to be made about the degree
of controllability and observability of the various inputs and outputs.
A set of ( A, B, C, D ) matrices can be normalized using diagonal scaling
matrices N u , N x , and N y to scale u, x, and y.
u = Nu un
x = Nx xn
y = Ny yn
so the normalized system is
x· n = A n x n + B n u n
yn = Cn xn + Dn un
where
–1
A n = N x ANx
–1
C n = N y CN x
–1
B n = N x BN u
–1
D n = N y DN u
Choose the diagonal scaling matrices according to some appropriate
normalization procedure. One criterion is to choose the maximum range of each
of the input, state, and output variables. This method originated in the days of
analog simulation computers when u n , x n , and y n were forced to be between
±10 Volts. A second method is to form scaling matrices where the diagonal
entries are the smallest deviations that are significant to each variable. An
10-15
10
Reliable Computations
excellent discussion of scaling is given in the introduction to the LINPACK
Users’ Guide, [1].
Choose scaling based upon physical insight to the problem at hand. If you
choose not to scale, and for many small problems scaling is not necessary, be
aware that this choice affects the accuracy of your answers.
Finally, note that the function ssbal performs automatic scaling of the state
vector. Specifically, it seeks to minimize the norm of
–1
–1
N x AN x
Nx B
CNx
0
by using diagonal scaling matrices N x . Such diagonal scaling is an economical
way to compress the numerical range and improve the conditioning of
subsequent state-space computations.
10-16
Summary
Summary
This chapter has described numerous things that can go wrong when
performing numerical computations. You won’t encounter most of these
difficulties when you solve practical lower-order problems. The problems
described here pertain to all computer analysis packages. MATLAB has some
of the best algorithms available, and, where possible, notifies you when there
are difficulties. The important points to remember are
• State-space models are, in general, the most reliable models for subsequent
computations.
• Scaling model data can improve the accuracy of your results.
• Numerical computing is a tricky business, and virtually all computer tools
can fail under certain conditions.
10-17
10
Reliable Computations
References
[1] Dongarra, J.J., J.R. Bunch, C.B. Moler, and G.W. Stewart, LINPACK User’s
Guide, SIAM Publications, Philadelphia, PA, 1978.
[2] Franklin, G.F. and J.D. Powell, Digital Control of Dynamic Systems,
Addison-Wesley, 1980.
[3] Kailath, T., Linear Systems, Prentice-Hall, 1980.
[4] Laub, A.J., “Numerical Linear Algebra Aspects of Control Design
Computations,” IEEE Transactions on Automatic Control, Vol. AC-30, No. 2,
February 1985, pp. 97-108.
[5] Wilkinson, J.H., Rounding Errors in Algebraic Processes, Prentice-Hall,
1963.
[6] Wilkinson, J.H., The Algebraic Eigenvalue Problem, Oxford University
Press, 1965.
10-18
11
Tool and Viewer
Quick Start
Introduction . . . . . . . . . . . . . . . . . . . . 11-2
SISO Design Tool . . . . . . .
Importing and Exporting Models .
Configuring the Feedback Structure
Tuning Compensators . . . . . .
Viewing Loop Responses . . . . .
Viewing System Data . . . . . .
Storing and Retrieving Designs . .
Customizing the SISO Design Tool
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. 11-3
. 11-4
. 11-7
. 11-8
11-13
11-14
11-15
11-16
LTI Viewer . . . . . . . . .
Right-Click Menu . . . . . .
LTI Viewer Toolbar . . . . . .
Basic LTI Viewer Tasks . . . .
Importing and Exporting Models
Selecting Response Types . . .
Analyzing MIMO Models . . .
Customizing the LTI Viewer . .
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11-18
11-18
11-19
11-19
11-20
11-22
11-23
11-26
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11
Tool and Viewer Quick Start
Introduction
The following two sections are a brief introduction to the basics of the tools
provided with the Control System Toolbox. The graphical user interfaces
(GUI’s) covered are as follows:
• “SISO Design Tool”— A GUI for designing prefilters and compensators for
SISO systems
• “LTI Viewer” — A tool for viewing and analyzing LTI system data
11-2
SISO Design Tool
SISO Design Tool
The SISO Design Tool is a graphical user interface (GUI) that allows you to
analyze and tune SISO feedback control systems. Using the SISO Design Tool,
you can graphically tune the gains and dynamics of the compensator (C) and
prefilter (F) using a mix of root locus and loop shaping techniques. For
example, you can use the root locus view to stabilize the feedback loop and
enforce some minimum damping, and use the Bode diagrams to adjust the
bandwidth, check the gain and phase margins, or add a notch filter for
disturbance rejection. You can also bring up an open-loop Nichols view or Bode
diagram of the prefilter (F) by selecting these items from the View menu. All
views are dynamically linked; for example, if you change the gain in the root
locus, it immediately affects the Bode diagrams as well.
Current Compensator panel
Root Locus
Feedback structure
Open-loop Bode diagram
Status panel
The SISO Design Tool is designed to work closely with the LTI Viewer,
allowing you to rapidly iterate on your design and immediately see the results
in the LTI Viewer. When you make a change in your compensator, the LTI
Viewer associated with your SISO Design Tool automatically updates the
response plots that you have chosen. By default, the SISO Design Tool displays
11-3
11
Tool and Viewer Quick Start
the root locus and open-loop Bode diagrams for your imported systems. You can
also bring up an open-loop Nichols view or prefilter Bode diagram by selecting
these items in the View menu.
Imported systems can include any of elements of the feedback structure
diagram located to the right of the Current Compensator panel. You cannot
change imported plant (G) or sensor (H) models, but you can use the SISO
Design Tool for designing a new (or modifying an existing) prefilter (F) or
compensator (C) for your imported plant and sensor configuration.
For a quick discussion of basic tasks you can do with the SISO Design Tool, see
the following:
• “Importing and Exporting Models”
• “Configuring the Feedback Structure”
• “Tuning Compensators”
- “Root-Locus”
- “Open-Loop Bode Diagram”
- “Open-Loop Nichols Plot”
• “Viewing Loop Responses”
• “Viewing Loop Responses”
• “Storing and Retrieving Designs”
• “Customizing the SISO Design Tool”
In addition, there is an extensive discussion of how to use the SISO Design Tool
in Designing Compensators, chapter 4 in Getting Started with the Control
System Toolbox. See SISO Design Tool for a description of all the features
available.
Importing and Exporting Models
The SISO Design Tool provides graphical user interfaces to facilitate importing
and exporting of linear models.
11-4
SISO Design Tool
Importing Models
To import models into the SISO Design Tool, select Import under the File
menu. This opens the Import System Data window.
To import a model:
• Specify whether you want to import a SISO model from the MATLAB
workspace, a MAT-file, or from a Simulink model. The window lists the
available models for each format under SISO Models.
• Click on the desired model.
• Click a right arrow to specify whether you want to import the model as the
plant (G), Sensor (H), Prefilter (F), or Compensator (C).
Click the OK Button
Alternatively, you can directly import a model into the SISO Design Tool using
the sisotool function, as in
sisotool(modelname)
In this case, modelname is imported as the plant (G). See the sisotool function
for more information.
11-5
11
Tool and Viewer Quick Start
Exporting Models
Use Export in the File menu to open the SISO Tool Export window.
Selecting Models to Export. The SISO Tool Export window not only lists all the
models displayed in your SISO Design Tool, but also contains various transfer
functions associated with standard control analysis techniques. These include
open and closed loop transfer functions, input and output sensitivity functions,
and the state-space model of the overall feedback loop.
To select a model for export, left-click on the row containing the component
name. To specify a different export name, double-click on the Export As cell for
the component. This makes the name in the cell editable.
You can export models back to the MATLAB workspace or to disk. In the latter
case, the models are saved in a MAT-file.
Exporting to Workspace. To export models to the MATLAB workspace, simply
click Export to Workspace.
11-6
SISO Design Tool
Exporting to Disk. If you choose Export to Disk, this window opens.
The Export to Disk window provides a default file name. If you want to change
the name, specify the new name for your model(s) and click Save. Your models
are stored in a MAT-file.
Exporting Multiple Models . There are two ways to export multiple models:
• If the models are adjacent in the model selection list, hold down the Shift key
while selecting the models with your mouse.
• If the models are nonadjacent, hold down the Ctrl key and select the models
by left-clicking
Configuring the Feedback Structure
The Feedback Structure panel displays the current configuration of these
components:
• Compensator (C)
• Prefilter (F)
• Plant (G)
• Sensor (H)
The default configuration is shown below.
11-7
11
Tool and Viewer Quick Start
To cycle through the feedback structures, click the FS button. This figure
shows the alternate feedback structures.
Compensator in the feedback path
Feedforward controller
Cascade configuration with filter F in the
minor loop
Clicking the +/- button toggles between positive and negative feedback signs.
Negative feedback is the default.
Tuning Compensators
The SISO Design Tool simplifies the task of designing compensators.
Regardless of which views you have open--root-locus, Bode diagrams, or
Nichols plots--there are three ways to alter compensator designs:
• Interactive graphics let you tune compensator gains and adjust dynamics
(poles and zeros).
• Right-click menus allow you to add/remove dynamics and customize the view
(for example, add a grid, zoom in/out, add design constraints, or customize
plot properties).
• The Edit Compensator window is a GUI with fields for keyboard entry of gain
values and pole/zero locations.
You can perform any of these tasks in the root locus, open-loop and prefilter
Bode diagrams, or Nichols plots in the SISO Design Tool. Once you've added
dynamics to your compensator, you can dynamically update pole and zero
11-8
SISO Design Tool
locations by dragging them. The SISO Design Tool is designed so that a change
in any one view is automatically reflected in other views in the GUI. In
particular, the Current Compensator panel always reflects the current
compensator design. The next sections discuss some of the ways you can tune
compensators in different SISO Design Tool views.
You can tune compensators in any of the views in the SISO Design Tool. These
views include the following:
• “Root-Locus”
• “Open-Loop Bode Diagram”
• “Open-Loop Nichols Plot”
• “Prefilter Bode Diagram”
Root-Locus
You can tune your compensator using root-locus techniques. This figure shows
an imported compensator and plant model; use the right-click menus and
11-9
11
Tool and Viewer Quick Start
interactive graphics features to add, adjust, and remove compensator
dynamics.
Move any red square to
change the compensator
gain.
You can move the
compensator poles and
zeros (red x’s and o’s,
respectively) with your
mouse.
Use the right-click menu to
add poles and zeros to your
compensator design.
The blue x’s and o’s
represent plant poles and
zeros. You cannot move
them.
The Status panel displays tips
on how to use the SISO
Design Tool.
See Root Locus Design in Getting Started with the Control System Toolbox for
an example of how to use root locus design techniques.
Open-Loop Bode Diagram
The SISO Design Tool supports the open-loop Bode diagram view of your
system. You can use the right-click menu and interactive graphics features to
11-10
SISO Design Tool
add, adjust, and remove compensator dynamics. This figure shows some of the
features of the open-loop Bode diagram.
Drag the bode magnitude
curve up and down to adjust
the compensator gain.
You can adjust notch
filter parameters by
moving them with
your mouse.
You can move compensator
poles and zeros (red x’s and
o’s, respectively) with your
mouse.
The blue x’s and o’s represent
plant poles and zeros. You
cannot move them.
Use the right-click
menu to add poles
and zeros to your
compensator.
The Status panel displays tips
on how to use the SISO Design
Tool.
See Bode Diagram Design in Getting Started with the Control System Toolbox
for an example of how to use Bode diagram design techniques.
Open-Loop Nichols Plot
An alternative method for compensator design is the open-loop Nichols plot.
You can use the right-click menu and interactive graphics features to add,
11-11
11
Tool and Viewer Quick Start
adjust, and remove compensator dynamics. This figure shows some of the
features of the open-loop Nichols plot.
Use the right-click menu to
add poles and zeros to your
compensator.
The brown stems
display the gain and
phase margins.
The blue x’s and o’s
represent plant poles
and zeros. You cannot
move them.
You can move compensator
poles and zeros (red x’s and
o’s, respectively) with your
mouse.
The Status panel displays
tips on how to use the SISO
Design Tool.
See Nichols Plot Design in Getting Started with the Control System Toolbox for
an example of how to use Nichols plot design techniques.
Prefilter Bode Diagram
The SISO Design Tool supports the prefilter Bode diagram view. You can add
dynamics to your prefilter design using the right-click menu, and you can
11-12
SISO Design Tool
adjust dynamics by dragging poles and zeros with your mouse. This figure
shows some of the features.
The green curve is the
prefilter Bode magnitude.
Move it up and down to
adjust the prefilter gain.
Use the right-click menu to
add poles and zeros to your
prefilter design.
The magenta curve is the
closed-loop response from
the prefilter input to the
plant output.
The green x is a prefilter
pole. Use your mouse to
move this pole to a new
location.
The Status panel displays tips
on how to use the SISO Design
Tool.
Viewing Loop Responses
The SISO Design Tool provides support for viewing loop responses for your
system. To see available frequency and time domain responses, pull down the
Analysis menu.
Choose the response you want to see; an LTI Viewer opens with the view
plotted. For example, if you select Response to Step Command, you will see a
closed-loop step response of your system.
11-13
11
Tool and Viewer Quick Start
If you select Other, the Response Plot Setup window opens.
This window allows for more specialized loop responses. Click here for more
information.
Viewing System Data
You can view data about your model by selecting System Data under the View
menu.
The System Data window displays the poles and zeros of your imported plant
and sensor models. Click Show Transfer Function to see the associated
11-14
SISO Design Tool
transfer function. For example, this picture shows the Gservo plant's transfer
function.
Type
load ltiexamples
at the MATLAB prompt to load the Gservo plant model.
Storing and Retrieving Designs
The SISO Design Tool provides a graphical user interface (GUI's) for storing
and retrieving compensator designs. Each design consists of a pair (C, F) of
compensator and prefilter models
11-15
11
Tool and Viewer Quick Start
To open the Compensator Design Archive window, select Store/Retrieve
from the Compensators menu.
You can use this window both to store and retrieve compensator designs.
To store a design, specify the name you want to save it under and click Store.
To retrieve any of the prefilter and/or compensator designs that you have
created during a SISO Design Tool session, right-clicking on the Design Name
you want to retrieve. Click Retrieve and the design is sent back to the SISO
Design Tool.
Customizing the SISO Design Tool
The SISO Design Tool provides a graphical user interface (GUI), called the
SISO Tool Preferences window, for customizing units, linestyles, axes
foreground and linestyle colors, fonts, compensator format, and Bode plot
options. Any options you set in this window apply to all the plots of the current
instance of the SISO Design Tool. If you open another instance of the SISO
Design Tool, it inherits its options from the Toolbox Preferences Editor.
11-16
SISO Design Tool
Opening the SISO Tool Preferences Window
To open the SISO Tool Preferences window, select SISO Tool Preferences
from the Edit menu.
There are four panes in the SISO Tool Preferences window:
• Units
• Style
• Options
• Line Colors
For complete descriptions of options available on the four panes, click on the
links.
11-17
11
Tool and Viewer Quick Start
LTI Viewer
LTI Viewer is a graphical user interface (GUI) that simplifies the analysis of
linear, time-invariant systems. You use the LTI Viewer to view and compare
the response plots of SISO and MIMO systems, or of several linear models at
the same time. You can generate time and frequency response plots to inspect
key response parameters, such as rise time, maximum overshoot, and stability
margins.
The LTI Viewer can display up to seven different plot types simultaneously,
including step, impulse, Bode (magnitude and phase or magnitude only),
Nyquist, Nichols, sigma, pole/zero, and I/O pole/zero.
Right-Click Menu
Using right-click menu options, you can access several LTI Viewer controls and
options, including:
• Plot Type — Changes the plot type
11-18
LTI Viewer
• Systems — Selects or deselects any of the models loaded in the LTI Viewer
• Characteristics — Displays key response characteristics and parameters
• Grid — Adds grids to your plot
• Properties — Opens the Property Editor, where you can customize plot
attributes
In addition to right-click menus, all response plots include data markers. These
allow you to scan the plot data, identify key data, and determine the source
system for a given plot.
LTI Viewer Toolbar
The LTI Viewer has a tool bar that you can use to do the following:
• Open a new LTI Viewer
• Print
• Zoom in and out
Basic LTI Viewer Tasks
For descriptions of basic tasks you can perform with the LTI Viewer, see the
other Help menu items:
• “Importing and Exporting Models”
• “Selecting Response Types”
• “Analyzing MIMO Models”
• “Customizing the LTI Viewer”
For examples of how to use the LTI Viewer, see Analyzing Models in Getting
Started with the Control System Toolbox. See LTI Viewer for descriptions of all
the features available in the LTI Viewer.
See ltiview for help on the function that opens an LTI Viewer.
11-19
11
Tool and Viewer Quick Start
Importing and Exporting Models
To import models into the LTI Viewer, select Import under the Edit menu.
This opens the LTI Browser, shown below.
Use the LTI Browser to import LTI models into or from the LTI Viewer
workspace.
To import a model:
• Click on the desired model in the LTI Browser List. To perform multiple
selections:
- Hold the Control key and click on nonadjacent models.
- Hold the Shift key while clicking to select multiple adjacent models.
• Click the OK or Apply Button
Note that the LTI Browser lists only the LTI models in the MATLAB
workspace.
Alternatively, you can directly import a model into the LTI Viewer using the
ltiview function, as in
ltiview({'step', 'bode'}, modelname)
See the ltiview function for more information.
11-20
LTI Viewer
Exporting Models
Use Export in the File menu to open the LTI Viewer Export window, shown
below.
The LTI Viewer Export window lists all the models with responses currently
displayed in your LTI Viewer. You can export models back to the MATLAB
workspace or to disk. In the latter case, the Control System Toolbox saves the
files as MAT-files.
To export single or multiple models, follow the steps described in the importing
models section above. If you choose Export to Disk, this window opens.
Choose a name for your model(s) and click Save. Your models are stored in a
MAT-file.
11-21
11
Tool and Viewer Quick Start
Selecting Response Types
There are two methods for selecting response plots in the LTI Viewer:
• Selecting Plot Type from the right-click menus
• Opening the Plot Configurations window
Right Click Menu: Plot Type
If you have a plot open in the LTI Viewer, you can switch to any other response
plot available by selecting Plot Type from the right click menu.
To change the response plot, select the new plot type from the Plot Type
submenu. The LTI Viewer automatically displays the new response plot.
Plot Configurations Window
The Plot Type feature of the right-click menu works on existing plots, but you
can also add plots to an LTI Viewer by using the Plot Configurations window.
By default, the LTI Viewer opens with a closed-loop step response. To
reconfigure an open viewer, select Plot Configuration in the Edit menu.
11-22
LTI Viewer
Use the radio buttons to select the number of plots you want displayed in your
LTI Viewer. For each plot, select a response type from the menus located on the
right-hand side of the window.
It's possible to configure a single LTI Viewer to contain up to six response plots.
Available response plots include: step, impulse, Bode (magnitude and phase, or
magnitude only), Nyquist, Nichols, sigma, pole/zero maps, and I/O pole/zero
maps.
Analyzing MIMO Models
If you import a MIMO system, or an LTI array containing multiple linear
models, you can use special features of the right-click menu to group the
response plots by input/output (I/O) pairs or select individual plots for display.
For example, if you randomly generate a 3-input, 3-output MIMO system,
sys_mimo=rss(3,3,3);
and open an LTI Viewer,
ltiview(sys_mimo);
11-23
11
Tool and Viewer Quick Start
the default is an unwrapped set of 9 plots, one from each input to each output.
I/O Grouping
You can group this by inputs, by outputs, or both by selecting I/O Grouping
and then Inputs, Outputs, or All, respectively, from the right-click menu.
11-24
LTI Viewer
For example, if you select Outputs, the LTI Viewer reconfigures into 3 plots,
one for each input.
Selecting None returns to the default configuration, where all I/O pairs are
displayed individually.
I/O Selector
Another way to organize MIMO system information is to choose I/O Selector
from the right-click menu, which opens the I/O Selector window.
11-25
11
Tool and Viewer Quick Start
This window automatically configures to the number of I/O pairs in your MIMO
system. You can select:
• Any individual plot (only one at a time) by clicking on a button
• Any row or column by clicking on Y(*) or U(*)
• All of the plots by clicking [all]
Using these options, you can inspect individual I/O pairs, or look at particular
I/O channels in detail.
Customizing the LTI Viewer
The LTI Viewer has a tool preferences editor, which allows you to set default
characteristics for specific instances of LTI Viewers. If you open a new instance
of either, each defaults to the characteristics specified in the Toolbox
Preferences editor.
LTI Viewer Preferences Editor
Select Viewer Preferences in the Edit menu of the LTI Viewer to open the LTI
Viewer Preferences editor. This figure shows the editor open to its first pane.
The LTI Viewer Preferences editor contains four panes:
• Units--Convert between various units, including rad/sec and Hertz
• Style--Customize grids, fonts, and colors
11-26
LTI Viewer
• Characteristics--Specify response plot characteristics, such as settling time
tolerance
• Parameters--Set time and frequency ranges, stop times, and time step size
If you want to customize the settings for all instances of LTI Viewers, see the
Toolbox Preferences editor.
11-27
11
Tool and Viewer Quick Start
11-28
Index
A
addition of LTI models 2-11
scalar 2-12
adjoint. See pertransposition
append 2-16, 4-28
array dimensions 4-7
arrays. See LTI arrays
state-space, to 1-40
TF model, to 1-39
ZPK model, to 1-39
covariance
error 9-56
customizing plots 8-1
customizing subplots 8-10
B
balancing realizations 3-8
building LTI arrays 4-12
C
canonical realizations 3-8
cell array 1-10
classical control 9-3
closed loop. See feedback
concatenation, model 1-10
horizontal 2-16
LTI arrays 4-15
conditioning, state-space models 10-5
connection
feedback 9-12
parallel 2-12
series 2-13
constructor functions, LTI objects 1-4
continuous-time 3-3
conversion, model
automatic 1-40
between model types 1-39
discrete to continuous (d2c) 1-34
with negative real poles 2-21
FRD model, to 1-39
resampling 2-26
SS model, to 1-39
D
d2d 2-26
delays
combining 1-51
discrete-time models 1-49
discretization 2-23
I/O 1-25
information, retrieving 1-51
input 1-25
output 1-25
Padé approximation 1-51
supported functionality 1-42
deletion
parts of LTI arrays 4-23
parts of LTI models 2-9
denominator
property 1-27
specification 1-8
value 1-23
descriptor systems. See state-space models,
descriptor
design
classical 9-3
Kalman estimator 9-36
LQG 9-31
regulators 9-31
robustness 9-28
I-1
Index
root locus 9-9
digital filter
filt 1-22
specification 1-21
dimensions
array 4-7
I/O 4-7
discrete-time models 3-3
Kalman estimator 9-50
resampling 2-26
See also LTI models
discretization 1-34
delay systems 2-23
first-order hold 2-22
matched poles/zeros 2-23
Tustin method 2-22
zero-order hold 2-20
dual. See transposition
E
error covariance 9-56
extraction
LTI arrays, in 4-21
LTI models, in 2-5
F
feedback 9-12
feedthrough gain 1-27
filt 1-22
filtering. See Kalman estimator
first-order hold (FOH) 2-22
with delays 2-23
FRD (frequency response data) objects
conversion to 1-39
frequencies
I-2
indexing by 2-7
referencing by 2-7
uses 1-3
frequency response 1-17
G
gain 1-11
feedthrough 1-27
property
LTI properties gain 1-27
gain margins 9-28
get 1-30
group. See I/O groups
H
hasdelay 1-51
I
I/O
concatenation 2-16
delays 1-25
dimensions 3-3
LTI arrays 4-7
groups 1-25
referencing models by group name 2-8
names 1-25, 1-35
conflicts, naming 2-4
referencing models by 2-8
relation 2-5
indexing into LTI arrays 4-20
single index access 4-20
inheritance 2-3
input 1-2
delays 1-25
Index
groups 1-25
names 1-25
number of inputs 3-3
InputDelay. See delays
InputGroup 1-25
conflicts, naming 2-4
See also I/O groups
InputName 1-32
conflicts, naming 2-4
See also I/O names
inversion
model 2-13
ioDelayMatrix. See delay
K
Kalman
filtering 9-50
Kalman estimator
continuous 9-36
discrete 9-50
L
LQG (linear quadratic-gaussian) method
continuous LQ regulator 9-36
cost function 9-36
design 9-31, 9-46
LQ-optimal gain 9-36
regulator 9-31
LTI (linear time-invariant) 1-2
LTI arrays 4-1
accessing models 4-20
analysis functions 4-29
array dimensions 4-7
building 4-15
building LTI arrays 4-12
building with rss 4-12
building with tf, zpk, ss, and frd 4-17
concatenation 4-15
conversion, model.See conversion
deleting parts of 4-23
dimensions, size, and shape 4-7
extracting subsystems 4-21
indexing into 4-20
interconnection functions 4-24
model dimensions 4-7
operations on 4-24
dimension requirements 4-26
special cases 4-26
reassigning parts of 4-22
size 4-7
stack 4-15
LTI models
addition 2-11
scalar 2-12
building 2-16
characteristics 3-3
continuous 3-3
conversion 1-39, 2-3
See also conversion, model
creating 1-8
discrete 1-19, 3-3
discretization, matched poles/zeros 2-23
empty 1-11, 3-3
functions, analysis 3-5
I/O group or channel name, referencing by 2-8
interconnection functions 2-16
inversion 2-13
model data, accessing 1-23
modifying 2-5
multiplication 2-13
operations 2-1
precedence rules 2-3
I-3
Index
See also operations
proper transfer function 3-3
resizing 2-9
subsystem, modifying 2-9
subtraction 2-12
type 3-3
LTI objects 1-25, 1-31
constructing 1-4
methods 1-4
properties. See LTI properties
See also LTI models
LTI properties 1-4, 1-25, 1-32
accessing property values (get) 1-30
displaying properties 1-31
generic properties 1-25
I/O groups. See I/O, groups
I/O names. See I/O, names
inheritance 2-3
model-specific properties 1-26
online help (ltiprops) 1-25
property names 1-25
property values 1-25
setting 1-28
LTI Viewer Preferences Editor 7-2
M
map, I/O 2-5
margins, gain and phase 9-28
methods 1-4
MIMO 1-2
model building 2-16
feedback connection 9-12
parallel connection 2-12
series connection 2-13
model dynamics, function list 3-5
modeling. See model building
I-4
multiplication 2-13
scalar 2-13
N
Notes 1-26
numerator
property 1-27
specification 1-8, 1-10
value 1-23
numerical stability 10-7
O
object-oriented programming 1-4
objects. See LTI objects
operations on LTI models
addition 2-11
arithmetic 2-11
concatenation 1-10
extracting a subsystem 1-6
inversion 2-13
multiplication 2-13
overloaded 1-4
pertransposition 2-14
precedence 2-3
resizing 2-9
subsystem, extraction 2-5
subtraction 2-12
transposition 2-14
output 1-2
delays 1-25
groups 1-25
names 1-25
number of outputs 3-3
OutputDelay. See delays
OutputGroup 1-25
Index
group names, conflicts 2-4
See also I/O, groups
OutputName 1-32
conflicts, naming 2-4
See also I/O, names
P
Padé approximation (pade) 1-51
parallel connection 2-12
pertransposition 2-14
phase margins 9-28
plot customization 8-1
poles 1-12
property 1-27
precedence rules 1-5
preferences and properties 5-2
proper transfer function 3-3
properties and preferences 5-2
properties. See LTI properties
Property Editor 8-3
R
realization
state coordinate transformation 3-8
realizations 3-8
balanced 3-8
canonical 3-8
regulation 9-31
resampling 2-26
response, I/O 2-5
robustness 9-28
root locus
design 9-9
rss
building an LTI array with 4-12
S
sample time 1-19
accessing 1-23
resampling 2-26
setting 1-34
unspecified 1-26
scaling 10-16
series connection 2-13
set 1-28
SISO 1-2, 3-3
SISO Design Tool
customizing plots 8-11
SISO Tool Preferences Editor 7-6
SS 2-14
ss 1-14
SS models 2-14
stability
numerical 10-7
stack 4-15
state 1-14
matrix 1-27
names 1-27
transformation 3-8
vector 1-2
state-space models 1-2
balancing 3-8
conditioning 10-5
conversion to 1-39
See also conversion
descriptor 1-16, 1-23
matrices 1-14
model data 1-14
quick data retrieval 1-23
realizations 3-8
scaling 10-16
specification 1-14
ss 1-14
I-5
Index
transfer functions of 1-39
subplot customization 8-10
subsystem 1-6, 2-5
subsystem operations on LTI models
subsystem, modifying 2-9
subtraction 2-12
T
Td. See delays
tf 1-8
tfdata
output, form of 1-23
time delays. See delays
Toolbox Preferences Editor 6-2
totaldelay 1-51
transfer functions 1-2
constructing with rational expressions 1-9
conversion to 1-39
denominator 1-8
discrete-time 1-19, 1-21
DSP convention 1-21
filt 1-22
MIMO 1-10
numerator 1-8
quick data retrieval 1-23
specification 1-8
static gain 1-11
tf 1-8
TF object, display for 1-9
variable property 1-27
transposition 2-14
triangle approximation 2-22
Ts. See sample time
Tustin approximation 2-22
with frequency prewarping 2-23
I-6
U
Userdata 1-26
Z
zero-order hold (ZOH) 2-20
with delays 2-23
zero-pole-gain (ZPK) models 1-2
conversion to 1-39
MIMO 1-13
quick data retrieval 1-23
specification 1-12
zpk 1-12
zeros 1-12
property 1-27
zpk 1-12
zpkdata
output, form of 1-23
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