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LMI control toolbox.pdf
Preface
About the Authors
Acknowledgments
Introduction
Linear Matrix Inequalities
Toolbox Features
LMIs and LMI Problems
The Three Generic LMI Problems
Further Mathematical Background
References
Uncertain Dynamical Systems
Linear Time-Invariant Systems
SYSTEM Matrix
Time and Frequency Response Plots
Interconnections of Linear Systems
Model Uncertainty
Uncertain State-Space Models
Polytopic Models
Affine Parameter-Dependent Models
Quantification of Parameter Uncertainty
Simulation of Parameter-Dependent Systems
From Affine to Polytopic Models
Example
Linear-Fractional Models of Uncertainty
How to Derive Such Models
Specification of the Uncertainty
From Affine to Linear-Fractional Models
References
Robustness Analysis
Quadratic Lyapunov Functions
LMI Formulation
Quadratic Stability
Maximizing the Quadratic Stability Region
Decay Rate
Quadratic H• Performance
Parameter-Dependent Lyapunov Functions
Stability Analysis
µ Analysis
Structured Singular Value
Robust Stability Analysis
Robust Performance
The Popov Criterion
Real Parameter Uncertainty
Example
References
State-Feedback Synthesis
Multi-Objective State-Feedback
Pole Placement in LMI Regions
LMI Formulation
Extension to the Multi-Model Case
The Function msfsyn
Design Example
References
Synthesis of H• Controllers
H• Control
Riccati- and LMI-Based Approaches
H• Synthesis
Validation of the Closed-Loop System
Multi-Objective H• Synthesis
LMI Formulation
The Function hinfmix
Loop-Shaping Design with hinfmix
References
Loop Shaping
The Loop-Shaping Methodology
The Loop-Shaping Methodology
Design Example
Specification of the Shaping Filters
Nonproper Filters and sderiv
Specification of the Control Structure
Controller Synthesis and Validation
Practical Considerations
Loop Shaping with Regional Pole Placement
References
Robust Gain-Scheduled Controllers
Gain-Scheduled Control
Synthesis of Gain-Scheduled H• Controllers
Simulation of Gain-Scheduled Control Systems
Design Example
References
The LMI Lab
Background and Terminology
Overview of the LMI Lab
Specifying a System of LMIs
A Simple Example
setlmis and getlmis
lmivar
lmiterm
The LMI Editor lmiedit
How It All Works
Retrieving Information
lmiinfo
lminbr and matnbr
LMI Solvers
From Decision to Matrix Variables and Vice Versa
Validating Results
Modifying a System of LMIs
dellmi
dellmi
setmvar
Advanced Topics
Structured Matrix Variables
Complex-Valued LMIs
Specifying cTx Objectives for mincx
Feasibility Radius
Well-Posedness Issues
Semi-Definite B(x) in gevp Problems
Efficiency and Complexity Issues
Solving M + PTXQ + QTXTP < 0
References
Command Reference
List of Functions
H• Control and Loop Shaping
LMI Lab: Specifying and Solving LMIs
LMI Lab: Additional Facilities
LMI Control Toolbox
For Use with MATLAB
®
Pascal Gahinet
Arkadi Nemirovski
Alan J. Laub
Mahmoud Chilali
User’s Guide
Version 1
How to Contact The MathWorks:
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Web
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The MathWorks, Inc.
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For contact information about worldwide offices, see the MathWorks Web site.
LMI Control Toolbox User’s Guide
COPYRIGHT 1995 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 repro-
duced in any form without prior written consent from The MathWorks, Inc.
FEDERAL ACQUISITION: This provision applies to all acquisitions of the Program and Documentation by
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hereby agrees that this software qualifies as "commercial" computer software within the meaning of FAR
Part 12.212, DFARS Part 227.7202-1, DFARS Part 227.7202-3, DFARS Part 252.227-7013, and DFARS Part
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to the government’s use and disclosure of the Program and Documentation, and shall supersede any
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is inconsistent in any respect with federal procurement law, the government agrees to return the Program
and Documentation, unused, to MathWorks.
MATLAB, Simulink, Stateflow, Handle Graphics, and Real-Time Workshop are registered trademarks, and
TargetBox is a trademark of The MathWorks, Inc.
Other product or brand names are trademarks or registered trademarks of their respective holders.
Printing History: May 1995
First printing New for Version 1
Contents
Preface
About the Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1
2
Introduction
Linear Matrix Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2
Toolbox Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3
LMIs and LMI Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4
The Three Generic LMI Problems . . . . . . . . . . . . . . . . . . . . . . . 1-5
Further Mathematical Background . . . . . . . . . . . . . . . . . . . . . 1-9
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-10
Uncertain Dynamical Systems
Linear Time-Invariant Systems . . . . . . . . . . . . . . . . . . . . . . . . 2-3
SYSTEM Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3
Time and Frequency Response Plots . . . . . . . . . . . . . . . . . . . 2-6
Interconnections of Linear Systems . . . . . . . . . . . . . . . . . . . . 2-9
i
Model Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12
Uncertain State-Space Models . . . . . . . . . . . . . . . . . . . . . . . . . 2-14
Polytopic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14
Affine Parameter-Dependent Models . . . . . . . . . . . . . . . . . . . . 2-15
Quantification of Parameter Uncertainty . . . . . . . . . . . . . . . . 2-17
Simulation of Parameter-Dependent Systems . . . . . . . . . . . . . 2-19
From Affine to Polytopic Models . . . . . . . . . . . . . . . . . . . . . . . . 2-20
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-21
Linear-Fractional Models of Uncertainty . . . . . . . . . . . . . . . 2-23
How to Derive Such Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-23
Specification of the Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . 2-26
From Affine to Linear-Fractional Models . . . . . . . . . . . . . . . . . 2-32
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-35
3
Robustness Analysis
Quadratic Lyapunov Functions . . . . . . . . . . . . . . . . . . . . . . . . 3-3
LMI Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-4
Quadratic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6
Maximizing the Quadratic Stability Region . . . . . . . . . . . . . . . . 3-8
Decay Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9
Quadratic H∞ Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-10
Parameter-Dependent Lyapunov Functions . . . . . . . . . . . . 3-12
Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-14
µ Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-17
Structured Singular Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-17
Robust Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-19
Robust Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-21
The Popov Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-24
Real Parameter Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-25
ii
Contents
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-28
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-32
4
5
State-Feedback Synthesis
Multi-Objective State-Feedback . . . . . . . . . . . . . . . . . . . . . . . . 4-3
Pole Placement in LMI Regions . . . . . . . . . . . . . . . . . . . . . . . . 4-5
LMI Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7
Extension to the Multi-Model Case . . . . . . . . . . . . . . . . . . . . . . . 4-9
The Function msfsyn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-11
Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-13
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-18
Synthesis of H∞ Controllers
H∞ Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3
Riccati- and LMI-Based Approaches . . . . . . . . . . . . . . . . . . . . . . 5-7
H∞ Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-10
Validation of the Closed-Loop System . . . . . . . . . . . . . . . . . . . 5-13
Multi-Objective H∞ Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . 5-15
LMI Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-16
The Function hinfmix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-20
Loop-Shaping Design with hinfmix . . . . . . . . . . . . . . . . . . . . . . 5-20
iii
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-22
6
7
Loop Shaping
The Loop-Shaping Methodology . . . . . . . . . . . . . . . . . . . . . . . . 6-2
The Loop-Shaping Methodology . . . . . . . . . . . . . . . . . . . . . . . . 6-3
Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5
Specification of the Shaping Filters . . . . . . . . . . . . . . . . . . . . 6-10
Nonproper Filters and sderiv . . . . . . . . . . . . . . . . . . . . . . . . . . 6-12
Specification of the Control Structure . . . . . . . . . . . . . . . . . 6-14
Controller Synthesis and Validation . . . . . . . . . . . . . . . . . . . 6-16
Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-18
Loop Shaping with Regional Pole Placement . . . . . . . . . . . 6-19
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-24
Robust Gain-Scheduled Controllers
Gain-Scheduled Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-3
Synthesis of Gain-Scheduled H• Controllers . . . . . . . . . . . . . 7-7
Simulation of Gain-Scheduled Control Systems . . . . . . . . . . 7-9
Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-10
iv
Contents
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-15
8
The LMI Lab
Background and Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3
Overview of the LMI Lab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-6
Specifying a System of LMIs . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-8
A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-9
setlmis and getlmis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-11
lmivar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-11
lmiterm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-13
The LMI Editor lmiedit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-16
How It All Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-18
Retrieving Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-21
lmiinfo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-21
lminbr and matnbr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-21
LMI Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-22
From Decision to Matrix Variables and Vice Versa . . . . . . 8-28
Validating Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-29
Modifying a System of LMIs . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-30
dellmi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-30
dellmi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-30
setmvar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-31
Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-33
Structured Matrix Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-33
Complex-Valued LMIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-35
Specifying cTx Objectives for mincx . . . . . . . . . . . . . . . . . . . . . 8-38
Feasibility Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-39
v
Well-Posedness Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-40
Semi-Definite B(x) in gevp Problems . . . . . . . . . . . . . . . . . . . . 8-41
Efficiency and Complexity Issues . . . . . . . . . . . . . . . . . . . . . . . 8-41
Solving M + PTXQ + QTXTP < 0 . . . . . . . . . . . . . . . . . . . . . . . . 8-42
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-44
9
Command Reference
List of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-3
H∞ Control and Loop Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . 9-6
LMI Lab: Specifying and Solving LMIs . . . . . . . . . . . . . . . . . . 9-7
LMI Lab: Additional Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . 9-8
vi
Contents
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