《自动控制原理》课程教学资源（The MathWorks - MATLAB 相关电子书籍）04 LMI Control Toolbox For Use with MATLAB User’s Guide Version 1

LMI Control Toolbox For Use with MATLAB Pascal Gahinet Arkadi nemirouski Alan J. Laub Mahmoud Chilali Computation Visualization Programming User's Guide The MathWorks Version 1

For Use with MATLAB Pascal Gahinet Arkadi Nemirovski Alan J. Laub Mahmoud Chilali ® User’s Guide Version 1 LMI Control Toolbox

Contents Preface About the authors Acknowledgments Introduction Linear Matrix Inequalities 1-2 Toolbox Features LMIs and lmi problems The Three Generic LMi Problems Further Mathematical Background 1-9 References Uncertain Dynamical Systems Linear Time-Invariant Systems SYSTEM Matrix 2-3 Time and Frequency Response Plots Interconnections of Linear Systems

i Contents Preface About the Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 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 2 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

Model Uncertainty Uncertain State-Space Models Affine Parameter-Dependent Models 2-15 Quantification of Parameter Uncertainty Simulation of Parameter-Dependent Systems From Affine to Polytopic Models Example 21 Linear-Fractional Models of Uncertainty 2-23 How to derive such models Specification of the Uncertainty From Affine to linear-Fractional models References Robustness analysis 3 Quadratic Lyapunov Functions 3-3 LMI Formulation Quadratic Stability 36 Maximizing the quadratic Stability region 3-8 Quadratic Hoo Performance 3-9 3-10 Parameter-Dependent Lyapunov Functions Stability analysis 3-14 alvis Structured Singular value .3-17 Robust Stability analysis 3-19 Robust Performance 3-24 Real Parameter Uncertainty l1 Contents

ii Contents 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

Example 3-28 References 3-32 State-Feedback Synthesis Multi-Objective State-Feedback Pole Placement in LMI Regions 45 LMI Formulation 4-7 Extension to the multi-Model case 4-9 The Function msfsyn 413 References 4-18 Synthesis of Hoo Controllers Hoo Control 53 Riccati- and LMI-Based approaches Hoo Synthesis Validation of the Closed-Loop System Multi-Objective Hoo Synthesis LMI Formulation 516 The Function hinfmix 520 Loop-Shaping Design with hinfmix

iii Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-28 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-32 4 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 5 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

Loop Shaping 6 The Loop-Shaping Methodology 6-2 The Loop-Shaping Methodology Example 6-5 f the sh Nonproper Filters and sderiv 6-12 Specification of the Control Structure 6-14 Controller Synthesis and Validation 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 Simulation of Gain-Scheduled Control Systems 7-9 Design Example 7-10

iv Contents References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-22 6 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 7 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

References The lMii lab Background and terminology Overview of the lmi lab 8-6 Specifying a System of LMIs A Simple Example 8-9 setlmis and getlmis Imilar Limiter The lmi Editor lmiedit How It all work 8-18 Retrieving Information 8-21 Imiinfo 8-21 Iminbr and matnbr 8-21 LMI Solvers From Decision to Matrix Variables and vice versa 8-28 Validating Results Modifying a System of LMIs delli 8-3 setmvar Advanced Topics 8-83 Structured matrix variables 8-33 Complex-Valued LMIs Specifying c" Objectives for mincx Feasibility radius 8-39

v 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

Well-Posedness issues Semi-Definite B(x)in gevp Problems Efficiency and Complexity Issues 4红2 Solving M PTXQ+QTXTP<0 References 8-44 Command reference List of functions 93 Hoo Control and Loop Shaping II Lab: Specifying and Solving LMIs 97 LMI Lab: Additional Facilities vI Contents

vi Contents 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

Prefe Ab。 ut the authors Dr Pascal Gahinet is a reserach fellow at INRIA Rocquencourt, France. His research interests include robust control theory, linear matrix inequalities numerical linear algebra, and numerical software for control Prof Arkadi Nemirovski is with the Faculty of Industrial Engineering and Management at Technion, Haifa, Israel. His research interests include convex optimization, complexity theory, and non-parametric statistics Prof Alan J. Laub is with the Electrical and Computer engineering Department of the University of California at Santa Barbara, USA. His research interests are in numerical analysis, mathematical software scientific computation, computer-aided control system design, and linear and large-scale trol and filtering Mahmoud Chilali is completing his Ph D at INRIA Rocquencourt, france His thesis is on the theory and applications of linear matrix inequalities

Preface viii About the Authors Dr. Pascal Gahinet is a reserach fellow at INRIA Rocquencourt, France. His research interests include robust control theory, linear matrix inequalities, numerical linear algebra, and numerical software for control. Prof. Arkadi Nemirovski is with the Faculty of Industrial Engineering and Management at Technion, Haifa, Israel. His research interests include convex optimization, complexity theory, and non-parametric statistics. Prof. Alan J. Laub is with the Electrical and Computer Engineering Department of the University of California at Santa Barbara, USA. His research interests are in numerical analysis, mathematical software, scientific computation, computer-aided control system design, and linear and large-scale control and filtering theory. Mahmoud Chilali is completing his Ph.D. at INRIA Rocquencourt, France. His thesis is on the theory and applications of linear matrix inequalities in control

Acknowledgments Acknowledgments The authors wish to express their gratitude to all colleagues who directly or indirectly contributed to the making of the LMI Control Toolbox. Special chanks to Pierre Apkarian, gregory Becker, Hiroyuki Kajiwara, and Anca Ignat for their help and contribution. Many thanks also to those who tested and helped refine the software, including bobby bodenheimer, Markus Lu, Roy Lurie, Jason Ly, John Morris, Ravi Prasanth, Michael Safonov bo Brandstetter. Eric Feron, K.C. Goh. Anders Helmersson. Ted Iwasaki. Ji Carsten Scherer, Andy Sparks, Mario Rotea, Matthew Lamont Tyler, Jim Tung, and John Wen. apologies, finally to those we may have omitted The work of Pascal Gahinet was supported in part by INRIA

Acknowledgments ix Acknowledgments The authors wish to express their gratitude to all colleagues who directly or indirectly contributed to the making of the LMI Control Toolbox. Special thanks to Pierre Apkarian, Gregory Becker, Hiroyuki Kajiwara, and Anca Ignat for their help and contribution. Many thanks also to those who tested and helped refine the software, including Bobby Bodenheimer, Markus Brandstetter, Eric Feron, K.C. Goh, Anders Helmersson, Ted Iwasaki, Jianbo Lu, Roy Lurie, Jason Ly, John Morris, Ravi Prasanth, Michael Safonov, Carsten Scherer, Andy Sparks, Mario Rotea, Matthew Lamont Tyler, Jim Tung, and John Wen. Apologies, finally, to those we may have omitted. The work of Pascal Gahinet was supported in part by INRIA

Introduction Linear Matrix Inequalities Toolbox Features LMIIs and lmi Problems 1-4 The Three Generic LMi Problems Further Mathematical Background References 1-10

1 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