《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