Analysis and Synthesis of Nonlinear Control Systems: A Convex Optimisation Approach

Description

This book presents a modern perspective on the modelling, analysis, and synthesis ideas behind convex-optimisation-based control of nonlinear systems: it embeds them in models with convex structures. Analysis and Synthesis of Nonlinear Control Systems begins with an introduction to the topic and a discussion of the problems to be solved. It then explores modelling via convex structures, including quasi-linear parameter-varying, Takagi–Sugeno models, and linear fractional transformation structures. The authors cover stability analysis, addressing Lyapunov functions and the stability of polynomial models, as well as the performance and robustness of the models. With detailed examples, simulations, and programming code, this book will be useful to instructors, researchers, and graduate students interested in nonlinear control systems. Contents Notation and Abbreviations Conventions Symbols and Notation General Notation Dynamic Systems TS Models 1 Introduction References 2 Problems to Be Solved and Scope of the Book 2.1 Some Useful Classes of Dynamical Systems 2.2 Control Objectives 2.3 Models 2.3.1 Model Approximations 2.3.2 Models with Convex Structure 2.4 The Convex-Structure Approach to Control: Motivation 2.5 Summary and Conclusions References 3 Modelling via Convex Structures 3.1 Introduction 3.2 Takagi–Sugeno Models 3.2.1 The Sector Nonlinearity Approach 3.2.2 Approximate Models 3.2.3 Piecewise and Affine Models 3.3 Polynomial Representations 3.4 Descriptor Representations 3.5 The Linear Fractional Transformation Structure 3.5.1 LFT in Physical Modelling 3.5.2 Structure of the Uncertainty/Nonlinearity Block Δ 3.5.3 Relationship with Descriptor Models 3.5.4 Uncertain Models 3.6 Summary and Conclusions References 4 Stability Analysis 4.1 Introduction 4.2 Quadratic Stability of TS Models 4.2.1 The Continuous-Time Case 4.2.2 The Discrete-Time Case 4.2.3 Shape-Dependent Quadratic Stability 4.3 Piecewise Lyapunov Functions 4.3.1 The Continuous-Time Case 4.3.2 The Discrete-Time Case 4.4 Parameter-Dependent Lyapunov Functions 4.4.1 The Continuous-Time Case: Box LPV Model 4.4.2 The Continuous-Time Case for Takagi–Sugeno Models 4.4.3 The Discrete-Time Case for Takagi–Sugeno Models 4.5 Stability of Polynomial Models 4.5.1 Locality Issues in the Polynomial Approach 4.6 Stability of Descriptor Models 4.7 Quadratic Stability of LFT Systems 4.7.1 Non-Augmented LMIs 4.7.2 Multiplier-Based Conditions 4.7.3 Choice of Multipliers 4.7.4 Relation to Polytopic/TS Quadratic Stability Conditions 4.8 Non-Quadratic Stability of LFT Models 4.9 Summary and Conclusions References 5 State Feedback, Performance, and Robustness 5.1 Introduction 5.2 Quadratic Stabilisation of TS Models 5.3 Parameter-Dependent Lyapunov Functions 5.3.1 The Continuous-Time Case 5.3.2 The Discrete-Time Case 5.4 Stabilisation of Polynomial Models 5.5 Stabilisation of Descriptor Models 5.6 Performance Specifications and Robustness 5.6.1 Constraints on the System Input and Output 5.6.2 mathcalHinfty Attenuation 5.6.3 Robust Control 5.7 Application Case Study: Wheelchair Swing-Up 5.8 Summary and Conclusions References 6 Observation and Output Feedback 6.1 Introduction 6.2 Observer Design 6.2.1 Observer Design for TS Models 6.2.2 Observer Design for Descriptor Models 6.2.3 Observer Design: Unmeasurable Membership Functions 6.3 Observer-Based Stabilisation 6.4 Output Feedback for LFT Models 6.4.1 Closed-Loop Equation 6.4.2 Convexification of the Synthesis Conditions 6.4.3 Controller Reconstruction 6.5 Application Case Study: Sitting Control for People with Spinal Injury 6.6 Application Case Study: Observer-Based Control for Wheelchairs 6.7 Summary and Conclusions References 7 Conclusions and Perspectives Appendix A Useful Matrix and Norm Results A.1 Column, Row and Null Spaces A.2 Matrix Inversion A.3 Eigenvalues and Eigenvectors A.4 Singular Values and Matrix Gains A.5 Properties Usually Employed to Derive LMI Conditions A.6 Matrix Exponential A.7 Signal and System Norms A.8 BIBO Stability and the Small-Gain Theorem Appendix B Lyapunov Stability B.1 Time-Invariant Continuous-Time Systems B.2 Time-Variant Continuous-Time Systems B.3 Time-Invariant Discrete-Time Systems B.4 Time-Variant Discrete-Time Systems Appendix C Convex Optimisation Tools C.1 Linear Matrix Inequalities C.2 Sum of Squares Appendix D Convex Sum Relaxations D.1 Summations Over the Standard Simplex D.2 Sufficient Shape-Independent Conditions D.3 Asymptotically Exact Conditions D.3.1 Without Additional Variables D.3.2 Additional Variables D.4 Exploiting Tensor-Product Structure References Index