Nonsmooth Lyapunov Analysis in Finite and Infinite Dimensions (Communications and Control Engineering)

Description

Nonsmooth Lyapunov Analysis in Finite and Infinite Dimensions provides helpful tools for the treatment of a broad class of dynamical systems that are governed, not only by ordinary differential equations but also by partial and functional differential equations. Existing Lyapunov constructions are extended to discontinuous systems―those with variable structure and impact―by the involvement of nonsmooth Lyapunov functions. The general theoretical presentation is illustrated by control-related applications; the nonsmooth Lyapunov construction is particularly applied to the tuning of sliding-mode controllers in the presence of mismatched disturbances and to orbital stabilization of the bipedal gate. The nonsmooth construction is readily extendible to the control and identification of distributed-parameter and time-delay systems. The first part of the book outlines the relevant fundamentals of benchmark models and mathematical basics. The second concentrates on the construction of nonsmooth Lyapunov functions. Part III covers design and applications material. This book will benefit the academic research and graduate student interested in the mathematics of Lyapunov equations and variable-structure control, stability analysis and robust feedback design for discontinuous systems. It will also serve the practitioner working with applications of such systems. The reader should have some knowledge of dynamical systems theory, but no background in discontinuous systems is required―they are thoroughly introduced in both finite- and infinite-dimensional settings. Foreword Preface Contents Abbreviations Part I Introduction 1 Benchmark Models 1.1 Variable Structure Systems 1.1.1 First- and Higher-Order Sliding Modes 1.1.2 Chattering Phenomenon and Discrete-Time Sliding Modes 1.1.3 Infinite-Dimensional Sliding Modes 1.2 Dynamics Under Unilateral Constraints 1.2.1 Zhuravlev–Ivanov Transformation 1.2.2 Bouncing Ball: State Resets and Zeno Behavior 1.2.3 Constrained Van der Pol Oscillator: Limit Cycles and Hopf Bifurcation 1.3 Concluding Remarks References 2 Mathematical Background 2.1 Comparison Principle and Barbalat's Lemma 2.2 Discontinuous and Multi-valued Vector Fields 2.2.1 Filippov Solutions 2.2.2 Equivalent Control Method and Other Solution Concepts 2.2.3 Ambiguous Sliding Modes 2.2.4 Uniqueness of Sliding Modes in Affine Systems 2.2.5 Regularization of Discontinuous Systems in Hilbert Space 2.3 Complementarity Formulation of Constrained Lagrange Dynamics 2.3.1 Implicit Euler Integration of Sliding Modes 2.4 Hopf Bifurcation of Discontinuous Limit Cycles: Case Study 2.4.1 Constrained Van der Pol Oscillator 2.4.2 Existence of a Constrained Limit Cycle 2.4.3 Numerical Analysis of Phenomenological Behaviors 2.4.4 Hopf Bifurcation Analysis via Poincaré Method 2.4.5 Constrained Van der Pol Oscillator with Manipulated Parameters 2.5 Concluding Remarks References 3 Mathematical Tools of Dynamic Systems in Hilbert Spaces 3.1 Sobolev Spaces and Instrumental Inequalities 3.2 Linear Partial Differential Equations 3.2.1 Linear Differential Operators 3.2.2 Parabolic, Elliptic, and Hyperbolic Operators 3.2.3 Green Function and Mild Solutions 3.2.4 Weak Solutions 3.3 Sturm–Liouville Operators and Their Properties 3.3.1 Eigenvalue Estimates 3.3.2 Uniform Boundedness of the Eigenfunctions 3.4 Separation of Variables 3.4.1 Parabolic Case Study 3.4.2 Hyperbolic Case Study 3.5 Nonlinear First-Order Partial Differential Equations 3.5.1 Viscosity Solutions of First-Order PDEs 3.5.2 Discontinuous Strict Hamilton–Jacobi Inequality and Its Proximal Solutions 3.6 Stability in Euclidean and Hilbert Spaces 3.6.1 Abstract Dynamic Systems and Relevant Stability Concepts 3.6.2 Robust Stability of Uncertain Dynamic Systems: Basic Definitions 3.6.3 Sliding Mode Dynamics in Hilbert Space 3.6.4 Hilbert Space-Valued Dynamics with Delay 3.6.5 Homogeneous Differential Inclusions and Their Finite Time Stability 3.7 Concluding Remarks References Part II Construction of Nonsmooth Lyapunov Functions 4 Modern Lyapunov Tools 4.1 Strict Lyapunov Functionals 4.1.1 Multiple Lyapunov Functionals 4.1.2 Semi-global Lyapunov Functionals 4.1.3 Finite Time Stable Lyapunov Functionals 4.1.4 Homogeneous Lyapunov Functions 4.1.5 Input-to-State Stable Lyapunov Functions 4.2 Non-strict Lyapunov Functionals 4.2.1 Invariance Principle 4.2.2 Invariance Principle Extension 4.3 Lyapunov Functionals Under Unilateral Constraints 4.4 Concluding Remarks References 5 Control Lyapunov Functions 5.1 Lyapunov Algebraic Equation and Quadratic Forms 5.2 Generalized Forms 5.2.1 Semiglobal Strict Lyapunov Functions of Twisting VSS 5.2.2 Strict Lyapunov Functions of Supertwisting VSS 5.2.3 GF Lyapunov Functions of Homogeneous Systems and Their LMI-Based Construction 5.3 Construction of Multiple FTS Lyapunov Functions via Solving Lyapunov Gradient Equation 5.4 Lyapunov Minmax Approach and Speed Gradient Method 5.5 Construction of Lyapunov Functions Using Proximal Solutions of Hamilton–Jacobi PDI 5.6 Concluding Remarks References Part III Lyapunov Redesign 6 Lyapunov-Based Tuning 6.1 mathcalL2-Gain Tuning of First-Order Sliding Modes 6.1.1 Tuning Under Full-State Information 6.1.2 Tuning of SM Estimator Gains 6.1.3 Tuning Under Incomplete State Information 6.2 mathcalL2-Gain Tuning of Second-Order Sliding Modes 6.2.1 Tuning of Twisting Controller 6.2.2 Tuning of Supertwisting Estimator 6.2.3 Output Feedback Tuning 6.3 Settling Time Tuning of Enforced Double Integrator 6.3.1 Switched Control Synthesis 6.3.2 Reaching Time Estimate of Linear Feedback 6.3.3 Settling Time Estimate of Twisting Controller 6.3.4 Settling Time Tuning 6.4 ISS Point-Wise Feedback Synthesis of Parabolic Systems 6.4.1 Control Synthesis 6.4.2 Existence of Closed-Loop Solutions 6.4.3 ISS Analysis and Tuning 6.4.4 Supporting Simulation 6.5 Concluding Remarks References 7 Lyapunov Approach to Adaptive Identification and Control in Infinite-Dimensional Setting 7.1 Lyapunov–Razumikhin Redesign and Identification of Linear Time-Delay Systems 7.1.1 State-Space Representation and Weak Controllability 7.1.2 Identifiability Analysis 7.1.3 Razumikhin-Based Adaptive Identifier Design 7.1.4 SISO Case Study 7.1.5 Application to Engine Transient Fuel Identification 7.2 Lyapunov–Krasovskii Redesign and Identification of Linear Parabolic PDE 7.2.1 Identification over In-Domain Sensing 7.2.2 Identification over Boundary Sensing 7.2.3 Simulation Results 7.3 Concluding Remarks References 8 Control Applications 8.1 Synthesis of Mechanical Systems Under Unilateral Constraints 8.1.1 Robust Tracking Problem and Hybrid Error Dynamics 8.1.2 Pre-feedback Design and mathcalHinfty Synthesis 8.1.3 mathcalHinfty-Control of Mass–Spring–Barrier System 8.1.4 mathcalHinfty Tracking of a Periodical Bipedal Gait 8.2 Energy Control of Continuum of Oscillators 8.2.1 Sine-Gordon Nonlinear PDE Model and Problem Statement 8.2.2 Control Objective 8.2.3 Energy Control Synthesis Using State Feedack 8.2.4 Luenberger Observer Design 8.2.5 Energy Control Synthesis Using Output Feedback 8.2.6 Numerical Study 8.3 Concluding Remarks References Index