Nonsmooth Lyapunov analysis in finite and infinite dimensions /

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Bibliographic Details
Author / Creator:Orlov, I︠U︡. V. (I︠U︡riĭ Vladimirovich)
Imprint:Cham : Springer, 2020.
Description:1 online resource (351 pages)
Series:Communications and control engineering
Communications and control engineering.
Subject:Lyapunov functions.
Lyapunov stability.
Lyapunov functions.
Lyapunov stability.
Electronic books.
Format: E-Resource Book
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Notes:4.1.5 Input-to-State Stable Lyapunov Functions
Includes bibliographical references.
Print version record.
Summary: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.
Other form:Print version: Orlov, Yury. Nonsmooth Lyapunov Analysis in Finite and Infinite Dimensions. Cham : Springer, ©2020 9783030376246
Table of Contents:
  • Intro
  • 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