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© 2009

Constructions of Strict Lyapunov Functions

Book

Part of the Communications and Control Engineering book series (CCE)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Background

    1. Front Matter
      Pages 1-1
  3. Time-Invariant Case

  4. Time-Varying Case

  5. Systems with Multiple Time Scales

    1. Front Matter
      Pages 273-273
  6. Back Matter
    Pages 343-386

About this book

Introduction

The construction of strict Lyapunov functions is a challenging problem that is of significant ongoing research interest. Although converse Lyapunov function theory guarantees the existence of strict Lyapunov functions in many situations, the Lyapunov functions that converse theory provides are often abstract and nonexplicit, and therefore may not lend themselves to engineering applications. Often, even when a system is known to be stable, one still needs explicit Lyapunov functions; however, once an appropriate strict Lyapunov function has been constructed, many robustness and stabilization problems can be solved almost immediately through standard feedback designs or robustness arguments. By contrast, non-strict Lyapunov functions are often readily constructed, e.g., from passivity, backstepping, or forwarding (especially in the time varying context), or by using the Hamiltonian in Euler–Lagrange systems.

Constructions of Strict Lyapunov Functions contains a broad repertoire of Lyapunov constructions for nonlinear systems, focusing on methods for transforming non-strict Lyapunov functions into strict ones. Many important classes of dynamics are covered: Jurdjevic–Quinn systems; time-varying systems satisfying LaSalle or Matrosov conditions; slowly and rapidly time-varying systems; adaptively controlled dynamics; and hybrid systems. The explicitness and simplicity of the constructions make them suitable for feedback design, and for quantifying the effects of uncertainty. Readers will benefit from the authors’ mathematical rigor and unifying, design-oriented approach, as well as the numerous worked examples, covering several applications that are of compelling interest including the adaptive control of chemostats and the stabilization of underactuated ships.

Researchers from applied-mathematical and engineering backgrounds working in nonlinear and dynamical systems will find this monograph to be most valuable and for graduate students of control theory it will also be an authoritative source of information on a very important subject.

Keywords

Lyapunov Analysis Lyapunov Constructions Time-varying Systems adaptive control control control theory dynamical system feedback nonlinear control nonlinear system robust control stability stabilization system uncertainty

Authors and affiliations

  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUSA
  2. 2.UMR Analyse des Systèmes et BiométrieINRIA Sophia-AntopolisMontpellier Cedex 1France

Bibliographic information

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Reviews

From the reviews:

“This monograph covers a wide range of nonlinear dynamical systems, including Jurdjevic-Quinn systems, time-varying systems satisfying LaSalle or Matrosov conditions, adaptively controlled dynamics, slowly and rapidly time-varying systems, and hybrid time-varying systems. … The book will be useful to researchers and graduate students in various areas of applied mathematics and control theory and engineering.” (Vladimir Sobolev, Zentralblatt MATH, Vol. 1186, 2010)

“This is an important and painstaking chore. … The book is a combination of a research monograph and an expository text. … those who need to analyze or to perform a detailed stability or stabilization analysis of concrete equations will find the book a very good source of ideas and references. … should be a desired addition to all libraries that serve mathematics departments and control engineering faculties. … could be used successfully as reading material for advanced graduate students interested in the subject matter.” (Zvi Artstein, SIAM Review, Vol. 53 (1), 2011)