Nonlinear Systems Stability Analysis

Nonlinear Systems Stability Analysis: Lyapunov-Based Approach

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  • Helps engineers design stable systems using practice-oriented methods
  • Brings a body of research together in one volume
  • Gives coherence and structure to the available sources
  • Forms the basis for a graduate course in a range of engineering disciplines
  • Touches on new topics, opening up new research possibilities


The equations used to describe dynamic properties of physical systems are often nonlinear, and it is rarely possible to find their solutions. Although numerical solutions are impractical and graphical techniques are not useful for many types of systems, there are different theorems and methods that are useful regarding qualitative properties of nonlinear systems and their solutions—system stability being the most crucial property. Without stability, a system will not have value.

Nonlinear Systems Stability Analysis: Lyapunov-Based Approach introduces advanced tools for stability analysis of nonlinear systems. It presents the most recent progress in stability analysis and provides a complete review of the dynamic systems stability analysis methods using Lyapunov approaches. The author discusses standard stability techniques, highlighting their shortcomings, and also describes recent developments in stability analysis that can improve applicability of the standard methods. The text covers mostly new topics such as stability of homogonous nonlinear systems and higher order Lyapunov functions derivatives for stability analysis. It also addresses special classes of nonlinear systems including time-delayed and fuzzy systems.

Presenting new methods, this book provides a nearly complete set of methods for constructing Lyapunov functions in both autonomous and nonautonomous systems, touching on new topics that open up novel research possibilities. Gathering a body of research into one volume, this text offers information to help engineers design stable systems using practice-oriented methods and can be used for graduate courses in a range of engineering disciplines.

Table of Contents

Basic Concepts
Mathematical Model for Nonlinear Systems
Existence and Uniqueness of Solutions
Qualitative Behavior of Second-Order Linear Time-Invariant Systems

Stability Analysis of Autonomous Systems
System Preliminaries
Lyapunov’s Second Method for Autonomous Systems
Lyapunov Function Generation for Linear Systems
Lyapunov Function Generation for Nonlinear Autonomous Systems
Relaxed Lyapunov Stability Conditions
New Stability Theorems
Lyapunov Stability Analysis of a Transformed Nonlinear System

Stability Analysis of Nonautonomous Systems
Relaxed Lyapunov Stability Conditions
New Stability Theorems
Application of Partial Stability Theory in Nonlinear Nonautonomous System Stability Analysis

Stability Analysis of Time-Delayed Systems
Stability Analysis of Linear Time-Delayed Systems
Delay-Dependent Stability Analysis of Nonlinear Time-Delayed Systems

An Introduction to Stability Analysis of Linguistic Fuzzy Dynamic Systems
TSK Fuzzy Model System’s Stability Analysis
Linguistic Fuzzy Stability Analysis Using a Fuzzy Petri Net
Linguistic Model Stability Analysis
Stability Analysis of Fuzzy Relational Dynamic Systems
Asymptotic Stability in a Sum-Prod FRDS
Asymptotic Convergence to the Equilibrium State

Author Bio(s)

Seyyed Kamaleddin Yadavar Nikravesh, Ph.D., is a professor in the electrical engineering department at Amirkabir University of Technology. His research interests include dynamic and biomedical modeling, system stability, and system optimization. He has published five different books on electrical circuit analysis, optimal control systems, industrial control system analysis, industrial control system synthesis and design, and system stability analysis: Lyapounov-based approach. He has also published more than 180 journal and conference papers, mostly in systems modeling, and system stability analysis and synthesis, which form the main structure of his present book.

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