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.
Preface
Part I : Basic Concepts
Chapter 1 : Mathematical Model for Nonlinear Systems
Chapter 2 : Existence and Uniqueness of Solutions
Chapter 3 : Qualitative Behavior of Second-Order Linear Time-Invariant Systems
Part II : Stability Analysis of Autonomous Systems
Chapter 4 : System Preliminaries
Chapter 5 : Lyapunov’s Second Method for Autonomous Systems
Chapter 6 : Lyapunov Function Generation for Linear Systems
Chapter 7 : Lyapunov Function Generation for Nonlinear Autonomous Systems
Chapter 8 : Relaxed Lyapunov Stability Conditions
Chapter 9 : New Stability Theorems
Chapter 10 : Lyapunov Stability Analysis of a Transformed Nonlinear System
Chapter 11 : Endnotes
Part III : Stability Analysis of Nonautonomous Systems
Chapter 12 : Preliminaries
Chapter 13 : Relaxed Lyapunov Stability Conditions
Chapter 14 : New Stability Theorems
Chapter 15 : Application of Partial Stability Theory in Nonlinear Nonautonomous System Stability Analysis
Part IV : Stability Analysis of Time-Delayed Systems
Chapter 16 : Preliminaries
Chapter 17 : Stability Analysis of Linear Time-Delayed Systems
Chapter 18 : Delay-Dependent Stability Analysis of Nonlinear Time-Delayed Systems
Part V : An Introduction to Stability Analysis of Linguistic Fuzzy Dynamic Systems
Chapter 19 : TSK Fuzzy Model System’s Stability Analysis
Chapter 20 : Linguistic Fuzzy Stability Analysis Using a Fuzzy Petri Net
Chapter 21 : Linguistic Model Stability Analysis
Chapter 22 : Stability Analysis of Fuzzy Relational Dynamic Systems
Chapter 23 : Asymptotic Stability in a Sum-Prod FRDS
Chapter 24 : Asymptotic Convergence to the Equilibrium State
Index
