# Dissipative Systems and Stability

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## Table of Contents

Preface

Chapter 1. Introduction

- Background
- The notion of a dynamical system
- The property of passivity
- Taking stored energy into account
- Passivity and stability
- Finite gain systems
- The extension to dissipative systems

Chapter 2. Defining dissipativeness

- Overview
- The mathematical setting
- Signal spaces
- The definition of dissipativeness
- Ultimate virtual dissipativeness and weak dissipativeness

Chapter 3. Storage Functions
- Dissipativeness as a state-space property
- The state-space model
- Storage functions
- The relationship between external and internal
dissipativeness
- Bounds on storage functions
- Positive storage functions
- Lossless systems
- More general energy functions

Chapter 4. Passive Systems
- Overview
- Passivity and stability
- Internal and external passivity
- Passivity in the frequency domain
- Strong passivity
- The single-loop stability result

Chapter 5. Algebraic conditions for dissipativeness
- Overview
- A class of nonlinear continuous-time systems
- The main result for continuous-time systems
- The question of differentiability
- More general nonlinear systems
- Linear continuous-time systems
- Linear discrete-time systems

Chapter 6. Stability
- Overview
- The basic stability results
- Interconnected systems
- Neutral interconnections
- Single-loop
feedback systems
- Passive subsystems
- A small gain theorem
- Conic subsystems
- Examples

Chapter 7. Instability
- Criteria for instability
- The basic input-output instability results
- A state-space instability result
- Interconnected systems

Chapter 8. Frequency domain tests
- Introduction
- General frequency domain criteria
- Graphical tests: the scalar case
- Graphical tests: the multivariable case
- Results using multipliers
- Discrete-time systems
- Notes and references

Chapter 9.Simple Nonlinear Systems
- Introduction
- The class of interesting (Q,S,R) triples
- General conditions for cyclodissipativeness
- Conditions for dissipativeness
- Systems with linear dynamics

Chapter 10. Additional results
- Relaxed stability tests
- Connective stability
- An optimal control problem
- Dissipation delay
- A structure result

Appendix A. Some useful matrix results
- A duality result
- Matrices with positive principal minors
- Quasidominant matrices
- M-matrices
- Transformations

Bibliography

Index

## The author

Peter Moylan is a retired academic in Electrical and Computer
Engineering.

If you want to contact him, see the home
page for
this web site.

This
page last
updated 21 July 2022.