# Feedback Systems: Input-output Properties

## 1st Edition

Editors:
eBook ISBN: 9780323157797
Published Date: 28th January 1975
Page Count: 2
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## Description

Feedback Systems: Input-output Properties deals with the basic input-output properties of feedback systems. Emphasis is placed on multiinput-multioutput feedback systems made of distributed subsystems, particularly continuous-time systems. Topics range from memoryless nonlinearities to linear systems, the small gain theorem, and passivity. Norms and general theorems are also considered. This book is comprised of six chapters and begins with an overview of a few simple facts about feedback systems and simple examples of nonlinear systems that illustrate the important distinction between the questions of existence, uniqueness, continuous dependence, and boundedness with respect to bounded input and output. The next chapter describes a number of useful properties of norms and induced norms and of normed spaces. Several theorems are then presented, along with the main results concerning linear systems. These results are used to illustrate the applications of the small gain theorem to different classes of systems. The final chapter outlines the framework necessary to discuss passivity and demonstrate the applications of the passivity theorem. This monograph will be a useful resource for mathematically inclined engineers interested in feedback systems, as well as undergraduate engineering students.

Preface

Acknowledgments

List of Symbols

I Memoryless Nonlinearities

1 Sector Conditions

2 Linear Feedback around a Nonlinearity (Memoryless Case)

3 Multiple Nonlinearities

Notes and References

II Norms

1 Norms: Definitions and Examples

2 Equivalent Norms

3 Relations between Normed Spaces

4 Geometric Interpretation of Norms

5 Induced Norms of Linear Maps

6 Two Examples

8 The Measure of a Matrix

Notes and References

III General Theorems

1 Setting of the Problem

2 Small Gain Theorem

3 Small Gain Theorem: Incremental Form

4 A Boundedness Result

5 An Existence and Uniqueness Theorem

6 Loop Transformation Theorem

7 L Stability

8 General Feedback Formula

Notes and References

IV Linear Systems

0 Introduction

1 Linear Feedback Systems with Rational Transfer Functions

2 Necessary and Sufficient Conditions: Factorization Method

3 Linear Feedback Systems with Dynamics in the Feedback Path (Rational Transfer Functions Case)

4 Convolution Feedback Systems

5 Graphical Test

6 Discrete-Time Systems

7 Linear Time-Varying Systems

8 Slowly Varying Systems

9 Linearization

Notes and References

V Applications of the Small Gain Theorem

1 Continuous-Time Systems—LP Stability

2 L2 Stability—Circle Criterion

3 Exponential Weighting—L∞ Stability

4 Discrete-Time Systems—LP Stability

5 Slowly-Varying Linear Systems

6 Nonlinear Circuit Example

7 Existence of Periodic Solutions

8 Popov Criterion

9 Instability

Notes and References

VI Passivity

0 Introduction

1 Motivation from Circuit Theory

2 Scalar Products

3 Formal Framework

4 Passive Systems: Definition and Examples

5 Passivity Theorem

6 The Popov Criterion

7 Discrete-Time Case

8 Average Logarithmic Variation Criterion

9 Multiplier Theory

10 Relation between the Passivity Theorem and the Small Gain Theorem

11 Invertibility of I + H

12 Instability Theorems

Notes and References

Appendixes

A Integrals and Series

A.1 Regulated Functions

A.2 Integrals

A.3 Series

B Fourier Transforms

B.1 L1 Theory

B.2 L2 Theory

B.3 Laplace Transform

C Convolution

C.1 Introduction

C.2 Convolution of Functions

C.3 Convolution of a Measure and a Function

C.4 Convolution of Sequences

D Algebras

D.1 Algebras

D.2 Ideals

D.3 Inverses in A

E Bellman-Gronwall Lemma

References

Index

No. of pages:
2
Language:
English