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1st Edition - January 28, 1973

**Editor:** Kumpati S. Narendra

eBook ISBN:

9 7 8 - 0 - 3 2 3 - 1 6 2 3 4 - 0

Frequency Domain Criteria for Absolute Stability presents some generalizations of the well-known Popov solution to the absolute stability problem proposed by Lur'e and Postnikov… Read more

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Frequency Domain Criteria for Absolute Stability presents some generalizations of the well-known Popov solution to the absolute stability problem proposed by Lur'e and Postnikov in 1944. This book is divided into nine chapters that focus on the application of Lyapunov's direct method to generate frequency domain criteria for stability. The first eight chapters explore the systems with a single nonlinear function or time-varying parameter. These chapters also discuss the development of stability criteria for these systems, the sufficiency theorems, and Lyapunov function. Some of the theorems applied to a damped version of the Mathieu equation and to a nonlinear equation derived from it are also covered. The concluding chapter deals with systems with multiple nonlinearities or time-varying gains. This chapter also outlines the basic definitions and tools, as well as the derivation of stability criteria. This work will serve as a reference for research courses concerning stability problems related to the absolute stability problem of Lur'e and Postnikov. Engineers and applied mathematicians will also find this book invaluable.

Foreword

Preface

Acknowledgments

Special Notation

I. Introduction

1. The System

2. Stability of Motion

3. Lyapunov's Direct Method

4. The Quadratic Lyapunov Function

5. Some Problems in Stability

6. The Conjectures of Aizerman and Kalman

7. The Absolute Stability Problem

8. The Criterion of Popov

9. Synopsis

II. Problem Statement

1. System Definition

2. Definitions of Stability

3. Formal Problem Statement

III. Mathematical Preliminaries

1. Sufficiency Theorems

2. The Absolute Lyapunov Function Candidates

3. Restated Stability Theorems

4. The Kalman-Yakubovich Lemma

5. Positive Real Functions

6. Existence Theorems

IV. Linear Time-Invariant Systems and Absolute Stability

1. Relations between Linear Time-Invariant and Nonlinear Time-Varying Systems

2. The Existence of the Quadratic Lyapunov Function xTPx and the Hurwitz Condition

3. The Existence of the Quadratic Lyapunov Function xΊΡx + κxTΜx and the Nyquist Criterion

V. Stability of Nonlinear Systems

1. The Popov Stability Criterion

2. Stability Criteria for Monotonic Nonlinearities

3. Linear Systems

4. Odd Monotonic Gains

5. The General Finite Sector Problem

VI. Stability of Nonlinear Time-Varying Systems

1. The Circle Criterion

2. An Extension of the Popov Criterion—Point Conditions

3. Stability Criteria for Restricted Nonlinear Behavior—Point Conditions

4. The Gentral Finite Sector Problem

5. Periodic Nonlinear Time-Varying Gains

6. Extension of the Popov Criterion—Integral Conditions

7. Integral Conditions for Restricted Nonlinear and Linear Gains

8. Integral Conditions for Linear Time-Varying Systems

VII. Geometric Stability Criteria

1. Linear Time-Invariant Systems

2. The Circle Criterion

3. The Popov Criterion

4. Monotonic Nonlinearities: An Off-Axis Circle Criterion

5. Further Geometric Interpretations for Time-Varying Systems

VIII. The Mathieu Equation: An Example

1. Solutions of the Mathieu Equation

2. Linear Case (a ≈ 1): A Perturbation Analysis

3. Linear Case (a ≈ 4): A Floquet Analysis

4. Application of Stability Criteria to the Linear Case

5. Application of Stability Criteria to the Nonlinear Case

IX. Absolute Stability of Systems with Multiple Nonlinear Time-Varying Gains

1. Introduction

2. Problem Statement

3. Mathematical Preliminaries

4. Linear Time-Invariant Systems and Absolute Stability

5. Stability of Nonlinear Systems

6. Stability of Nonlinear Time-Varying Systems

Appendix. Matrix Version of the Kalman-Yakubovich Lemma

References

INDEX

- No. of pages: 268
- Language: English
- Published: January 28, 1973
- Imprint: Academic Press
- eBook ISBN: 9780323162340