Stability and Oscillations of Nonlinear

This book gives a comprehensive treatment of mathematical methods for the study of stability and oscillations in control systems with various types of pulse modulations. The original approaches developed by the authors include the averaging method which enables readers to extend the absolute stability theory and the fixed-point approach for the study of forced oscillations for pulse modulated systems.

Preface

xi

(4)

List of Notations

xv

1 Mathematical Description of Pulse-Modulated Systems

1

(28)

1.1 Pulse Modulation. General Concepts

1

(5)

1.2 Types of Pulse Modulation with Pulses of Finite Duration

6

(7)

1.2.1 Pulse-amplitude modulation

6

(1)

1.2.2 Pulse-frequency modulation of the first kind

7

(1)

1.2.3 Pulse-frequency modulation of the second kind

7

(1)

1.2.4 Pulse-width modulation of the first kind

7

(1)

1.2.5 Pulse-width modulation of the second kind

8

(1)

1.2.6 Combined pulse modulation

8

(1)

1.2.7 Integral pulse-width modulation

8

(1)

1.2.8 Linear integral pulse-width modulation

9

(1)

1.2.9 Pulse-width modulation of the second kind with a sawtooth characteristic

10

(1)

1.2.10 Phase modulation with nonautonomous pulse generation

11

(1)

1.2.11 Frequency-phase conversion with nonautonomous pulse generation and without reset in the modulator

12

(1)

1.3 Types of Pulse Modulation with Pulses of Zero Duration (Impulses)

13

(2)

1.3.1 Pulse-amplitude modulation

13

(1)

1.3.2 Pulse-frequency modulation of the first kind

13

(1)

1.3.3 Pulse-frequency modulation of the second kind

14

(1)

1.3.4 Integral pulse-frequency modulation

14

(1)

1.3.5 Sigma pulse-frequency modulation

15

(1)

1.4 Lumped-Parameter Continuous Linear Part

15

(3)

1.5 Distributed-Parameter Continuous Linear Part

18

(5)

1.6 Reduction to Discrete-Time Systems

23

(1)

1.7 Reduction to Relay Systems

24

(5)

2 Stability of Equilibria. Miscellaneous Methods

29

(22)

2.1 Introduction

29

(1)

2.2 Method of Integral Quadratic Bounds

30

(3)

2.3 Method Based on Properties of Positive Kernels

33

(3)

2.4 Method of Direct Integral Estimates

36

(1)

2.5 Direct Lyapunov Method. Approach of Kuntsevich and Chekhovoi

37

(4)

2.6 Direct Lyapunov Method. Approach of Gelig and Churilov

41

(10)

3 Stability of Equilibria. Averaging Method

51

(46)

3.1 Introduction

51

(4)

3.2 Problem Setting

55

(8)

3.3 Analogue of Popov Stability Criterion. General Case

63

(5)

3.4 Analogue of the Circle Criterion. General Case

68

(4)

3.5 Stability of Systems with Pulses of a Special Type

72

(2)

3.6 Proofs of Theorems from Section 3.3

74

(9)

3.7 Proofs of Theorems from Section 3.4

83

(4)

3.8 Proofs of Theorems from Section 3.5

87

(1)

3.9 Stability of Integral Pulse-Width Modulated Systems

88

(9)

4 Stability of Processes. Averaging Method

97

(32)

4.1 Problem Setting

97

(1)

4.2 Pulse-Amplitude Modulation

98

(2)

4.3 Pulse-Width Modulation of the First Kind

100

(1)

4.4 Pulse-Width Modulation of the Second Kind

101

(4)

4.5 PWM-2 with a Sawtooth Characteristic

105

(3)

4.6 Integral Pulse-Width Modulation

108

(7)

4.7 Linear Integral Pulse-Width Modulation

115

(6)

4.8 Pulse-Phase Modulation of the First Kind

121

(3)

4.9 Pulse-Phase Modulation of the Second Kind

124

(2)

4.10 Frequency-Phase Transform

126

(3)

5 Forced Periodic Oscillations. Method of Equations of Periods

129

(40)

5.1 Definition of a Periodic Mode

129

(8)

5.2 Pulse-Width Modulation of the First Kind

137

(5)

5.3 Pulse-Width Modulation of the Second Kind

142

(8)

5.4 Integral Pulse-Width Modulation

150

(4)

5.5 Linear Integral Pulse-Width Modulation

154

(5)

5.6 Pulse-Frequency Modulation of the First Kind

159

(4)

5.7 Pulse-Frequency Modulation of the Second Kind

163

(2)

5.8 Sigma Pulse-Frequency Modulation

165

(4)

6 Oscillations of PWM Systems. Fixed-Point Method

169

(42)

6.1 Problem Setting

169

(2)

6.2 Pulse-Width Modulation of the First Kind

171

(6)

6.3 Pulse-Width Modulation of the Second Kind

177

(7)

6.4 Integral Pulse-Width Modulation

184

(4)

6.5 Linear Integral Pulse-Width Modulation

188

(7)

6.6 Systems with a Variable Structure of a Linear Part

195

(16)

7 Oscillations of PFM Systems. Fixed-Point Method

211

(22)

7.1 Forced Oscillations in a General Case

211

(8)

7.2 Case of a Dominant Eigenvalue for PFM-1

219

(11)

7.3 Case of a Dominant Eigenvalue for PFM-2

230

(3)

8 Method of Harmonic Balance and Its Justification

233

(18)

8.1 Method of Harmonic Balance

233

(4)

8.2 Periodic Modes in a PFM System with a Stable CLP

237

(5)

8.3 Periodic Modes in a PFM System with a Neutral CLP

242

(9)

9 Auto-Oscillations in Pulse Modulated Systems

251

(20)

9.1 Problem Setting

251

(2)

9.2 Conditions for Auto-Oscillations

253

(4)

9.3 Proofs of the Theorems on Auto-Oscillations

257

(14)

10 Pulse-Width Modulated Systems of Phase Synchronization

271

(28)

10.1 Example from Electromechanics

271

(2)

10.2 Mathematical Setting of the Problem

273

(2)

10.3 Stationary Modes of a Synchronization System

275

(3)

10.4 Conditions of Quasi-Synchronism

278

(2)

10.5 Proofs of the Theorems on a Stationary Mode

280

(4)

10.6 Proofs of the Theorems on Quasi-Synchronism

284

(15)

Appendix

299

(44)

A.1 Controllability, Observability, Nondegeneracy, Hurwitz Property

299

(3)

A.2 Lyapunov Matrix Equation and Inertia Theorems

302

(1)

A.3 Kalman-Yakubovich Lemma

303

(2)

A.4 Auxiliary Statements Related to the Kalman-Yakubovich Lemma

305

(6)

A.5 Estimates of a Functional

311

(5)

A.6 Solvability of Matrix Inequalities (PWM Systems)

316

(7)

A.7 Solvability of Matrix Inequalities (PFM Systems)

323

(4)

A.8 Positive Kernels of Integral Equations

327

(6)

A.9 Stability of Systems with Integral Quadratic Bounds

333

(4)

A.10 Miscellaneous Statements

337

(6)

References

343

(18)

Index

361

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Stability and Oscillations of Nonlinear Pulse-Modulated Systems

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