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"An Introduction to Nonlinearity in Control Systems" by Derek Atherton

BoBoCoAe, DA& Ventus Publishing ApS | 2011 | ISBN: 8776817909 9788776817909 | 176 pages | PDF | 11 MB

The book is intended to provide an introduction to the effects of nonlinear elements in feedback control systems. The book is concerned with the effects of nonlinearity in feedback control systems and techniques which can be used to design feedback loops containing nonlinear elements.

A central topic is the use of the Describing Function (DF) method since in combination with simulation it provides an excellent approach for the practicing engineer and follows on logically from a first course in classical control, such as the companion volume in this series. Some of the basic material on the topic can be found in my earlier book which is frequently referenced throughout the text.

Contents

Preface

1 Introduction

1.1 What is nonlinearity?

1.2 Forms of nonlinearity

1.3 Structure and Behaviour

1.4 Overview of contents

1.5 References

2 The Phase Plane Method

2.1 Introduction

2.2 Basic Principles

2.2.1 The Linear Case

2.2.2 The Nonlinear Case

2.2.3 An Example

2.3 The Phase Plane for Systems with Linear Segmented Nonlinearities

2.3.1 Example 1 - Nonlinear Output Derivative Feedback

2.3.2 Example 2 - Relay Position Control

2.3.3 Example 3 - Position Control with Torque Saturation

2. 4 Conclusions

2.5 References

2.6 Bibliography

3 The Describing Function

3.1 Introduction

3.2 The Sinusoidal Describing Function

3.3 Some Properties of the DF

3.4 The Evaluation of some DFs

3.4.1 Cubic Nonlinearity

3.4.2 Saturation Nonlinearity

3.4.3 Relay with Dead Zone and Hysteresis

3.5 Nonlinear Models and DFs

3.6 Harmonic Outputs

3.7 Sine plus Bias DF and the IDF

3.8 Conclusions

3.9 References

3.10 Bibliography

3.11 Appendix -Tables of Describing Functions

4 Stability and Limit Cycles using the DF

4.1 Introduction

4.2 Limit Cycle Evaluation

4.3 Stability of a Predicted Limit Cycle

4.4 DF Accuracy

4.5 Some Examples of Limit Cycle Evaluation

4.5.1 The Van der Pol Equation

4.5.2 Feedback Loop Containing a Relay with Dead Zone

4.5.3 Feedback Loop with On Off Relay

4.5.4 Use of the IDF

4.6 More than one Nonlinear Element

4.7 Applications of SBDF to find Limit Cycles

4.7.1 Asymmetrical nonlinearity

4.7.2 Effect of Bias with Odd Symmetrical Nonlinearity

4.8 Conclusions

4.9 Bibliography

5 The SSDF and Harmonically Forced Systems

5.1 Introduction

5.2 The Cubic Nonlinearity with Two Sinusoidal Inputs

5.3 Modified Nonlinearities and the SSDF

5.3.1 Power Law Characteristic

5.3.2 Harmonic Nonlinearity

5.3.3 Ideal Relay

5.4 The IDF for Related Signals

5.5 More Accurate Determination of Limit Cycles

5.6 Closed Loop Frequency Response

5.7 Jump Resonance

5.7.1 Jump resonance region- use of the IDF

5.7.2 Some Examples of Jump Resonance

5.8 Conclusions

5.9 References

6 Limit cycles in relay systems

6.1 Introduction

6.2 The Frequency Domain Approach

6.3 Properties and Evaluation of A loci

6.4 Solving for Limit Cycles

6.4.1 Relay with no Dead Zone

6.4.2 Relay with Dead Zone

6.5 Limit Cycle Stability

6.6 Some Interesting Limit Cycle Problems

6.6.1 Example 1 - Invalid Continuity Condition

6.6.2 Example 2 - Multipulse Limit Cycles

6.6.3 Example 3 - Limit Cycle with a Sliding Mode

6.6.4 Example 4 - Multiple Limit Cycle Solutions

6.6.5 Example 5 - Chaotic Motion

6.6.6 Example 6 - Unstable Limit Cycles and Simulation

6.7 Forced oscillations

6.8 Conclusions

6.9 References

6.10 Appendix

7 Controller Tuning from Relay Produced Limit Cycles

7.1 Introduction

7.2 Knowledge from the Limit Cycle

7.2.1 Relay Autotuning

7.2.2 Plant Parameter Identification

7.3 Tuning the Controller

7.3.1 An Example

7.4 Autotuning using the Relay in Parallel with the Controller

7.4.1 An Example

7.5 Conclusions

7.6 References

8 Absolute Stability Results

8.1 Introduction

8.2 Lyapunov’s Method

8.2.1 Definitions of Stability

8.2.2 Positive Definite Functions

8.2.3 Lyapunov’s Theorems

8.3 Application of Lyapunov’s Method

8.3.1 Linear System

8.3.2 Nonlinear System

8.4 Definitions and Loop Transformations

8.4.2 Loop Transformations

8.5 Frequency Domain Criteria

8.5.1 Popov Theorem

8.5.2 The Circle Criterion

8.5.3 The Off-axis Circle Criterion

8.6 Examples

8.7 Conclusions

8.8 References

8.9 Bibliography

9 Design of Nonlinear Control Systems

9.1 Introduction

9.2 Linearization

9.3 Frequency Response Shaping

9.4 Nonlinear Compensation

9.5 Compensation using DF Models

9.6 Time Optimal Control

9.7 Sliding Mode Control

9.7.1 A Second Order System

9.7.2 Further Comments

9.8 Fuzzy Logic

9.9 Neural Networks

9.10 Exact Linearization

9.10.1 Relative Degree

9.10.2 Input-Output Linearization

9.10.3 Exact State Linearization

9.10.4 Examples

9.10.5 Further Comments

9.11 General Comments

9.12 References

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