Robust Adaptive Control for Fractional-order Systems With Disturbance and Saturation

A treatise on investigating tracking control and synchronization control of fractional-order nonlinear systems with system uncertainties, external disturbance, and input saturation

Robust Adaptive Control for Fractional-Order Systems, with Disturbance and Saturation provides the reader with a good understanding on how to achieve tracking control and synchronization control of fractional-order nonlinear systems with system uncertainties, external disturbance, and input saturation. Although some texts have touched upon control of fractional-order systems, the issues of input saturation and disturbances have rarely been considered together.

This book offers chapter coverage of fractional calculus and fractional-order systems; fractional-order PID controller and fractional-order disturbance observer; design of fractional-order controllers for nonlinear chaotic systems and some applications; sliding mode control for fractional-order nonlinear systems based on disturbance observer; disturbance observer based neural control for an uncertain fractional-order rotational mechanical system; adaptive neural tracking control for uncertain fractional-order chaotic systems subject to input saturation and disturbance; stabilization control of continuous-time fractional positive systems based on disturbance observer; sliding mode synchronization control for fractional-order chaotic systems with disturbance; and more.

• Based on the approximation ability of the neural network (NN), the adaptive neural control schemes are reported for uncertain fractional-order nonlinear systems
• Covers the disturbance estimation techniques that have been developed to alleviate the restriction faced by traditional feedforward control and reject the effect of external disturbances for uncertain fractional-order nonlinear systems
• By combining the NN with the disturbance observer, the disturbance observer based adaptive neural control schemes have been studied for uncertain fractional-order nonlinear systems with unknown disturbances
• Considers, together, the issue of input saturation and the disturbance for the control of fractional-order nonlinear systems in the present of system uncertainty, external disturbance, and input saturation

Robust Adaptive Control for Fractional-Order Systems, with Disturbance and Saturation can be used as a reference for the academic research on fractional-order nonlinear systems or used in Ph.D. study of control theory and engineering.

Mou Chen, PhD is a Professor at the College of Automation Engineering at Nanjing University of Aeronautics and Astronautics, China. He also serves as an associate editor for IEEE access and neurocomputing.

Shuyi Shao is working toward a Ph.D. degree with a major in control theory and control engineering from the College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, China.

Peng Shi, PhD is a Professor and Chair in systems and control, at the University of Adelaide, and Victoria University, Australia. He is also an IEEE Distinguished Lecturer, and is a Member of the College of Expert, Australian Research Council.

Preface xi

Series Preface xv

Symbols and Acronyms xvii

1 Introduction 1

2 Fractional Calculus and Fractional-Order Systems 9

2.1 Fractional Calculus 9

2.1.1 Several Important Functions of Fractional Calculus 9

2.1.2 Fractional Integral and Derivatives 11

2.1.3 Some Important Lemmas 12

2.2 Some Typical Fractional-Order Systems 16

2.2.1 Fractional-Order Lorenz System 16

2.2.2 Fractional-Order Van Der Pol Oscillator 18

2.2.3 Fractional-Order Genesio–Tesi System 18

2.2.4 Fractional-Order Arneodo System 20

2.2.5 Fractional-Order Lotka–Volterra System 21

2.2.6 Fractional-Order Financial System 23

2.2.7 Fractional-Order Newton–Leipnik System 25

2.2.8 Fractional-Order Duffing System 27

2.2.9 Fractional-Order Lü System 29

2.2.10 Fractional-Order Three-Dimensional System 33

2.2.11 Fractional-Order Hyperchaotic Oscillator 35

2.2.12 Fractional-Order Four-Dimensional Hyperchaotic System 37

2.2.13 Fractional-Order Hyperchaotic Cellular Neural Network 39

2.3 Conclusion 41

3 Fractional-Order PID Controller and Fractional-Order Disturbance Observer 43

3.1 Problem Statement 43

3.2 Fractional-Order PID Controller 44

3.2.1 Integer-Order PID Controller 44

3.2.2 Fractional-Order PI;;D;; Controller 44

3.2.3 Control Based on Fractional-Order PI;;D;; Controller 45

3.3 Frequency-Domain Fractional-Order Disturbance Observer 48

3.3.1 Classical Integer-Order Disturbance Observer 48

3.3.2 Fractional-Order Disturbance Observer 49

3.3.3 Estimation Performance of Fractional-Order Disturbance Observer 51

3.3.4 Control Based on Fractional-Order Disturbance Observer 52

3.4 Conclusion 53

4 Design of Fractional-Order Controllers for Nonlinear Chaotic Systems and Some Applications 55

4.1 Fractional-Order Control for a Novel Chaotic SystemWithout Equilibrium 55

4.1.1 Problem Statement 55

4.1.2 Design of Chaotic System and Circuit Implementation 56

4.1.2.1 A Novel Chaotic System 56

4.1.2.2 Circuit Implementation 58

4.1.3 Design of Fractional-Order Controller and Stability Analysis 59

4.1.4 Numerical Simulation 62

4.1.4.1 Novel Chaotic System 62

4.1.4.2 Chaotic Systems with Equilibrium 63

4.2 Application of Chaotic System without Equilibrium in Image Encryption 68

4.2.1 Image Encryption Scheme 69

4.2.2 Histogram Analysis 69

4.2.3 Correlation of Two Adjacent Pixels 71

4.2.4 Anti-Attack Ability of Image Encryption Scheme 71

4.2.5 Sensitivity Analysis of Key 71

4.3 Synchronization Control for Fractional-Order Nonlinear Chaotic Systems 73

4.3.1 Problem Description 73

4.3.2 Design of Synchronization Controller 73

4.3.3 Simulation Examples 75

4.3.3.1 Fractional-Order Chen System 76

4.3.3.2 Fractional-Order Lorenz System 79

4.3.4 Application of Synchronization Control Scheme in Secure Communication 82

4.4 Conclusion 83

5 Sliding-Mode Control for Fractional-Order Nonlinear Systems Based on Disturbance Observer 85

5.1 Problem Statement 85

5.2 Adaptive Control Design Based on Fractional-Order Sliding-Mode Disturbance Observer 86

5.2.1 Design of Fractional-Order Sliding-Mode Disturbance Observer 86

5.2.2 Controller Design and Stability Analysis 87

5.3 Simulation Examples 89

5.3.1 Example 1 89

5.3.2 Example 2 91

5.4 Conclusion 94

6 Disturbance-Observer-Based Neural Control for Uncertain Fractional-Order Rotational Mechanical System 95

6.1 Problem Statement 95

6.2 Adaptive Neural Control Design 96

6.2.1 Design of Fractional-Order Disturbance Observer 96

6.2.2 Controller Design and Stability Analysis 97

6.3 Simulation Example 101

6.4 Conclusion 105

7 Adaptive Neural Tracking Control for Uncertain Fractional-Order Chaotic Systems Subject to Input Saturation and Disturbance 107

7.1 Problem Statement 107

7.2 Adaptive Neural Control Design Based on Fractional-Order Disturbance Observer 108

7.3 Simulation Examples 115

7.3.1 Fractional-Order Chaotic Electronic Oscillator 116

7.3.2 Fractional-OrderModified Jerk System 118

7.4 Conclusion 121

8 Stabilization Control of Continuous-Time Fractional Positive Systems Based on Disturbance Observer 123

8.1 Problem Statement 123

8.1.1 Notation and Definitions 123

8.1.2 Preliminaries 123

8.2 Main Results 126

8.2.1 Fractional Disturbance Observer 126

8.2.2 Stabilization Control of Fractional Positive System 128

8.2.3 Simulation of Fractional Positive System 130

8.2.4 Stabilization Control of Fractional Bounded Positive System 131

8.2.5 Simulation of Fractional Bounded Positive System 133

8.3 Conclusion 137

9 Sliding-Mode Synchronization Control for Fractional-Order Chaotic Systems with Disturbance 139

9.1 Problem Statement 139

9.2 Design of Fractional-Order Disturbance Observer 139

9.3 Disturbance-Observer-Based Synchronization Control of Fractional-Order Chaotic Systems 141

9.4 Simulation Examples 144

9.4.1 Synchronization Control of Modified Fractional-Order Jerk System 144

9.4.2 Synchronization Control of Fractional-Order Liu System 148

9.5 Conclusion 152

10 Anti-Synchronization Control for Fractional-Order Nonlinear Systems Using Disturbance Observer and Neural Networks 153

10.1 Problem Statement 153

10.2 Design of Disturbance Observer 153

10.3 Anti-Synchronization Control of Fractional-Order Nonlinear Systems 155

10.4 Simulation Examples 158

10.4.1 Anti-Synchronization Control of Fractional-Order Lorenz System 159

10.4.2 Anti-Synchronization Control of Fractional-Order Lü System 161

10.5 Conclusion 167

11 Synchronization Control for Fractional-Order Systems Subjected to Input Saturation 169

11.1 Problem Statement 169

11.2 Synchronization Control Design of Fractional-Order Systems with Input Saturation 170

11.3 Simulation Examples 172

11.3.1 Fractional-OrderModified Chua’s Circuit with Sine Function 172

11.3.2 Fractional-Order Four-Dimensional Modified Chua’s Circuit 174

11.4 Conclusion 179

12 Synchronization Control for Fractional-Order Chaotic Systems with Input Saturation and Disturbance 181

12.1 Problem Statement 181

12.2 Design of Fractional-Order Disturbance Observer 181

12.3 Design of Synchronization Control 183

12.4 Simulation Examples 185

12.4.1 Fractional-Order Chua’s Circuit 185

12.4.2 Fractional-Order Hyperchaos Chua’s Circuit 189

12.5 Conclusion 197

Appendix A Fractional Derivatives of Some Functions 199

A.1 Fractional Derivative of Constant 199

A.2 Fractional Derivative of the Power Function 199

A.3 Fractional Derivative of the Exponential Function 200

A.4 Fractional Derivatives of Sine and Cosine Functions 201

Appendix B Table of Caputo Derivatives 203

Appendix C Laplace Transforms Involving Fractional Operations 205

C.1 Laplace Transforms 205

C.2 Special Functions for Laplace Transforms 205

C.3 Laplace Transform Tables 205

References 211

Index 227

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