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 System Without 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-Order Modified 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-Order Modified 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
