简介
"Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation" presents a study of fractional-order chaotic systems accompanied by Matlab programs for simulating their state space trajectories, which are shown in the illustrations in the book. Description of the chaotic systems is clearly presented and their analysis and numerical solution are done in an easy-to-follow manner. Simulink models for the selected fractional-order systems are also presented. The readers will understand the fundamentals of the fractional calculus, how real dynamical systems can be described using fractional derivatives and fractional differential equations,聽how such equations can be solved, and how to simulate and explore chaotic systems of fractional order. The book addresses to mathematicians, physicists, engineers, and other scientists interested in chaos phenomena or in fractional-order systems. It can be used in courses on dynamical systems, control theory, and applied mathematics at graduate or postgraduate level.聽聽 Ivo Petr谩拧 is an Associate Professor of automatic control and the Director of the Institute of Control and Informatization of Production Processes, Faculty of BERG, Technical University of Ko拧ice, Slovak Republic. His main research interests include control systems, industrial automation, and applied mathematics.
目录
Title Page 3
Copyright Page 4
Preface 6
Acknowledgements 7
Table of Contents 9
Acronyms 12
Chapter 1 Introduction 14
References 16
Chapter 2 Fractional Calculus 19
2.1 Special Functions 19
2.2 Definitions of Fractional Derivatives and Integrals 21
2.3 Grunwald-Letnikov Fractional Integrals and Derivatives 21
2.4 Riemann-Liouville Fractional Integrals and Derivatives 23
2.5 Caputo Fractional Derivatives 24
2.6 Laplace Transform Method 24
2.6.1 Basic Facts about the Laplace Transform 24
2.6.2 Laplace Transform of Fractional Integrals 26
2.6.3 Laplace Transform of Fractional Derivatives 26
2.7 Fourier Transform Method 27
2.7.1 Basic Facts about the Fourier Transform 27
2.7.2 Fourier Transform of Fractional Integrals 28
2.7.3 Fourier Transform of Fractional Derivatives 29
2.8 Some Properties of Fractional Derivatives and Integrals 30
2.9 Numerical Methods for Calculation of Fractional Derivatives and Integrals 31
2.10 Fractional Calculus and Electricity 41
2.10.1 Analogue Fractional-Order Circuits 47
2.10.2 Experimental Measurement 48
2.10.3 Additional Remarks 49
References 50
Chapter 3 Fractional-Order Systems 55
3.1 Fractional LTI Systems 55
3.2 Fractional Nonlinear Systems 59
3.3 Fractional-Order Controllers 59
3.3.1 Definition of Fractional-Order Controllers 59
3.3.2 Properties and Characteristics of Controller 61
3.3.3 Design of Controller Parameters and Implementation 63
References 64
Chapter 4 Stability of Fractional-Order Systems 67
4.1 Preliminary Consideration 67
4.2 Stability of Fractional LTI Systems 75
4.3 Stability of Fractional Nonlinear Systems 90
4.4 Robust Stability of Fractional-Order LTI Systems 94
4.4.1 Stability Check When the Fractional Orders Are Crisp and Commensurate 94
4.4.2 Stability Check When the Fractional Orders Are Also Interval Real Numbers 99
4.5 Stability of Fractional-Order Nonlinear Uncertain Systems 109
References 110
Chapter 5 Fractional-Order Chaotic Systems 114
5.1 Introduction to Chaotic Dynamics 114
5.2 Concept of Chua\u2019s Circuit 115
5.2.1 Classical Chua\u2019s Oscillator 115
5.2.2 Fractional-Order Chua\u2019s Oscillator 118
5.2.2.1 Experimental Measurements 119
5.2.2.2 Simulation Results 121
5.2.3 Fractional-Order Chua-Podlubny\u2019s Oscillator 124
5.2.4 Fractional-Order Chua-Hartley\u2019s Oscillator 125
5.2.5 Fractional-Order Memristor-Based Chua\u2019s Oscillator 125
5.3 Fractional-Order Van der Pol Oscillator 138
5.4 Fractional-Order Duffing\u2019s Oscillator 141
5.5 Fractional-Order Lorenz\u2019s System 145
5.6 Fractional-Order Chen\u2019s System 149
5.7 Fractional-Order Lu\u2019s System 151
5.8 Fractional-Order Liu\u2019s System 153
5.9 Fractional-Order Genesio-Tesi\u2019s System 156
5.10 Fractional-Order Arneodo\u2019s System 159
5.11 Fractional-Order Rossler\u2019s System 162
5.12 Fractional-Order Newton-Leipnik\u2019s System 165
5.13 Fractional-Order Lotka-Volterra System 171
5.14 Fractional-Order Financial System 176
5.15 Fractional-Order CNN 179
5.16 Fractional-Order Volta\u2019s System 182
References 192
Chapter 6 Control of Fractional-Order Chaotic Systems 196
6.1 Preliminary Considerations 196
6.2 A Survey of Control Strategies 198
6.3 Examples: Feed-Back Control of Chaotic Systems 199
6.3.1 Sampled-Data Control of Chua\u2019s Oscillator 199
6.3.2 Sliding Mode Control of the Economical System 203
References 208
Chapter 7 Conclusion 211
References 213
Appendix A A List of Matlab Functions 216
Appendix B Laplace and Inverse Laplace Transforms 219
Glossary 223
Index 225
Copyright Page 4
Preface 6
Acknowledgements 7
Table of Contents 9
Acronyms 12
Chapter 1 Introduction 14
References 16
Chapter 2 Fractional Calculus 19
2.1 Special Functions 19
2.2 Definitions of Fractional Derivatives and Integrals 21
2.3 Grunwald-Letnikov Fractional Integrals and Derivatives 21
2.4 Riemann-Liouville Fractional Integrals and Derivatives 23
2.5 Caputo Fractional Derivatives 24
2.6 Laplace Transform Method 24
2.6.1 Basic Facts about the Laplace Transform 24
2.6.2 Laplace Transform of Fractional Integrals 26
2.6.3 Laplace Transform of Fractional Derivatives 26
2.7 Fourier Transform Method 27
2.7.1 Basic Facts about the Fourier Transform 27
2.7.2 Fourier Transform of Fractional Integrals 28
2.7.3 Fourier Transform of Fractional Derivatives 29
2.8 Some Properties of Fractional Derivatives and Integrals 30
2.9 Numerical Methods for Calculation of Fractional Derivatives and Integrals 31
2.10 Fractional Calculus and Electricity 41
2.10.1 Analogue Fractional-Order Circuits 47
2.10.2 Experimental Measurement 48
2.10.3 Additional Remarks 49
References 50
Chapter 3 Fractional-Order Systems 55
3.1 Fractional LTI Systems 55
3.2 Fractional Nonlinear Systems 59
3.3 Fractional-Order Controllers 59
3.3.1 Definition of Fractional-Order Controllers 59
3.3.2 Properties and Characteristics of Controller 61
3.3.3 Design of Controller Parameters and Implementation 63
References 64
Chapter 4 Stability of Fractional-Order Systems 67
4.1 Preliminary Consideration 67
4.2 Stability of Fractional LTI Systems 75
4.3 Stability of Fractional Nonlinear Systems 90
4.4 Robust Stability of Fractional-Order LTI Systems 94
4.4.1 Stability Check When the Fractional Orders Are Crisp and Commensurate 94
4.4.2 Stability Check When the Fractional Orders Are Also Interval Real Numbers 99
4.5 Stability of Fractional-Order Nonlinear Uncertain Systems 109
References 110
Chapter 5 Fractional-Order Chaotic Systems 114
5.1 Introduction to Chaotic Dynamics 114
5.2 Concept of Chua\u2019s Circuit 115
5.2.1 Classical Chua\u2019s Oscillator 115
5.2.2 Fractional-Order Chua\u2019s Oscillator 118
5.2.2.1 Experimental Measurements 119
5.2.2.2 Simulation Results 121
5.2.3 Fractional-Order Chua-Podlubny\u2019s Oscillator 124
5.2.4 Fractional-Order Chua-Hartley\u2019s Oscillator 125
5.2.5 Fractional-Order Memristor-Based Chua\u2019s Oscillator 125
5.3 Fractional-Order Van der Pol Oscillator 138
5.4 Fractional-Order Duffing\u2019s Oscillator 141
5.5 Fractional-Order Lorenz\u2019s System 145
5.6 Fractional-Order Chen\u2019s System 149
5.7 Fractional-Order Lu\u2019s System 151
5.8 Fractional-Order Liu\u2019s System 153
5.9 Fractional-Order Genesio-Tesi\u2019s System 156
5.10 Fractional-Order Arneodo\u2019s System 159
5.11 Fractional-Order Rossler\u2019s System 162
5.12 Fractional-Order Newton-Leipnik\u2019s System 165
5.13 Fractional-Order Lotka-Volterra System 171
5.14 Fractional-Order Financial System 176
5.15 Fractional-Order CNN 179
5.16 Fractional-Order Volta\u2019s System 182
References 192
Chapter 6 Control of Fractional-Order Chaotic Systems 196
6.1 Preliminary Considerations 196
6.2 A Survey of Control Strategies 198
6.3 Examples: Feed-Back Control of Chaotic Systems 199
6.3.1 Sampled-Data Control of Chua\u2019s Oscillator 199
6.3.2 Sliding Mode Control of the Economical System 203
References 208
Chapter 7 Conclusion 211
References 213
Appendix A A List of Matlab Functions 216
Appendix B Laplace and Inverse Laplace Transforms 219
Glossary 223
Index 225
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