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研究生:王俊傑
研究生(外文):Chun-Chieh Wang
論文名稱:適應性可變結構控制於渾沌控制與渾沌同步之研究
論文名稱(外文):Adaptive Variable Structure Approaches to Control and Synchronization of Chaotic Systems
指導教授:蘇仲鵬蘇仲鵬引用關係
學位類別:博士
校院名稱:國立雲林科技大學
系所名稱:工程科技研究所博士班
學門:工程學門
學類:綜合工程學類
論文種類:學術論文
論文出版年:2004
畢業學年度:92
語文別:英文
論文頁數:112
中文關鍵詞:互補式可變結構控制適應性互補式可變結構控制渾沌系統渾沌同步安全通訊Chua電路Lorenz系統ssler系統Duffing-Holmes oscillatorBonhoeffe
外文關鍵詞:adaptive complementary variable structure controomplementary variable structure controlchaotic synchronizationchaotic systemssecure communicationChua’s circuitssler systemDuffing-Holmes oscillatorBonhoeffer-van der Pol oscillator.Lorenz system
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本論文提出了一新的互補式可變結構控制法則來處理具有不確定性的渾沌控制以及渾沌電路同步系統之問題。
就渾沌控制方面而言,此一新的控制法則成功抑制了系統中不想要的渾沌現象,並且改善了系統的暫態與穩態響應。文中以具有不確定性的杜芬系統、Lorenz系統及Chua電路等為範例設計控制器。理論推導與模擬結果皆證實此一互補式可變結構控制即使在針對高維系統亦能有效的抑制掉系統中的渾沌現象,進而獲得較佳的性能。
另外,基於上述之互補式可變結構控制概念,本論文又提出了一新的適應性互補式可變結構控制針對渾沌同步問題來設計控制器。藉由Lyapunov穩定性分析及Babalat輔助定理證明本文所提出的控制法則能使得傳送端及接收端的渾沌系統在具有外加干擾下亦能達到同步且收斂至零點附近。最後再以具有不確定參數之Rössler系統、Chua電路以及杜芬系統為例,證實此一新的適應性控制法則的確能有效改善同步響應之誤差,並且還能進一步成功的應用於安全通訊系統上。
This dissertation presents a novel complementary variable structure control for controlling chaos and synchronization of chaos with uncertainties.
In controlling chaos, the newly proposed control law was shown to have remarkably improved the transient response as well as steady-state response. To illustrate the effectiveness of the design, a Duffing-Holmes oscillator with uncertainties, the Lorenz chaotic system and the Chua''s circuit with uncertainties were used as simulated examples. Both theoretical and simulation results reveal that, even for complex high-dimensional systems, the proposed control scheme is promising for controlling uncertain chaotic dynamics.
Based on the concept of complementary variable structure control we also presented an adaptive complementary variable structure control scheme for synchronization of a class of cascade-connected chaotic systems. By applying of Lyapunov synthesis method and Babalat''s lemma, the proposed control scheme has been shown to result in a closed-loop system that all signals are uniformly bounded and the synchronous error converges to a neighborhood of zero. In this study, not only two chaotic systems of the same type with unmatched uncertain parameters, but also two completely different chaotic systems were considered. The synchronization and secure communication of Rössler system, Chua''s circuit and Duffing-Holmes oscillator were used as illustrative examples to demonstrate the design. Both theoretical and simulation results have shown that the proposed novel adaptive variable structure control is effective for synchronizing chaotic dynamics with unknown parameters and bounded disturbances.
Contents


Chapter 1 Introduction 1
1.1 Literature Survey and Motivation 1
1.2 Dissertation Outline 5
Chapter 2 Chaotic Systems 6
2.1 Concept of Chaos-What is Chaos? 6
2.2 Benchmark Examples of Chaotic Systems 8
2.2.1 Chua''s Circuit 8
2.2.2 Rössler System 9
2.2.3 Lorenz System 10
2.2.4 Duffing-Holmes Oscillator 10
2.2.5 Bonhoeffer-van der Pol Oscillator (BVP) 11
Chapter 3 Fundamentals: A Review of Basic Theories 13
3.1 Variable Structure Control (VSC) 13
3.2 Sliding Mode Control 18
3.2 Adaptive Control 20
3.2.1 Model-Reference Adaptive Systems (MRAS) 21
3.2.2 Self-Tuning Regulators (STR) 22
3.3 Input-to-State Stability 24
Chapter 4 Control of Chaos 27
4.1 Complementary Variable Structure Control of Chaos 27
4.1.1 Problem Formulation 27
4.1.2 Case Study 33
4.1.2.1 A Duffing-Holmes Oscillator 33
4.1.2.2 A Cascade-Connected Lorenz System 34
4.1.2.3 A Cascade-Connected Chua''s Circuit 36
4.1.2.4 The Control of Unmodelled Chaos 38
4.2 Summary 38
Chapter 5 Synchronization of Chaos 53
5.1 Chaotic Synchronization via Complementary Variable Structure
Control Scheme 53
5.1.1 Problem Formulation 53
5.1.2 Case Study 55
5.1.2.1 Synchronization of Cascade-Connected Chua''s Circuit 55
5.1.2.2 Synchronization of Cascade-Connected Rössler System 56
5.1.2.3 Synchronization of Cascade-Connected Lorenz System 57
5.2 Adaptive Complementary Variable Structure Control of Synchronization
of Chaotic Systems 64
5.2.1 Problem Statement 64
5.2.2 Case Study 70
5.2.2.1 A Duffing-Holmes Oscillator 70
5.2.2.2 Synchronization of the Rössler System and the Duffing-
Holmes Oscillator 72
5.3 Adaptive Complementary Variable Structure Control of Synchronization
of Cascade-Connected Chaotic Systems 80
5.3.1 Problem Formulation 80
5.3.2 Case Study 88
5.3.2.1 A Cascade-Connected Rössler System 89
5.3.2.2 A Cascade-Connected Chua''s Circuit 91
5.3 Summary 99
Chapter 6 Conclusions 100
6.1 Conclusions 100
6.2 Future Research 101
Bibliography 103
Autobiography 109




List of Figures


Figure 2.1 Chaotic trajectories of the Lorenz system with parameters and 11
Figure 2.2 Chua’s circuit and the V-I characteristics of the Chua’s diode 12
Figure 3.1 Phase portrait of simple harmonic motion with 14
Figure 3.2 Phase portrait of simple harmonic motion with 15
Figure 3.3 Phase portrait of the system under VSC 15
Figure 3.4 Switching lines 16
Figure 3.5 Phase portrait of an unstable focus 17
Figure 3.6 Phase portrait of an unstable saddle 17
Figure 3.7 Phase portrait of the VSC stable system 18
Figure 3.8 Chattering phenomenon 20
Figure 3.9 Block diagram of an adaptive system 23
Figure 3.10 Block diagram of a MRAS 23
Figure 3.11 Block diagram of a STR 24
Figure 4.1 Time responses of the Duffing-Holmes system without a control signal 39
Figure 4.2 Time responses of the BVP system without a control signal 39
Figure 4.3 Tracking error of Duffing-Holmes system under control 40
Figure 4.4 Time responses of Duffing-Holmes system under control 40
Figure 4.5 Time responses of under control 41
Figure 4.6 Boundedness of control action 41
Figure 4.7 Chaotic trajectories of the Lorenz system without a control signal 42
Figure 4.8 Time responses of the Lorenz system without a control signal 42
Figure 4.9 Time responses under control for sudden changes of parameters 43
Figure 4.10 Control input for Case A 43
Figure 4.11 Time responses under control for sudden changes of reference signals 44
Figure 4.12 Control input for Case B 44
Figure 4.13 Time responses under control for and 45
Figure 4.14 Control input for and 45
Figure 4.15 Time responses under control for and 46
Figure 4.16 Control input for and 46
Figure 4.17 Time responses under control for 47
Figure 4.18 Control input for 47
Figure 4.19 Trajectories of the Chua’s circuit: period-1 attractor 48
Figure 4.20 Trajectories of the Chua’s circuit: double-scroll attractor 48
Figure 4.21 Time responses of and the reference signal 49
Figure 4.22 Tracking error of the controlled Chua’s circuit 49
Figure 4.23 Tracking error of the controlled Chua’s circuit in another scale 50
Figure 4.24 of the controlled Chua’s circuit 50
Figure 4.25 Control input of the controlled Chua’s ciruit 51
Figure 4.26 Tracking error of the unmodelled chaotic system under control 51
Figure 4.27 Boundedness of control action 52
Figure 5.1 Chaotic synchronization systems 55
Figure 5.2 Synchronization errors in two Chua’s circuits 58
Figure 5.3 Synchronization errors in another scale 58
Figure 5.4 Synchronization errors in two Chua’s circuits [23] 59
Figure 5.5 The trajectories of and 59
Figure 5.6 The trajectories of and 60
Figure 5.7 The trajectories of and 60
Figure 5.8 Synchronization errors in two Rössler Systems 61
Figure 5.9 The disturbance 61
Figure 5.10 The information signal 62
Figure 5.11 The error between and 62
Figure 5.12 Synchronization errors in two Lorenz system 63
Figure 5.13 Synchronization errors in another scale 63
Figure 5.14 Block diagram of an adaptive synchronization of chaotic systems 66
Figure 5.15 The trajectories of and when 74
Figure 5.16 Synchronization error in two Duffing-Holmes oscillator 75
Figure 5.17 The disturbance 75
Figure 5.18 The response of and when and 76
Figure 5.19 Synchronization error in two Duffing-Holmes oscillator 76
Figure 5.20 The error between and ( ) 77
Figure 5.21 The controller 77
Figure 5.22 The noise 78
Figure 5.23 The error between and ( ) 78
Figure 5.24 The error between and 79
Figure 5.25 The white Gaussian noise 79
Figure 5.26 The error between and 80
Figure 5.27 The time responses of and without disturbances 93
Figure 5.28 The white Gaussian noise 94
Figure 5.29 The time responses of and with a white Gaussian noise 94
Figure 5.30 The information signal 95
Figure 5.31 The error between and without disturbances 95
Figure 5.32 The error between and with a white Gaussian noise 96
Figure 5.33 The time responses of and without disturbances 96
Figure 5.34 The white Gaussian noise 97
Figure 5.35 The time responses of and with a white Gaussian noise 97
Figure 5.36 The information signal 98
Figure 5.37 The error between and without disturbances 98
Figure 5.38 The error between and with a white Gaussian noise 99
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