具雜訊混沌系統之控制

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陳冠宇(Kuan-Yu Chen) 查詢紙本館藏

畢業系所

機械工程學系

論文名稱

具雜訊混沌系統之控制
(Controlling Chaos in a Noisy Chaotic System)

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摘要(中)

獨立成份分析是一種訊號處理和統計的方法，在僅有未知訊號源的觀察
或量測之混合資料時，假設未知訊號源均為非高斯分佈且彼此互相獨立，
則獨立成份分析可以自混合資料中分離出獨立訊號；混沌非線性系統的控
制在許多工業應用上顯得越來越重要，加上主從式混沌系統間的混沌同步
在秘密通訊應用上的重要性，使得混沌控制成為相當受到矚目的領域；狀
態回授控制對於消除控制系統的干擾和非線性已具備系統化且發展完善；
基於上述的技術，本文提出一個新的控制方案結合獨立成份分析自量測白
雜訊中分離出混沌訊號，以及運用狀態回授控制消除混沌系統的非線性。
首先在Lur’e 系統的基礎上發展狀態回授控制的系統化程序，用以分析
二個具隨機白雜訊混沌系統的同步現象，藉由提出的改良式獨立成份分析
法的幫助，當混沌訊號遭受隨機白雜訊的汙染時，仍能自雜訊源中取出真
實的混沌訊號，並可以任意設計同步時間且保證穩定，即使系統的輸出具
有量測雜訊。其次，發展結合改良式獨立成份分析過濾雜訊和狀態回授消
除系統非線性的混沌控制方案，統御具雜訊之混沌系統，在系統動態已知
的情況下，此混沌控制方案易於理解並實現，本文提出二個範例展示此方
案的效能。最後，本文的模擬結果顯示狀態回授和獨立成份分析的技術應
用於具雜訊混沌系統的同步和控制問題具有相當的成效。此新方案是第一
次用於具量測雜訊之控制系統上可以取代傳統的Kalman 濾波器。

摘要(英)

Independent component analysis (ICA) is a signal processing and statistical
method designed to separate independent sources given only observed or
measured data that are mixtures of some unknown sources. These unknown
sources are assumed to be non-Gaussian and mutually independent. In addition,
controlling chaos of chaotic nonlinear systems has been received much attention
and becomes more important for many industrial applications. Furthermore,
chaos synchronization between master and slave chaotic systems has been
attractive topic for its potential applications for secure communications. State
feedback control for canceling disturbance and nonlinearity of control systems
has been systematic and well developed. In this dissertation, based upon these
techniques as mention above, a new scheme has been proposed to combine the
ICA method for separating chaotic signals from measured white noise with a
state feedback control for cancelling nonlinearity of chaotic system.
In this study, we first develop a systematic procedure of state feedback
control, based on a Lur’e-type system, to analyze the synchronization of two
chaotic systems in the presence of random white noise. With the aid of the
proposed modified independent component analysis, the real chaotic signal can
be extracted from a noisy source where the chaotic signal has been contaminated
by random white noise. The synchronization time can be arbitrarily designed
to guarantee stability, even if the system’s output is corrupted by measurement
noise. Secondly, we combine a modified independent component analysis
approach with an approach for feedback cancellation of nonlinear terms. This
approach to engineering control can be utilized to efficiently govern a noisy
chaotic system. The methodology is easy to comprehend and to implement,
but previous knowledge of the system dynamics is needed. Two examples are
provided to show the effectiveness of the proposed scheme. Finally, the results
of the thesis demonstrate the fruitfulness of the state feedback and the ICA
theory application to the synchronization and control problems for noisy chaotic
systems. The new scheme is first used for control systems with measurement
noise which can replace the conventional Kalman filter.

關鍵字(中)

★ 狀態回授
★ Rössler方程式
★ 混沌同步
★ 混沌控制
★ 獨立成份分析
★ Duffing方程式

關鍵字(英)

★ chaos synchronization
★ controlling chaos
★ independent component analysis
★ state feedback
★ Duffing equation
★ Rössler equation

論文目次

摘要 I
謝誌 IV
一、緒論一
1.1 研究動機一
1.2 文獻回顧一
1.3 論文架構六
二、混沌控制七
2.1 微分方程式中的混沌七
2.1.1 Lorenz吸子七
2.1.2 Duffing振子七
2.1.3 Rössler吸子七
2.1.4 Chua電路八
2.2 混沌控制的方法八
2.2.1 OGY法九
2.2.2 OPF控制器九
2.3 混沌控制的演算法九
2.4 混沌同步九
三、狀態回授控制一一
四、獨立成份分析一二
五、應用改良式ICA及狀態回授於具雜訊混沌系統之控制一三
六、結論一四
Abstract i
Contents iii
List of Figures v
List of Tables ix
Abbreviations and Symbols x
I. Introduction 1
1.1 Motivation 1
1.2 Literature survey 1
1.3 Dissertation organization 7
II. Chaos Control 8
2.1 Chaos in differential equations 8
2.1.1 Lorenz attractor 8
2.1.2 Duffing oscillator 11
2.1.3 Rössler attractor 15
2.1.4 Chua’s circuit 19
2.2 Methods of chaos control 25
2.2.1 The OGY method 27
2.2.2 The OPF controller 32
2.3 Chaotic control algorithm 33
2.4 Chaos synchronization 37
2.4.1 Synchronization of coupled scalar equations 38
2.4.2 Synchronization of chaotic Lorenz attractors 40
2.4.3 Synchronization of chaotic Rössler systems 46
III. State Feedback Control 49
3.1 Input-state linearization 49
3.2 Input-state stabilization 50
IV. Independent Component Analysis (ICA) 55
4.1 Definition of ICA 55
4.2 ICA by maximization of non-Gaussianity 57
4.2.1 General concepts of probability theory 58
4.2.2 Maximizing the non-Gaussianity 63
4.2.3 Measuring non-Gaussianity by kurtosis 64
4.2.4 Measuring non-Gaussianity by negentropy 65
4.3 The fixed-point FastICA algorithm 67
V. Controlling Chaos via Modified ICA and SFC in a Noisy Chaotic System 74
5.1 Chaos synchronization in the presence of noise 74
5.1.1 Output feedback coupling 74
5.1.2 Modified ICA 78
5.1.3 Example 81
5.2 Controlling chaos in a noisy system 89
5.2.1 State feedback control 89
5.2.2 Modified ICA 90
5.2.3 Example 93
VI. Conclusion 98
References 99

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指導教授

董必正(Pi-Cheng Tung)

審核日期

2008-7-31

推文