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研究生:蔡循恆
論文名稱:渾沌系統之非線性控制
指導教授:董 必 正張 江 南
學位類別:博士
校院名稱:國立中央大學
系所名稱:機械工程研究所
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:2002
畢業學年度:90
語文別:中文
論文頁數:85
中文關鍵詞:渾沌控制非線性
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本論文針對非線性渾沌系統,提出幾種控制器的設計方法。首先針對一受外力激發渾沌現像之時變系統,我們提出一種新的滑動模式控制方法,使系統狀態能控制至任意軌跡,有別於目前一般渾沌控制方法將軌跡收斂至內嵌軌跡(Embedded orbit)上。其次針對OGY之控制方法,我們提出一系列之改良方案,使系統不僅收斂速度增快,控制器啟動等待時間縮短,而且在系統與欲收斂軌跡均具不確定因素下,仍然可以成功的達到穩定的目的。最後我們利用微分幾何法取代線性化,擴大控制器的操作範圍,不僅成功地將渾沌運動控制到一個穩定狀態,並能追蹤我們所期望的週期性軌跡。
Abstract
Physical systems are inherently nonlinear, and one special phenomenon of nonlinear systems is chaotic motion. The dissertation is devoted to control chaotic systems to regular motions. Numerous controlling chaos cases are studied separately in the dissertation. First, we show that one can control a chaotic system under external force excitation to arbitrary trajectories, even the desired trajectories are not located on the embedded orbits of a chaotic system. The method utilizes a newly developed sliding mode controller with a time varying manifold dynamic, to offer a feedback control in compensation with the external excitation, and drive the system orbits to any desired states. The proposed controller does not need high gain to suppress the external force, meanwhile, keeps robustness against parameter uncertainty and noise disturbance as the traditional sliding mode control. Simulations are provided to illustrate the performance of the controller.
Second, a simple and efficient method for controlling high dimensional discrete-time chaotic systems is proposed. This method is implemented similar to the OGY method, and is feasible for practical experiments. The key component is to assign the eigenvalues of a linearized map by using the well-known pole placement technique. According to the Cayley-Hamilton theorem, the trajectory will converge to the desired fixed point after iterations at most ( is the dimension of the map), if the real trajectory of the chaotic system falls within the neighborhood of the desired fixed point. The proposed approach improves the convergence rate and the robustness of the OGY method, especially for the case where the modulus of the stable eigenvalue is close to unity. The simulations illustrate the performance of our presented controller for controlling a chaotic system compared to the OGY method.
Third, a universal approach for controlling high dimensional chaotic systems is proposed, in which the controllability assumption can be relaxed and only the stabilization condition is required. The main feature of the proposed method is that all of the controllable unstable eigenvalues of the linear approximation assigned to be zero; the remained stable eigenvalues may be uncontrollable. Only small parameter perturbations are required to stabilize chaotic situation when the trajectory falls in the neighborhood of the desired fixed point, the region of attraction. We estimate the region of attraction to determine the moment of acting the controller, and this will reduce the tedious waiting time. However, to ensure zero steady state control error in the presence of uncertainty, the robustness of regulation under integral control is intuitively developed.
Finally, we present the differential geometric method to feedback linearization, which allows us to characterize the class of feedback linearizable system by geometric condition. The approach differs entirely from conventional linearization as the Jacobian linearization, in that feedback linearization is achieved by exact state transformations and feedback, rather than by linear approximation of the dynamics. As map functions are discrete, an approach for controlling discrete-time chaotic systems by feedback linearization is proposed. This method can not only stabilize unstable periodic orbits embedded in a strange attractor, but also can be applied even if the real trajectory is far from the target one. A Hénon map with different operation conditions is implemented to demonstrate the feasibility of the proposed method.
Contents
Page
Abstract……………………………………………………………………………..I
List of Figures……………………………………………………………………….III
Nomenclature……………………………………………………………………..…VI
Contents………………………………………………………………...…….……VIII
1. Introduction…………….………………………………………………………..1
1.1 Problem under investigation……………………...……….………………….1
1.2 Literature survey………..…………………………………………………….2
1.2.1 Variable structure control………..……………………….…………….…3
1.2.2 Pole placement method………….……………………….…………….…4
1.2.3 Region of attraction of a desired fixed point…………..…………………...4
1.2.4 Integral control……………………………………………………………...5
1.2.3 Differential geometry method………….……………….…………….…5
1.3 Organization of the dissertation……………………………………………….6
2. A robust controller for chaotic systems under external force excitation ……………..………………………….….…………………………9
2.1 Sliding mode control design…………………………..…………………...…..9
2.2 Numerical simulation…………………………………………………..……....14
3.Fast Convergence Control Approach for High Dimensional Chaotic Systems…………………………………………………………………………….18
3.1 Criteria for control……………………………………………………………...18
3.2 Numerical example……………………………………………………………..23
4.A Universal Control Approach for High Dimensional Chaotic Systems………………………………………………………………..…………..31
4.1 Controller design……………………………………………………………….31
4.2 Controllability…………….………………………………….….………….….33
4.3 Region of attraction of a desired fixed point……………..…...……………….36
4.4 Numerical examples…………………………………………………………..37
4.5 Robust Control…………………………………………………………….……40
5.Control of Discrete-Time Chaotic Systems via Feedback Linearization………………………….…………………………………………51
5.1 Differential Geometric Approach…………………………………………….50
5.2. Numerical examples…………………………………………………………..56
6. Discussion and Conclusion……………………………………………………...65
References……………………………..…………………………………………..69
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