以LMI為基礎之模糊與非線性系統的精確與近似控制

本論文分別對模糊系統與非線性系統提出以LMI為基礎之精確與近似控制理論，並用以達成各種常見之控制目標。首先，將非線性系統以數個具線性系統為後件部的模糊子系統表示之，再融合所有子系統精確表示原非線性系統。針對此精確模糊受控系統，設計具相同前件部之模糊狀態迴授控制器，並利用穩定性分析方法找出閉迴路系統穩定之充分條件。將充分條件轉換為LMI條件後，控制器達成目的所需之控制增益等等，可由電腦以數值方式運算出。本論文所達控制目標有(一) 穩定；(二) 調節；(三) 輸出追蹤/調節；(四) 非線性模式追蹤；(五) 輸出迴授追蹤控制。所解決之問題如(一) 受控系統與參考模式系統之系統矩陣間有參數差異；(二) 有無法量測狀態；(三) 利用強健法則壓制干擾。另一方面，本論文也深入研究建立模糊模式過程對LMI有解與否之影響，並利用Kharitonov’s定理找出系統穩定之充分必要條件。針對精確表示法可能造成子系統不可控之問題，提出一近似模式化與控制法則解決。論文最後，提出將非線性系統模式化為具與狀態相關參數之似線性系統。針對此一新表示法，提出一動態輸出迴授控制器。再者，利用線性輸出調節器觀念，達成非線性系統之輸出追蹤與調節。在數值模擬方面，以著名之連續與離散混沌系統、非完整約束跳動機械人、非線性質量彈簧系統、與直流對直流降壓型轉換器做為驗證理論所推結果。在實作方面，以DSP為基礎做混信號量測再次驗證與理論結果相符。

關鍵字

控制；非線性系統；模糊系統；線性矩陣不等式

並列摘要

n this dissertation, various control objectives for general nonlinear systems are achieved in a unified manner. First, the nonlinear system is represented by several fuzzy subsystems where the consequent part are linear dynamical systems. Then by blending these rules, we exactly represent the original nonlinear system. Following the modeling stage, a fuzzy state feedback controller for each linear subsystem is designed with same premise variables as that of the fuzzy plant model representation. Using Lyapunov's direct method, the stability analysis is carried out on the overall closed-loop system. The sufficient conditions arising from the stability analysis is then formulated into linear matrix inequalities (LMIs). Finally, using powerful numerical toolboxes, the linear matrix inequalities are solved and controller gains obtained. In the process of controller synthesis, if only partial states are known, an observer is designed to estimate immeasurable states. The control objectives achieved, in this dissertation, are i) stabilization; ii) regulation; iii) output tracking/regulation iv) nonlinear model following; and v) output feedback tracking control. The problems coped with are i) mismatched parameters existing between system matrices of plant and reference model; ii) immeasurable states; and iii) attenuation of disturbances using robust criterions. Another main issue in this dissertation is the discussion of fuzzy modeling effects on LMI feasibility using Kharitonov's method. Interval polynomials provide insight to the sufficient and necessary conditions of the overall system. Proposing an approximate modeling and control method, we are able to control systems which would have had uncontrollable modes if using the exact modeling representation. At the end of the dissertation, we propose a methodology to represent nonlinear systems into linear-like systems with state-dependent parameters. Once the new representation is obtained, we propose a dynamic output feedback controller for stabilization and adopt linear regulator theory for robust output tracking/regulation. For numerical simulations and DSP-based experiments, we use well-known continuous-time and discrete-time chaotic systems, a hopping robot, and DC-DC PWM buck converter as examples to further verify the theoretical derivations.

並列關鍵字

nonlinear systems ； Takagi-Sugeno ； fuzzy systems ； LMI

參考文獻

[1] Takagi, T. and Sugeno, M., [1985] "Fuzzy identification of systems and its applications to modeling and control," IEEE Trans. Syst., Man, Cybern. vol. 15, no. 1, pp. 116-132.

[2] Tanaka, K., and Sugeno, M., [1992] "Stability analysis and design of fuzzy control systems," Fuzzy Sets Syst. no. 45, pp. 135-156.

[3] Tanaka, K., Ikeda, T. and Wang, H. O., [1998] "A unified approach to controlling chaos via an LMI-based fuzzy control system design," IEEE Trans. Circuits Syst. I, vol. 45, no. 10, pp. 1021-1040.

[4] Tanaka, K., Ikeda, T. and Wang, H. O.,[1998] "Fuzzy regulators and fuzzy observers: Relaxed stability conditions and LMI-based designs," IEEE Trans. Fuzzy Syst. vol. 6, no. 2, pp. 250-265.

[5] Tanaka, K., and Wang, H. O., [2001], Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach}, John Wiley and Sons, Canada.

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以LMI為基礎之模糊與非線性系統的精確與近似控制

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