以線性矩陣不等式及疊代線性矩陣不等式分析大型模糊系統的穩定度

近年來T-S 模糊模型(T-S fuzzy model) 被大量使用於處理非線性系統的控制穩定分析方法之一。此方法由Takagi和Sugeno兩位學者在1985年所提出，T-S模糊模型(T-S fuzzy model)能將非線性系統以IF-THEN的規則化成許多的線性系統來表示，以便於我們分析設計。至於在控制器方面的設計上Takagi和Sugeno兩位學者也提出了一種所謂平行分佈補償(Parallel Distributed Compensation)方法的概念去設計模糊控制器，其控制規則(control rule)所使用的前件部與歸屬函數和T-S模糊模型(T-S fuzzy model)所使用的前件部與歸屬函數是相同的。而模糊控制器的回授方式可使用狀態回授控制和輸出回授控制。我們根據李亞普諾(Lyapunov direct method)穩定準則來分析T-S模糊系統模型穩定條件。最後我們可以把所有穩定條件都將它轉換成線性矩陣不等式(Linear Matrix Inequalities)的問題，並使用Matlab LMI Toolbox工具箱方法去求解。在本論文中，我們基於典型T-S 模糊模型(T-S fuzzy model)與平行分佈補償(Parallel Distributed Compensation)方法為基礎去建構出大型T-S模糊系統模型，而以李亞普諾(Lyapunov direct method)準則為根據來分析大型T-S模糊系統模型穩定度，並將穩定條件利用Schur complement的技巧把線性矩陣不等式轉換成LMIs問題去探討。由於不容易以Matlab LMI Toolbox最佳化演算法方法去求解大型T-S模糊系統穩定條件線性矩陣不等式問題，於是我們提出了一個基於ILMIs (Iterative Linear Matrix Inequalities) 方法來分析大型T-S模糊系統穩定度的問題。ILMIs (Iterative Linear Matrix Inequalities)也是一種有效率近似得到大型T-S模糊系統穩定條件方法之一。我們相信本論文的研究成果，對未來分析非線性系統上有相當的助益。

關鍵字

線性矩陣不等式；大型模糊系統

並列摘要

Recently, the Takagi Sugeno (TS) fuzzy model have been a large number used in nonlinear control approaches and is one of way to analyze stabilization method. The method was proposed by Takagi and Sugeno in 1985. The T-S fuzzy dynamic model is described by fuzzy IF-Then rules in which the consequent parts represent local linear models. Once a fuzzy representation of a nonlinear system is described by if-then rules, the control problem then becomes to find a local linear/nonlinear compensator to achieve the desired objective. The controller design of the T-S fuzzy system is carried out via the so called parallel distributed compensation (PDC) approach. In the PDC design, each control rule is designed from the corresponding rule of a T-S fuzzy model. The designed fuzzy controller shares the same fuzzy sets with the fuzzy model in the premise parts. The fuzzy control rules have a linear state feedback law controllers or a linear output feedback law controllers in the consequent parts. We also can derive stabilization condition by Lyapunov direct method. Finally, stabilization condition can be converted nonlinear inequalities into LMIs and solved using Matlab LMI Toolbox convex optimization technique. In this thesis, we used T-S fuzzy model and parallel distributed compensation (PDC) methods to construct large-scale T-S fuzzy model. And large-scale T-S fuzzy model is based on Lyapunov direct method criterion to analyze stabilization condition. Due to we can’t easily use Matlab LMI Toolbox convex optimization algorithm to solve large-scale T-S fuzzy model nonconvex nonlinear matrix inequalities problem. So we proposed a new iterative linear matrix inequalities method to solve nonconvex nonlinear matrix inequalities problem. Iterative linear matrix inequalities is also one of effectiveness way method to approaches large-scale T-S fuzzy system. It is believed that the results of our study in this thesis will be help to analyze nonlinear systems in the future.

並列關鍵字

TS fuzzy system ； linear matrix inequality

參考文獻

[38]張智星﹐MATLAB程式設計與應用，清蔚科技股份有限公司1999.

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[5] Hung T. Nguyen, and Nadipuram R. Prasad, Fuzzy Modeling and Control: Selected Works of M. Sugeno, North-Holland. 1999.

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以線性矩陣不等式及疊代線性矩陣不等式分析大型模糊系統的穩定度

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