主要分為三大部分。第一部份介紹李亞普諾夫(Lyapunov~stability)判斷以及數學推導，再來介紹寬鬆變數以及波雅Polya定理的結合，系統是不是更容易求解(寬鬆性)，第三部份則是加入了前件部不同的控制器來做討論，最後以平方和寬鬆方法(sum of squares， SOS)為主，線性矩陣不等式(Linear matrix inequalities， LMI)為輔作為判斷工具。第一部份所探討的為一般所熟悉的現有成果。在較早的 Takagi-Sugeno (T-S) 模糊控制文獻中，大部分研究都只著重於找出滿足二次穩定 (quadratically stable) 的共同李雅普諾夫函數(common/single/global $P$)，2000年左右由於寬鬆變數矩陣(slack matrix) 的概念出現，加速了求解過程；2005年波雅理論的發展已趨成熟，當隨著波雅冪次(P卅’{o}lya''s exponent)增加到足夠大時，可使模糊系統穩定度滿足充分條件，對寬鬆性有很大幫助；在2008年時，萬嘉仁學長的研究中，將寬鬆變數矩陣概念及波雅理論加以結合，模擬結果顯示所需的波雅冪次小於波雅理論所建議的值，並且展現了更大的解空間，但隨著波雅冪次的增加，寬鬆變數量會呈指數遞增，造成電腦運算上的負擔，因此，提出了平方和寬鬆法以解決變數上的問題，並探討其寬鬆性。但系統矩陣中始終都是常數，但實際範例中可能並非如此，故我們在系統矩陣中加入了$x$，故為此篇論文的主軸。再來是前件部不對稱的部份，因為前件部的不同，所以歸屬函數也會不同，故我們做了一個轉換，使得兩個前件部有所關聯，再進一步排成大矩陣的形式，放入電腦求解。

In this thesis, three topics are addressed First, we investigate a general control problem via the Lyapunov quadratic stability, and the system matrix elements contains x; second we investigate slack variables and the Polya''s theorem; third we investigate combinational of different membership functions(imperfect matching) to tackle the stability problem, in the final use LMI(Linear matrix inequalities)-toolbox and SOS(sum of squares)-toolbox to slove

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