本篇論文主要研究連續時間模糊(fuzzy)系統及離散時間模糊(fuzzy)系統的二次穩定寬鬆條件，我們利用波雅定理(P''olya Theorem)的代數性質加上寬鬆矩陣變數(slack matrix variables)，再利用激發強度為基礎之多項式排列的控制器與觀測器來做控制與估測之相關分析，利用這些條件來建立一組寬鬆的線性矩陣不等式(LMI)，因為上述的這些條件已可以將求解保守性降低不少，但本篇論文還有一個很重大的貢獻，即是將以往加入寬鬆矩陣變數與波雅定理的線性矩陣不等式以多項式矩陣型態來表示，在判斷式子中因加入了寬鬆矩陣變數，如此可應用多項式矩陣型態之特性，將同階數的元素放在矩陣對角線上或同階數之非對角線上作變化，這將會使判斷式保守度更加降低，多項式矩陣型態可由第二章範例中了解其意義，這些改善將會以例子來證明了解其優點。

In this thesis, we investigate quadratic relaxation for continuous-time and discreate-time fuzzy systems, which are characterized by parameter-dependent LMIs (PD-LMIs), comprising the algebric property of P''olya Theorem to construct a family of finite-dimensional LMI relaxations with right-hand side slack matrices and matrix-values HPPD function of degree g that release conservatism. Lastly, numerical experiments to illustrate the advantage of relaxations, being less conservative and effective, are provided.

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