本論文主要研究離散 Takagi-Sugeno (T-S) 模糊控制系統的非二次
(non-quadratic) 寬鬆穩定條件分析，透過波雅定理 (Pólya Theorem)
的代數性質結合寬鬆矩陣變數 (slack matrix variables) 的激發強度排
列來建立一組寬鬆的線性矩陣不等式 (LMI)，由於非二次穩定的分
析加上波雅定理 (Pólya Theorem) 以及寬鬆矩陣變數 (slack matrix
variables); 使得系統求解的保守性大大的降低。
此外，本文另一貢獻在於寬鬆矩陣變數的架構與以往文獻不同;
即是將波雅定理結合寬鬆矩陣變數所產生的線性矩陣不等式以多項式
矩陣型態來表示，透過多項式矩陣型態之特性，同階數的激發強度所
對應的元素可放在矩陣對角線上或同階數之非對角線上做變化，如此
一來可使求解的保守度進一步降低。最後舉幾個例子來呈現本文所提
出的理論之優點。

In this thesis, we investigate a non-quadratic stabilization problem of discrete-time Takagi- Sugeno (T-S) fuzzy systems by means of homogeneous polynomially parameter-dependent (HPPD) functions, exploiting the algebraic property of Pólya to construct a family of matrixvalued HPPD functions that releases conservatism, assuring existence to non-quadratic Lyapunov functions. The obtained stabilization conditions, characterized by parameter-dependent LMIs (PD-LMIs), are further relaxed by using the proposed right-hand side slackness. A solution technique is proposed through the SOS decomposition of positive semidefinite matrixvalued polynomials. That is, we transform the PD-LMIs based on non-quadratic Lyapunov method into SOS matrix polynomials and then apply matrix RHS relaxation with semi-definite programming searching for a feasible solution to PD-LMIs. Lastly, numerical experiments to illustrate the advantage of RHS relaxation, being less conservative and effective, are provided.

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