離散型T-S模糊系統的穩定條件放寬之研究

Relaxed Stability Criteria for T-S Fuzzy Discrete Systems

孫崇訓(Chung-Hsun Sun)

指導教授 ： 王文俊

國立中央大學/資訊電機學院/電機工程學系/博士(2006年)

摘要

基於李亞普諾夫穩定定理（Lyapunov stability criterion），保證離散型T-S模糊系統穩定的充分條件，就是找到一個共同P矩陣滿足系統中所有李亞普諾夫不等式（Lyapunov inequality）。一般而言，這個共同P矩陣可藉由線性矩陣不等式（LMI）的求解法所求得。然而，當模糊系統中的規則數過多時，即使是使用線性矩陣不等式的軟體工具，也不一定能求得這個共同P矩陣。近來，為了解決這個問題，許多學者改以片段連續型李亞普諾夫二次式（piecewise quadratic Lyapunov function）和權重相依型李亞普諾夫二次式（weighting dependent Lyapunov function）來推導離散型T-S模糊系統的穩定條件。理論上來說，以這兩種李亞普諾夫二次式所推導出的穩定條件會較為寬鬆，但是這兩種李亞普諾夫二次式卻會導至較多的李亞普諾夫不等式。所以，在本篇論文中，我們藉由分析模糊系統中的狀態位置資訊來減少李亞普諾夫不等式的數量，以達到穩定條件放寬之目的。
在第三章和第四章中，藉由觸發規則群的觀念和狀態間距概念的分析，我們可獲知以前學者所提出的穩定條件中，某些李亞普諾夫不等式其實是不必要的，因此若能扣除這些多餘的李亞普諾夫不等式，則可得到較寬鬆的穩定條件。在第五章中，除了之前提到的兩個概念外，又引入頂點表示法（vertex expression）來描述觸發規則群的空間大小，這使我們能夠找出更多不需要的李亞普諾夫不等式，更放寬了原有的穩定條件。此外，在第三、四、五章的最後，藉由數值的例子與定理的比較，可證明我們所提出的理論確實具有放寬穩定條件的效能。

關鍵字

T-S 模糊系統 ； 穩定性 ； 條件放寬

並列摘要

It is well known that the first stability criterion for Takagi-Sugeno (T-S) fuzzy discrete system is derived from the common quadratic Lyapunov function. That is to find a common matrix P to satisfy all Lyapunov inequalities. Then the stability of T-S fuzzy discrete systems can be guaranteed. In general, the common matrix can be found by means of linear matrix inequalities (LMI) method. However, if the number of rules of a fuzzy system is large, the common matrix P may not exist or may not be found even using LMI. Recently, the piecewise quadratic Lyapunov function and the weighting dependent Lyapunov function are employed instead of the common quadratic Lyapunov function. It is believed that the above two Lyapunov functions derive more relaxed stability criteria than the common quadratic Lyapunov function does. However they induce more Lyapunov inequalities to be satisfied. In this dissertation, more relaxed stability criteria for T-S fuzzy discrete systems are proposed. They are based on the piecewise quadratic Lyapunov function and the weighting dependent Lyapunov function respectively. This dissertation combines the ideas of group-fired rules, the width of states step and the vertex expression together, so that the relaxed stability criteria need to satisfy fewer Lyapunov inequalities. In each chapter, some numerical examples and comparisons are presented to show the effectiveness of this work.

並列關鍵字

stability ； relaxed conditions ； T-S fuzzy system

參考文獻

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