連續模糊系統平方和穩定性分析-尤拉齊次多項式定理

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林峯億(Feng-yi Lin) 查詢紙本館藏

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機械工程學系

論文名稱

連續模糊系統平方和穩定性分析-尤拉齊次多項式定理
(Stabilization Analysis of Polynomial Fuzzy Systems via SOS - Euler’s Theorem for Homogeneous Functions)

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★ H∞模糊系統控制-多凸面法	★ H∞模糊系統控制-寬鬆變數法
★ 時間延遲 T-S 模糊系統之強健 H2/H(Infinity) 控制與估測	★ 寬鬆耗散性模糊控制-波雅定理

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摘要(中)

本論文主要研究連續模糊控制系統的非二次穩定(non-quadratic
stability) 條件，關於擴展狀態決定於高階的非二次李亞普諾夫函數，其函數形式是V(x) = 1/2(x^TP^(-1)(x)x)，其中條件P^(-1)(x) > 0取決於P(x)x 是一正定的梯度向量(gradient vector)。遺憾的是，此梯度向量P(x)x 是一非凸面體(nonconvex) 的問題。因此可控制的模糊系統之穩定性檢測條件，需要使用尤拉齊次多項式定理，並使用其定理之齊次性質與波雅定理(Pólya theorem) 之代數性質，以平方和方法(sum of squares) 去檢驗非凸面體問題，使得其模擬系統之空間解更寬鬆。最後，模擬其多項式模糊系統，表現出本論文提出之方法是有效的。

摘要(英)

Extension of the state dependent Riccati inequalities to non-quadratic Lyapunov function of the form V(x) = 1/2(x^TP^(-1)(x)x), with P^(-1)(x) > 0 requires that P(x)x is a gradient of positive definite function. Unfortunately, the test of P^(-1)(x)x is nonconvex problem. Thus this thesis studies stabilization problems of the polynomial fuzzy systems via homogeneous Lyapunov functions exploiting the Euler’s homogeneity property and algebraic property of Pólya to construct a family of SOS polynomials that solves the nonconvexity problem and releases conservatism as well. Lastly, examples of polynomial fuzzy systems are demonstrated to show the proposed method being effective and effective.

關鍵字(中)

★ Takagi-Sugeno模糊系統
★ 參數相依齊次多項式
★ 尤拉齊次多項式定理
★ 平方和
★ 非二次穩定

關鍵字(英)

論文目次

一、簡介............. 1
1.1 文獻回顧 . . . . . . . . 1
1.2 研究動機 . . . . . . . . 2
1.3 論文結構 . . . . . . . . 3
1.4 符號標記 . . . . . . . . 4
1.5 預備定理 . . . . . . . . 6
二、連續系統架構與穩定度條件 ........... 7
2.1 連續系統架構 . . . . . . . 7
2.2 連續模糊閉迴路系統之穩定檢測條件（一） . . . 8
2.3 尤拉齊次多項式定理 . . . . . . . 11
2.4 連續模糊閉迴路系統之穩定檢測條件（二） . . . 12
三、穩定性檢測條件之寬鬆性.......... 17
3.1 波雅定理 . . . . . . . . 17
3.2 增加激發強度 µ 之冪次 . . . . . . 18
3.3 平方和寬鬆法 . . . . . . . 19
3.4 平方和寬鬆法穩定性檢測條件（一） . . . 22
3.5 平方和寬鬆法穩定性檢測條件（二） . . . 23
四、電腦模擬................ 27
4.1 模擬題目（一） . . . . . . . . 27
4.2 模擬題目（二） . . . . . . . . 31
4.3 模擬題目（三） . . . . . . . . 35
4.4 模擬題目（四） . . . . . . . . 38
4.5 模擬題目（五） . . . . . . . . 43
4.6 模擬題目（六） . . . . . . . . 48
4.7 模擬題目（七） . . . . . . . . 54
五、結論與未來方向.......... 59
5.1 結論 . . . . . . . . . . 59
5.2 未來研究方向 . . . . . . . 60
附錄一 ................ 61
A.1 結合波雅定理與寬鬆變數矩陣 . . . . 61
A.2 平方和寬鬆法穩定性檢測條件（二） . . . 64
參考文獻.................. 69

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指導教授

羅吉昌(Ji-chang Lo)

審核日期

2013-7-16

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