藉由偕正寬鬆確保李雅普諾夫二次穩定

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姓名

劉明哲(Ming-zhe Liu) 查詢紙本館藏

畢業系所

機械工程學系

論文名稱

藉由偕正寬鬆確保李雅普諾夫二次穩定
(Copositive Relaxations Assuring Lyapunov Quadratic Stability)

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摘要(中)

主要分為三大部分。第一部分討論李雅普諾夫穩定度判斷，以及考慮波雅理論後，系統是否較容易求解(寬鬆性)；第二部分考慮性能指標 H_∞，也就是系統受雜訊干擾時的穩定度檢測條件，最後則以平方和寬鬆方法去達到完整解，使系統穩定度滿足充分條件，並以模擬結果加以驗證。
前兩部分所探討的為一般所熟悉的現有成果，採循序漸近的方式加以回顧。在較早的模糊控制文獻中，大部分研究都只著重於找出滿足二次穩定的共同李雅普諾夫函數，2000年左右由於寬鬆變數矩陣的概念出現，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[1]，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。另一方面，也提出了以芬斯勒定理為架構的穩定度條件加以比較，第二部分主要探討考慮H∞性能在一般及芬斯勒架構下的穩定度檢測條件。
第三部分，我們將一個考慮閉迴路的模糊系統轉換成偕正矩陣，並且以平方和分解去近似偕正矩陣的錐體，使變數相依的線性矩陣不等式達到完整解(非保守解)。經由系統化的挑選單項式，我們即可以偕正矩陣去建立一個偕正寬鬆家族，使系統達漸近穩定且收斂。最後的數值分析結果顯示，平方和寬鬆法與線性矩陣不等式寬鬆法皆有相同的寬鬆結果，但是值得注意的是，偕正寬鬆方法並不會因寬鬆階數的提高，而增加額外的寬鬆變數，故偕正寬鬆法有其優點，值得推廣。

摘要(英)

In this thesis, we convert a double fuzzy summation into cone of copositive matrices, proposing an SOS decomposition to approximate the cone of copositive matrices, leading to exact relaxation for fuzzy PD-LMIs. Based on suitable monomials which can be generated systematically, we show that one can work with copositive matrices to construct asymptotically exact copositive relaxation families with certificates of convergence. Numerical experiments show that both LMI and SOS relaxations reach the exact solutions, but the copositive relaxation demands NO slack variables when the level of relaxation increases.

關鍵字(中)

★ 線性矩陣不等式
★ 偕正矩陣
★ 齊次多項式
★ T-S模糊系統
★ 波雅理論
★ 二次寬鬆

關鍵字(英)

★ PD-LMIs
★ Quadratic relaxation
★ Homogeneous polynomials
★ Copositive matrix
★ LMI

論文目次

論文摘要 i
Abstract iii
誌謝 iv
圖目 viii
表目 ix
第一章簡介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............................................7
第二章系統架構與穩定度條件9
2.1 模糊系統架構........................................9
2.2 共同P檢測條件......................................10
2.2.1 純系統.........................................10
2.2.2 狀態回饋控制系統...............................11
2.3 Finsler架構........................................12
2.3.1 純系統.........................................12
2.3.2 狀態回饋控制系統...............................13
2.4 寬鬆性演變.........................................15
第三章波雅定理之代數應用18
3.1 波雅定理(P′olya's Theorem).........................18
3.2 矩陣型波雅定理(Matrix-valued P′olya's Theorem).....19
3.3 寬鬆變數矩陣.......................................21
第四章H_∞性能指標26
4.1 系統架構...........................................26
4.2 H_∞定義...........................................28
4.3 共同P檢測條件......................................30
4.3.1 純系統.........................................30
4.3.2 狀態回饋控制系統...............................34
4.3.3 Finsler架構....................................39
第五章偕正矩陣寬鬆方法41
5.1 SOS基本定義........................................41
5.2 共同P檢測條件......................................45
5.3 範例及LMI比較......................................46
5.4 LMI寬鬆家族........................................49
第六章電腦模擬52
6.1 解空間比較.........................................52
6.1.1 連續系統.......................................52
6.1.2 離散系統.......................................56
6.2 H_∞性能比較.......................................59
6.2.1 連續系統.......................................59
6.2.2 離散系統.......................................61
第七章結論與未來方向63
7.1 總結...............................................63
7.2 未來研究方向.......................................64
參考文獻 65
著作目錄 69

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指導教授

羅吉昌(Ji-Chang Lo)

審核日期

2010-6-19

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