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研究生:萬嘉仁
研究生(外文):Jia-ren Wan
論文名稱:寬鬆耗散性模糊控制-波雅定理
論文名稱(外文):Relaxed, dissipative fuzzy control - Polya theorem
指導教授:羅吉昌
指導教授(外文):Ji-chang Lo
學位類別:碩士
校院名稱:國立中央大學
系所名稱:機械工程研究所
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:2008
畢業學年度:96
語文別:中文
論文頁數:100
中文關鍵詞:線性矩陣不等式有界非線性輸入耗散性控制波雅定理寬鬆環境
外文關鍵詞:Dissipative controlT-S modelPolya''''s theoremRelaxed conditionLinear matrix inequality
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1. 使用Takagi-Sugeno 模糊模型和耗散性控制(dissipative control) 解決系統輸入受到有界非線性項影響的問題。
2. 使用波雅定理(P′olya''s theorem) 的代數性質來建立一組線性矩陣不等式, 此組線性矩陣不等式可求得出一二次穩定之不保守解, 進而漸進至系統穩定之必要條件。
3. 結合波雅定理的代數性質與寬鬆矩陣變數(slack matrix variables) 方法來建立一寬鬆環境, 藉由此寬鬆環境可提升工具程式求解的性能(降低耗時、擴大解空間)。
第一部分,Takagi-Sugeno 模糊模型可完整地轉換原始的非線性系統, 並且可藉由Lyapunov 定理將系統穩定的檢測條件轉為線性矩陣不等式, 此一簡單且兼具數理基礎和系統化步驟的特色成為本篇文章使用的主要原因。承上, 我們先建立一具耗散性的模糊系統, 接著導入耗散性控制可藉由選取補充率(supply rate) 的特點來處理各種性能問題, 其中我們專注於系統輸入受到有界非線性項影響的處理方法, 首先對於一有界區域非線性項(sector-bounded nonlinearilities) 的輸入拆解成線性區和擾動區, 再設計一分散平行補償控制器(Parallel distributed compensation) 利用狀態回饋控制來控制系統。此外若希望加強系統對於有界非線性輸入的強健性, 我們必須要擴大對有界區域非線性輸入項的容忍, 但這意味著求解困難, 此一問題在第二部分獲得一解決方案。
第二部分, 在最近幾年的模糊控制文獻中, 大部分研究主要著重於找出一個共同P矩陣來滿足二次李亞普諾夫函數(quadratic Lyapunov function), 此一方法為充分但非必要條件, 且求解較保守(conservatism) 。在此我們使用了波雅定理(P′olya''s theorem) 的代數性質來建立一組線性矩陣不等式, 此組線性矩陣不等式可求得出一二次穩定之不保守解(less conservative solution), 進而漸進至系統穩定之必要條件, 在數理方面證明了雙向的充要條件, 以工程的角度則可設計出使系統性能變好的控制器, 經模擬顯示, 由於軟硬體的限制使得程式無法真正得到最佳(或較佳) 的解(例如: 遲緩過程(slow progress), 迴圈上限, 耗時), 故而我們在第三部分提供一寬鬆環境來處理這個問題。
第三部分, 在閱讀文獻[1]-[5] 後, 我們將加入寬鬆矩陣變數的概念跟波雅定理做結合, 並以模擬的結果展現此一寬鬆環境可以加速求解過程, 並且給予的d值小於波雅定理所建議的d值卻可得到更大的解空間, 大大地增強以波雅定理設計控制器的可行性。
In this thesis, we propose a general quadratic dissipative state feedback control method to solve a stabilization problem for fuzzy system with sector-bounded type nonlinearities at the input.
The problem covers the bounded real, positive real and sector-bounded performance as a special case by choosing the corresponding quadratic supply rate.
Moreover, we also prove necessary and sufficient conditions to state feedback controllers ensuring quadratic stability for Takagi-Sugeno fuzzy systems in theory.
But our main objective is to generate a family of linear matrix inequalities based on an extension of Polya theorem (a.k.a. Matrix-valued Polya theorem).
The proposed conditions are stated as progressively less conservative sets of linear matrix inequalities,
allowing us to obtain a solution for the quadratic stabilizability problem whenever a solution exists.
At last, an additional relaxed condition is also provided, relying on the use of slack matrix variables.
All proposed methods will be shown via theoretical analysis and numerical simulations.
論文摘要 i
Abstract iv
誌謝 v
圖目 x
第一章 簡介 1
1.1 文獻回顧 1
1.2 研究動機 3
1.3 論文結構 4
1.4 符號標記 5
1.5 預備定理 7
1.6 耗散性之物理意義 8
第一部份:耗散性控制(Dissipativecontrol) 13
第二章 系統架構與耗散性檢測條件 13
2.1 系統架構 13
2.1.1 廣義非線性系統 13
2.1.2 非線性模糊系統 15
2.2 檢測耗散性條件 16
第三章 有界非線性輸入控制器之設計 23
3.1 有界非線性輸入系統 23
3.2 狀態回饋控制器 25
第四章 電腦模擬一 32
4.1 連續系統 32
4.1.1 系統模型 32
4.1.2 求解 34
4.2 離散系統 40
4.2.1 系統模型 40
4.2.2 求解 43
第二部份:波雅定理之代數應用 50
第五章 模糊閉迴路系統之充要條件 50
5.1 波雅定理(P′olya’sTheorem) 50
5.2 矩陣波雅定理(Matrix-valued P′olya''s Theorem) 52
5.3 範例及寬鬆性 56
第六章 電腦模擬二 58
6.1 解空間 58
6.2 倒單擺系統 63
6.2.1 系統描述 63
6.2.2 求解 63
第三部份:具鬆弛矩陣變數之寬鬆環境 68
第七章 波雅定理之寬鬆環境 69
7.1 鬆弛矩陣變數 69
7.2 範例 76
第八章 電腦模擬三 82
8.1 解空間比較 82
8.1.1 連續系統 82
8.1.2 離散系統 86
8.2 耗時比較 90
第九章 結論與未來研究方向 91
9.1 總結 91
9.2 未來研究方向 92
參考文獻 94
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