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系統識別號 U0002-0508201016001100
中文論文名稱 以座標轉換為基礎的多目標控制器合成方法
英文論文名稱 Coordinate- Transformation-based Multi-objective Controller Synthesis
校院名稱 淡江大學
系所名稱(中) 電機工程學系碩士班
系所名稱(英) Department of Electrical Engineering
學年度 98
學期 2
出版年 99
研究生中文姓名 朱彥碩
研究生英文姓名 Yen-Shou Chu
學號 697460136
學位類別 碩士
語文別 中文
口試日期 2010-07-14
論文頁數 162頁
口試委員 指導教授-周永山
委員-容志輝
委員-吳政郎
中文關鍵字 座標轉換  H2/H∞ 控制  降階  多目標  線性矩陣不等式 
英文關鍵字 coordinate transformation  H2/H∞ control  reduced order  multi-objective  linear matrix inequality 
學科別分類 學科別應用科學電機及電子
中文摘要 本論文主要研究H2/H∞多目標控制器合成問題。1997年Scherer等人所提出的連續時域LMI解法是為人稱道的有效方法,但其受限於單一的李亞普諾夫變數,有所謂的共同P問題。目前的最新技術,在離散時域有學者Oliveira等人,引進額外的變數,即所謂的參數相關李亞普諾夫函數方法,證實可減少多目標控制問題LMI解法的保守性,但並沒有完全脫離共同P問題的窘境。另外也有演化計算的方法,利用大量的計算,來逼近全域最佳解。此方法雖然沒有上述共同P問題,但是必須耗費大量的運算時間與成本。
本論文提出一種新型方法,稱為「以座標轉換為基礎的多目標控制器合成方法。此方法無上述共同變數的限制,而與一般熟知的雙線性矩陣不等式(Bilinear Matrix Inequality, BMI)解法相比較,求解新控制器時,有一額外的自由變數可協同求解,故可搜尋其未能搜尋的解空間。當以這些現存方法之最佳解作為本文方法之初始控制器,此方法可作為提昇現有方法性能之用。另外,本論文提出的座標轉換技術,在很容易滿足的假設條件下,可將BMI問題轉換為含特殊格式座標轉換的“LMI問題”,可作為各種系統性能的新型有解條件。
本論文提供數個數值例子,包括連續時域H2/H∞多目標控制與H∞降階控制、離散時域 多目標控制問題。模擬結果驗證了本文方法使用不同初始控制器的影響、效能,並與前述方法作完整的比較。
英文摘要 This thesis revisits the H2/H∞ multi-objective control problem. The continuous-time LMI method developed by Scherer at al. in 1997 is well recognized as a powerful tool for solving these sorts of problems. However, this approach suffers from the constraint of a single common Lyapunov function for the multiple performance specifications, known as the common P problem. One of the state-of-the-art techniques includes the method developed by Oliveira et al.,known as the Parameter-Dependent Lyapunov Function method. The method introduces a slack variable into the condition which has been proven to be useful for reducing conservatism. But it is known that the problem remains. Alternatively, there is an approach using evolution algorithm to search for a good solution at the expense of a large amount of computation cost, though no aforementioned problem exists.
The thesis proposes a new method, named Coordinate-Transformation-based Multi-objective Controller Synthesis (CTMCS), which is free of the aforementioned common variable constraint. When compared with the well-known BMI method, an additional free variable exists in the step of searching for new controllers. This degree of freedom is useful for exploring the solution space which has never been touched by the aforementioned LMI methods. As a result, the proposed method can serve as a performance-enhanced approach when the solutions of the aforementioned methods are used as initial controllers.

In addition, this thesis presents a coordinate-transformation technique which, under mild assumption, transforms a BMI problem into a LMI problem. Consequently, the resulting conditions serve as new solvability conditions for the various system performance analysis problems.
In this thesis several numerical examples involving the continuous-time H2/H∞ control problem, H∞ educed-order controller design, and the discrete-time control problem are provided for illustration. A thorough comparison with the aforementioned methods is made.
論文目次 目錄

中文摘要............................................................................................................I
英文摘要..........................................................................................................II
目錄.................................................................................................................IV
圖目錄.............................................................................................................VI
表目錄...............................................................................................................X
第一章 緒論.....................................................................................................1
第二章 背景知識、問題敘述與重要解法......................................................5
2.1 符號與重要引理..................................................................................5
2.2 連續時域的強健控制問題…............................................................10
2.2.1 連續時域的H2與H∞性能分析式..........................................13
2.2.2 連續時域的H2與H∞性能式合成式......................................14
2.3 離散時域的強健控制問題…............................................................16
2.3.1 離散時域的H2與H∞性能分析式..........................................18
2.3.2 離散時域的H2與H∞性能合成式..........................................22
2.4 H2/H∞動態輸出回授控制問題與解法..........................................30
2.4.1 演化法解法..............................................................................31
2.4.2 LMI解法..................................................................................35
第三章 連續時域的多目標控制器設計.......................................................39
3.1 問題敘述............................................................................................41
3.2 連續時域CTMCS的核心理論.........................................................42
3.3 以座標轉換為基礎的H2/H∞控制器設計法..................................51
3.3.1 CTMCS-1方法......................................................................58
3.3.2 CTMCS-2方法......................................................................60
3.4 連續時域的H∞降階控制器設計......................................................69
3.5 數值例子............................................................................................72
第四章 離散時域的多目標控制器設計.....................................................129
4.1 前言..................................................................................................129
4.2 H2/H∞多目標控制問題與CTMCS的核心理論........................130
4.3 離散時域CTMCS的設計步驟.......................................................137
4.4 數值例子..........................................................................................140
第五章 結論...............................................................................................159
參考文獻.......................................................................................................161



圖目錄

圖2.1標準P-K架構.......................................................................................10
圖2.2 P-K架構圖............................................................................................30
圖2.3演化法於H2/H∞控制問題之解..........................................................31
圖2.4演化法階級比較圖...............................................................................32
圖2.5演化法圖形中最佳控制器的位置.......................................................33
圖2.6 最佳單目標控制器在多目標圖形中的位置......................................34
圖3.1 CTMCS演算法流程圖.........................................................................57
圖3.2 混合式H2/H∞控制問題解空間.........................................................58
圖3.3 CTMCS-1受限解空間與實際解空間的比較......................................59
圖3.4 CTMCS-1方法在 較大時的搜尋範圍.............................................60
圖3.5 CTMCS-2方法的搜尋方式及範圍......................................................61
圖3.6拋物線v2 - mr = copti,i=1,2;(實線i=1;虛線i=2。 m>0,copt1>copt2>0)............................................................................................62
圖3.7 拋物線v2 - mr = copti,i=1,2;(實線i=1,虛線i=2;copt>0,m1>m2)..........................................................................................................63
圖3.8 CTMCS-2合成控制器的跳躍示意圖..................................................64圖3.9 CTMCS-2合成控制器的跳躍情形(m = 1) ........................................64
圖3.10 CTMCS-2解空間之極限....................................................................65
圖3.11 CTMCS-2求解的極限........................................................................66
圖3.12虛擬H∞範數邊界之作用...................................................................67
圖3.13 mincx指令超出解空間使程式中斷的情形......................................68
圖3.14 H∞降階控制性能提昇設計流程圖..................................................71
圖3.15 例3.1問題之解空間..........................................................................73
圖3.16 例3.1問題,r∞ = 0.65時之解空間....................................................76
圖3.17 r∞ = 0.65,初始控制器為IC-1,方法CTMCS-1的結果..................80
圖3.18 r∞ = 0.65,初始控制器為IC-1,方法CTMCS-1的軌跡圖..............81
圖3.19 圖3.18方形區域放大圖....................................................................82
圖3.20 r∞ = 0.65,初始控制器為IC-1,方法CTMCS-2的結果..................85
圖3.21 r∞ = 0.65,初始控制器為IC-1,CTMCS - 2的軌跡圖.....................86
圖3.22 圖3.21長方形區域放大圖................................................................87
圖3.23 r∞ = 0.65,初始控制器為IC-2,方法CTMCS-1的結果..................91
圖3.24 r∞ = 0.65,初始控制器為IC-2,方法CTMCS-1的軌跡圖..............92
圖3.25 圖3.24長方形區域放大圖................................................................93
圖3.26 r∞ = 0.65,初始控制器為IC-2,方法CTMCS-2的結果..................96
圖3.27 r∞ = 0.65,初始控制器為IC-2,方法CTMCS-2的軌跡圖..............97
圖3.28 圖3.27長方形區域放大圖................................................................98
圖3.29 r∞ = 0.65,初始控制器為IC-2,可行解方法的結果......................101
圖3.30 r∞ = 0.65,初始控制器為IC-2,可行解方法的軌跡圖..................102
圖3.31圖3.30長方形區域放大圖...............................................................103
圖3.32 不同r∞值設定.................................................................................105
圖3.33 不同H∞範數邊界r∞下,Matlab指令hinfmix造成的H∞性能與r∞的差距...........................................................................................................106
圖3.34 r∞ = 0.75,初始控制器為hinfmix,方法CTMCS-2的結果...........109
圖3.35 r∞ = 0.75,初始控制器為IC-1,方法CTMCS - 2的軌跡圖..........110
圖3.36 圖3.35長方形區域放大圖..............................................................111
圖3.37 r∞ = 1,初始控制器為IC-1,方法CTMCS-2的結果......................114
圖3.38 r∞ = 1,初始控制器為IC-1,CTMCS - 2的軌跡圖........................115
圖3.39 圖3.38方形區域放大圖..................................................................116
圖3.40 ||Tz2w2||2^2 + ||Tz∞w∞||∞^2最小化問題,hinfmix、演化法(EA)、CTMCS-WSS的結果.......................................................................................................... 123
圖3.41 圖3.40的局部放大圖......................................................................124
圖3.42 CTMCS-H∞-R方法的遞迴次數與相對效能 .......................127
圖4.1 離散時域CTMCS演算法流程圖.....................................................139
圖4.2 4.4節例子的Pareto Front 示意圖.....................................................142
圖4.3 4.4節例子的Pareto optimal front......................................................143
圖4.4 r∞ = 32.38,初始控制器IC-D1,CTMCS-2與[2]、改良[2]比較之圖形...............................................................................................................147
圖4.5 r∞ = 32.38,初始控制器IC-D2,CTMCS-2與[2]比較之圖形.........150
圖4.6 r∞ = 15,初始控制器IC-D1,CTMCS與改良[2]比較之圖形..........154
圖4.7 r∞ = 15,初始控制器IC-D3,CTMCS-2與改良[2]比較之圖形......157



















表目錄

表3.1 演化法的參數.......................................................................................74
表3.2 r∞ = 0.65,初始控制器為IC-1,方法CTMCS-1之結果....................78
表3.3 r∞ = 0.65,初始控制器為IC-1,方法CTMCS-2之結果....................83
表3.4 r∞ = 0.65,初始控制器為IC-2,方法CTMCS-1的結果....................89
表3.5 r∞ = 0.65,初始控制器為IC-2,方法CTMCS-2的結果....................94
表3.6 r∞ = 0.65,初始控制器為IC-2,可行解方法的結果..........................99
表3.7 初始控制器IC-1/IC-2,不同CTMCS方法相對於hinfmix指令的改良效果...........................................................................................................104
表3.8 Matlab指令hinfmix 造成的H∞性能||Tz∞w∞||∞與r∞的差距(%).........106
表3.9 r∞ = 0.75,初始控制器為IC-1,方法CTMCS-2的結果..................107
表3.10 r∞ = 1,初始控制器為IC-1,方法CTMCS-2的結果......................112
表3.11 r∞ = 2.7,例3.2使用CTMCS的結果.............................................119
表3.12 例3.3使用CTMCS-WSS的結果...................................................121
表3.13 hinfmix、演化法(EA)、CTMCS的結果............................................123
表3.14 CTMCS-H∞-R求解一階控制器的結果.........................................126
表4.1 演化法的參數.....................................................................................143
表4.2 r∞ = 32.38,初始控制器IC-D1,CTMCS-2與[2]、改良[2]之比較...................................................................................................................145
表4.3 r∞ = 32.38,初始控制器IC-D2,CTMCS-2與[2]、改良[2]之比較...................................................................................................................148
表4.4 r∞ = 32.38 > r2*,CTMCS-2與v2*之比較...........................................151
表4.5 r∞ = 15,初始控制器IC-D1,CTMCS-2與[2]、改良[2]之比較........152
表4.6 r∞ = 15,初始控制器IC-D3,CTMCS-2與[2]、改良[2]之比較........155
參考文獻 參考文獻

[1] C.W. Scherer, P. Gahinet, and M. Chilali, “Multiobjective output- feedback control via LMI optimization,” IEEE Transactions Automatica Control, vol. 42, no.7, 1997 pp. 896-911.
[2] M.C. de Oliveira, J.C. Geromel, and J. Bernussou, “An LMI optimization approach to multi-objective controller design for discrete-time systems,” Proceedings of 38th IEEE Conference on Decision and Control, Phoenix, AZ, USA, 1999, pp. 3611-3616.
[3] C. Fonseca , Multi-objective Genetic Algorithms with Application to Control Engineering Problems. Ph.D. thesis , Departement of Automatic Control and Systems Engineering. University of Sheffield, 1995.
[4] N. Srinivas and K. Deb, “Multi-objective Optimization Using Nondominated Sorting in Genetic Algorithms”, Evolution Computation, No.3, 1994, 221-248.
[5] K. Deb, Multi-objective optimization using Evolutionary Algorithms. John Wiley and Sons Ltd., 2001.
[6] K. Zhou and J.C. Doyle, Essentials of Robust Control, Prentice-Hall, 1998.
[7] C.A. Desoer and M. Vidyasagar , Feedback Systems: Input-Output Properties, Academic Press, New York, 1975.
[8] C.F. Yung and C.Y. Cheng, “A game theoretic approach to strictly positive real control,” Proceedings of the 36th IEEE Conference on Decision & control, San Diego, California USA, vol. 3, Dec. 1997, pp. 2115-2120.
[9] B.M. Chen, “Direct computation of infimum in discrete-time H∞-optimization using measurement feedback,” System & Control Letter vol. 35, Dec. 1998, pp. 269-278.
[10] C.W. Scherer, “An efficient solution to multi-objective control problem with LMI objectives,” System & Control Letter, vol. 40, 2000, pp. 43-57.
[11] Z. Duan, J. Zhang, C. Zhang, and E. Mosca, “Robust H2 and H∞ filtering for uncertain linear systems,” Automatica, vol. 42, 2006, pp. 1919-1926.
[12] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, Manual of LMI Control Toolbox, the Math Works, Inc, 1995.
[13] P. Apkarian, D. Noll, and H.D. Tuan, “Fixed-order H∞ control design via partially augmented Lagrangian method,” International Journal of Robust and Nonlinear Control, 2003, pp. 1137-1148.
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