本文提出一以多目標粒子群聚最佳化法(MOPSO)為基礎之混合式演算法，藉由結合NM單體搜尋法(NM simplex search)與粒子群聚最佳化法，針對多目標最佳化問題求得可能的Pareto最佳解。作法上係將NM單體搜尋法嵌入粒子群聚最佳化演算法中，充分利用粒子群聚最佳化法在搜尋空間中做探索(exploration)搜尋，輔以改良式NM單體搜尋法在區域鑽探(exploitation)搜尋的能力，進一步提升多目標最佳化演算法之多樣性及精確性。在控制系統方面，我們以MOPSO應用在解決最佳PID控制器設計的問題。作法上係對PID控制器參數作實數編碼後，再與受控系統作結合，以閉迴路系統之性能做考慮，並以平方誤差與時間乘積之積分(ITSE)以及干擾拒絕(disturbance rejection)為適合度評定機制之指標，找出理想的PID控制器參數。

This paper proposes a hybrid multi-objective particle swarm optimizer incorporation (MOPSO) an enhanced Nelder-Mead simplex search scheme to solve multi-objective optimization problems. Because of the strength of PSO in explorative search and NM simplex search in exploitative search to locate promising particles closest to the optimum during the optimization process, both diversity and accuracy of the proposed optimization algorithm can be significantly improved. To demonstrate its application on solving control system problem, the proposed MOPSO is applied to design an optimal PID controller, optimizing the ITSE and disturbance objection specifications.

目錄.......................................................I
圖目錄.....................................................IV
表目錄.....................................................VI
第一章、緒論…………………………………………………………………1
       1.1 前言…………………………………..…………………………1
       1.2 研究動機……………………………..…………………………2
       1.3 文獻回顧……………………………..…………………………3
       1.4 研究方法與目的………………………………………………..4
       1.5 論文架構……………………………..…………………………5
第二章、多目標最佳化法.........................................6
       2.1 多目標最佳化之定義…………………………………………..6
           2.1.1 多目標演算法之定義…………………………………...6
           2.1.2 不受支配解與支配的概念……………………………...6
           2.1.3 Pareto最佳化的概念……………………………………7

           2.1.4 擁擠距離值之計算……………………………………...9
       2.2 多目標最佳化法之文獻回顧…………………………………10
       2.3多目標最佳化法的效能評估機制……………………………..14
第三章、以PSO為基礎之多目標最佳化法………………………………18
       3.1 粒子群聚最佳化法……………………………………………18
       3.2 多目標粒子群聚最佳化法作法敘述…………………………21
           3.2.1 演算法流程簡介……………………………………….21
           3.2.2 找出 之方法………………………………………..23
第四章、結合NM單體搜尋法與PSO之多目標最佳化法……………..25
       4.1 NM單體搜尋法……………………………………………….25
       4.2改良式NM單體搜尋法………………………………………31
       4.3 MO-NMPSO作法敘述………………………………….……..34
第五章、模擬結果…………………………………………….……………37
       5.1 模擬結果………………………………………………………38
       5.2 結果討論………………………………………………………48
第六章、多目標最佳化演算法應用於控制器之設計…………………….52
       6.1最佳控制器之設計問題……………………………………….53
           6.1.1 以最小ITSE為基礎之最佳控制器設計………………53
           6.1.2 以最小干擾拒絕為基礎之最佳控制器設計………….53
           6.1.3 穩定性判斷與懲罰函數之定義……………………….54
           6.1.4 多目標最佳控制器之設計…………………………….55
       6.2 MOPSO應用於最佳PID控制器之設計…………………….55
第七章、結論與未來研究方向……………………………………………..59
       7.1 結論……………………………………..……………………..59
       7.2 未來研究方向…………………………………………………60
參考文獻………………………………………………………………….…61
附錄………………………………………………………………………….66

圖目錄
圖2.1 範例之Pareto front………………………….…………………………9
圖2.2 計算擁擠距離值示意圖……………………………………………10
圖2.3 動態搜尋法之示意圖…………..……..……………………………..11
圖2.4 格子方法之示意圖………………………………..…………………12
圖2.5 sigma方法之示意圖……………………………………...…………..13
圖2.6 支配樹方法示意圖…..…………………………………………..…..14
圖3.1 粒子群聚最佳化法流程圖…………….…………………………….21
圖3.2 多目標粒子群聚最佳化法流程圖………..…………………………23
圖3.3 找尋 之示意圖…….…………………..…………………………24
圖4.1 Nelder-Mead 樞軸運算之反射程序示意圖…..……….…………….27
圖4.2 Nelder-Mead 樞軸運算之擴張程序示意圖………..……..…………28
圖4.3 (a)NM單體搜尋法向外收縮程序示意圖…..……….……………….29
圖4.3 (b)NM單體搜尋法向內收縮程序示意圖…..……….……………….29
圖4.4 (a)當 更好於 時之NM單體搜尋法之縮小程序示意圖….…30
圖4.4 (b)當 更好於 時之NM單體搜尋法之縮小程序示意圖……30
圖4.5 二次擴張 (Second expansion) 程序之示意圖……..…..………..…32
圖4.6 連續擴張 (Continuous expansion) 程序之示意圖…....……………33
圖5.1 FUN.1結果圖……………………………………….…..…………….39
圖5.2 FUN.2結果圖…………………………………………………………39
圖5.3 FUN.3結果圖…………………………………………………………39
圖5.4 FUN.4結果圖………………..………………………………………..40
圖5.5 FUN.5結果圖………………..………………………………………..40
圖5.6 FUN.6結果圖………………………..………………………………..40
圖5.7 FUN.7結果圖………………………….……………………………...41
圖5.8 FUN.8結果圖…………………………………………………………41
圖5.9 FUN.9結果圖…………………………………………………………42
圖5.10 FUN.10結果圖………………………………………………………42
圖5.11 FUN.11結果圖………………………………………………………42
圖5.12 FUN.12結果圖………………………………………………………43
圖5.13 FUN.13結果圖………………………………………………………43
圖5.14 FUN.14結果圖……….…………………………………………….44
圖5.15 FUN.15結果圖……………………………………………….…….44
圖6.1 控制系統示意圖…………………………………………………….52
圖6.2 控制器設計之不受支配解.………….…………………………….56
圖6.3 所求得控制器之單位步階響應…………………………..…………58
表目錄
表5.1 測試函數之規格表………….…………….…………………………38
表5.2 MOPSO針對15個測試函數之性能統計…………….………………45
表5.3 MO-NMPSO針對15個測試函數之性能統計(1)……………………46
表5.4 MO-NMPSO針對15個測試函數之性能統計(2)…………………….47
表5.5 Coello提出之方法 [2]針對3個測試函數之性能統計………………49
表5.6 Liu提出之方法 [30]針對3個測試函數之性能統計….……………..50
表5.7 Deb提出之方法 [19]針對9個測試函數之性能統計...............51
表6.1 PID控制器設計的不受支配解之參數與適應值.……………………57
表6.2 所提出方法與參考文獻之適應值比較…….………………………58

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