近二十年來專家們致力於研究複雜非線性系統之控制問題，而很多研究指出模糊控制器控制具有效能突出，且設計方法更加直覺簡易等優點，同時已被證明具有解決各類複雜的非線性系統的能力，然而目前的方法皆僅能求得保守解(conservation)。本論文針對非線性系統為取得更加寬鬆的解來進行控制器設計，提出了一新型的保證成本控制方法 (guaranteed cost control)，藉由路徑追隨方法求解非凸函數，該方法結合多項式模糊控制器以及保性能控制器，同時引入路徑追隨方法藉由不斷最小化由保證成本能控制器推得的效能函數的上界，藉此獲得更加寬鬆的解。本論文有以下三點貢獻：第一點，使用路徑追隨方法求解非凸函數。由於一般以轉換函數求解非凸函數的方法在轉換的過程中會導致最終僅能求得保守解，因此本方法透過路徑追蹤得已直接對非凸函數求解因而取得更加寬鬆的解；透過偕正寬鬆 (copositive relaxation)理論得以再次寬鬆解；本論文使用之方法相對以往路徑追隨方法只需最小化路徑追隨α之上界外，還需同時最小化效能函數λ的上界，故相較以往方法在設計上顯得更加複雜與困難。第二點，路徑追隨方法是藉由不斷迭代取得最佳解之最小上界，為加快迭代速度本方法引進了二分搜尋法。本論文使用 MATLAB 之模擬環境並使用所提供的 SOSOPT 工具箱對多項式不等式迅速求得一合適解。第三點，本論文比較使用保能控制、路徑追隨方法與本論文提出之方法，對基準非線性問題 (benchmark nonlinear system) 與三維非線性混沌系統求解，從模擬結果可獲得本論文所提出之方法得以取得相較其他方法更加寬鬆的解。

This thesis presents a novel algorithm of guaranteed cost control method via path-following approach for solving complex nonlinear systems. In chapter 2, We introduced solving nonlinear system by guaranteed cost control, which derives the convex condition into nonconvex condition via typical transfer function. And chapter 3 introduced solving convex condition directly via path-following, which can evade conservation issue due to transfer function. In chapter 4, we design a novel method by merging guaranteed cost control and path-following approach. There are three key features in this work. First, we directly solve nonconvex SOS design condition without applying the typical transformation. Secondly by introducing the so-called copositivity concept provides additional relaxations. The third feature is compare to regular path-following method the proposed method has to minimizing two object variables at the same time, which are α from  path-following approach and λ from guaranteed control; Thereby, the structure of novel method is more complicate. Finally, two complex nonlinear system examples are employed to illustrate the validity and applicability of the proposed method, which can obtain more relax solution than methods in previous chapters.

Acknowledgements i
Chinese Abstract ii
English Abstract iii
Contents iv
List of Figures vi
List of Tables viii
Chapter 1 Introduction 1
1.1 Introduction 1
1.2 Chapters Guide 2
Chapter 2 Guaranteed Cost Control 4
2.1 Polynomial Fuzzy Model and Controller 4
2.2 Concept of Guaranteed Cost Control via Sum-of-Square  7
Chapter 3 Path-Following Approach 12
3.1 Concept of Path-Following 12
3.2 Design Processes 16
Chapter 4 Guaranteed Cost Control via Path-Following Approach 20
4.1 Guaranteed Cost Control via Path-Following 20
4.2 Design Processes  22
Chapter 5 Design Examples 28
5.1 A Three-Dimension Polynomial Chaotic System 28
5.2 A Complicated Nonlinear System 36
6 Conclusion and Future Work 43
Bibliography 44

List of Figures
3.1 Outline of Polynomial fuzzy control via path-following algorithm 17
4.1 Outline of the proposed method 26
4.2 The relationship among a3, a4, and l according to each iteration N 27
5.1 Projection of three-dimension chaotic system 30
5.2 The controlled trajectory of the controlled complicated nonlinear system
with three controller 34
5.3 The time responses of control output u of three-dimension polynomial
chaotic system by three different controller 35
5.4 The behavior of the uncontrolled complicated nonlinear system 37
5.5 The controlled behavior of the complicated nonlinear system by guaranteed
cost control 38
5.6 The controlled behavior of the complicated nonlinear system by polynomial
fuzzy controller via path-following approach 39
5.7 The controlled behavior of the complicated nonlinear system by guaranteed
cost control via path-following approach 39
5.8 The controlled trajectory with respect to three different controller 40
5.9 The time responses of states x1 and x2 with three different controller 41
5.10 The control output u with different controller 41

List of Tables
5.1 Performance function J value of three-dimension chaotic system with
respect to different controller 33
5.2 Performance function J value of the complicated nonlinear system with
respect to different controller 42

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