摘要
本論文的主旨是討論未確定多時間尺度系統在含有參數擾動的影響下的特徵值分析，以及設計分散式強健控制器來達到極點安置於特定區域的目的。由於一個多時間尺度系統的特徵值分成幾個部分叢集在複數平面上，且可以將其分解成若干個彼此獨立的子系統。因此吾人可以分別討論各子系統的特徵值的特性，並分別設計控制器，再將其合併為一分散式控制器。
在動態系統中，特徵值的位置決定了系統的穩定性和工作性能，由於系統的參數擾動，故無法得到未確定系統特徵值的精確位置。因此本文將討論系統的特徵值在不同的型式的參數擾動下，叢集在特定區域的充分條件，其中參數擾動的型式分別為非結構型和結構型。接著提出一個設計的法則，來設計分散式控制器，並將極點安置於吾人期望的區域，而文中考慮的區域有半平面和圓盤區域。其次，本文亦討論了利用分散式控制理論，並配合最佳控制理論中，線性二次調節器(LQR)控制以及線性二次高斯(LQG)控制的設計，應用於受到擾動的多時間尺度系統上，以期得到系統的最佳強健工作性能。
而在多時間尺度系統中，奇異擾動參數的界值會影響系統的強健穩定性。因此在本文中，將利用李亞普諾夫穩定法則和範數原理，針對不同的子系統，求出所對應的奇異擾動參數的界值。接著將其理論延伸，求出當子系統的極點被安置在特定區域時，所對應的奇異擾動參數的界值，以及推導系統的的極點在此限制下強健穩定的充分條件，並提出強健控制器設計法則，使系統可承受較大的擾動。
在本文的各個討論的主題中，皆附有範例來說明文中所提出的充分條件，和設計法則的應用，其結果都令人滿意。

Abstract
In this dissertation, the eigenvalue analysis and decentralized robust controller design of uncertain multi-time-scale system with parametrical perturbations are considered. Because the eigenvalues of the multi-time-scale systems cluster in some difference regions of the complex plane, we can use the singular perturbation method to separate the systems into some subsystems. These subsystems are independent to each other. We can discuss the properties of eigenvalues and design controller for these subsystem respectively, then we composite these controllers to a decentralized controller.
The eigenvalue positions dominate the stability and the performance of the dynamic system. However, we cannot obtain the precise position of the eigenvalues from the influence of parametrical perturbations. The sufficient conditions of the eigenvalues clustering for the multi-time-scale systems will be discussed. The uncertainties consider as unstructured and structured perturbations are taken into considerations. The design algorithm provides for designing a decentralized controller that can assign the poles to our respect regions. The specified regions are half-plane and circular disk.
Furthermore, the concepts of decentralized control and optimal control are used to design the linear quadratic regulator (LQR) controller and linear quadratic Gaussian (LQG) controller for the perturbed multi-time-scale systems. That is, the system can get the optimal robust performance.
The bound of the singular perturbation parameter would influence the robust stability of the multi-time-scale systems. Finally, the sufficient condition to obtain the upper bound of the singular perturbation parameter presented by the Lyapunov method and matrix norm. The condition also extends for the pole assignment in the specified regions of each subsystem respectively.
The illustrative examples are presented behind each topic. They show the applicability of the proposed theorems, and the results are satisfactory.

目錄
目錄 i
圖表引索 iv
符號說明 vi
論文摘要(中文) viii
論文摘要(英文) x
第一章 緒論 1
1.1 研究動機與目的 1
1.2 文獻回顧 4
1.2.1 多時間尺度系統的控制 4
1.2.2 特徵值叢集與極點安置 5
1.2.3 強健線性二次控制 6
1.2.4 奇異擾動參數和穩定性分析 7
1.3 章節組織 8
第二章 系統描述與基本定理 11
2.1 多時間尺度系統的基本模式 11
2.2 矩陣基本定義與定理 14
2.3 向量與矩陣的範數 15
2.4 特徵值落在特定區域的相關定理 18
2.5 一些有用的輔助定理 20
第三章 特徵值分析與強健極點安置 28
3.1 特徵值叢集於複數平面中的代數區域 29
3.1.1 慢速系統於一次區域 29
3.1.2 快速系統於一次區域 31
3.1.3 慢速系統於二次區域 32
3.1.4 快速系統於二次區域 33
3.2 特例：半平面和圓盤區域的討論 34
3.2.1 慢速系統於半平面區域 34
3.2.2 快速系統於半平面區域 35
3.2.3 慢速系統於圓盤區域 36
3.2.4 快速系統於圓盤區域 37
3.3 控制器設計 38
3.4 範例 40
附錄3A 44
附錄3B 49
第四章 強健線性二次控制 59
4.1 線性二次調節器(LQR)控制 59
4.1.1 二次調節器(LQR)控制的基本原理 60
4.1.2 慢速系統為漸進穩定的條件 61
4.1.3 快速系統為漸進穩定的條件 63
4.1.4 控制器設計 64
4.2 利用LQR設計將極點安置於圓盤區域 66
4.2.1 慢速系統的充分條件 68
4.2.2 快速系統的充分條件 69
4.2.3控制器設計 70
4.3 線性二次高斯控制(LQG)控制 72
4.3.1 二次高斯控制的基本原理 74
4.3.2 強健高斯控制 77
附錄4A 82
附錄4B 86
第五章 奇異擾動參數為有界的強健控制 94
5.1 狀態回授下的強健穩定條件 95
5.2 極點安置於特定區域的強健穩定條件 100
5.2.1 半平面區域 100
5.2.2 圓盤區域 103
5.3 控制器設計 105
5.4 範例 106
附錄5 110
第六章 結論 116
6.1 總結 116
6.2 未來展望 117
參考文獻 119

圖表索引
圖1-1 四分之一車的主動式懸吊裝置 10
圖2-1 多時間尺度系統之示意圖 24
圖2-2 兩個半平面區域 和 25
圖2-3 圓盤區域 26
圖3-1 範例3.1的原始閉迴路系統在最佳增益下的特徵值分佈 54
圖3-2 範例3.1的閉迴路系統其慢速部分在最佳增益下的特徵值位置 55
圖3-3 範例3.2的閉迴路系統其慢速部分在初始增益 下的極點位置 57
圖3-4 範例3.2的閉迴路系統的慢速部分在最佳增益 下的極點位置 57
圖3-5 範例3.2的原始閉迴路系統在最佳增益下的極點分佈 58
圖4-1 範例4.2的原始閉迴路系統在最佳增益下的極點分佈 92
圖4-2 範例4.2的閉迴路系統的慢速部分在最佳增益下的極點位置 93
圖5-1 範例5.2的系統在強健迴授增益下整體的極點分佈 113
圖5-2 範例5.3的系統在強健迴授增益下整體的極點分佈 114
圖5-3 範例5.3的系統在強健迴授增益下各子系統的極點分佈 115
表2-1 一些常見的 區域 27

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