臺灣博碩士論文加值系統

本論文中，主要是研究有奇異矩陣領導的二階向量線性微分系統的指數穩定分析與控制，本論文用歸納範數(induced norm)、矩陣測度(matrix measure)和一些不等式的觀念，推導出了一個新的充份指數穩定條件，並且舉例說明來證實提出的定理比一些文獻的結果更不具保守性。我們以定理一的穩定條件和最佳化方法對此系統設計控制器，使系統從不穩定變穩定。
系統有適當的相位邊限(phase margin)以及增益邊限(gain margin)的話，將會使系統有良好的強健性，但對於多輸入多輸出(MIMO) 系統要使得整個系統達到所要求的規格將會是非常複雜而困難的，於是本論文就以相位邊限的規格要求（增益邊限規格可以類推適用），利用Gershgorin定理設計比例-微分控制器(Proportional-Derivative Controller)、比例-積分控制器(Proportional-Integral Controller)、以及相位領先或落後補償器(Phase Lead or Lag Compensator)。

This thesis is concerned with exponential stability analysis and design of linear systems represented by the second-order vector differential equations with singular leading coefficient matrices. A new sufficient condition for exponential stability is derived. By illustrative examples, it is shown that the proposed criterion is less conservative as compared with some results in the literature. Then, we use the developed criterion and an optimization method to design controllers to make the considered systems stable.
In control theorems, the gain margin and the phase margin are important robustness specifications for the design of practical control systems. This thesis also considers the design of time-invariant systems with the specified phase margin. The Gershgorin theorem is used to design a proportional-derivative (PD) controller, or a proportional-integral (PI) controller, or a phase lead compensator, or a lag compensator to achieve the required phase margin. Examples are also given.

List of Figures...........................................................................................................................III
List of Tables..............................................................................................................................V

Chapter 1 Introduction.............................................................................................................1
1-1 Background and the Motivation....................................................................................1
1-2 Organization of this Thesis............................................................................................3

Chapter 2 Fundamental Concept..............................................................................................4
2-1 Induced Norms...............................................................................................................4
2-2 Matrix Measures............................................................................................................6
2-3 Gain Margin and Phase Margin.....................................................................................8
2-3-1 Gain Margin.............................................................................................................8
2-3-2 Phase Margin..........................................................................................................11
2-4 Gershgorin Theorem....................................................................................................13

Chapter 3 Stability Analysis and Illustrative Examples.........................................................16
3-1 Stability Analysis.........................................................................................................16
3-2 Illustrative Examples...................................................................................................22

Chapter 4 Design of Controllers............................................................................................31
4-1 Time-domain Design...................................................................................................31
4-2 Frequency-domain Design...........................................................................................37
4-2-1 PD Controller.........................................................................................................38
4-2-2 PI Controller...........................................................................................................48
4-2-3 Phase Lead or Lag Compensator............................................................................55

Chapter 5 Conclusions...........................................................................................................63

References.................................................................................................................................64

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