Riccati方程式矩陣邊界解方法之研究｜國立中央大學博碩士論文系統

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研究生：	張志禮 Chih-lee Chang
論文名稱：	Riccati方程式矩陣邊界解方法之研究 Matrix Bounds of the Solution for the Algebraic Riccati Equations
指導教授：	莊堯棠 Yau-Tarng Juang
口試委員:
學位類別：	碩士 Master
系所名稱：	資訊電機學院 - 電機工程學系 Department of Electrical Engineering
畢業學年度：	89
語文別：	中文
論文頁數：	55
中文關鍵詞：	離散雷卡提方程式、連續雷卡提方程式、矩陣邊界解、離散李亞普諾夫方程式、連續李亞普諾夫方程式
外文關鍵詞：	Discrete Riccati Equation, Continuous Riccati Equation, Matrix Bound, Discrete Lyapunov Equation, Continuous Lyapunov Equation
相關次數：	點閱：5 下載：0
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就離散系統而言，以解離散李亞普諾夫(Lyapunov)方程式為中心，我們對離散Riccati方程式，提出新矩陣邊界解，在此方法中只要離散的Lyapunov方程式的解存在，我們必可求得所要的矩陣邊界解。因此，我們不需要 BB''>0 or Q>0 的限制條件，另一方面，如果 BB''>0 or Q>0 的條件成立下，在某些情形下我們的方法表現出，即使Lyapunov方程式唯一解不一定存在(亦即狀態矩陣不穩定時)，本文的方法亦可求得Riccati 方程式的邊界解。
而在連續的系統方面，亦即以連續Lyapunov方程式為中心，本論文提出一組矩陣邊界解。文中提到並證明如何以最佳控制理論，得到最初始上界，同時我們利用新的方法來證明Riccati 方程式，可經由Lyapunov方程式的遞歸解來求得真實解，其中的條件只要控制系統為可穩定系統。本章中，我們的主要研究方向，在於探討連續Riccati方程式的遞歸解的速度及精確度，基於穩定問題限制，只能推展出上邊界的遞歸解，對於下邊界只能求出其對應上邊界的矩陣近似解。在本論文中我們將探討，藉由矩陣邊界來解離與連續散雷卡提(Riccati)方程式的新方法，本文中所提的新法將改善現存文獻中的限制，並可獲得較接近精確值的解，且可使收斂速度加快。
就離散系統而言，以解離散李亞普諾夫(Lyapunov)方程式為中心，我們對離散Riccati方程式，提出新矩陣邊界解，在此方法中只要離散的Lyapunov方程式的解存在，我們必可求得所要的矩陣邊界解。因此，我們不需要 BB''>0 or Q>0 的限制條件，另一方面，如果 BB''>0 or Q>0 的條件成立下，在某些情形下我們的方法表現出，即使Lyapunov方程式唯一解不一定存在(亦即狀態矩陣不穩定時)，本文的方法亦可求得Riccati 方程式的邊界解。
而在連續的系統方面，亦即以連續Lyapunov方程式為中心，本論文提出一組矩陣邊界解。文中提到並證明如何以最佳控制理論，得到最初始上界，同時我們利用新的方法來證明Riccati 方程式，可經由Lyapunov方程式的遞歸解來求得真實解，其中的條件只要控制系統為可穩定系統。本章中，我們的主要研究方向，在於探討連續Riccati方程式的遞歸解的速度及精確度，基於穩定問題限制，只能推展出上邊界的遞歸解，對於下邊界只能求出其對應上邊界的矩陣近似解。

In this thesis, we will presents new matrix bounds of the solution for the discrete and continuous Riccati equations. Based on the solution of certain continuous and discrete Lyapunov equations, the improved upper and lower matrix bounds are obtained. In the discrete systems, the upper and lower matrix bounds always exist if the DLE solution exists. Then, further improvements on the bounds are presented. On other hand, these upper and lower matrix bounds of solution for the continuous Riccati equation are always exist if the system is stabilizable. Numerical examples are given to show that our methods are less conservative and less restrictive than some recent results.

Contents
Abstract …………………………………………………………………… I
Contents ………………………………………………………………… II
Notation ………………………………………………………………… IV
Chapter 1 Introduction ……………………………………………… 1
1-1. Motivation …………………………………………………1
1-2. Literature Survey ……………………………………… 2
1-3. Outline of This Thesis ……………………………… 3
Chapter 2 Upper and Lower Matrix Bounds of the Solution for the Discrete Riccati Equation …………………………………… 4
2-1. Introduction ……………………………………………… 4
2-2. Preliminaries …………………………………………… 4
2-3. Main Results ……………………………………………… 6
2-4. Illustrative Examples ………………………………… 13
2-5. Conclusions ……………………………………………… 20
Chapter 3 Matrix Bounds of Solution for the Continuous Riccati Equation ……………………………………………………… 21
3-1. Introduction …………………………………………… 21
3-2. Preliminaries …………………………………………… 21
3-3. Main Results …………………………………………… 23
3-4. Illustrative Examples ………………………………… 28
3-5. Conclusions ……………………………………………… 31
Chapter 4 Conclusions and Suggestions ………………………… 32
4-1. Conclusions ……………………………………………… 32
4-2. Suggestions for Further Research ………………… 32
Reference ………………………………………………………………… 33

                                

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