臺灣博碩士論文加值系統

Riccati方程式在最佳控制中的線性最佳控制問題與估計中kalman filter扮演的角色。在本文利用三種演算法－Schur演算法、特徵向量解symplectic pencil法和Hamiltonian+Square-root演算法，來分析討論Riccati方程式。並利用Matlab程式語言撰寫程式。用例子來討論三種演算法的結果。

Riccati equation plays a role in the linear optimal control problem of optimal control and kalman filter of optimal estimation.
In this thesis，we use three kind of algorithms－Schur algorithm、eigenvector to solve symplectic pencil and Hamiltonian+Square-root algorithm to analyze and discuss.
An M-script program implementing algorithms is developed by MATLAB language. Numerical examples are used to discuss the results of three kind of algorithms.

摘要………………………………………………………………………Ι
ABSTRACT…………………………………………………………………Ι
目錄&圖目錄&表目錄………………………………………………….ΙΙ
第一章 緒論
1.1 引言…………………………………………………………….1-1
1.2 研究動機和背景……………………………………………….1-3
1.3 論文架構……………………………………………………….1-4
第二章 代數Riccati 方程式的數學推導
2.1 前言…………………………………………………………….2-1
2.2 代數的Riccati方程式……………………………………….2-2
2.3 DT和CT的Riccati方程式…………………………………….2-8
2.3.1 Discrete-time 代數Riccati方程式……………………….2-8
2.3.2 Continuous-time代數Riccati方程式………………………2-9
第三章 頻譜分解推導Riccati方程式
3.1 引言……………………………………………………………3-1
3.2 頻譜分解推導Riccati方程式………………………………3-2
3.3 線性矩陣不等式推導Riccati方程式………………………3-5
3.4 結論……………………………………………………………3-7
第四章 特徵值的演算法
4.1 前言…………………………………………………………….4-1
4.2 特徵向量演算法….……………………………………………4-2
4.3 例題說明……………………………………………………….4-4
4.4 結果討論……………………………………………………….4-7
第五章Schur 演算法
5.1 前言…………………………………………………………….5-1
5.2 Schur 演算法………………………………………………….5-2
5.3 例題說明與討論……………………………………………….5-6
5.4 Schur 法的優缺點…………………………………………….5-8
第六章 Hamiltonian和Square-root演算法
6.1 前言…………………………………………………………….6-1
6.2 方程式與矩陣的基本定義…………………………………….6-1
6.3 演算法………………………………………………………….6-3
6.4 例題說明與討論……………………………………………….6-5
6.5 結果討論……………………………………………………….6-9
第七章 代數Riccati方程式數值解的誤差分析
7.1 前言…………………………………………………………….7-1
7.2 ARE模型與解法的整理…………………………………………7-2
7.3 Norm 和Condition number……………………………………7-3
7.4 例題說明……………………………………………………….7-6
7.5 結果討論……………………………………………………….7-9
第八章 結論…………………………………….……………………8-1
參考文獻
附錄
圖(2-1) 最佳暫態觀測器……………………………………………2-5
表6-1 演算法比較……………………………………………………6-6
表6-2 演算法比較……………………………………………………6-7
表6-3 演算法比較……………………………………………………6-8
表7-1 CARE數值解的誤差分析…………………………………….7-7
表7-2 DARE數值解的誤差分析…………………………………….7-8

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