臺灣博碩士論文加值系統

本論文以探討判別系統穩定度最著名且方便的演算法--路斯演算法為基礎
，進而直接克服在路斯表格中的殊異情形。根在虛軸上的總數及重根分佈
情形可被決定出，如此可區分出條件穩定和不穩定。本論文所推衍之延伸
路斯演算法可決定更多種區間的根分佈情形，而非局限於右半平面；如此
其可成為判別系統特性的有效方法。原本只被用於處理實數係數多項式的
路斯表格也同時被推廣至複數係數多項式。基於原始路斯演算法，作者成
功地提出嶄新的表格以判斷系統穩定度，並對於其殊異現象有詳盡討論；
另外對於由跋克斯所提出的複數路斯表格也提出另一種證明並提出克服其
殊異現象方法。相對於線性離散系統之穩定度分析，朱利表格為最著名的
工具，藉此可得知根分佈於單位圓的情形。本論文提出有效的方法以克服
其殊異現象，並判別根位 於單位圓上的總數及重根情形，進而可區別出
條件穩定或不穩定情形。本論文對於適用於多變數系統之矩陣路斯演算法
有深入的探討。矩陣連分數演算法可視為其廣義演算法，本論文以此法提
出有效的程序來處理在多變數系統研究領域中一些重要的問題，如戴逢第
方程式之解，兩多項式矩陣之最大共同左/右公因矩陣，右(左)矩陣分數
表示式轉換至左(右)矩陣分數表示式等。將以上結果，應用於多變數控制
單一回授架構中控制器之設計而達到根設定的效果。本論文也附著路斯演
算法百年發展至今仍不墜之因由，並將所發展出眾多的應用作一綜觀性的
整理分析，期能成為一本顧及歷史及現代發展的完整論文。

The purpose of this dissertation is to discuss and further
investigate the most famous and convenient algorithm --- the
Routh algorithm, for determining stability. In particular,
modified procedures are developed to treat singular cases. The
information of simple and repeated roots with respective orders
will be obtained; therefore, one can distinguish the situation
of conditional stability or instability. The extended Routh
algorithm is developed to be applicable to more kinds of
regions. The original Routh table dealing with the root
distribution of a real polynomial is extended for the case of a
complex polynomial. A new tabular form for determining the root
distribution of a complex polynomial with respect to the
imaginary axis is proposed based on the original Routh
algorithm. Based on the tabular form proposed by Parks,
modified procedures for treating singular cases are also
proposed. Concerning a linear discrete system, the Jury table
is the most famous one for determining the root distribution
with respect to the unit circle. Efficient procedures are
developed in this dissertation to overcome the singular cases
of the Jury algorithm. Theorems for obtaining the information
of simple and repeated roots on the unit cicrle with their
respective orders are also proposed. To extend the applications
of the Routh algorithm to the area of multivariable systems, we
discuss matrix Routh algorithm widely. Based on the general
one of the matrix Routh algorithm -- the matrix continued-
fraction algorithm, problems in the research area of
multivariable systems are investigated.

摘要
Abstract
誌謝
Publication List
Contents
List of Tables
List of Figures
List of Symbols
1 Introduction
1.1 Background Review
1.2 Dissertation Outline
2 Root Distribution of a Polynomial in Subregions of the Complex Plane
2.1 Introduction
2.2 Modified Processes for Singular Cases of Routh Table
2.3 The Extended Sturm Test and the Extended Routh Theorem
2.4 Proofs
2.5 Illustrative Examples
2.6 Conclusions
3 A New Tabular Form for Determining Root Distribution of a Complex Polynomial with Respect to the Imaginary Axis
3.1 Introduction
3.2 Complex Routh Algorithm
3.3 Modified Proceses for Singular Cases
3.4 Proofs
3.5 Illustrative Examples
3.6 Conclusions
4 On the Singular Cases of the Complex Routh''s Algorithm for Stability Text
4.1 Introduction
4.2 Preliminaries
4.3 Modified Processes for Singular Cases
4.4 Results and Stability Criteria
4.5 Proofs
4.6 Illustrative Examples
4.7 Conclusions
5 On the Singular Cases of Jury''s Algorithms for Stability Test of Linear Discrete Systems
5.1 Introduction
5.2 Preliminaries
5.3 Singular Cases of Jury''s Algorithms
5.4 Proofs
5.5 Illustrative Examples
5.6 Conclusions
6 Generalized Matrix Routh Algorithms for Solving Diophantine Equations and Associated Problems
6.1 Introduction
6.2 Preliminaties
6.3 The Greatest Common Right (Left) Divisor
6.4 The Diophantine Equation and Conversions of Right (Left) MFD to Left (Right) MFD
6.5 Modified Procedures for Non-Square Problems
6.6 Multivariable Pole-Assignment Problems
6.7 Illustrative Examples
6.8 Conclusions
7 Conclusions
Appendix A: Edward John Routh
Appendix B: Applications of Routh Algorithm and Modern Approach results
Appendix C: Algorithm for Transforming a Non-Singular Polynomial Matrix to a Co-Regular One
Bibliography