本論文旨在說明滯後型時變時滯線性系統的穩定性分析，主要基於Lyapunov定理與線性矩陣不等式的數學方法來探討穩定性與穩定化的條件。藉由建立一個新穎的增廣型Lyapunov泛函，其中包含了雙重積分項、三重積分項和一個帶有高次方純量的二次式。在推導過程中，我們使用了自由權重矩陣、詹森不等式和其延伸的引理等有效技巧來盡量減少穩定條件的保守度，時滯相關的穩定性條件最終皆以表示成線性矩陣不等式的形式。此外，我們利用這個穩定條件延伸至多面體不確定性系統進而推導出一個強健穩定性條件。最後，從條件中也可以設計出可以使系統穩定的狀態回授控制器。由數值範例可以來證實本論文所提出的方法確實改善以往文獻中的結果。

This thesis considers the stability of linear systems with retarded time-varying delay. It is mainly based on Lyapunov-Krasovskii theory and linear matrix inequality (LMI) methodology to investigate the stability and stabilization criteria. By constructing a new augmented Lyapunov functional which contains double-integral terms, a triple-integral term and a higher degree scalar quadratic function, a delay-dependent stability criterion is presented in an LMI form. In derivation process, we use some effective techniques like free-weighting matrix approach, Jensen inequality and its extended lemma to reduce the conservatism of the stability criteria. Moreover, we make this criterion extend to the polytopic uncertainty system and then propose a robust stability criterion. Finally, the stability criterion is also used to design a stabilizing state-feedback controller. Numerical examples are given to show that our results achieve by far the best result in the literatures.

Contents

摘 要 i
ABSTRACT ii
誌 謝 iii
Contents iv
List of Tables vi
List of Figures vii
Chapter 1 Introduction 1
1.1 Background 1
1.2 Motivation 5
1.3 Review of Some Methods for Deriving Stability Conditions 7
1.3.1 Fixed Model Transformation 8
1.3.2 Free-Weighting Matrix Approach 10
1.3.3 Jensen Integral Inequality Approach 11
1.4 Purpose and Contribution 13
1.5 Organization of the Thesis 14
Chapter 2 Preliminary 16
2.1 Stability of Time-delay Systems 16
2.1.1 Time-delayed Mathematical Models 16
2.1.2 Concept of Stability 17
2.1.3 Lyapunov-Krasovskii Stability Theorem 19
2.2 LMI Method 20
2.2.1 Common Specifications of LMIs 21
2.2.2 Standard LMI Problems 22
2.3 Facts and Lemmas 23
Chapter 3 New Stability Criteria of Retarded Time-Varying Delay Systems 25
3.1 Problem Formulation 25
3.2 Construction of Lyapunov Functional 26
3.3 Main Results for Stability Criterion 28
3.4 Verification of Delay Decomposition approach 37
3.5 An Illustrative Example 40
Chapter 4 Robust Stability Criteria for Linear Time-Delay Systems with Polytopic Uncertainties 42
4.1 Problem Formulation 43
4.2 Main Results for Robust Stability Criterion 44
4.3 An Illustrative Example 47
Chapter 5 Stabilizing Controller Design for Retarded Time-Varying Delay Systems 49
5.1 Two Methods of Controller Design 49
5.2 Problem Formulation 50
5.3 Main Results for Stabilization Criterion 51
5.4 Another Parameter-Tuning Method 56
5.5 An Illustrative Example 60
Chapter 6 Conclusions and Future Work 63
References 65
Notations 70

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