# 臺灣博碩士論文加值系統

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 本論文以Lyapunov-Krasovskii方法針對區間時變時滯線性系統，推導與區間延遲相關的穩定性。推導過程中適時使用自由權重法、Jensen積分不等式及凸集合技術來儘可能放寬穩定條件，並以線性矩陣不等式的形式表示最終的結果。由數值範例的比較，明確地呈現出本論文所提出的方法確實能夠達到文獻上迄今最好的結果。為取得創新的結果，首先我們提出一個新穎的增廣Lyapunov泛涵，其特色是將區間時變時滯線性系統下界的時滯區間分成多個等距的子區間，接著在每個子區間及時滯區間構築含有三重積分在內的增廣Lyapunov泛涵。接著，利用自由權重法及Jensen積分不等式來處理該增廣Lyapunov泛涵對時間取導數後產生的積分項。然後為了避免以往因過度界定時變時滯項而造成不可避免的保守性發生，我們引入凸集合技術來完成最終的推導，於是區間時變時滯線性系統的穩定條件可以線性矩陣不等式的型式表示之。最後，由數值範例來證實本論文所提出的方法確實比以往的結果顯著的改善了保守性。
 In this thesis, the delay-range-dependent stability of linear systems with interval time-varying delay will be studied. It is mainly based on Lyapunov-Krasovskii methodology. The derivation techniques include the delay free weighting matrix (FWM) method, Jensen''s integral inequality, the convex combination technique, and the linear matrix inequality (LMI). Stability criteria are given in terms of LMIs, and numerical examples are given to illustrate the effectiveness and the merits of the results. To this end, we first decompose the lower bound delay interval into multiple equidistant subintervals, then construct augmented Lyapunov functionals which contains some triple-integral terms on these intervals and the time-varying delay interval. Consequently, a novel augmented Lyapunov functional is proposed. Secondly, by using FWM approach, Jensen''s integral inequality and the convex combination technique, some new stability criteria for interval time-varying delay systems are derived in terms of LMIs. Finally, numerical examples are presented to demonstrate the less conservatism of the obtained results, when compared to existing results.
 摘　要 iABSTRACT ii致　謝 iiiContents ivList of Tables viList of Figures viiChapter 1 Introduction 11.1 Background 11.2 Motivation 31.3 Review of Delay-Dependent Stability Analysis for Interval Time-Delay Systems 41.4 Purpose and Contribution 61.5 Organization of the Thesis 7Chapter 2 Preliminary 92.1 Stability of Time-Delay Systems 92.1.1 Time-Delayed Mathematical Models 92.1.2 Stability Concept 102.1.3 Lyapunov-Krasovskii Stability Theorem 112.2Linear Matrix Inequality (LMI) Method 122.2.1 General Form of LMIs 132.2.2 A Simple Example of LMI Method 132.3 Facts and Lemmas 14Chapter 3 Main Skills for Deriving Stability Conditions 173.1 Free-Weighting-Matrix (FWM) Approach 173.2 Jenson Integral Inequality Approach 213.3 Delay Decomposition Approach 223.4 A Revisit to The Stability Conditions of Systems with Interval Time-Varying Delay 25Chapter 4 New Stability Criteria of Interval Time-Varying Delay Systems 304.1 Problem Formulation 304.2 Construction of Lyapunov Functional 314.3 Main Results for Stability Conditions 33Chapter 5 Illustrative Examples 53Chapter 6 Conclusions and Future Work 56References 58Notations 63
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