本論文將討論線性非週期取樣系統的穩定度分析並採取連續時間的觀點，亦被稱為『輸入延遲方法』，來處理此問題。吾人將非週期取樣行為用『平均延遲差分』操作子來描述。線性非週期取樣線性系統將被視為一線性時變系統和一『平均延遲差分』操作子的回授連結。藉由『平均延遲差分』操作子滿足之二次積分限制，吾人可以用二次積分限制理論來推導穩定性準則進而判斷線性非週期取樣系統的穩定度。吾人在本論文提出『平均延遲差分』操作子滿足之時變二次積分限制，其中一些二次積分限制可以對應至其他文獻提出的李亞普諾夫函數。吾人證明運用李亞普諾夫理論所推導出的穩定性準則和運用這些二次積分限制理論所推導出的穩定性準則是等價的。若運用所有吾人提出的二次積分限制，吾人得到一較文獻中所提更低保守性的穩定性準則。其有效性將由一些數值例子來驗證。

The thesis is concerned with stability analysis of sampled-data systems with nonuniform samplings. The stability problem is trackled from a continuous-time point of view, via so-called ``input delay approach', where the ``aperiodic sampling operation' is modelled by a ``average-delay-difference' operator for which characterization based on integral quadratic
constrains (IQC) is identified. The system is then viewed as feedback interconnection of a stable linear time-varying sytem and the ``average-delay-difference' operator. We propose time-varying IQCs for the ``average-delay-difference' operator. It is shown that some IQCs we propose corresponds to certain Lyapunov functionals proposed in the literature. The equivalence between the stability criteria by the IQC theory and those by the Lyapunov theory is proven. With all the IQCs we proposed, a less conservative stability criterion is drived. Results of numerical tests are given to illustrate the effectiveness of the proposed approch.

中文摘要 i
Abstract ii
Contents iii
List of Figures v
List of Figures vi
Chapter 1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Brief Sketch of the Contents . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Notation and Terminology . . . . . . . . . . . . . . . . . . . . . . . . . 4
Chapter 2 Preliminaries 7
2.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Stability analysis via IQC . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Stability of asynchronous sampling systems: methodology . . . . 10
Chapter 3 The driving of IQCs 14
Chapter 4 The equivalence between Lyapunov and IQC conditions 20
4.1 Stability criteria by the IQC theory . . . . . . . . . . . . . . . . . . . . . 20
4.1.1 Stability criteria drived with Proposition 2 . . . . . . . . . . . . . 20
4.1.2 Stability criteria drived with Proposition 2-5 . . . . . . . . . . . . 24
4.2 Equivalence between Lyapunov and IQC conditions . . . . . . . . . . . . 28
4.2.1 The equivalent between Corollary 1 and Lemma 3 . . . . . . . . 29
4.2.2 The equivalence between Theorem 3 and Lemma 4 . . . . . . . . 32
Chapter 5 A new stability criterion 34
5.1 A less conservative stability criterion . . . . . . . . . . . . . . . . . . . . 34
5.2 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Chapter 6 Conclusions 39
Bibliography 40

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