摘要

在本論文裡，基於李亞普諾函數，新的強健穩定性準則將針對帶有時變延遲基因調控網路而被推導。SUM調控函數被用於表示基因調控機制。針對不確定性之帶有時變延遲的基因調控網路，結構與非結構的不確定非線性函數兩者將用來表示建模的誤差。基於李亞普諾的充要條件之強健穩定的達成，是由不等式邊界技巧和自由矩陣的導入，而不須嚴格定義的變數則是由牛頓-萊布尼茲公式導入。我們藉由選定合適的李亞普諾函數來建構參數相依的強健性穩定準則，而李亞普諾函數可將相關參數直接納入，如延遲參數、不確定性參數…等。更進一步，我們考慮將延遲函數的微分上界加入Lyapunov-Krasovskii函數來降低準則的保守性。根據傑生積分公式和自由分布參數推導一個較不保守的充要條件，是用來增加可延遲區間和降低延遲的微分項在線性矩陣不等式裡的正向貢獻；再者，針對帶有快速變動的時變延遲基因調控網路也可直接處理，而我們可以確保強健性穩定在允許的區間內。所有提出的穩定性條件都可以藉由Schur公式輕易地將李卡地不等式寫成線性矩陣不等式。最後，數值模擬的例子將展現出定理的效果。

關鍵字：基因調控網路、線性矩陣不等式、傑生積分公式、強健穩定、模型不確定性

Abstract

In this thesis, new robust stability criteria of genetic regulatory networks with time-varying delays based on Lyapunov functional will be derived. The SUM regulatory functions are utilized to model the genetic regulatory mechanism. For the uncertain genetic regulatory network with time-varying delays, both of unstructured and structured nonlinear uncertainty functions are employed to present model uncertainties. Lyapunov based sufficient conditions for robust stability are obtained by introducing inequality bounding techniques and free weighting matrices. Furthermore, the uncertain genetic regulatory network with large fast time-varying delays will be explored. The upper bound of the derivative of delay function will be incorporated into Lyapunov-Krasovskii functional such that the conservatism of the criteria can be reduced and the robust stability of the uncertain genetic regulatory network with time-varying delays will be guaranteed. A less conservative sufficient condition has been derived by Jenson’s integral and free distribution parameters for increasing allowable delay interval and reducing positive contribution of the derivative of delays in the LMIs. All the proposed stability conditions written as Riccati inequality can be easily expressed in terms of linear matrix inequality (LMIs) by Schur complements. Finally, numerical examples are given to demonstrate the effectiveness of the proposed methods.

Keywords: Genetic regulatory network, delay, linear matrix inequality, Jenson’s integral, robust stability, model uncertainty

Contents

Abstract II

List of Figures IV

1. Introduction 1
1.1: Genetic Regulatory Networks 1
1.2: Motivations 1
1.3: Overview of Previous Research 2
1.4: Contributions of the Thesis 3
1.5: Organization of the Thesis 4

2. Models of Genetic Regulatory Networks 5

3. Stability Analysis of Genetic Regulatory Networks with Time-varying Delays 9
3.1: Introduction 9
3.2: Stability Analysis 11
3.3: Illustrative Example 30
3.4: Summary 40

4. Robust Stability of Genetic Regulatory Networks with Fast Time-Varying Delays 41
4.1: Introduction 41
4.2: Stability Analysis 42
4.3: Illustrative example 64
4.4: Summary 70

5. Conclusions 72

References

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