本論文針對區間時變延遲不確定性基因調控網路提出強健耗散性穩定度分析與狀態估測器設計之問題探討。在耗散性穩定度分析的部分，針對區間時變延遲不確定性基因調控網路進行全域耗散性穩定度分析。其中，激勵函數假設為有界及全域性利普希茨(Lipschitz)連續。藉由李亞普諾夫函數(Lyapunov-Krasovskii functional)及線性矩陣不等式(linear matrix inequality)技巧，並透過使用改善的上邊界技巧，可推得時延相關全域強健耗散性穩定度條件，同時引入自由加權矩陣，以降低系統保守性，並且時延微分不需要限制小於1。

在設計狀態估測器方面，主要目標是設計一個狀態估測器使得區間時變延遲基因調控網路的狀態誤差達全域漸近穩定。為了避免複雜的數學推導與過於保守的結果，藉由使用混合式田口基因演算法(Hybrid-Taguchi genetic algorithm)並結合線性矩陣不等式技巧來尋找估測器增益並滿足李亞普諾夫函數(Lyapunov-Krasovskii functional)穩定度不等式，可求解得到所需要設計的狀態估測器。最後，經由一些數值範例證明所提理論的可應用性與有效性。

This thesis presents the complete study of robust passivity stability analysis and state estimator design. The system is focused on uncertain genetic regulatory networks with interval time-varying delays. For a passivity analysis problem, the goal is to develop globally robust passivity stability for uncertain genetic regulatory networks with interval time-varying delays. The activation functions are assumed to be bounded and globally Lipschitz continuous. Based on the Lyapunov-Krasovskii functional approaches, linear matrix inequality (LMI) techniques and an improved bounding technique, the globally delay-dependent robust passivity stability criteria for the uncertain genetic regulatory networks with interval time-varying delays are established, and some free-weighting matrices are introduced to reduce system conservativeness. The derivatives of the delays are no longer less than 1 required in our analysis.

For the estimator design problem, state estimations for genetic regulatory networks with interval time-varying delays is investigated. Attention was focused on the design of state estimators which ensure the globally asymptotical stability of the estimation state errors. To avoid complex mathematical derivations and conservative results, the hybrid Taguchi-genetic algorithm (HTGA) method is integrated with a linear matrix inequality (LMI) method to seek the estimator gain matrices that satisfy the Lyapunov-Krasovskii functional stability inequality. When this LMI is feasible, the expression of desired state estimators is also presented. Finally, some illustrative examples are provided to demonstrate the applicability and effectiveness of the proposed approach.

中文摘要 I
Abstract II
Acknowledgment IV
Contents VI
List of Figures VIII
List of Tables X

Chapter 1 Introduction 1
1.1 Genetic Regulatory Networks 1
1.2 Linear Matrix Inequalities (LMIs) 6
1.3 Hybrid Taguchi-Genetic Algorithm (HTGA) 13
1.4 Passivity Systems 13
1.5 Time Delay Systems 16
1.6 Delay-Dependent / Delay-Independent Conditions 19
1.7 Contribution of the Thesis 20
1.8 Brief Sketch of the Contents 21

Chapter 2 Delay-Dependent Robust Passivity Stability Analysis for Uncertain Genetic Regulatory Networks with Interval Time-Varying Delays 22
2.1 Introduction 22
2.2 Problem formulation 25
2.3 Mathematical formulation 28
2.4 Examples 41
2.5 Summary 50

Chapter 3 Design of State Estimator for Genetic Regulatory Networks with Time-varying Interval Delays Using LMI-Based Approach and Hybrid Taguchi-Genetic Algorithm 52
3.1 Introduction 52
3.2 Problem formulation 55
3.3 Mathematical formulation 60
3.4 Examples 69
3.5 Summary 83

Chapter 4 Conclusions and Future Research 84
4.1 Conclusions 84
4.2 Further Research Directions 84

References 86
Biography 95

List of Figures
Fig. 1.1 The process of gene transcription and translation 2
Fig. 1.2 The type of Biological chips 4
Fig. 1.3 Gene network with Boolean model 5
Fig. 1.4 Genetic regulatory network with a feedback loop for transcription and splicing processes 6
Fig. 1.5 Single input and output with time delay system block diagram 17
Fig. 2.1 Trajectories of mi(t)(i=1,2,3) 44
Fig. 2.2 Trajectories of pi(t)(i=1,2,3) 44
Fig. 2.3 Trajectories of mi(t)(i=1,2,...,5) 49
Fig. 2.4 Trajectories of pi(t)(i=1,2,...,5) 49
Fig. 3.1 Response of the true state m1(t)(solid) and its estimation ~m1(t)(dash-dot) 72
Fig. 3.2 Response of the true state m2(t)(solid) and its estimation ~m2(t)(dash-dot) 73
Fig. 3.3 Response of the true state m3(t)(solid) and its estimation ~m3(t)(dash-dot) 73
Fig. 3.4 Response of the true state p1(t)(solid) and its estimation ~p1(t)(dash-dot) 74
Fig. 3.5 Response of the true state p2(t)(solid) and its estimation ~p2(t)(dash-dot) 74
Fig. 3.6 Response of the true state p3(t)(solid) and its estimation ~p3(t)(dash-dot) 75
Fig. 3.7 Response of the estimation errors emi(t)(i=1,2,3)75
Fig. 3.8 Response of the estimation errors epi(t)(i=1,2,3)76
Fig. 3.9 Response of the true state m1(t)(solid) and its estimation ~m1(t)(dash-dot) 79
Fig. 3.10 Response of the true state m2(t)(solid) and its estimation ~m2(t)(dash-dot) 79
Fig. 3.11 Response of the true state p1(t)(solid) and its estimation ~p1(t)(dash-dot) 80
Fig. 3.12 Response of the true state p2(t)(solid) and its estimation ~p2(t)(dash-dot) 80
Fig. 3.13 Response of the estimation errors em1(t) and em2(t) 81
Fig. 3.14 Response of the estimation errors ep1(t) and ep2(t) 81

List of Tables
Table 2.1 The derivatives of the delays dm and dp are larger than 1 50
Table 3.1 The derivatives of the delays dm and dp are larger than 1 82

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