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 本文主要研究探討以下幾個問題：連續二維系統之狀態空間實現、離散二維線性系統的最佳控制、離散二維線性系統的最佳追蹤器、連續時間二維系統之離散化二次最佳化控制，及利用連續時間最佳追蹤器之二維線性系統之數位再設計。 本文的貢獻在於(一)藉由已知的轉移函數，提出連續二維系統狀態空間實現的準則，(二)根據羅莎(Roesser)模型，吾人推導出二維系統的一維等效模型，並提出具有可變係數及自由邊界條件之離散二維系統之最佳控制方法，(三)建構一個控制律，並藉由閉迴路控制在完整時間內使離散二維系統追蹤想要的軌跡，(四)提出連續二維系統之離散化二次最佳控制，利用新的狀態向量直接轉換原始連續二維二次成本函數為耦合之離散型態，以建構具有新虛擬控制輸入的新虛擬離散二維模型，間接的求出連續二維系統的離散化二次最佳控制及(五)提出新的二維系統的最佳數位再設計方法，得到動態數位控制法則。
 In this dissertation, several researches are presented: state-space realization from a given transfer function of continuous-time two-dimensional systems, optimal regulator for discrete-time two-dimensional linear systems, discrete linear quadratic tracker for two-dimensional linear systems, discretized quadratic optimal regulator for continuous- time two-dimensional systems and digital redesign of continuous-time suboptimal tracker for two-dimensional systems. This includes the following novel features: (i) An algorithm is presented for the state-space realization of continuous-time two-dimensional systems via a given transfer function, (ii) Based on the Roesser’s model, an equivalent general one-dimensional model of the two-dimensional system has been presented. Then, an optimal control method for discrete-time two-dimensional linear systems with variable coefficients and free boundary conditions is developed, (iii) Construct a control scheme to make the discrete-time two-dimensional linear systems follow the desired trajectories over the entire time intervals by using a closed-loop control law, (iv) A discretized quadratic optimal control for continuous-time two-dimensional system is proposed. A new state vector to directly convert the original continuous-time two-dimensional quadratic cost function into a decoupled discretized form is newly induced. Then, a new virtual discrete-time two-dimensional model with the new virtual control input is constructed for indirectly finding the desired discretized quadratic optimal regulator for the continuous-time two-dimensional system, and (v) Present a new optimal digital redesign technique of two-dimensional system for finding a dynamic digital control law from the given continuous-time two-dimensional systems by minimizing a quadratic cost function.
 Chapter 1 Introduction 1.1 Two-dimensional systems and applications 1.2 State-space models of two-dimensional linear systems 1.3 Relations between the models 1.4 Optimal control problem of two-dimensional systems 1.5 Organization of the dissertation Chapter 2 Mathematical Preliminaries 2.1 Transition matrix and general response formula for Roesser’s model 2.2 Controllability and observability of two-dimensional systems 2.3 Stability of two-dimensional systems Chapter 3 State-Space Realization of Continuous-Time Two-dimensional Systems 3.1 Introduction 3.2 Background 3.3 Realization of continuous-time two-dimensional systems 3.4 An illustrative example 3.5 Summary Chapter 4 Optimal Control for Discrete-Time Two-Dimensional Linear Systems with Variable Coefficients 4.1 Introduction 4.2 An equivalent one-dimensional state-space model for discrete-time two-dimensional linear systems 4.3 Optimal control law for discrete-time two-dimensional linear systems 4.4 An illustrative example 4.5 Summary Chapter 5 Discrete Linear Quadratic Tracker for Discrete-Time Two-Dimensional Linear Systems 5.1 Introduction 5.2 An equivalent one-dimensional state-space model for discrete-time two-dimensional linear systems 5.3 Discrete linear quadratic tracker for discrete-time two-dimensional systems 5.4 An illustrative example 5.5 Summary Chapter 6 Discretized Quadratic Optimal Control for Continuous-Time Two-Dimensional Systems 6.1 Introduction 6.2 The discretized cost function 6.3 Optimal control of discretized continuous-time two-dimensional systems 6.4 An illustrative example 6.5 Summary Chapter 7 Alternative Digital Redesign of Continuous-Time Suboptimal Tracker for Two-Dimensional Systems 7.1 Introduction 7.2 The discretized cost function 7.3 Linear quadratic tracker for discretized continuous-time two-dimensional systems 7.4 An illustrative example 7.5 Summary Chapter 8 Conclusions
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