A Sampled-Data Formulation for Boundary Temperature Control of A Threadlike Metal Wire

本篇論文在提出一種解決金屬細線邊界溫度控制問題之分析方法。藉由邊界條件滿足司徒姆-呂維耳問題(Strum-Liouville problem)之規則形式，使得有限維之熱傳導方程式被格式化成一無限次序之離散時間控制問題；並由傅立葉級數最小平方差定理取得有限維離散時間系統之近似解，同時證明滿足與特徵值相關之可控制及可觀察性。為驗証本論文之分析推導結果，則採用兩種Matlab方法，一為利用pdepe.m功能直接解出系統偏微分方程式，另一為利用本論文提出之有限維常微分方程式方法來進行數值模擬。This thesis presents an analytic solution to the boundary temperature control problem for a threadlike metal wire. With boundary conditions satisfying the regular form of Strum-Liouville problem, the infinite-dimensional linear heat-conduction equation can be formulated into an infinite sequence of discrete-time control problems. The finite-dimensional approximation of the discrete-time system is obtained by the minimum square error theorem of Fourier series. The controllability- observability conditions are related to the associated eigenvalues. As a verification to the proposed analytic solution, numerical simulations are conducted using both a Matlab partial differential equation solver, pdepe.m , and the proposed finite dimensional ordinary differential equation approximation.Chinese Abstract………………………………………….……...…i English Abstract………………………………………….…...……ii Acknowledgement…………………………………………....……iii Contents………………………………………….………...………iv List of Figures………………………………………….………….vi List of Tables………………………………………….…...……...vii Nomenclature….………………………………….……………...viii Chapter 1: Introduction……………………………………….…..1 1.1 Motivation…………………………………………………….…………1 1.2 Introduction……………………………………………………………...2 1.3 Mathematical Description of Heat Equation….…………………………5 1.4 Goal of the Thesis………………………………………………………..9 1.5 Organization of the Thesis……………………………………………….9 Chapter 2: Problem Statement….……………………….………...11 2.1 System Modeling………………….………………………….………...11 2.2 Initial and Boundary Condition..………………... ………….…………12 2.3 Solution to Constant Input…………..………………………………….15 Chapter 3: Discrete-Time Representation………………….…...26 3.1 Discretization through Zero-Order-Hold….……………………………26 3.2 Finite State Approximation ……...………………….…………………28 3.3 Sampled-Data Formulation..…………………………………………...32 Chapter 4: A Numerical Example……………..…………………38 4.1 Condition of Heat-Transfer Coefficient h=1...........................................39 4.1.1 Solving the Parabolic Equation by MATLAB……………………39 4.1.2 The Sampled-Data Solution......…………………………………..40 4.2 Condition of Heat-Transfer Coefficient h=2…………………………...43 4.2.1 Solving the Parabolic Equation by MATLAB……………………43 4.2.2 The Sampled-Data Solution……..………………………………..45 Chapter 5: Conclusion and Future Work……….………………48 5.1 Conclusion…………………………………………………………...…48 5.2 Future Work…………………………………………………………….49 Reference…………………………………………………………..5