臺灣博碩士論文加值系統

　　本論文係以轉移式勒甘德（Shifted Legendre）多項式函數之積分轉換應用於動態系統（Dynamic Systems）上。其所研究之動態系統範圍包括結晶程序之模擬及控制系統之設計。
　　首先利用轉移式勒甘德多項式函數之特性，導出三個轉移式勒甘德多項式連乘積之積分式，並且利用所發展出之回遞運算策略（Recursive Algorithm），運用在計算結晶程序之方程式時，可簡化計算程式，同時計算機運算所消耗之時間亦大量減速少。再者，本文以轉移式勒甘德多項式函數所導出之積分運算矩陣（Operational Matrix For Integration），尋求控制系統之研究和兩點邊界值問題（Two-Point Boundary Valure Problems）時，均簡易有效地獲得其解。
　　模擬結晶程序系統中，由顆粒均衡方程式（Population Balance Equation）可獲得在混懸連續式結晶槽之穩定狀況晶粒分佈常微分方程式，瞬時晶粒分佈偏微分方程式以及批式晶粒分佈偏微分方程式，和由碎裂模式（Breakage Model）所得之曲伸泛函微分方程式（Strectched Functional Differential Equation）。運算方法之重點在將晶粒密度函數（Population Density Function）以轉移式勒甘德多項式函數展開，再運用積分轉換，可將偏微分（或常微分）方程式轉換成一系列之常微分（或代數式）方程式。利用轉移式勒甘德多項式函數之三個連乘積公式，不但可節省計算時間和使問題簡化，同時獲得正確之結果。
　　利用轉移式勒甘德多項式函數於控制系統之研究，包括參數識別（Parameter Identification），變分問題（Variational Problems），模式簡化（Model Reduction），控制系統設計（Control System Design）和動態系統模擬等。利用積分運算矩陣和最小平方誤差估計法，將系統的輸入輸出資料轉換成轉移式勒和最小平方誤差估計法，將系統的輸入輸出資料轉換成轉移式，由計算方法，亦皆得令人滿意之結果。
　　本文對於由工程問題所引發之兩點邊界值問題亦加以研究。由於轉移式勒甘德多項式函數之特性，可得對兩點邊界值問題和起始值問題（Initial Value Problems）施於相同計算方法。因此，兩點邊界值問題之運算大為簡化，計算機運算所需時間亦大為降低。

　　In this dissertation, the integral transformation of the shifted Legendre functions is employed to study the problems of dynamic systems which are related to the crystallization processes and control models.
　　The recursive algorithm of the integration of the triple multiplication of the shifted Legendre functions is first developed. The calculation procedure is simplified, hence the computation time needed is greatly reduced in the simulation of crystallization processed. Furthermore, the operational matrix for the integration of the shifted Legendre functions are also first derived and applied to the dynamic systems and the two-point boundary value problems. Staisfactory results are obtained.
　　The ordinary differential equation, partial differential equations and stretched differential (or integro-differential equations of the population density function are obtained during modeling the crystallization processes through various operating conditions. The key method for simulating such equations is that the population density function is experessed by a series of the shifted Legendre polynomial functios. The partial differential equation (or ordinary) is transformed into a series of ordinary differential equations (or algebric) of expansion coefficients by applying the shifted Legendre functions. Using the characteristics of the integration of the triple multiplication of the shifted Legendre polynomial functions. not only the computationsla time is greatly saved as well as the problem is simplifed, but also the computational results are satisfactory.
　　Using the operational matrix of the shifted Legendre polynomial functions and the minimization of the least square error to approach the control problem, state and control variables are converted into a series of expansion coefficients of the shifted Legendre functions. The cases study are parameter identification, variational problems, model reduction and control system design. The solutions are obtained by solving the expansion coefficients. The computational results are accurate.
　　In the study of two-point boundary value problems which are concerned with the engineering systems, the same calculation procedure as that of the initial value problems are obtained by the operational matrix of the shifted vegendre polynomial functions. Thus, the computational algorithm of the two-point boundary value problems can be simplifed and the computation time is also greatly reduced.

目錄
圖目錄
表目錄
符號說明
摘要
第一章 緒論
1-1 前言
1-2 文獻回顧
1-3 章節及組織
第二章 勒甘德多項式函數之特性
2-1 轉移式勒甘德多項式函數之積性
2-2 轉移式勒甘德多項式函數的運算矩陣
2-3 勒甘德－高斯數值積分法
第三章 轉移式勒甘德多項式函數在結晶程序上之應用
3-1 解常微分方程式及在穩定連續結晶程序上之應用
3-2 轉移式勒甘德多項式函數在偏微分方程式之應用
3-3 模擬實際瞬間結晶程序方程式之解
3-4 批式結晶程序模擬－冰－盬水系統
第四章 應用轉移式勒甘德多項式函數解曲伸方程式
4-1 晶粒均衡方程式
4-2 轉移式勒甘德近似法之修飾
第五章 轉移式勒甘德多項式函數在動態系統之應用
5-1 動態系統方程式之轉移式勒甘德函數近似解
5-2 運算矩陣在動態系統分析上之應用
5-3 邊界值問題之狀態方程式解
5-4 分佈參數系統之最佳控制
第六章 轉移式勒甘德多項式函數之積分運算矩陣在控制系統上之應用
6-1 參數鑑別
6-2 模式簡化
6-3 控制系統設計
6-4 變分問題之直接法
第七章 結論
附錄A 公式(4-9)之衍導
參考文獻
自傳
謝誌
英文摘要