# 臺灣博碩士論文加值系統

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 In this dissertation, operational matrix methods are used to study theproblems of analysis, parameter identification, model simplification,control system design, and optimization of dynamic systems. The approachis the expansion of the variables of the systems into series of orthogonalfunctions. The key concept is the representation of the mathematicaloperations of the vectors with the orthogonal functions as elements byoperational matrices. The mathematical operations include differentation,integration, delay, and stretching of the independent variable of afunction. Block pulse functions and laguerre polynomials are chosen as theorthogonal functions in this dissertation. This is due to their peculiarforms of the operational matrices.Block pulse functions are a set of functions which are obtained by (1)partitioning the problem interval into sub-intervals, and (2) definingfunctions which vanish everywhere except on a chosen sub-interval where avalue of unity is assumed. The set of functions is naturally orthogonal.The integration operational matrix which relates block pulse functions totheir integrals is an upper trianglar matrix that consists of diagonalelements h/2 and the other elements being h; where h is the sub-intervallength. By taking advantage of this peculiar arrangement of theoperational matrix of block pulse functions, an effective and recursivealgorithm is first developed for solving the inverse Laplace transform.The presentation of stretch operational matrix of block pulse functionsenables one to apply block pulse functions to solve the population balanceequations which arise in the description of particulate processes.The problems of analysis, parameter estimation, and optimization of thelinear time-delay systems are also considered via a coefficient shiftoperational matrix which converts the original delay differentalinput-output model to a linear algebraic (or regression) model, which isin recursive and is convenient for a least squares estimation of unknownparameters. In order to save compuation time and preserve accuary inapplying block pulse fucntions to stiff differential equations, theordinary block pulse functions are extended to a more general fashion withadjustable pulse width.The operational matrices which similiar to those of the block pulsefucntions also exist for Laguerre polynomials. Thus, Laguerre polynomialscan also be applied to almost of the abovd mentioned problems.The integration operational matrix of Laguerre polynomials is generalizedto analyze distributed systems characterized by irrational transferfunctions. The special form of the convolution operational matrix is usedto solve Volterra type integral equations. A recursive formula ispresented to generate the stretch operational matrix for Laguerrepolynomials. By using this stretch operational matrix, functionaldifferential equation of the type dy(x)/dx=y(x) can be handled viaLaguerre polynomials.In parameter identification, the original input-output data are convertedto Laguerre spectral which are good information-bearing quantities.Combining Laguerre spectral and integration operational matrix of Laguerrepolynomials, an identifying algorithm is proposed. An important feature ofthis method is that it permits the identification of unknown initalconditions simultaneously with the parameter identification.No matter using block pulse functions or Laguerre polynomials, theproposed algorithms appear to be computationally recursive and convenient.This is owing to the elegant properties of the these functions. Besides,the algorithms are suitable for computer programming and provide thedesired accuracy solutions with resonable computation time.本論文係利用運算矩陣法有關動態系統之分析、參數鑑別、模式簡化、控制系統設計及最佳化等問題。文中所採用的研究方法，是將系統的變數以正交函數的級數展開。關鍵性的觀念，在於由正交函數為元素所構成的向量之數學運算用運算矩陣來表示，這些數學運算包括函數對其自變數的微分、積分、延遲及伸縮。本論文分別選用了方塊脈波函數及Laguerre多項式作為系統變數以正交級數展開式表示的陸底，這是因為這兩種函數各具有其特殊形式的運算矩陣。方塊脈波函數是經由以下兩步驟定義之：(1) 所將的問題區間分割成許多小區間，(2) 定函數值在所選定的一小區間內為1 ，而在其他小區間為0 ，因此這組函數自然是正交。關連方塊脈波函數及其積分的積分運算矩陣是一個上三角的矩陣，其中對角線上的元素為h ，而他的元素為h/2 ；h 是小區間的長度。利用此運算矩陣的特殊排列的優點，首先導出一個簡單又有效的遞迴方式，以求反Laplace轉換的數值解。其次提出了一個方塊脈波函數的伸縮運算矩陣，以解描述顆粒程序的顆數平衡方程式。應用係數轉移運算矩陣，將方塊脈波函數的應用延伸到線性延遲系統之分析，鑑別及最佳化的問題。轉移運算矩陣及積分運算矩陣，將原延遲微分模式改變成一組聯立線性的代數式，此組代數式具有遞迴性，此可以很方便地解出延遲系統的分憂常數近似解，並可利用最小平方誤差估計法，求得系統的未知參數。傳統等區間長度的方塊脈波函數，被延伸到區間長度可調整的形式。用這些通用式的方塊脈波函數來解Sitff微分方程式，不但節省計算時間，而且維持準確度。Laguerre多項式亦具有類似於方塊脈波函數的運算矩陣，因此Laguerre多項式可應用到前面所提的大部份問題。Laguerre積分運算矩陣，可用於解狀態方程式，並將此積分運算矩陣加以推廣，用來分析分配系統的無理轉移函數。利用Laguerre多項式所具有的特殊褶積運算矩陣，導出了解Volterra形式的積分方程式。此外利用由遞迴公式所產生的伸縮運算矩陣解dy(x)/dx=y( λx)形式的泛函微分方程式。在參數鑑別方面，首先將系統的軟輸出資料轉換成Laguerre頻譜。利用Laguerre多項式的積分運算矩陣及Laguerre頻譜，提出了一個鑑別的方法，這個方法的主要特性是可以同時鑑別系統的起始條件及系統常數。最後將這方法應用在模式簡化及控制系統的設計。不論用方塊脈波函數或Laguerre多項式，所提出的絕大多數運算矩陣法，在計算上都具有遞迴性，不但適於計算機的程式化，而且用合理的計算時間即可得到希望的準確度。
 COVER,摘要,ACKNOWLEDGEMENTS,CONTENTSCHAPTER 1 INTRODUCTION1.1 GENERAL BACKGROUND1.2 ALGEBRAIC METHODS1.3 OPERATIONAL MATRIX APPROACH1.4 SCOPE AND ORGANIZATIONCHAPTER 2 BLOCK PULSE FUNCTIONS AND THEIR APPLICATIONS IN ANALYSIS2.1 BLOCK PULSE FUNCTIONS2.2 INVERSION OF THE RATIONAL LAPLACH TRANSFORM2.3 SOLUTIONS OF POPULATION BALANCE EOUATIONS2.4 REMARKSCHAPTER 3 APPLICATION OF BLOCK PULSE FUNCTIONS IN TIME-DELAY SYSTEMS3.1 BLOCK PULSE APPROXIMATION OF DELAY SYSTEMS3.2 PARAMETER ESTIMATION3.3 SUBOPTIMAL CONTROL OF TIME-DELAY SYSTEMS3.4 REMARKSCHAPTER 4 SOLUTIONS OF STIFF DIFFERENTIAL EQUATIONS VIA GENERALIZED BLOCK PULSE FUNCTIONS4.1 GENERALIZED BLOCK PULSE FUNCTIONS4.2 ANALYSIS OF LINEAR SYSTEMS4.3 NONLINEAR DIFFERENTIAL EQUATIONS4.4 REMARKSCHAPTER 5 APPLICATIONS OF LAGUERRE POLYNOMIALS IN ANALYSIS OF DYNANIC SYSTENS5.1 LAGURRE POLYNOMIALS AND THEIR RELEVANT OPERATIONAL MATRICES5.2 SOLUTION OF STATE EQUATIONS5.3 SOLUTIONS OF INTEGRAL EQUATIONS5.4 SOLUTION OF FUNCTIONAL DIFFERENTIAL EQUATIONS5.5 REMARKSCHAPTER 6 LAGUERRE OPERATIONAL MATRICES FOR FRACTIONAL CALCULUS AND APPLICATION6.1 OPERATIONAL MATRIX OF INTEGRATION6.2 OPERATIONAL MATRIX OF DIFFERENTIATION6.3 OPERATIONAL MATRICES FOR FRACTIONAL CALCULUS6.4 APPLICATION6.5 REMARKSCHAPTER 7 APPLICATIONS OF LAGUERRE POLYNOMIALS IN MATCHING PROBLEMS7.1 PARANETER IDENTIFICATION7.2 MODEL SIMPLIFICATION7.3 CONTROL SYSTEM DESIGN7.4 REMARKSCHAPTER 8 CONCLUSIONS8.1 SUMMAPY8.2 RECOMMENDATIONS FOR FURTHER RESEARCHESAPPENDIX RECURSIVE FORMULASREFERENCESVITA
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