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研究生:黃奇君
研究生(外文):Huang, Qi-Jun
論文名稱:運算矩陣法應用在動態系統上之研究
指導教授:石延平石延平引用關係
指導教授(外文):Shi, Yan-Ping
學位類別:博士
校院名稱:國立成功大學
系所名稱:化學工程研究所
學門:工程學門
學類:化學工程學類
論文種類:學術論文
畢業學年度:70
語文別:中文
論文頁數:152
中文關鍵詞:運算矩陣法動態系統線性延遲系統最小平方誤差估計遞迴公式正交級數VOLTERRA形式化學工程化學
外文關鍵詞:LAGUERRECHEMICAL-ENGINEERINGCHEMISTRY
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In this dissertation, operational matrix methods are used to study the
problems of analysis, parameter identification, model simplification,
control system design, and optimization of dynamic systems. The approach
is the expansion of the variables of the systems into series of orthogonal
functions. The key concept is the representation of the mathematical
operations of the vectors with the orthogonal functions as elements by
operational matrices. The mathematical operations include differentation,
integration, delay, and stretching of the independent variable of a
function. Block pulse functions and laguerre polynomials are chosen as the
orthogonal functions in this dissertation. This is due to their peculiar
forms 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) defining
functions which vanish everywhere except on a chosen sub-interval where a
value of unity is assumed. The set of functions is naturally orthogonal.
The integration operational matrix which relates block pulse functions to
their integrals is an upper trianglar matrix that consists of diagonal
elements h/2 and the other elements being h; where h is the sub-interval
length. By taking advantage of this peculiar arrangement of the
operational matrix of block pulse functions, an effective and recursive
algorithm is first developed for solving the inverse Laplace transform.
The presentation of stretch operational matrix of block pulse functions
enables one to apply block pulse functions to solve the population balance
equations which arise in the description of particulate processes.
The problems of analysis, parameter estimation, and optimization of the
linear time-delay systems are also considered via a coefficient shift
operational matrix which converts the original delay differental
input-output model to a linear algebraic (or regression) model, which is
in recursive and is convenient for a least squares estimation of unknown
parameters. In order to save compuation time and preserve accuary in
applying block pulse fucntions to stiff differential equations, the
ordinary block pulse functions are extended to a more general fashion with
adjustable pulse width.
The operational matrices which similiar to those of the block pulse
fucntions also exist for Laguerre polynomials. Thus, Laguerre polynomials
can also be applied to almost of the abovd mentioned problems.
The integration operational matrix of Laguerre polynomials is generalized
to analyze distributed systems characterized by irrational transfer
functions. The special form of the convolution operational matrix is used
to solve Volterra type integral equations. A recursive formula is
presented to generate the stretch operational matrix for Laguerre
polynomials. By using this stretch operational matrix, functional
differential equation of the type dy(x)/dx=y(x) can be handled via
Laguerre polynomials.
In parameter identification, the original input-output data are converted
to Laguerre spectral which are good information-bearing quantities.
Combining Laguerre spectral and integration operational matrix of Laguerre
polynomials, an identifying algorithm is proposed. An important feature of
this method is that it permits the identification of unknown inital
conditions simultaneously with the parameter identification.
No matter using block pulse functions or Laguerre polynomials, the
proposed 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 the
desired 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,CONTENTS
CHAPTER 1 INTRODUCTION
1.1 GENERAL BACKGROUND
1.2 ALGEBRAIC METHODS
1.3 OPERATIONAL MATRIX APPROACH
1.4 SCOPE AND ORGANIZATION
CHAPTER 2 BLOCK PULSE FUNCTIONS AND THEIR APPLICATIONS IN ANALYSIS
2.1 BLOCK PULSE FUNCTIONS
2.2 INVERSION OF THE RATIONAL LAPLACH TRANSFORM
2.3 SOLUTIONS OF POPULATION BALANCE EOUATIONS
2.4 REMARKS
CHAPTER 3 APPLICATION OF BLOCK PULSE FUNCTIONS IN TIME-DELAY SYSTEMS
3.1 BLOCK PULSE APPROXIMATION OF DELAY SYSTEMS
3.2 PARAMETER ESTIMATION
3.3 SUBOPTIMAL CONTROL OF TIME-DELAY SYSTEMS
3.4 REMARKS
CHAPTER 4 SOLUTIONS OF STIFF DIFFERENTIAL EQUATIONS VIA GENERALIZED BLOCK PULSE FUNCTIONS
4.1 GENERALIZED BLOCK PULSE FUNCTIONS
4.2 ANALYSIS OF LINEAR SYSTEMS
4.3 NONLINEAR DIFFERENTIAL EQUATIONS
4.4 REMARKS
CHAPTER 5 APPLICATIONS OF LAGUERRE POLYNOMIALS IN ANALYSIS OF DYNANIC SYSTENS
5.1 LAGURRE POLYNOMIALS AND THEIR RELEVANT OPERATIONAL MATRICES
5.2 SOLUTION OF STATE EQUATIONS
5.3 SOLUTIONS OF INTEGRAL EQUATIONS
5.4 SOLUTION OF FUNCTIONAL DIFFERENTIAL EQUATIONS
5.5 REMARKS
CHAPTER 6 LAGUERRE OPERATIONAL MATRICES FOR FRACTIONAL CALCULUS AND APPLICATION
6.1 OPERATIONAL MATRIX OF INTEGRATION
6.2 OPERATIONAL MATRIX OF DIFFERENTIATION
6.3 OPERATIONAL MATRICES FOR FRACTIONAL CALCULUS
6.4 APPLICATION
6.5 REMARKS
CHAPTER 7 APPLICATIONS OF LAGUERRE POLYNOMIALS IN MATCHING PROBLEMS
7.1 PARANETER IDENTIFICATION
7.2 MODEL SIMPLIFICATION
7.3 CONTROL SYSTEM DESIGN
7.4 REMARKS
CHAPTER 8 CONCLUSIONS
8.1 SUMMAPY
8.2 RECOMMENDATIONS FOR FURTHER RESEARCHES
APPENDIX RECURSIVE FORMULAS
REFERENCES
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