General Order Matrix Thiele-Padé Approximation And Its Applications In Control Theory

For the sake of the complexity of the nature,non linear approximation is popular toacademic circle.This article enriched the theory and algorithms of bivariate matrix Pad′eapproximation.It establishes a Matrix Thiele-PadéApproximation based on the generalizedmatrix inverse, provides its recurrence algorithm, and employed in control systems;This papergives a kind of Bivariate symmetrical Thiele matrix rational interpolation,and discussed someapproximation properties.The paper includes four chapters. Chapter one is introduction.It mainly introduces thebackground,classical results and my ideas.The second chapter reviews one variable Thiele matrix rational interpolation whichwill be generalized.It includes the generalized matrix inverse,the background of one variableThiele matrix rational interpolation,recurrence algorithms and properties.The last section inthis chapter introduces the definition of bivariate Thiele matrix rational interpolation and itsproperties.Chapter three and four is the primary part of this paper.In section one of chapter threethe author uses continued fraction(CF)to set up a new method of matrix valued rationalinterpolation-bivariate symmetrical Thiele matrix valued rational interpolation.The methodfirst defines bivariate matrix inverse diferences by matrix generalized inverse,so give thetheorem about bivariate matrix valued interpolating CF,and a numerical example followingthe proof.In section two we probe into its existence,divisibility and orders.Owing to the closed relations between interpolations and approximations,we discussedMatrix Thiele-PadéApproximation in chapter four. Section one reviewed existing bivariatematrix Padéapproximation methods.In section two we give the recurrence algorithm of bi-variate symmetrical Thiele matrix CF expansion which is the generalization of one variable.Itsmain idea is to get the limit of bivariate inverse diferences,we define the limit by bivariateinverse derivatives.We can provide the bivariate symmetrical Thiele matrix CF expansion assoon as these bivariate inverse derivatives are found.We searched the subtle relations betweenone variable and two variable inverse diferences,so we find the algorithm.In section three weshow such approximation’s definition,following an elementary application in control systems.