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A Class of Continuous Nonsymmetric Coupled Riccati Equations

Author: LuoFangFang
Tutor: LiuJianZhou
School: Xiangtan University
Course: Applied Mathematics
Keywords: Coupled Nonsymmetric Continuous Algebraic RiccatiEquations M-matrix Newton’s iteration method fixed-basic iteration method
CLC: O241.6
Type: Master's thesis
Year: 2011
Downloads: 14
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Abstract


In the Jump Linear(Linear-Quadratic )control systems, the optimal controlis one of the most concerned questions. Since the control equations are alwayscomplex in the realistic exist, it is very hard for the researchers to solve the con-trol parameter point-blank、directly. In order to solve the questions conveniently,we need optimize the condition in the allowable scope in which the system couldhold in intrinsic state. Then we could translate it into the equivalent questionand continue the study. Among the equivalent questions, Nonsymmetric Cou-pled Riccati Equations is an important class of the control problems. In recentyears, the Riccati Equations and Lyapnov Equations were always the hot topicsin control field.In this paper, we use Newton’s iteration and fixed-basic iteration respectivelyto study the existence of the positive solutions of the coupled Riccati equations,which arising in Jump Linear(Linear-Quadratic )control system. And we alsogive some analysis about the method convergence. In the same time we provethat once the equations have positive solutions, then both methods could findtheir minimum positive solutions. The paper is constructed with following parts:In chapter one, we first present some background knowledge and recent worksabout the coupled Riccati equations. Next we introduce a new class equations,i. e: The Continuous Nonsymmetric Coupled Riccati Equations and describe theapplication of the methods of Newton’s iteration and fixed-basic iteration. In theend , we give the main work.In chapter two, we adopt the first Fre′chet partial derivative and the secondFre′chet partial derivative to give the Newton’s iteration, and based on the itera-tion we prove the existence of the positive solutions and analysis the convergence.emulation examples illustrate the result e?ectiveness.In chapter three, we combine the reasonable splittings of matrices and splitthe linear coe?cient matrices into two parts, but the front matrices still keep ini-tial probabilities. And then we give the fixed-basic iteration in accordance withthe splitting matrices equality, moreover, we also give the proof of existence ofthe positive solutions and state the method convergence . emulation examplesillustrate the method feasibility and e?ectiveness.In chapter four, in allusion to the respective characteristic of Newton’smethod and fixed-basic method, we synthetically use their strong points to dis- cuss the thought of utilizing the two iteration formes in one iteration procedure,and give the corresponding algorithm. already examples illustrate the advanta-geous.

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CLC: > Mathematical sciences and chemical > Mathematics > Computational Mathematics > Numerical Analysis > Linear algebra method of calculating
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