The Upper And Lower Matrix Bounds For The Solution Of The Discrete And Discrete Coupled Algebraic Riccati Matrix Equation And Their Application In The Multi-agent System

In engineering calculation,biological engineering and automatic control fields,many problems can often be converted into studying related matrix equations.A large number of scholars have made a thorough study of the algebraic Riccati equa-tion in recent years because of its wide application in the control system.Among them,the discrete algebraic Riccati equation plays a major role in the analysis and design of many control systems,especially in the optimal control.And the dis-crete coupled algebraic Riccati equation plays a fundamental role in the jump linear quadratic optimal control problem.Therefore,the discussion of these equations not only has important practical significance but also has high application value.We discuss the solution bounds of the discrete algebraic Riccati matrix and their some applications in the multi-agent system.And this thesis gives the matrix bounds and an existence uniqueness condition,an iteration algorithm for the solution of the discrete coupled algebraic Riccati matrix equation.In chapter one,we introduce some research background and research status of the algebraic Riccati equation,the main work of this paper,and some of the preparatory knowledge to be used later.In chapter two,under the assumption of the existence of the solution for the discrete algebraic Riccati equation we get matrix bounds for this equation by using the controllability of the matrix pair(A,B)to construct the positive semi-definite matrix and using majorization inequalities and eigenvalue inequalities.And an iter-ative algorithm for the solution of the discrete algebraic Riccati equation is proposed using the lower bound and the Monotone Convergence Theorem of positive oper-ators.Then,the distributed state feedback design of the consensus of multi-agent system is given by using the upper and lower matrix bounds of the discrete alge-braic Riccati equation.Finally,two corresponding numerical examples are used to illustrate the effectiveness of the derived results.In chapter three,we improved a upper matrix bound for the solution of the discrete coupled algebraic Riccati equation by using the properties of M-matrix and its inverse matrix,combining scaling techniques of matrix inequalities and eigenvalue inequalities.We offer new upper and lower matrix bounds of the discrete coupled algebraic Riccati equation by using the matrix Schur complement to construct the equivalent form of the discrete coupled algebraic Riccati equation,in terms of the improved upper matrix bound.Then,using a contraction map and a fixed point theorem we discussed an existence uniqueness condition and an iteration algorithm for the solution of this equation.Finally,we give the corresponding numerical examples to show that our results are valid.