Estimation Of Upper And Lower Bounds For Solutions Of The Unified Algebraic Riccati Matrix Equations Based On Delta Operator

With the fast improvement and progress of modern science and technology,the research and application of the control system has been paid more and more attention by scholars.The three vital properties of the modern systems of control are the system’s stability,controllability and observability,and the study of these properties can usually be transformed into the study of the positive semi-definite solution of the Riccati matrix equation corresponding to the relevant system.However,it is quite difficult to solve the positive semi-definite solution of the Riccati matrix equation at present,and we usually only need to discuss the bound estimation of its solution in practical problems.The unified algebraic Riccati matrix equation based on the delta operator can unify the discrete algebraic Riccati equation and the continuous algebraic Riccati equation.Therefore,it is particularly important to study the bound estimation of the unified algebraic Riccati equation based on the delta operator,and some of conclusion have been got.This paper mainly concentrates on the estimation of the upper and lower bounds for solutions of the unified algebraic Riccati matrix equation.In this paper,by using simple matrix inequality scaling and properties of positive definite matrices,we discuss upper bound and eigenvalues upper bound for the solution of a class of unified algebraic Riccati equation,Numerical examples to verify the validity of the conclusions obtained under the same conditions.By using the properties of positive definite matrices and the scaling of eigenvalue inequality,we propose two upper bounds and eigenvalues upper bounds for the solution of the unified algebraic Riccati equation under two different conditions.This upper bound improves the existing results under the same conditions.Numerical examples are prove that the three different upper bounds under three different conditions have their own advantages.Finally we construct the matrix sequence,we discuss the progressive iterative upper bound for the solutions of the unified algebraic Riccati equation.By using mathematical induction and the monotone bounded theorem,we proved that the constructed sequence converges and converges to the solution of the unified algebraic Riccati matrix equation.Finally,by using the properties of positive definite matrices and the scaling of eigenvalue inequality,we show two lower bounds and eigenvalues for the solution of the unified algebraic Riccati equation,which improves and extends the existing research results,and numerical examples to prove the two lower bounds have their own advantages.Construct a sequence of matrices based on the obtained lower bound,we show the progressive iterative lower bound for the solutions of the unified algebraic Riccati equation.By using mathematical induction and the monotonic bounded theorem,we proved that the constructed sequence converges and converges to the solution of the unified algebraic Riccati matrix equation.