The Upper Bound For The Solution Of The Continuous Algebraic Riccati Matrix Equation And Its Application

With the rapid expansion and progress of modern science and technology,the control system has been paid much attention by many researchers.When analyzing a control system,we first consider the three corresponding properties,namely its stability,controllability,and observability,and the study of these nature can be converted into solving the solution of the corresponding Riccati matrix equation or estimating the bounds of its solution.Therefore,in recent years,the solution of Riccati matrix equation and the bounds of its solution has attracted much attention.This paper focuses on the estimation of the upper bound of the solution of the CARE corresponding to the linear system,and then studies the stability of a class of time-delay system and the redundancy control problem.First,in line with properties of the constrain matrix Riccati equation,a semi-positive definite matrix is constructed by the method of continuous transition.Then,by singular value decomposition of the coefficient matrix of the constraint matrix Riccati equation,matrix identity transformation and congruent transformation,we transform the CARE into a matrix equation with two parameters.Next,by taking advantage of the necessary and sufficient conditions that the real part of the matrix is positive definite,we obtain an upper bound of the solution of the CARE and demonstrate the obtained upper bounds ameliorate some recent results.In addition,we give some numerical examples to verify our consequence.In this paper,basing on previous results,we construct a Lyapunov function by utilizing the positive semi-definiteness of solutions of the CARE.Then,by using matrix inequalities and the definition of Lyapunov stability,a simple condition for stability of time-delay systems is obtained.Finally,by using the properties of matrix trace,the properties of matrix spectral norm and matrix inequality,a condition for slowing the saturation velocity of the actuator is obtained.