The Bounds For The Solution Of The Continuous Coupled Algebraic Lyapunov Matrix Equation And Its Application In A Class Of Time-delay System

With the rapid development of modern technology and mechanical engineering,the research and application of control systems has received unprecedented attention from scholars.When studying control systems,people often need to consider and study their stability analysis and optimization problems.In many cases,the problem can be transformed into the solution of the corresponding Lyapunov matrix equation and the upper and lower bounds of its solution.Therefore,the research on the semi-positive definite solution of Lyapunov matrix equations and its application have attracted the attention of many scholars,and in the theory and specific applications,many research results have been obtained.In this paper,we use the Kronecker product of the matrix,the non-negative of the M-matrix inverse and the inequality of the matrix to transform the continuous coupled algebraic Lyapunov matrix equation,and use the identity of the matrix and the prop-erties of the positive definite matrix,the two upper bounds containing the parameters.Then,by constructing the equivalent form of the continuous coupled algebraic Lyapunov matrix equation,and then using the properties of the matrix eigenvalues,the identity transformation of the matrix,combined with the scaling of the matrix inequality,the lower bound of the continuous coupled algebraic Lyapunov matrix equation is obtained.Moreover,the validity of the results obtained is verified by specific numerical examples,and their accuracy is compared.Finally,according to the upper bound of the solution of the continuous coupling algebra Lyapunov matrix equation obtained by the paper,combined with the matrix identity transformation and the inequality scaling and selecting the appropriate Lya-punov function,the conditions for stabilizing a class of time-delay systems are obtained,and use specific numerical examples to illustrate the validity of the results.