Estimation Of The Upper And Lower Bound For The Solutions Of The Unified Algebraic Lyapunov Equations Based On The Delta Domain And Their Numerical Algorithms

In the performance analysis of the system,stability,measability and energy control are all important criteria for analyzing the performance of the constructed system,and the discussion of these problems often can be equivalent into the solution of Lyapunov matrix equation and the bound of the solution.The Lyapunov equation can be divided into s-domain,z-domain,δ-domain,so far,for the continuous algebraic Lyapunov matrix equation(s-domain)and discrete algebra Lyapunov matrix equation(z-domain)and the bound have achieved systematic results.However,there are few finding about the upper and lower bounds of the unified algebraic lyapunov equation solution and its solution based on the δ-domain.In this paper,using the correlation properties of the Schur complement and matrix inequality,make some extensions of the upper and lower bound estimates of the solution of the unified algebraic Lyapunov equation on the δ-domain,and give two iterative algorithms to solve the unified algebraic Lyapunov equation.At the same time,the convergence of the given iterative algorithm is proved theoretically.Finally,the given upper and lower bounds and the feasibility of the algorithm are verified with numerical examples.In the first chapter,the background and research status of the unified algebraic Lyapunov equations based on the δ-domain are mainly introduced,along with the relevant marks and preparatory knowledge used in this paper.In the second chapter,combining matrix inequalities,Schur complements and the correlation properties of positive definite matrices give a new upper and lower bound estimate of the solutions of unified algebraic Lyapunov equations through rigorous theoretical derivation procedures.We also prove theoretically the existence and uniqueness of positive definite solutions of the unified algebraic Lyapunov equation under certain conditions by using the fixed point theorem and the properties of continuous functions.In the third chapter,we first give the iterative algorithm of solving the unified algebraic Lyapunov equation with a positive definite solution combining the straightening operator,the correlation properties of the Kronecker product,and strictly prove the convergence of the algorithm.The optimal selection method for the parameter αis also presented.Then,combining the fixed point theorem,the definition of serial convergence and the Cauchy convergence law give the fixed point iteration algorithm and the convergence proof of the unified generation Lyapunov equation.Finally,we give the accelerated iteration algorithm of the fixed point iteration algorithm using the ideas of the weighted average.In the last chapter,corresponding numerical examples are presented to verify the feasibility of the given upper,lower bounds and the algorithm.