Iterative solution to linear matrix inequality arising from time delay descriptor systems

Linear matrix inequalities (LMIs) are widely used to analyze the stability or performance of time delay descriptor systems (TDDSs). They are solved by the well known interior-point method (IPM) via minimizing a strictly convex function by transforming the matrix variable into an expanded vector variable. Newton’s method is used to get the unique minimizer of the strictly convex function by the iteration involving its gradient and Hessian. The obvious disadvantage of the IPM is the high storage requirement for the Hessian. Hence, this often renders that the IPM cannot solve “large” LMI problems due to finite memory limit. To overcome this shortcoming, for the first time, an iterative algorithm based on the steepest descent method (SDM) is proposed to solve LMIs by keeping matrix variable form instead of transforming it to an expanded vector and without using Hessian matrix. The gradient of the proposed objective function is explicitly given by a matrix function with the same dimension of the original matrix variable. The efficiency of the proposed method is verified with numerical examples.