Iterative solvers for periodic matrix equations and model reduction for periodic control systems

Abstract

Analysis and synthesis of periodic control systems for various engineering and mechanical problems are strongly related to the well known matrix equations, i.e., Lyapunov or Riccati equations. This paper presents the iterative methods for solving large-scale sparse projected discrete-time periodic Lyapunov equations which arise in periodic state feedback control problems and in model reduction of periodic descriptor systems. We extend the alternating direction implicit method and the Smith method to such equations and discuss their applications to the model reduction procedure for periodic descriptor systems. Numerical results are given to illustrate the efficiency and accuracy of the proposed methods.

Matrix form of Biconjugate Residual Algorithm to Solve the Discrete‐Time Periodic Sylvester Matrix Equations

“…can be given by the discrete-time periodic Sylvester matrix equations [6]. Various methods have been proposed to solve the linear matrix equations [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. New variants of the squared Smith iteration and Krylov subspace based methods were proposed for the approximate solution of discrete-time periodic Lyapunov matrix equations [23].…”

“…Also, the cyclic lifted representations of the linear discrete‐time periodic descriptor system with time‐varying dimensions can be given by the discrete‐time periodic Sylvester matrix equations . Various methods have been proposed to solve the linear matrix equations . New variants of the squared Smith iteration and Krylov subspace based methods were proposed for the approximate solution of discrete‐time periodic Lyapunov matrix equations .…”

Gradient Based Iterative Algorithm to Solve General Coupled Discrete-Time Periodic Matrix Equations Over Generalized Reflexive Matrices

“…The applications of linear matrix equations have motivated both mathematicians and engineers to construct methods catering to solve linear matrix equations [1,4,6,7,8,9,19,23,25]. Based on Smith iterations [24], iterative methods were developed for periodic standard Lyapunov matrix equations and projected generalized Lyapunov matrix equations [27,28].…”