Abstract

The discrete-time periodic matrix equations are encountered in periodic state feedback problems and model reduction of periodic descriptor systems. The aim of this paper is to compute the generalized reflexive solutions of the general coupled discrete-time periodic matrix equations. We introduce a gradient-based iterative (GI) algorithm for finding the generalized reflexive solutions of the general coupled discretetime periodic matrix equations. It is shown that the introduced GI algorithm always converges to the generalized reflexive solutions for any initial generalized reflexive matrices. Finally, two numerical examples are investigated to confirm the efficiency of GI algorithm.

Highlights

  • Let us begin with some notations and definitions

  • We propose a gradient-based iterative (GI) algorithm to find the generalized reflexive solutions of the general coupled discrete-time periodic matrix equations

  • The results show that Algorithm 1 can quickly obtain the solutions of the discrete-time periodic matrix equations

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Summary

Introduction

Let us begin with some notations and definitions. The symbols AT , tr(A) and A will stand for the transpose, the trace and the Frobenius norm of a matrix A ∈ Rm×n, respectively. If A = P AQ A ∈ Rn×n is called a generalized reflexive matrix with respect to (P, Q) [5]. Eixi+1 = Aixi + Biui, yi = Cixi, ∀i ∈ Z we need to solve the following generalized projected periodic discrete-time algebraic Lyapunov matrix equations [2, 6, 27]. Kressner introduced new variants of the squared Smith iteration and Krylov subspace based methods for the approximate solution of discrete-time periodic Lyapunov matrix equations [20]. We propose a GI algorithm to find the generalized reflexive solutions of the general coupled discrete-time periodic matrix equations.

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