Gradient‐based iterative algorithm for a class of the coupled matrix equations related to control systems

Mathematical Modelling and Analysis

2016

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.

Section: Resultsmentioning

confidence: 99%

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

Mathematical Modelling and Analysis

2016

“…Iterative algorithms are one of the most successful techniques to solve linear systems and system identification [25][26][27][28][29][30]. During the last decade one can observe that by extending the iterative methods proposed for solving linear system of equations Mx = b, efficient iterative algorithms were proposed for solving several linear matrix equations [31][32][33][34][35]. For instance, a gradient based iterative algorithm was proposed for solving general linear matrix equations including the Sylvester-transpose matrix equation by extending the Jacobi iteration and by applying the hierarchical identification principle [36].…”

Section: Introductionmentioning

confidence: 99%

New Finite Algorithm for Solving the Generalized Nonhomogeneous Yakubovich‐Transpose Matrix Equation

Asian Journal of Control

2016

In this paper, the development of the conjugate direction (CD) method is constructed to solve the generalized nonhomogeneous Yakubovich‐transpose matrix equation AXB + CXTD + EYF = R. We prove that the constructed method can obtain the (least Frobenius norm) solution pair (X,Y) of the generalized nonhomogeneous Yakubovich‐transpose matrix equation for any (special) initial matrix pair within a finite number of iterations in the absence of round‐off errors. Finally, two numerical examples show that the constructed method is more efficient than other similar iterative methods in practical computation.

“…To find the solution of linear and nonlinear matrix equations, so far several direct and iterative methods have been introduced [7][8][9][10][11][12][13][14][15][16]. By a generalization of the Bartels-Stewart method and a extension of Hammarling's method, Penzl proposed two efficient methods for solving generalized Lyapunov matrix equations [17].…”

Section: Introductionmentioning

confidence: 99%

Convergence Results of the Biconjugate Residual Algorithm for Solving Generalized Sylvester Matrix Equation

Asian Journal of Control

2016

In this article, we investigate a variant of the biconjugate residual (BCR) algorithm to solve the generalized Sylvester matrix equation ∑i=1kAiXBi+∑j=1lCjYDj=E, which includes the well‐known Lyapunov, Stein and Sylvester matrix equations. We show that the BCR algorithm with any (special) initial matrix pair can smoothly compute the (least Frobenius norm) solution pair of the generalized Sylvester matrix equation within a finite number of iterations in the absence of round‐off errors. Finally the accuracy and effectiveness of the BCR algorithm in comparison to some existing algorithms are demonstrated by two numerical examples.