Gradient Based Iterative Algorithm to Solve General Coupled DiscreteTime Periodic Matrix Equations over Generalized Reflexive Matrices

AbstractThe discretetime 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 discretetime periodic matrix equations. We introduce a gradientbased iterative (GI) algorithm for finding the generalized reflexive solutions of the general coupled discrete time 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.

[1]  R. A. Smith.  Matrix Equation $XA + BX = C$ , 1968 .

[2]  P. Dooren,et al.  Periodic Schur forms and some matrix equations , 1994 .

[3]  Noah H. Rhee,et al.  Cyclic Schur and Hessenberg-Schur Numerical Methods for Solving Periodic Lyapunov and Sylvester Equa , 1995 .

[4]  S. Bittanti,et al.  Analysis of discrete-time linear periodic systems , 1996 .

[5]  A. Varga.  Periodic Lyapunov equations: Some applications and new algorithms , 1997 .

[6]  Hsin-Chu Chen.  Generalized Reflexive Matrices: Special Properties and Applications , 1998, SIAM J. Matrix Anal. Appl..

[7]  D. Kressner.  Large periodic Lyapunov equations: Algorithms and applications , 2003, 2003 European Control Conference (ECC).

[8]  Feng Ding,et al.  Iterative least-squares solutions of coupled Sylvester matrix equations , 2005, Syst. Control. Lett..

[9]  Feng Ding,et al.  Gradient Based Iterative Algorithms for Solving a Class of Matrix Equations , 2005, IEEE Trans. Autom. Control..

[10]  T. Stykel.  Low rank iterative methods for projected generalized Lyapunov equations , 2005 .

[11]  Feng Ding,et al.  On Iterative Solutions of General Coupled Matrix Equations , 2006, SIAM J. Control. Optim..

[12]  Tatjana Stykel,et al.  On some norms for descriptor systems , 2006, IEEE Transactions on Automatic Control.

[13]  Isak Jonsson,et al.  Recursive Blocked Algorithms for Solving Periodic Triangular Sylvester-Type Matrix Equations , 2006, PARA.

[14]  András Varga.  On computing minimal realizations of periodic descriptor systems , 2007, PSYCO.

[15]  Wen-Wei Lin,et al.  Projected Generalized Discrete-Time Periodic Lyapunov Equations and Balanced Realization of Periodic Descriptor Systems , 2007, SIAM J. Matrix Anal. Appl..

[16]  Feng Ding,et al.  Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle , 2008, Appl. Math. Comput..

[17]  Mehdi Dehghan,et al.  An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation , 2008, Appl. Math. Comput..

[18]  Guang-Ren Duan,et al.  Gradient based iterative algorithm for solving coupled matrix equations , 2009, Syst. Control. Lett..

[19]  Guang-Ren Duan,et al.  A parametric periodic Lyapunov equation with application in semi-global stabilization of discrete-time periodic systems subject to actuator saturation , 2010, Proceedings of the 2010 American Control Conference.

[20]  Mehdi Dehghan,et al.  An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices , 2010 .

[21]  M. Dehghan,et al.  The general coupled matrix equations over generalized bisymmetric matrices , 2010 .

[22]  Feng Ding,et al.  Iterative solutions to matrix equations of the form AiXBi=Fi , 2010, Comput. Math. Appl..

[23]  M. Dehghan,et al.  Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations , 2011 .

[24]  Peter Benner,et al.  Model Reduction of Periodic Descriptor Systems Using Balanced Truncation , 2011 .

[25]  Q. Niu,et al.  A relaxed gradient based algorithm for solving sylvester equations , 2011 .

[26]  Xiang Wang,et al.  The optimal convergence factor of the gradient based iterative algorithm for linear matrix equations , 2012 .

[27]  P. Benner,et al.  Iterative solvers for periodic matrix equations and model reduction for periodic control systems , 2012, 2012 7th International Conference on Electrical and Computer Engineering.

[28]  Lin Dai,et al.  A modified gradient based algorithm for solving Sylvester equations , 2012, Appl. Math. Comput..

[29]  Peter Benner,et al.  Low-rank iterative methods for periodic projected Lyapunov equations and their application in model reduction of periodic descriptor systems , 2014, Numerical Algorithms.

[30]  Feng Ding,et al.  Brief Paper - Gradient-based iterative algorithm for a class of the coupled matrix equations related to control systems , 2014 .

[31]  Masoud Hajarian.  Developing the CGLS algorithm for the least squares solutions of the general coupled matrix equations , 2014 .

[32]  D. K. Salkuyeh,et al.  minimum norm least-squares solution to general complex coupled linear matrix equations via iteration , 2015 .

[33]  Hui Zhang,et al.  Robust two-mode-dependent controller design for networked control systems with random delays modelled by Markov chains , 2015, Int. J. Control.

[34]  Zhuohua Peng,et al.  The (R, S)-symmetric least squares solutions of the general coupled matrix equations , 2015 .

[35]  Hui Zhang,et al.  NOx Sensor Ammonia-Cross-Sensitivity Factor Estimation in Diesel Engine Selective Catalytic Reduction Systems , 2015 .