# American Institute of Mathematical Sciences

January 2018, 14(1): 413-425. doi: 10.3934/jimo.2017053

## An iterative algorithm for periodic sylvester matrix equations

 1 Institute of Electric power, North China University of Water Resources and Electric Power, Zhengzhou 450011, China 2 Computer and Information Engineering College, Henan University, Kaifeng 475004, China

Corresponding author: Lei Zhang

Received  March 2016 Revised  September 2016 Published  June 2017

Fund Project: This work is supported by the Programs of National Natural Science Foundation of China (Nos. 11501200, U1604148,61402149), Innovative Talents of Higher Learning Institutions of Henan (No. 17HASTIT023), China Postdoctoral Science Foundation (No. 2016M592285), and Innovative Research Team in University of Henan Province (No. 16IRTSTHN017)

The problem of solving periodic Sylvester matrix equations is discussed in this paper. A new kind of iterative algorithm is proposed for constructing the least square solution for the equations. The basic idea is to develop the solution matrices in the least square sense. Two numerical examples are presented to illustrate the convergence and performance of the iterative method.

Citation: Lingling Lv, Zhe Zhang, Lei Zhang, Weishu Wang. An iterative algorithm for periodic sylvester matrix equations. Journal of Industrial & Management Optimization, 2018, 14 (1) : 413-425. doi: 10.3934/jimo.2017053
##### References:
 [1] P. Benner, M. S. Hossain and T. Stykel, Low-rank iterative methods for periodic projected Lyapunov equations and their application in model reduction of periodic descriptor systems, Springer Seminars in Immunopathology, 67 (2014), 669-690. doi: 10.1007/s11075-013-9816-6. [2] P. Benner and M. S. Hossain, Structure Preserving Iterative Solution of Periodic Projected Lyapunov Equations, Mathematical Modelling, 45 (2012), 276-281. [3] P. Benner, M. S. Hossain and T. Stykel, Model reduction of periodic descriptor systems using balanced truncation, Lecture Notes in Electrical Engineering, 74 (2010), 193-206. doi: 10.1007/978-94-007-0089-5_11. [4] C. Y. Chiang, On the Sylvester-like matrix equation $AX + f(X)B = C$, Journal of the Franklin Institute, 353 (2016), 1061-1074. doi: 10.1016/j.jfranklin.2015.03.024. [5] M. Dehghan and M. Hajarian, Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations, Applied Mathematical Modelling, 35 (2011), 3285-3300. doi: 10.1016/j.apm.2011.01.022. [6] M. Dehghan and M. Hajarian, The general coupled matrix equations over generalized bisymmetric matrices, Linear Algebra & Its Applications, 432 (2010), 1531-1552. doi: 10.1016/j.laa.2009.11.014. [7] M. Dehghan and M. Hajarian, Efficient iterative method for solving the second-order Sylvester matrix equation $EVF^{2}-AVF^{2}-CV = BW$, IET Control Theory & Applications, 3 (2009), 1401-1408. doi: 10.1049/iet-cta.2008.0450. [8] F. Ding and T. Chen, Gradient based iterative algorithms for solving a class of matrix equations, IEEE Transactions on Automatic Control, 50 (2005), 1216-1221. doi: 10.1109/TAC.2005.852558. [9] F. Ding and H. Zhang, Gradient-based iterative algorithm for a class of the coupled matrix equations related to control systems, IET Control Theory & Applications, 8 (2014), 1588-1595. doi: 10.1049/iet-cta.2013.1044. [10] M. Hajarian, Solving the general Sylvester discrete-time periodic matrix equations via the gradient based iterative method, Applied Mathematics Letters, 52 (2016), 87-95. doi: 10.1016/j.aml.2015.08.017. [11] M. Hajarian, Matrix iterative methods for solving the Sylvester-transpose and periodic Sylvester matrix equations, Journal of the Franklin Institute, 350 (2013), 3328-3341. doi: 10.1016/j.jfranklin.2013.07.008. [12] Z. Li, B. Zhou, Y. Wang and G. Duan, Numerical solution to linear matrix equation by finite steps iteration, IET Control Theory & Applications, 4 (2010), 1245-1253. doi: 10.1049/iet-cta.2009.0015. [13] S. Longhi and R. Zulli, A note on robust pole assignment for periodic systems, IEEE Transactions on Automatic Control, 41 (1996), 1493-1497. doi: 10.1109/9.539431. [14] L. Lv and L. Zhang, New iterative algorithms for coupled matrix equations, Journal of Computational Analysis and Applications, 19 (2015), 947-958. [15] E.-S. M. E. Mostafa, A nonlinear conjugate gradient method for a special class of matrix optimization problems, Journal of Industrial & Management Optimization, 10 (2014), 883-903. doi: 10.3934/jimo.2014.10.883. [16] W. J. Rugh, Linear system theory (2nd ed. ), Upper Saddle River, New Jersey: Prentice Hall, 1996. [17] Q. W. Wang, J. W. V. D. Woude and H. X. Chang, A system of real quaternion matrix equations with applications, Linear Algebra & Its Applications, 431 (2009), 2291-2303. doi: 10.1016/j.laa.2009.02.010. [18] Q. W. Wang and C. K. Li, Ranks and the least-norm of the general solution to a system of quaternion matrix equations, Linear Algebra & Its Applications, 430 (2009), 1626-1640. doi: 10.1016/j.laa.2008.05.031. [19] L. Xie, Y. Liu and H. Yang, Gradient based and least squares based iterative algorithms for matrix equations $AXB+CX^{\mathrm{T}}D=F$, Applied Mathematics & Computation, 217 (2010), 2191-2199. doi: 10.1016/j.amc.2010.07.019. [20] L. Xu, Application of the Newton iteration algorithm to the parameter estimation for dynamical systems, Journal of Computational & Applied Mathematics, 288 (2015), 33-43. doi: 10.1016/j.cam.2015.03.057. [21] L. Xu, A proportional differential control method for a time-delay system using the Taylor expansion approximation, Applied Mathematics & Computation, 236 (2014), 391-399. doi: 10.1016/j.amc.2014.02.087. [22] Y. Yang, An efficient algorithm for periodic Riccati equation with periodically time-varying input matrix, Automatica, 78 (2017), 103-109. doi: 10.1016/j.automatica.2016.12.028. [23] H. Zhang and F. Ding, A property of the eigenvalues of the symmetric positive definite matrix and the iterative algorithm for coupled Sylvester matrix equations, Journal of the Franklin Institute, 351 (2014), 340-357. doi: 10.1016/j.jfranklin.2013.08.023. [24] H. Zhang and F. Ding, Iterative algorithms for $X+A^{\mathrm{T} }X^{-1}A=I$, by using the hierarchical identification principle, Journal of the Franklin Institute, 353 (2015), 1132-1146. doi: 10.1016/j.jfranklin.2015.04.003. [25] L. Zhang, A. Zhu and A. Wu, Parametric solutions to the regulator-conjugate matrix equations, Journal of Industrial & Management Optimization, 13 (2017), 623-631. doi: 10.3934/jimo.2016036. [26] B. Zhou, On asymptotic stability of linear time-varying systems, Automatica, 68 (2016), 266-276. doi: 10.1016/j.automatica.2015.12.030. [27] B. Zhou and A. V. Egorov, Razumikhin and Krasovskii stability theorems for time-varying time-delay systems, Automatica, 71 (2016), 281-291. doi: 10.1016/j.automatica.2016.04.048. [28] B. Zhou, G. R. Duan and Z. Y. Li, Gradient based iterative algorithm for solving coupled matrix equations, Systems and Control Letters, 58 (2009), 327-333. doi: 10.1016/j.sysconle.2008.12.004.

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##### References:
 [1] P. Benner, M. S. Hossain and T. Stykel, Low-rank iterative methods for periodic projected Lyapunov equations and their application in model reduction of periodic descriptor systems, Springer Seminars in Immunopathology, 67 (2014), 669-690. doi: 10.1007/s11075-013-9816-6. [2] P. Benner and M. S. Hossain, Structure Preserving Iterative Solution of Periodic Projected Lyapunov Equations, Mathematical Modelling, 45 (2012), 276-281. [3] P. Benner, M. S. Hossain and T. Stykel, Model reduction of periodic descriptor systems using balanced truncation, Lecture Notes in Electrical Engineering, 74 (2010), 193-206. doi: 10.1007/978-94-007-0089-5_11. [4] C. Y. Chiang, On the Sylvester-like matrix equation $AX + f(X)B = C$, Journal of the Franklin Institute, 353 (2016), 1061-1074. doi: 10.1016/j.jfranklin.2015.03.024. [5] M. Dehghan and M. Hajarian, Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations, Applied Mathematical Modelling, 35 (2011), 3285-3300. doi: 10.1016/j.apm.2011.01.022. [6] M. Dehghan and M. Hajarian, The general coupled matrix equations over generalized bisymmetric matrices, Linear Algebra & Its Applications, 432 (2010), 1531-1552. doi: 10.1016/j.laa.2009.11.014. [7] M. Dehghan and M. Hajarian, Efficient iterative method for solving the second-order Sylvester matrix equation $EVF^{2}-AVF^{2}-CV = BW$, IET Control Theory & Applications, 3 (2009), 1401-1408. doi: 10.1049/iet-cta.2008.0450. [8] F. Ding and T. Chen, Gradient based iterative algorithms for solving a class of matrix equations, IEEE Transactions on Automatic Control, 50 (2005), 1216-1221. doi: 10.1109/TAC.2005.852558. [9] F. Ding and H. Zhang, Gradient-based iterative algorithm for a class of the coupled matrix equations related to control systems, IET Control Theory & Applications, 8 (2014), 1588-1595. doi: 10.1049/iet-cta.2013.1044. [10] M. Hajarian, Solving the general Sylvester discrete-time periodic matrix equations via the gradient based iterative method, Applied Mathematics Letters, 52 (2016), 87-95. doi: 10.1016/j.aml.2015.08.017. [11] M. Hajarian, Matrix iterative methods for solving the Sylvester-transpose and periodic Sylvester matrix equations, Journal of the Franklin Institute, 350 (2013), 3328-3341. doi: 10.1016/j.jfranklin.2013.07.008. [12] Z. Li, B. Zhou, Y. Wang and G. Duan, Numerical solution to linear matrix equation by finite steps iteration, IET Control Theory & Applications, 4 (2010), 1245-1253. doi: 10.1049/iet-cta.2009.0015. [13] S. Longhi and R. Zulli, A note on robust pole assignment for periodic systems, IEEE Transactions on Automatic Control, 41 (1996), 1493-1497. doi: 10.1109/9.539431. [14] L. Lv and L. Zhang, New iterative algorithms for coupled matrix equations, Journal of Computational Analysis and Applications, 19 (2015), 947-958. [15] E.-S. M. E. Mostafa, A nonlinear conjugate gradient method for a special class of matrix optimization problems, Journal of Industrial & Management Optimization, 10 (2014), 883-903. doi: 10.3934/jimo.2014.10.883. [16] W. J. Rugh, Linear system theory (2nd ed. ), Upper Saddle River, New Jersey: Prentice Hall, 1996. [17] Q. W. Wang, J. W. V. D. Woude and H. X. Chang, A system of real quaternion matrix equations with applications, Linear Algebra & Its Applications, 431 (2009), 2291-2303. doi: 10.1016/j.laa.2009.02.010. [18] Q. W. Wang and C. K. Li, Ranks and the least-norm of the general solution to a system of quaternion matrix equations, Linear Algebra & Its Applications, 430 (2009), 1626-1640. doi: 10.1016/j.laa.2008.05.031. [19] L. Xie, Y. Liu and H. Yang, Gradient based and least squares based iterative algorithms for matrix equations $AXB+CX^{\mathrm{T}}D=F$, Applied Mathematics & Computation, 217 (2010), 2191-2199. doi: 10.1016/j.amc.2010.07.019. [20] L. Xu, Application of the Newton iteration algorithm to the parameter estimation for dynamical systems, Journal of Computational & Applied Mathematics, 288 (2015), 33-43. doi: 10.1016/j.cam.2015.03.057. [21] L. Xu, A proportional differential control method for a time-delay system using the Taylor expansion approximation, Applied Mathematics & Computation, 236 (2014), 391-399. doi: 10.1016/j.amc.2014.02.087. [22] Y. Yang, An efficient algorithm for periodic Riccati equation with periodically time-varying input matrix, Automatica, 78 (2017), 103-109. doi: 10.1016/j.automatica.2016.12.028. [23] H. Zhang and F. Ding, A property of the eigenvalues of the symmetric positive definite matrix and the iterative algorithm for coupled Sylvester matrix equations, Journal of the Franklin Institute, 351 (2014), 340-357. doi: 10.1016/j.jfranklin.2013.08.023. [24] H. Zhang and F. Ding, Iterative algorithms for $X+A^{\mathrm{T} }X^{-1}A=I$, by using the hierarchical identification principle, Journal of the Franklin Institute, 353 (2015), 1132-1146. doi: 10.1016/j.jfranklin.2015.04.003. [25] L. Zhang, A. Zhu and A. Wu, Parametric solutions to the regulator-conjugate matrix equations, Journal of Industrial & Management Optimization, 13 (2017), 623-631. doi: 10.3934/jimo.2016036. [26] B. Zhou, On asymptotic stability of linear time-varying systems, Automatica, 68 (2016), 266-276. doi: 10.1016/j.automatica.2015.12.030. [27] B. Zhou and A. V. Egorov, Razumikhin and Krasovskii stability theorems for time-varying time-delay systems, Automatica, 71 (2016), 281-291. doi: 10.1016/j.automatica.2016.04.048. [28] B. Zhou, G. R. Duan and Z. Y. Li, Gradient based iterative algorithm for solving coupled matrix equations, Systems and Control Letters, 58 (2009), 327-333. doi: 10.1016/j.sysconle.2008.12.004.
The residuals for the iterative algorithm
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