# 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. Google Scholar [2] P. Benner and M. S. Hossain, Structure Preserving Iterative Solution of Periodic Projected Lyapunov Equations, Mathematical Modelling, 45 (2012), 276-281. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [14] L. Lv and L. Zhang, New iterative algorithms for coupled matrix equations, Journal of Computational Analysis and Applications, 19 (2015), 947-958. Google Scholar [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. Google Scholar [16] W. J. Rugh, Linear system theory (2nd ed. ), Upper Saddle River, New Jersey: Prentice Hall, 1996.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [26] B. Zhou, On asymptotic stability of linear time-varying systems, Automatica, 68 (2016), 266-276. doi: 10.1016/j.automatica.2015.12.030. Google Scholar [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. Google Scholar [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. Google Scholar

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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. Google Scholar [2] P. Benner and M. S. Hossain, Structure Preserving Iterative Solution of Periodic Projected Lyapunov Equations, Mathematical Modelling, 45 (2012), 276-281. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [14] L. Lv and L. Zhang, New iterative algorithms for coupled matrix equations, Journal of Computational Analysis and Applications, 19 (2015), 947-958. Google Scholar [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. Google Scholar [16] W. J. Rugh, Linear system theory (2nd ed. ), Upper Saddle River, New Jersey: Prentice Hall, 1996.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [26] B. Zhou, On asymptotic stability of linear time-varying systems, Automatica, 68 (2016), 266-276. doi: 10.1016/j.automatica.2015.12.030. Google Scholar [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. Google Scholar [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. Google Scholar
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