American Institute of Mathematical Sciences

Advanced
doi: 10.3934/jimo.2019093

Parametric Smith iterative algorithms for discrete Lyapunov matrix equations

Ai-Guo Wu , Ying Zhang , and  Hui-Jie Sun

School of Mechanical Engineering and Automation, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China

* Corresponding author: zhangyinghit@126.com

Received  November 2018 Revised  March 2019 Published  July 2019

Fund Project: The authors are supported by Shenzhen Municipal Basic Research Project for Discipline Layout with Project No.JCYJ20170811160715620, by the National Natural Science Foundation of China under Grant No. 61822305, by Guangdong Natural Science Foundation under Grant No. 2017A030313340, and by Shenzhen Municipal Project for International Cooperation with Project No. GJHZ20180420180849805

An iterative algorithm is established in this paper for solving the discrete Lyapunov matrix equations. The proposed algorithm contains a tunable parameter, and includes the Smith iteration as a special case, and thus is called the parametric Smith iterative algorithm. Some convergence conditions are developed for the proposed parametric Smith iterative algorithm. Moreover, the optimal parameter for the proposed algorithm to have the fastest convergence rate is also provided for a special case. Finally, numerical examples are employed to illustrate the effectiveness of the proposed algorithm.

Keywords: Lyapunov matrix equations, iterative algorithms, convergence rate.
Mathematics Subject Classification: Primary: 34D20, 15A06; Secondary: 15A24.
Citation: Ai-Guo Wu, Ying Zhang, Hui-Jie Sun. Parametric Smith iterative algorithms for discrete Lyapunov matrix equations. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019093
References:
[1]

S. Azou, P. Brehonnet, P. Vilbe and L. C. Calvez, A new discrete impulse response Gramian and its application to model reduction, IEEE Transactions on Automatic Control, 45 (2000), 533-537.  doi: 10.1109/9.847738.  Google Scholar

[2]

J. Bibby, Axiomatisations of the average and a further generalisation of monotonic sequences, Glasgow Math. Journal, 15 (1974), 63-65.  doi: 10.1017/S0017089500002135.  Google Scholar

[3]

M. Dehghan and M. Hajarian, The general coupled matrix equations over generalized bisymmetric matrices, Linear Algebra and its Appl., 432 (2010), 1531-1552.  doi: 10.1016/j.laa.2009.11.014.  Google Scholar

[4]

F. Ding and T. Chen, Gradient based iterative algorithms for solving a class of matrix equations, IEEE Trans. on Automat. Control, 50 (2005), 1216-1221.  doi: 10.1109/TAC.2005.852558.  Google Scholar

[5]

S. Hammarling, Numerical solution of the discrete-time, convergent, non-negative definite Lyapunov equation, Systems and Control Letters, 17 (1991), 137-139.  doi: 10.1016/0167-6911(91)90039-H.  Google Scholar

[6]

S. J. Hammarling, Numerical solution of the stable, nonnegative definite Lyapunov equation, IMA J. of Numerical Anal., 2 (1982), 303-323.  doi: 10.1093/imanum/2.3.303.  Google Scholar

[7]

T. Kailath, Linear Systems, Prentice-Hall, New Jersey, 1980.  Google Scholar

[8]

L. Lv and Z. Zhang, Finite iterative solutions to periodic Sylvester matrix equations, J. of the Franklin Institute, 354 (2017), 2358-2370.  doi: 10.1016/j.jfranklin.2017.01.004.  Google Scholar

[9]

Q. Niu, X. Wang and L.-Z. Lu, A relaxed gradient based algorithm for solving Sylvester equations, Asian Journal of Control, 13 (2011), 461-464.  doi: 10.1002/asjc.328.  Google Scholar

[10]

T. Penzl, A cyclic low-rank Smith method for large sparse Lyapunov equations, SIAM J. on Scientific Computing, 21 (1999), 1401-1408.  doi: 10.1137/S1064827598347666.  Google Scholar

[11]

V. Ptak, The discrete Lyapunov equation in controllable canonical form, IEEE Trans. on Auto. Control, 26 (1981), 580-581.  doi: 10.1109/TAC.1981.1102644.  Google Scholar

[12]

M. Sadkane and L. Grammont, A note on the Lyapunov stability of periodic discrete-time systems, J. of Comp. and Appl. Math., 176 (2005), 463-466.  doi: 10.1016/j.cam.2004.08.012.  Google Scholar

[13]

V. Sreeram and P. Agathoklis, Model reduction of linear discrete systems via weighted impulse response gramians, Int. J. of Control, 53 (1991), 129-144.  doi: 10.1080/00207179108953613.  Google Scholar

[14]

Z. Tian, C. M. Fan, Y. Deng and P. H. Wen, New explicit iteration algorithms for solving coupled continuous Markovian jump Lyapunov matrix equations, J. of the Franklin Institute, 355 (2018), 8346-8372.  doi: 10.1016/j.jfranklin.2018.09.027.  Google Scholar

[15]

Z. Tian and C. Gu, A numerical algorithm for Lyapunov equations, Appl. Math. and Comp., 202 (2008), 44-53.  doi: 10.1016/j.amc.2007.12.057.  Google Scholar

[16]

Z. Tian, M. Tian, C. Gu and X. Hao, An accelerated Jacobi-gradient based iterative algorithm for solving Sylvester matrix equations, Filomat, 8 (2017), 2381-2390.  doi: 10.2298/FIL1708381T.  Google Scholar

[17]

V. Varga, A note on Hammarling's algorithm for the discrete Lyapunov equation, Systems and Control Letters, 15 (1990), 273-275.  doi: 10.1016/0167-6911(90)90121-A.  Google Scholar

[18]

Y. Zhang, A. G. Wu and C. T. Shao, Implicit iterative algorithms with a tuning parameter for discrete stochastic Lyapunov matrix equations, IET Control Theory and Appl., 11 (2017), 1554-1560.  doi: 10.1049/iet-cta.2016.1601.  Google Scholar

[19]

Y. Zhang, A. G. Wu and H. J. Sun, An implicit iterative algorithm with a tuning parameter for Itô Lyapunov matrix equations, Int. J. of Systems Science, 49 (2018), 425-434.  doi: 10.1080/00207721.2017.1407009.  Google Scholar

show all references

References:
[1]

S. Azou, P. Brehonnet, P. Vilbe and L. C. Calvez, A new discrete impulse response Gramian and its application to model reduction, IEEE Transactions on Automatic Control, 45 (2000), 533-537.  doi: 10.1109/9.847738.  Google Scholar

[2]

J. Bibby, Axiomatisations of the average and a further generalisation of monotonic sequences, Glasgow Math. Journal, 15 (1974), 63-65.  doi: 10.1017/S0017089500002135.  Google Scholar

[3]

M. Dehghan and M. Hajarian, The general coupled matrix equations over generalized bisymmetric matrices, Linear Algebra and its Appl., 432 (2010), 1531-1552.  doi: 10.1016/j.laa.2009.11.014.  Google Scholar

[4]

F. Ding and T. Chen, Gradient based iterative algorithms for solving a class of matrix equations, IEEE Trans. on Automat. Control, 50 (2005), 1216-1221.  doi: 10.1109/TAC.2005.852558.  Google Scholar

[5]

S. Hammarling, Numerical solution of the discrete-time, convergent, non-negative definite Lyapunov equation, Systems and Control Letters, 17 (1991), 137-139.  doi: 10.1016/0167-6911(91)90039-H.  Google Scholar

[6]

S. J. Hammarling, Numerical solution of the stable, nonnegative definite Lyapunov equation, IMA J. of Numerical Anal., 2 (1982), 303-323.  doi: 10.1093/imanum/2.3.303.  Google Scholar

[7]

T. Kailath, Linear Systems, Prentice-Hall, New Jersey, 1980.  Google Scholar

[8]

L. Lv and Z. Zhang, Finite iterative solutions to periodic Sylvester matrix equations, J. of the Franklin Institute, 354 (2017), 2358-2370.  doi: 10.1016/j.jfranklin.2017.01.004.  Google Scholar

[9]

Q. Niu, X. Wang and L.-Z. Lu, A relaxed gradient based algorithm for solving Sylvester equations, Asian Journal of Control, 13 (2011), 461-464.  doi: 10.1002/asjc.328.  Google Scholar

[10]

T. Penzl, A cyclic low-rank Smith method for large sparse Lyapunov equations, SIAM J. on Scientific Computing, 21 (1999), 1401-1408.  doi: 10.1137/S1064827598347666.  Google Scholar

[11]

V. Ptak, The discrete Lyapunov equation in controllable canonical form, IEEE Trans. on Auto. Control, 26 (1981), 580-581.  doi: 10.1109/TAC.1981.1102644.  Google Scholar

[12]

M. Sadkane and L. Grammont, A note on the Lyapunov stability of periodic discrete-time systems, J. of Comp. and Appl. Math., 176 (2005), 463-466.  doi: 10.1016/j.cam.2004.08.012.  Google Scholar

[13]

V. Sreeram and P. Agathoklis, Model reduction of linear discrete systems via weighted impulse response gramians, Int. J. of Control, 53 (1991), 129-144.  doi: 10.1080/00207179108953613.  Google Scholar

[14]

Z. Tian, C. M. Fan, Y. Deng and P. H. Wen, New explicit iteration algorithms for solving coupled continuous Markovian jump Lyapunov matrix equations, J. of the Franklin Institute, 355 (2018), 8346-8372.  doi: 10.1016/j.jfranklin.2018.09.027.  Google Scholar

[15]

Z. Tian and C. Gu, A numerical algorithm for Lyapunov equations, Appl. Math. and Comp., 202 (2008), 44-53.  doi: 10.1016/j.amc.2007.12.057.  Google Scholar

[16]

Z. Tian, M. Tian, C. Gu and X. Hao, An accelerated Jacobi-gradient based iterative algorithm for solving Sylvester matrix equations, Filomat, 8 (2017), 2381-2390.  doi: 10.2298/FIL1708381T.  Google Scholar

[17]

V. Varga, A note on Hammarling's algorithm for the discrete Lyapunov equation, Systems and Control Letters, 15 (1990), 273-275.  doi: 10.1016/0167-6911(90)90121-A.  Google Scholar

[18]

Y. Zhang, A. G. Wu and C. T. Shao, Implicit iterative algorithms with a tuning parameter for discrete stochastic Lyapunov matrix equations, IET Control Theory and Appl., 11 (2017), 1554-1560.  doi: 10.1049/iet-cta.2016.1601.  Google Scholar

[19]

Y. Zhang, A. G. Wu and H. J. Sun, An implicit iterative algorithm with a tuning parameter for Itô Lyapunov matrix equations, Int. J. of Systems Science, 49 (2018), 425-434.  doi: 10.1080/00207721.2017.1407009.  Google Scholar

Figure 1.  Convergence performance of the algorithm (3) for Example 1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Spectral radius of $ G $ for Example 1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Convergence curve of the algorithm (5) for Example 1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Convergence performance of the algorithm (3) for Example 2
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Spectral radius of $ G $ for Example 2
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Convergence curve of the algorithm (5) for Example 2
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Spectral radius of $ G $ for Example 3
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Convergence curve of the algorithm (5) for Example 3
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

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
[2]

M. Sumon Hossain, M. Monir Uddin. Iterative methods for solving large sparse Lyapunov equations and application to model reduction of index 1 differential-algebraic-equations. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 173-186. doi: 10.3934/naco.2019013
[3]

Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907
[4]

Giuseppe Marino, Hong-Kun Xu. Convergence of generalized proximal point algorithms. Communications on Pure & Applied Analysis, 2004, 3 (4) : 791-808. doi: 10.3934/cpaa.2004.3.791
[5]

Victor Kozyakin. Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 143-158. doi: 10.3934/dcdsb.2010.14.143
[6]

Tong Li, Hui Yin. Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux. Communications on Pure & Applied Analysis, 2014, 13 (2) : 835-858. doi: 10.3934/cpaa.2014.13.835
[7]

Boris Haspot, Ewelina Zatorska. From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3107-3123. doi: 10.3934/dcds.2016.36.3107
[8]

Yulan Lu, Minghui Song, Mingzhu Liu. Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 695-717. doi: 10.3934/dcdsb.2018203
[9]

Wilhelm Schlag. Regularity and convergence rates for the Lyapunov exponents of linear cocycles. Journal of Modern Dynamics, 2013, 7 (4) : 619-637. doi: 10.3934/jmd.2013.7.619
[10]

Yanxing Cui, Chuanlong Wang, Ruiping Wen. On the convergence of generalized parallel multisplitting iterative methods for semidefinite linear systems. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 863-873. doi: 10.3934/naco.2012.2.863
[11]

Yan Tang. Convergence analysis of a new iterative algorithm for solving split variational inclusion problems. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-20. doi: 10.3934/jimo.2018187
[12]

Jairo Bochi, Michal Rams. The entropy of Lyapunov-optimizing measures of some matrix cocycles. Journal of Modern Dynamics, 2016, 10: 255-286. doi: 10.3934/jmd.2016.10.255
[13]

Jinyan Fan, Jianyu Pan. On the convergence rate of the inexact Levenberg-Marquardt method. Journal of Industrial & Management Optimization, 2011, 7 (1) : 199-210. doi: 10.3934/jimo.2011.7.199
[14]

Shahad Al-azzawi, Jicheng Liu, Xianming Liu. Convergence rate of synchronization of systems with additive noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 227-245. doi: 10.3934/dcdsb.2017012
[15]

Yves Bourgault, Damien Broizat, Pierre-Emmanuel Jabin. Convergence rate for the method of moments with linear closure relations. Kinetic & Related Models, 2015, 8 (1) : 1-27. doi: 10.3934/krm.2015.8.1
[16]

Andriy Bondarenko, Guy Bouchitté, Luísa Mascarenhas, Rajesh Mahadevan. Rate of convergence for correctors in almost periodic homogenization. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 503-514. doi: 10.3934/dcds.2005.13.503
[17]

Luis Barreira, Claudia Valls. Stability of nonautonomous equations and Lyapunov functions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2631-2650. doi: 10.3934/dcds.2013.33.2631
[18]

Yong Zhang, Francis Y. L. Chin, Francis C. M. Lau, Haisheng Tan, Hing-Fung Ting. Constant competitive algorithms for unbounded one-Way trading under monotone hazard rate. Mathematical Foundations of Computing, 2018, 1 (4) : 383-392. doi: 10.3934/mfc.2018019
[19]

Fabio Camilli, Claudio Marchi. On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems. Networks & Heterogeneous Media, 2011, 6 (1) : 61-75. doi: 10.3934/nhm.2011.6.61
[20]

Oleg Makarenkov, Paolo Nistri. On the rate of convergence of periodic solutions in perturbed autonomous systems as the perturbation vanishes. Communications on Pure & Applied Analysis, 2008, 7 (1) : 49-61. doi: 10.3934/cpaa.2008.7.49

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (38)
  • HTML views (211)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences