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November  2020, 16(6): 3047-3063. 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  November 2020 Early access  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, 2020, 16 (6) : 3047-3063. 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
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Figure 2.  Spectral radius of $ G $ for Example 1
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Figure 3.  Convergence curve of the algorithm (5) for Example 1
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Figure 4.  Convergence performance of the algorithm (3) for Example 2
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Figure 5.  Spectral radius of $ G $ for Example 2
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Figure 6.  Convergence curve of the algorithm (5) for Example 2
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Figure 7.  Spectral radius of $ G $ for Example 3
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Figure 8.  Convergence curve of the algorithm (5) for Example 3
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