Figure 1.
Convergence performance of the algorithm (3) for Example 1
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November
2020, 16(6): 3047-3063.
doi: 10.3934/jimo.2019093
Parametric Smith iterative algorithms for discrete Lyapunov matrix equations
School of Mechanical Engineering and Automation, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China |
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.
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 and Management Optimization,
2020, 16
(6)
: 3047-3063.
doi: 10.3934/jimo.2019093
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
T. Kailath, Linear Systems, Prentice-Hall, New Jersey, 1980. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
T. Kailath, Linear Systems, Prentice-Hall, New Jersey, 1980. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
Figure 2.
Spectral radius of $ G $ for Example 1
Figure 3.
Convergence curve of the algorithm (5) for Example 1
Figure 4.
Convergence performance of the algorithm (3) for Example 2
Figure 5.
Spectral radius of $ G $ for Example 2
Figure 6.
Convergence curve of the algorithm (5) for Example 2
Figure 7.
Spectral radius of $ G $ for Example 3
Figure 8.
Convergence curve of the algorithm (5) for Example 3
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