|
|
耦合Sylvester转置矩阵方程的多迭代因子梯度算法
|
Abstract:
本文研究了耦合Sylvester转置矩阵方程的数值求解问题。通过充分利用方程的耦合性，引入了多个参数，并提出了多迭代因子梯度(MGI)算法。此外，还推导出了确保算法生成的迭代解对于任意初始矩阵都收敛到精确解的充要条件。最后，通过一个数值例子验证了所提算法的有效性。
This paper investigates the numerical solution of coupled Sylvester-transpose matrix equations. By fully exploiting the coupling of the equations, several parameters are introduced, and an multi-iterative factors gradient algorithm is proposed. Additionally, the necessary and sufficient conditions that ensure the convergence of the iterative solutions generated by the algorithm to the exact solution for any initial matrix are derived. Finally, the effectiveness of the proposed algorithm has been verified through a numerical example.
| [1] | Costa, O.L.V. and Fragoso, M.D. (1993) Stability Results for Discrete-Time Linear Systems with Markovian Jumping Parameters. Journal of Mathematical Analysis and Applications, 179, 154-178. https://doi.org/10.1006/jmaa.1993.1341 |
| [2] | Zhou, B. (2015) Global Stabilization of Periodic Linear Systems by Bounded Controls with Applications to Spacecraft Magnetic Attitude Control. Automatica, 60, 145-154. https://doi.org/10.1016/j.automatica.2015.07.003 |
| [3] | Ding, F. (2023) Least Squares Parameter Estimation and Multi-Innovation Least Squares Methods for Linear Fitting Problems from Noisy Data. Journal of Computational and Applied Mathematics, 426, Article ID: 115107. https://doi.org/10.1016/j.cam.2023.115107 |
| [4] | Zhou, B. (2020) Finite-Time Stabilization of Linear Systems by Bounded Linear Time-Varying Feedback. Automatica, 113, Article ID: 108760. https://doi.org/10.1016/j.automatica.2019.108760 |
| [5] | Dai, L. (1989) Singular Control Systems. Springer. |
| [6] | Ding, F. and Chen, T. (2005) Iterative Least-Squares Solutions of Coupled Sylvester Matrix Equations. Systems & Control Letters, 54, 95-107. https://doi.org/10.1016/j.sysconle.2004.06.008 |
| [7] | Ding, F., Liu, P.X. and Ding, J. (2008) Iterative Solutions of the Generalized Sylvester Matrix Equations by Using the Hierarchical Identification Principle. Applied Mathematics and Computation, 197, 41-50. https://doi.org/10.1016/j.amc.2007.07.040 |
| [8] | Xie, L., Ding, J. and Ding, F. (2009) Gradient Based Iterative Solutions for General Linear Matrix Equations. Computers & Mathematics with Applications, 58, 1441-1448. https://doi.org/10.1016/j.camwa.2009.06.047 |
| [9] | Niu, Q., Wang, X. and Lu, L. (2010) A Relaxed Gradient Based Algorithm for Solving Sylvester Equations. Asian Journal of Control, 13, 461-464. https://doi.org/10.1002/asjc.328 |
| [10] | Xie, Y. and Ma, C. (2016) The Accelerated Gradient Based Iterative Algorithm for Solving a Class of Generalized Sylvester-Transpose Matrix Equation. Applied Mathematics and Computation, 273, 1257-1269. https://doi.org/10.1016/j.amc.2015.07.022 |
| [11] | Li, Z., Wang, Y., Zhou, B. and Duan, G. (2010) Least Squares Solution with the Minimum-Norm to General Matrix Equations via Iteration. Applied Mathematics and Computation, 215, 3547-3562. https://doi.org/10.1016/j.amc.2009.10.052 |
| [12] | Song, C., Chen, G. and Zhao, L. (2011) Iterative Solutions to Coupled Sylvester-Transpose Matrix Equations. Applied Mathematical Modelling, 35, 4675-4683. https://doi.org/10.1016/j.apm.2011.03.038 |
| [13] | Al Zhour, Z. and Kiliçman, A. (2007) Some New Connections between Matrix Products for Partitioned and Non-Partitioned Matrices. Computers & Mathematics with Applications, 54, 763-784. https://doi.org/10.1016/j.camwa.2006.12.045 |
| [14] | Wang, W. and Song, C. (2021) Iterative Algorithms for Discrete-Time Periodic Sylvester Matrix Equations and Its Application in Antilinear Periodic System. Applied Numerical Mathematics, 168, 251-273. https://doi.org/10.1016/j.apnum.2021.06.006 |
| [15] | Khalil, I., Doyle, J. and Glover, K. (1996) Robust and Optimal Control. Vol. 2, Prentice Hall. |
| [16] | Huang, B. and Ma, C. (2022) On the Relaxed Gradient-Based Iterative Methods for the Generalized Coupled Sylvester-Transpose Matrix Equations. Journal of the Franklin Institute, 359, 10688-10725. https://doi.org/10.1016/j.jfranklin.2022.07.051 |
