基于离散时间系统交叉Gram矩阵的H2最优模型降阶方法
H2 Optimal Model Order Reduction Method Based on the Cross-Gramian for Discrete-Time Systems
摘要: 针对单输入单输出(SISO)离散时间系统，本文提出了基于交叉Gram矩阵的一阶必要条件。首先，应用交叉Gram矩阵，得到误差系统的H2范数；然后，根据交叉Gram矩阵所满足的Sylvester方程，得到了误差系统H2范数关于降阶系统系数矩阵的梯度；最后，根据误差系统H2范数的梯度，得到了基于交叉Gram矩阵的一阶必要条件。与此同时，得到降阶系统。
Abstract: In this paper, the first-order necessary conditions based on the cross-Gramian are presented for the discrete-time Single-Input-Single-Output (SISO) systems. First, by using the cross-Gramian, the H2-norm of the error system is obtained. Then, according to the Sylvester equations satisfied by the cross-Gramian, the gradients are obtained with respect to the coefficient matrices of the reduced order system. Finally, due to the gradients of the H2-norm of the error system, the first-order necessary conditions based on the cross-Gramian are achieved. Meanwhile, the reduced order system is accordingly obtained.
文章引用：王兆鸿, 李延鹏. 基于离散时间系统交叉Gram矩阵的H2最优模型降阶方法[J]. 应用数学进展, 2018, 7(4): 316-322. https://doi.org/10.12677/AAM.2018.74038

参考文献

[1] 蒋耀林. 模型降阶方法[M]. 北京: 科学出版社, 2010.
[2] Wang, X.L. and Jiang, Y.L. (2016) Model Reduction of Discrete-Time Bilinear Systems by a Laguerre Expansion Technique. Applied Mathematical Modeling, 40, 6650-6662. [Google Scholar] [CrossRef] 
[3] Moore, B. (1981) Principal Component Analysis in Linear Systems: Controllability, Observability, and Model Reduction. IEEE Transactions on Automatic Control, 26, 17-32. [Google Scholar] [CrossRef] 
[4] Wang, D. and Zilouchian, A. (2000) Model Reduction of Discrete Linear Systems via Frequency-Domain Balanced Structure. IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, 47, 830-837. [Google Scholar] [CrossRef] 
[5] Jaimoukha, I.M. and Kasenally, E.M. (1997) Implicitly Restarted Krylov Subspace Methods for Stable Partial Realizations. SIAM Journal on Matrix Analysis and Applications, 18, 633-652. [Google Scholar] [CrossRef] 
[6] Willcox, K. and Peraire, J. (2002) Balanced Model Reduction via the Proper Orthogonal Decomposition. AIAA Journal, 40, 2323-2330. [Google Scholar] [CrossRef] 
[7] Zhang, L.Q., Lam, J., Huang, B. and Yang, G.H. (2003) On Gramians and Balanced Truncation of Discrete-Time Bilinear Systems. International Journal of Control, 76, 414-427. [Google Scholar] [CrossRef] 
[8] Jazlan, A., Sreeram, V., Shaker, H.R. and Togneri, R. (2016) Frequency Interval Balanced Truncation of Discrete Time Bilinear Systems. Cogent Engineering, 3, Article ID: 1203082. [Google Scholar] [CrossRef] 
[9] Birouche, A., Mourllion, B. and Basset, M. (2012) Model Order Reduction for Discrete-Time Switched Linear Systems. International Journal of Systems Science, 43, 1753-1763. [Google Scholar] [CrossRef] 
[10] Wilson, D. (1970) Optimum Solution of Model-Reduction Problem. Proceedings of the Institution of Electrical Engineers, 117, 1161-1165. [Google Scholar] [CrossRef] 
[11] Hyland, D.C. and Bernstein, D.S. (1985) The Optimal Projection Equations for Model Reduction and the Relationship among the Methods of Wilson, Skelton and Moore. IEEE Transactions on Automatic Control, 30, 1201-1211. [Google Scholar] [CrossRef] 
[12] Bunse-Gerstner, A., Kubalińska, D., Vossen, G. and Wilczek, D. (2010) H2-Norm Optimal Model Reduction for Large Scale Discrete Dynamical MIMO Systems. Journal of Computational and Applied Mathematics, 233, 1202-1216. [Google Scholar] [CrossRef] 
[13] Van Dooren, P., Gallivan, K.A. and Absil, P.A. (2008) H2-Optimal Model Reduction of MIMO Systems. Applied Mathematics Letters, 21, 1267-1273. [Google Scholar] [CrossRef] 
[14] Fernando, K.V. and Nicholson, H. (1984) On a Fundamental Property of the Cross-Gramian Matrix. IEEE Transactions on Circuits and Systems, 31, 504-505. [Google Scholar] [CrossRef] 

为你推荐