# American Institute of Mathematical Sciences

2016, 6(3): 329-337. doi: 10.3934/naco.2016015

## Partial fraction expansion based frequency weighted model reduction for discrete-time systems

 1 Motilal Nehru National Institute of Technology Allahabad, Allahabad, Uttar Pradesh 211004, India 2 School of Electrical and Electronics Engineering, University of Western Australia, 35 Stirling Highway, WA 6009, Australia, Australia

Received  July 2015 Revised  September 2016 Published  September 2016

In this paper, a partial fraction expansion based frequency weighted model reduction algorithm is developed for discrete-time systems. The proposed method is an extension to the method by Sreeram et al. [13] and it yields stable reduced order models with both single and double sided weighting functions. Effectiveness of the proposed algorithm is demonstrated by a numerical example.
Citation: Deepak Kumar, Ahmad Jazlan, Victor Sreeram, Roberto Togneri. Partial fraction expansion based frequency weighted model reduction for discrete-time systems. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 329-337. doi: 10.3934/naco.2016015
##### References:
 [1] D. W. Ding, X. Du and X. Li, Finite-frequency model reduction of two-dimensional digital filters,, IEEE Trans. Autom. Control, 60 (2015), 1624.  doi: 10.1109/TAC.2014.2359305.  Google Scholar [2] X. Du, F. Fan, D. W. Ding and F. Liu, Finite-frequency model order reduction of discrete-time linear time-delayed systems,, Nonlinear Dynamics, 83 (2016), 2485.  doi: 10.1007/s11071-015-2496-0.  Google Scholar [3] D. Enns, Model reduction with balanced realizations: An error bound and a frequency weighted generalization,, In: Proceedings of the 23rd IEEE Conference on Decision and Control, (1984), 127.   Google Scholar [4] M. Imran and A. Ghafoor, Model reduction of descriptor systems using frequency limited gramians,, J. Franklin Inst., 352 (2015), 33.  doi: 10.1016/j.jfranklin.2014.10.013.  Google Scholar [5] X. Li, C. Yu and H. Gao, Frequency limited H∞ model reduction for positive systems,, IEEE Trans. Autom. Control, 60 (2015), 1093.  doi: 10.1109/TAC.2014.2352751.  Google Scholar [6] C. A. Lin and T. Y. Chiu, Model reduction via frequency weighted balanced realization,, Control Theory and Advanced Technology, 8 (1992), 341.   Google Scholar [7] Y. Liu and B. D. O. Anderson, Singular perturbation approximation of balanced systems,, International Journal of Control, 50 (1989), 1339.  doi: 10.1080/00207178908953437.  Google Scholar [8] B. C. Moore, Principal component analysis in linear system: Controllability, observability, and model reduction,, IEEE Trans. Automat. Contr, AC-26 (1981), 17.  doi: 10.1109/TAC.1981.1102568.  Google Scholar [9] U. M. Saggaf and G. F. Franklin, On model reduction,, Proc. of the 23rd IEEE Conf. on Decision and Control, (1986), 1064.   Google Scholar [10] U. M. Saggaf and G. F. Franklin, Model reduction via balanced realization,, IEEE Trans. on Autom. Control, AC-33 (1988), 687.  doi: 10.1109/9.1280.  Google Scholar [11] H. R. Shaker and M. Tahavori, Frequency interval model reduction of bilinear systems,, IEEE Trans. Autom. Control, 59 (2014), 1948.  doi: 10.1109/TAC.2013.2295661.  Google Scholar [12] V. Sreeram and B. D. O. Anderson, Frequency weighted balanced reduction technique: A generalization and an error bound,, Proceedings of the 34th IEEE Conference on Decision and Control, (1995), 3576.   Google Scholar [13] V. Sreeram, S. Sahlan, W. M. W Muda, T. Fernando and H. H. C. Iu, A generalised partial-fraction-expansion based frequency weighted balanced truncation technique,, International Journal of Control, 86 (2013), 833.  doi: 10.1080/00207179.2013.764017.  Google Scholar [14] T. Van-Gestel, B. Anderson and P. Van-Overschee, On frequency weighted balanced truncation: Hankel singular values and error bounds,, European Journal of Control, 7 (2001), 584.   Google Scholar [15] G. Wang, V. Sreeram and W. Q. Liu, A new frequency-weighted balanced truncation method and an error bound,, IEEE Transactions on Automatic Control, 44 (1999), 1734.  doi: 10.1109/9.788542.  Google Scholar

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##### References:
 [1] D. W. Ding, X. Du and X. Li, Finite-frequency model reduction of two-dimensional digital filters,, IEEE Trans. Autom. Control, 60 (2015), 1624.  doi: 10.1109/TAC.2014.2359305.  Google Scholar [2] X. Du, F. Fan, D. W. Ding and F. Liu, Finite-frequency model order reduction of discrete-time linear time-delayed systems,, Nonlinear Dynamics, 83 (2016), 2485.  doi: 10.1007/s11071-015-2496-0.  Google Scholar [3] D. Enns, Model reduction with balanced realizations: An error bound and a frequency weighted generalization,, In: Proceedings of the 23rd IEEE Conference on Decision and Control, (1984), 127.   Google Scholar [4] M. Imran and A. Ghafoor, Model reduction of descriptor systems using frequency limited gramians,, J. Franklin Inst., 352 (2015), 33.  doi: 10.1016/j.jfranklin.2014.10.013.  Google Scholar [5] X. Li, C. Yu and H. Gao, Frequency limited H∞ model reduction for positive systems,, IEEE Trans. Autom. Control, 60 (2015), 1093.  doi: 10.1109/TAC.2014.2352751.  Google Scholar [6] C. A. Lin and T. Y. Chiu, Model reduction via frequency weighted balanced realization,, Control Theory and Advanced Technology, 8 (1992), 341.   Google Scholar [7] Y. Liu and B. D. O. Anderson, Singular perturbation approximation of balanced systems,, International Journal of Control, 50 (1989), 1339.  doi: 10.1080/00207178908953437.  Google Scholar [8] B. C. Moore, Principal component analysis in linear system: Controllability, observability, and model reduction,, IEEE Trans. Automat. Contr, AC-26 (1981), 17.  doi: 10.1109/TAC.1981.1102568.  Google Scholar [9] U. M. Saggaf and G. F. Franklin, On model reduction,, Proc. of the 23rd IEEE Conf. on Decision and Control, (1986), 1064.   Google Scholar [10] U. M. Saggaf and G. F. Franklin, Model reduction via balanced realization,, IEEE Trans. on Autom. Control, AC-33 (1988), 687.  doi: 10.1109/9.1280.  Google Scholar [11] H. R. Shaker and M. Tahavori, Frequency interval model reduction of bilinear systems,, IEEE Trans. Autom. Control, 59 (2014), 1948.  doi: 10.1109/TAC.2013.2295661.  Google Scholar [12] V. Sreeram and B. D. O. Anderson, Frequency weighted balanced reduction technique: A generalization and an error bound,, Proceedings of the 34th IEEE Conference on Decision and Control, (1995), 3576.   Google Scholar [13] V. Sreeram, S. Sahlan, W. M. W Muda, T. Fernando and H. H. C. Iu, A generalised partial-fraction-expansion based frequency weighted balanced truncation technique,, International Journal of Control, 86 (2013), 833.  doi: 10.1080/00207179.2013.764017.  Google Scholar [14] T. Van-Gestel, B. Anderson and P. Van-Overschee, On frequency weighted balanced truncation: Hankel singular values and error bounds,, European Journal of Control, 7 (2001), 584.   Google Scholar [15] G. Wang, V. Sreeram and W. Q. Liu, A new frequency-weighted balanced truncation method and an error bound,, IEEE Transactions on Automatic Control, 44 (1999), 1734.  doi: 10.1109/9.788542.  Google Scholar
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