# 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 and 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), 1624-1629. doi: 10.1109/TAC.2014.2359305. [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), 2485-2496. doi: 10.1007/s11071-015-2496-0. [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, Las Vegas, USA, (1984), 127-132. [4] M. Imran and A. Ghafoor, Model reduction of descriptor systems using frequency limited gramians, J. Franklin Inst., 352 (2015), 33-51. doi: 10.1016/j.jfranklin.2014.10.013. [5] X. Li, C. Yu and H. Gao, Frequency limited H∞ model reduction for positive systems, IEEE Trans. Autom. Control, 60 (2015), 1093-1098. doi: 10.1109/TAC.2014.2352751. [6] C. A. Lin and T. Y. Chiu, Model reduction via frequency weighted balanced realization, Control Theory and Advanced Technology, 8 (1992), 341-451. [7] Y. Liu and B. D. O. Anderson, Singular perturbation approximation of balanced systems, International Journal of Control, 50 (1989), 1339-1405. doi: 10.1080/00207178908953437. [8] B. C. Moore, Principal component analysis in linear system: Controllability, observability, and model reduction, IEEE Trans. Automat. Contr, AC-26 (1981), 17-32. doi: 10.1109/TAC.1981.1102568. [9] U. M. Saggaf and G. F. Franklin, On model reduction, Proc. of the 23rd IEEE Conf. on Decision and Control, (1986), 1064-1069. [10] U. M. Saggaf and G. F. Franklin, Model reduction via balanced realization, IEEE Trans. on Autom. Control, AC-33 (1988), 687-692. doi: 10.1109/9.1280. [11] H. R. Shaker and M. Tahavori, Frequency interval model reduction of bilinear systems, IEEE Trans. Autom. Control, 59 (2014), 1948-1953. doi: 10.1109/TAC.2013.2295661. [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), 3576-3581. [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), 833-843. doi: 10.1080/00207179.2013.764017. [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), 584-592. [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), 1734-1737. doi: 10.1109/9.788542.

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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), 1624-1629. doi: 10.1109/TAC.2014.2359305. [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), 2485-2496. doi: 10.1007/s11071-015-2496-0. [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, Las Vegas, USA, (1984), 127-132. [4] M. Imran and A. Ghafoor, Model reduction of descriptor systems using frequency limited gramians, J. Franklin Inst., 352 (2015), 33-51. doi: 10.1016/j.jfranklin.2014.10.013. [5] X. Li, C. Yu and H. Gao, Frequency limited H∞ model reduction for positive systems, IEEE Trans. Autom. Control, 60 (2015), 1093-1098. doi: 10.1109/TAC.2014.2352751. [6] C. A. Lin and T. Y. Chiu, Model reduction via frequency weighted balanced realization, Control Theory and Advanced Technology, 8 (1992), 341-451. [7] Y. Liu and B. D. O. Anderson, Singular perturbation approximation of balanced systems, International Journal of Control, 50 (1989), 1339-1405. doi: 10.1080/00207178908953437. [8] B. C. Moore, Principal component analysis in linear system: Controllability, observability, and model reduction, IEEE Trans. Automat. Contr, AC-26 (1981), 17-32. doi: 10.1109/TAC.1981.1102568. [9] U. M. Saggaf and G. F. Franklin, On model reduction, Proc. of the 23rd IEEE Conf. on Decision and Control, (1986), 1064-1069. [10] U. M. Saggaf and G. F. Franklin, Model reduction via balanced realization, IEEE Trans. on Autom. Control, AC-33 (1988), 687-692. doi: 10.1109/9.1280. [11] H. R. Shaker and M. Tahavori, Frequency interval model reduction of bilinear systems, IEEE Trans. Autom. Control, 59 (2014), 1948-1953. doi: 10.1109/TAC.2013.2295661. [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), 3576-3581. [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), 833-843. doi: 10.1080/00207179.2013.764017. [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), 584-592. [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), 1734-1737. doi: 10.1109/9.788542.
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