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

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

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

Deepak Kumar ^1, , Ahmad Jazlan ^2, , Victor Sreeram ^2, and Roberto Togneri ^2,

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

Abstract
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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.

Keywords: Model Reduction, partial fraction expansion, weighted approximation error and free parameters..

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

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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