American Institute of Mathematical Sciences

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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

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 & 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

show all references

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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