Model Order Reduction of Discrete-Time Systems Using Fuzzy C-Means Clustering
A computationally simple approach of model order
reduction for single input single output (SISO) and linear timeinvariant
discrete systems modeled in frequency domain is proposed
in this paper. Denominator of the reduced order model is determined
using fuzzy C-means clustering while the numerator parameters are
found by matching time moments and Markov parameters of high
order system.
<p>[1] L.Fortuna, G. Nunnari, and A. Gallo, “Model Order Reduction
Techniques with applications in Electrical Engineering", Springer-
Verlag, Lomndon, 1992.
[2] Y Shamash, “Stable reduced-order models using Padé type
approximations”, IEEE Trans. Auto. Control, Vol. AC-19, No.5, pp.
615-616, October 1974.
[3] Y. Shamash, “Linear system reduction using Padé approximation to
allow retention of dominant modes", Int. J. Contr., Vo. 21, Issue. 2, pp.
257-272, 1975.
[4] Maurice F.Hutton and Bernard Friedland, “Routh-approximation for
reducing order of linear, time-invariant systems” IEEE Trans. Auto.
Contr. Vol. AC-20, No.3, pp. 329-337, June 1975.
[5] Vimal Singh, Dinesh Chandra, and Harnath Kar, “Improved Routh -
Padé Approximant: A Computer-aided approach” IEEE Trans. Auto.
Contr. Vol. AC-49, No.2, pp. 292-296, February 2004.
[6] Vimal Singh, “Obtaining Routh-Padé approximants using the Luus-
Jaakola algorithm”, IEE Proceedings, Control Theory Applications,
vol.152,no.2, March 2005.
[7] T.N.Lucas, “Scaled Impulse Energy approximation for Model
Reduction”, IEEE Trans. Auto. Contr. Vol. AC-33, No.8, pp. 791793,
August 1988.
[8] W. Krajewski, A. Lepschy, and U. Viaro, “Model reduction by matching
Markov parameters, time moments and impulse response energy”, IEEE
Trans. Auto. Contr. Vol. AC-40, No.5, pp. 949-953, May 1995.
[9] B. Salimbahrami, B. Lohmann, T. Bechtold, and J. Korvink, “A Two-
Sided Arnoldi-Algorithm with Stopping Criterion and an application in
Order Reduction of MEMS”, Mathematical and Computer Modelling of
Dynamical Systems, 11(1):79–93, 2005.
[10] W.H. Enright and M.S. Kamal, “On selecting a low order model using
dominant mode concept”, IEEE Trans. Auto. Contr. Vol. AC-25, No.5,
pp. 976-978, Oct. 1980.
[11] M.Gopal and S.I. Mehta, “On selection of eigen-values to be retained in
the reduced order models”, IEEE Trans. Auto. Contr. Vol. AC-27, No.3,
pp. 688-690, June 1982.
[12] Tai-Yih Chiu, “Model reduction by the low frequency approximation
balancing method for unstable systems”, IEEE Trans. Auto. Contr. Vol.
AC-41, No.7, pp. 995-997, July 1996.
[13] M. G. Safonov and R. Y. Chiang, “A Survey of model reduction by
balanced truncation and some new results”, Int. J. Control, 77(8):748–
766, 2004.
[14] C.B.Vishwakarma and R. Prasad, “Clustering method for Reduction
Order of Linear System using Pade approximation”, IETE Journal of
Research, vol 54, issue5, Oct 2008.
[15] M. Srinivasan, A. Krishnan, “Transformer Linear Section Model Order
Reduction with an Improved Pole Clustering”, European Journal of
Scintific Research, Vol. 44, No.4, pp.541-549, 2010.
[16] D. Napoleon and S. Pavalakodi, “ A New Method for Dimensionality
Reduction using K-Means Clustering Algorithm for high Dimensional
Data Set”, International Journal of Computer Applications, Vol.13,
No.7, January 2011.
[17] Y.P. Shih, and W.T. Wu., “Simplification of z-transfer functions by
continued fraction” Int. J. Contr., Vo. 17, pp. 1089-1094, 1973.
[18] Chuang C., “Homographic transformation for the simplification of
discrete-time functions by Padé approximation”, Int. J. Contr., Vo. 22,
pp. 721-729, 1975.
[19] R.Y. Hwang, and Y.P. Shih, “ Combined methods of model reduction
via discrete Laguerre polynomials” Int. J. Contr., Vo. 37, pp. 615-622,
1983.
[20] Khalid Hammouda, Fakhareddine Karray, “A Comparative Study of
Data Clustering Techniques, University of Waterloo, Ontario, Canada.
[21] Jesus Gonzalez, Ignacio Rojas, Hector Pomares, Julio Ortega, & Alberto
Prieto, “A New Clustering Technique for Function Approximation”
IEEE Transaction on Neural Networks, Vol.13, NO.1, January, 2002.
[22] Kai-Ming Tse, Chen-Chien Hsu, and Chi-Hsu Wang, “Discrete-time
model reduction of sampled systems using an enhanced
multiresolutional dynamic genetic algorithm”, IEEE Conference Proc.
On Systems, Man, and Cybernatics, Taipei, 2001.
[23] Chyi Hwang and Yen-Ping Shih, “On the time moments of discrete
systems”, Int. J. Contr., Vo. 34, No. 6, pp. 1227-1228, 1981.
[24] Chyi Hwang and Ching-Shieh Hsieh, “A new canonical expansion of ztransfer
function of reduced-order modeling of discrete-time systems”,
IEEE Transactions circuits and systems, vol. 36, No., December 1989.
[25] N.N.Puri and M.T. Lim, “Stable optimal model reduction of linear
discrete-time systems”, Transactions of the ASME, Measurement, and
Control,vol.119, pp.300-304, June 1997.
[26] Vinay Pratap Singh, and Dinesh Chandra, “Reduction of discrete
interval systems based on pole clustering and improved Pade
approximation: a computer aided approach”, AMO-Advanced Modeling
and Optimization, volume 14, Number 1, 2012.
[27] Shailendra K. Mittal, Dinesh Chandra, and Bhartiu Dwivedi, “VEGA
based Routh-Padé approximants for discrete-time systems: a computer
aided approach”, IACSIT International Journal of Engineering and
Technology Vol.1, No.5, December, 2009.
[28] C.M. Liaw, C.T.Pan, and M.Quyang, “Model reduction of discrete
system using the power decomposition method and the system
identification method”, IEE Proceeding, vol. 133, No.1, January, 1986.
[29] G.Saraswathi, “A modified method for order reduction of large scale
discrete systems”, International Journal of Advance Computer Science
and Applications, vol.2, No.6, 2011.
[30] M.Farsi, K.Warwick and M. Guilandoust, “Stable reduced-order models
for discrete-time system”, IEE proceedings, Vol. 133, pt D, No.3,
pp137-141, May 1986.
[31] S. Mukherjee, Satakshi, R.C. Mittal, “Discrete system order reduction
using multipoint step response matching”, Journal of Computational
and Applied Mathematics, 461-466, 2004.
[32] S.N.Deepa, G. Sugumaran, “MPSO based Model Order Formulation
Scheme for Discrete Time Linear System in State Space Form”,
European Journal of Scientific Research, vol.58 No.4, pp 444-454,
2011.</p>
<p>[1] L.Fortuna, G. Nunnari, and A. Gallo, “Model Order Reduction
Techniques with applications in Electrical Engineering", Springer-
Verlag, Lomndon, 1992.
[2] Y Shamash, “Stable reduced-order models using Padé type
approximations”, IEEE Trans. Auto. Control, Vol. AC-19, No.5, pp.
615-616, October 1974.
[3] Y. Shamash, “Linear system reduction using Padé approximation to
allow retention of dominant modes", Int. J. Contr., Vo. 21, Issue. 2, pp.
257-272, 1975.
[4] Maurice F.Hutton and Bernard Friedland, “Routh-approximation for
reducing order of linear, time-invariant systems” IEEE Trans. Auto.
Contr. Vol. AC-20, No.3, pp. 329-337, June 1975.
[5] Vimal Singh, Dinesh Chandra, and Harnath Kar, “Improved Routh -
Padé Approximant: A Computer-aided approach” IEEE Trans. Auto.
Contr. Vol. AC-49, No.2, pp. 292-296, February 2004.
[6] Vimal Singh, “Obtaining Routh-Padé approximants using the Luus-
Jaakola algorithm”, IEE Proceedings, Control Theory Applications,
vol.152,no.2, March 2005.
[7] T.N.Lucas, “Scaled Impulse Energy approximation for Model
Reduction”, IEEE Trans. Auto. Contr. Vol. AC-33, No.8, pp. 791793,
August 1988.
[8] W. Krajewski, A. Lepschy, and U. Viaro, “Model reduction by matching
Markov parameters, time moments and impulse response energy”, IEEE
Trans. Auto. Contr. Vol. AC-40, No.5, pp. 949-953, May 1995.
[9] B. Salimbahrami, B. Lohmann, T. Bechtold, and J. Korvink, “A Two-
Sided Arnoldi-Algorithm with Stopping Criterion and an application in
Order Reduction of MEMS”, Mathematical and Computer Modelling of
Dynamical Systems, 11(1):79–93, 2005.
[10] W.H. Enright and M.S. Kamal, “On selecting a low order model using
dominant mode concept”, IEEE Trans. Auto. Contr. Vol. AC-25, No.5,
pp. 976-978, Oct. 1980.
[11] M.Gopal and S.I. Mehta, “On selection of eigen-values to be retained in
the reduced order models”, IEEE Trans. Auto. Contr. Vol. AC-27, No.3,
pp. 688-690, June 1982.
[12] Tai-Yih Chiu, “Model reduction by the low frequency approximation
balancing method for unstable systems”, IEEE Trans. Auto. Contr. Vol.
AC-41, No.7, pp. 995-997, July 1996.
[13] M. G. Safonov and R. Y. Chiang, “A Survey of model reduction by
balanced truncation and some new results”, Int. J. Control, 77(8):748–
766, 2004.
[14] C.B.Vishwakarma and R. Prasad, “Clustering method for Reduction
Order of Linear System using Pade approximation”, IETE Journal of
Research, vol 54, issue5, Oct 2008.
[15] M. Srinivasan, A. Krishnan, “Transformer Linear Section Model Order
Reduction with an Improved Pole Clustering”, European Journal of
Scintific Research, Vol. 44, No.4, pp.541-549, 2010.
[16] D. Napoleon and S. Pavalakodi, “ A New Method for Dimensionality
Reduction using K-Means Clustering Algorithm for high Dimensional
Data Set”, International Journal of Computer Applications, Vol.13,
No.7, January 2011.
[17] Y.P. Shih, and W.T. Wu., “Simplification of z-transfer functions by
continued fraction” Int. J. Contr., Vo. 17, pp. 1089-1094, 1973.
[18] Chuang C., “Homographic transformation for the simplification of
discrete-time functions by Padé approximation”, Int. J. Contr., Vo. 22,
pp. 721-729, 1975.
[19] R.Y. Hwang, and Y.P. Shih, “ Combined methods of model reduction
via discrete Laguerre polynomials” Int. J. Contr., Vo. 37, pp. 615-622,
1983.
[20] Khalid Hammouda, Fakhareddine Karray, “A Comparative Study of
Data Clustering Techniques, University of Waterloo, Ontario, Canada.
[21] Jesus Gonzalez, Ignacio Rojas, Hector Pomares, Julio Ortega, & Alberto
Prieto, “A New Clustering Technique for Function Approximation”
IEEE Transaction on Neural Networks, Vol.13, NO.1, January, 2002.
[22] Kai-Ming Tse, Chen-Chien Hsu, and Chi-Hsu Wang, “Discrete-time
model reduction of sampled systems using an enhanced
multiresolutional dynamic genetic algorithm”, IEEE Conference Proc.
On Systems, Man, and Cybernatics, Taipei, 2001.
[23] Chyi Hwang and Yen-Ping Shih, “On the time moments of discrete
systems”, Int. J. Contr., Vo. 34, No. 6, pp. 1227-1228, 1981.
[24] Chyi Hwang and Ching-Shieh Hsieh, “A new canonical expansion of ztransfer
function of reduced-order modeling of discrete-time systems”,
IEEE Transactions circuits and systems, vol. 36, No., December 1989.
[25] N.N.Puri and M.T. Lim, “Stable optimal model reduction of linear
discrete-time systems”, Transactions of the ASME, Measurement, and
Control,vol.119, pp.300-304, June 1997.
[26] Vinay Pratap Singh, and Dinesh Chandra, “Reduction of discrete
interval systems based on pole clustering and improved Pade
approximation: a computer aided approach”, AMO-Advanced Modeling
and Optimization, volume 14, Number 1, 2012.
[27] Shailendra K. Mittal, Dinesh Chandra, and Bhartiu Dwivedi, “VEGA
based Routh-Padé approximants for discrete-time systems: a computer
aided approach”, IACSIT International Journal of Engineering and
Technology Vol.1, No.5, December, 2009.
[28] C.M. Liaw, C.T.Pan, and M.Quyang, “Model reduction of discrete
system using the power decomposition method and the system
identification method”, IEE Proceeding, vol. 133, No.1, January, 1986.
[29] G.Saraswathi, “A modified method for order reduction of large scale
discrete systems”, International Journal of Advance Computer Science
and Applications, vol.2, No.6, 2011.
[30] M.Farsi, K.Warwick and M. Guilandoust, “Stable reduced-order models
for discrete-time system”, IEE proceedings, Vol. 133, pt D, No.3,
pp137-141, May 1986.
[31] S. Mukherjee, Satakshi, R.C. Mittal, “Discrete system order reduction
using multipoint step response matching”, Journal of Computational
and Applied Mathematics, 461-466, 2004.
[32] S.N.Deepa, G. Sugumaran, “MPSO based Model Order Formulation
Scheme for Discrete Time Linear System in State Space Form”,
European Journal of Scientific Research, vol.58 No.4, pp 444-454,
2011.</p>
@article{"International Journal of Information, Control and Computer Sciences:51019", author = "Anirudha Narain and Dinesh Chandra and Ravindra K. S.", title = "Model Order Reduction of Discrete-Time Systems Using Fuzzy C-Means Clustering", abstract = "A computationally simple approach of model order
reduction for single input single output (SISO) and linear timeinvariant
discrete systems modeled in frequency domain is proposed
in this paper. Denominator of the reduced order model is determined
using fuzzy C-means clustering while the numerator parameters are
found by matching time moments and Markov parameters of high
order system.
", keywords = "Model Order reduction, Discrete-time system, Fuzzy
C-Means Clustering, Padé approximation.", volume = "7", number = "7", pages = "882-8", }
