A Combined Conventional and Differential Evolution Method for Model Order Reduction
In this paper a mixed method by combining an evolutionary and a conventional technique is proposed for reduction of Single Input Single Output (SISO) continuous systems into Reduced Order Model (ROM). In the conventional technique, the mixed advantages of Mihailov stability criterion and continued Fraction Expansions (CFE) technique is employed where the reduced denominator polynomial is derived using Mihailov stability criterion and the numerator is obtained by matching the quotients of the Cauer second form of Continued fraction expansions. Then, retaining the numerator polynomial, the denominator polynomial is recalculated by an evolutionary technique. In the evolutionary method, the recently proposed Differential Evolution (DE) optimization technique is employed. DE method is based on the minimization of the Integral Squared Error (ISE) between the transient responses of original higher order model and the reduced order model pertaining to a unit step input. The proposed method is illustrated through a numerical example and compared with ROM where both numerator and denominator polynomials are obtained by conventional method to show its superiority.
[1] M. J. Bosley and F. P. Lees, "A survey of simple transfer
function derivations from high order state variable models",
Automatica, Vol. 8, pp. 765-775, !978.
[2] M. F. Hutton and B. Fried land, "Routh approximations for
reducing order of linear time- invariant systems", IEEE Trans.
Auto. Control, Vol. 20, pp 329-337, 1975.
[3] R. K. Appiah, "Linear model reduction using Hurwitz
polynomial approximation", Int. J. Control, Vol. 28, no. 3, pp
477-488, 1978.
[4] T. C. Chen, C. Y. Chang and K. W. Han, "Reduction of transfer
functions by the stability equation method", Journal of Franklin
Institute, Vol. 308, pp 389-404, 1979.
[5] Y. Shamash, "Truncation method of reduction: a viable
alternative", Electronics Letters, Vol. 17, pp 97-99, 1981.
[6] P. O. Gutman, C. F. Mannerfelt and P. Molander, "Contributions
to the model reduction problem", IEEE Trans. Auto. Control,
Vol. 27, pp 454-455, 1982.
[7] Y. Shamash, "Model reduction using the Routh stability
criterion and the Pade approximation technique", Int. J. Control,
Vol. 21, pp 475-484, 1975.
[8] T. C. Chen, C. Y. Chang and K. W. Han, "Model Reduction
using the stability-equation method and the Pade approximation
method", Journal of Franklin Institute, Vol. 309, pp 473-490,
1980.
[9] Bai-Wu Wan, "Linear model reduction using Mihailov criterion
and Pade approximation technique", Int. J. Control, Vol. 33, pp
1073-1089, 1981.
[10] V. Singh, D. Chandra and H. Kar, "Improved Routh-Pade
Approximants: A Computer-Aided Approach", IEEE Trans.
Auto. Control, Vol. 49. No. 2, pp292-296, 2004.
[11] Stron Rainer and Price Kennth, Differential Evolution - "A
simple and efficient adaptive scheme forGlobal Optimization
over continuous spaces", Journal of Global Optimization,
Vol.11, pp. 341-359, 1997.
[12] Storn Rainer, Differential Evolution for Continuous Function
Optimization," http://www.icsi.berkeley.edu/-
storn/code.html,2005.
[13] DE bibliography, http://www.lut.fi/~jlampine/debiblio.htm
[14] S. Panda, S. K. Tomar, R. Prasad, C. Ardil, "Model Reduction
of Linear Systems by Conventional and Evolutionary
Techniques", International Journal of Computational and
Mathematical Sciences, Vol. 3, No. 1, pp. 28-34, 2009.
[15] S. Panda, S.C.Swain and A.K.Baliarsingh, "Power System
Stability Improvement by Differential Evolution Optimized
TCSC-Based Controller", Proceedings of International
Conference on Computing, (CIC 2008), Mexico City, Mexico,
Held on Dec., 3-5, 2008.
[16] S. Panda, S. K. Tomar, R. Prasad, C. Ardil, "Reduction of
Linear Time-Invariant Systems Using Routh-Approximation and
PSO", International Journal of Applied Mathematics and
Computer Sciences, Vol. 5, No. 2, pp. 82-89, 2009.
[17] S. Panda, J. S. Yadav, N. P. Patidar and C. Ardil, "Evolutionary
Techniques for Model Order Reduction of Large Scale Linear
Systems", International Journal of Applied Science, Engineering
and Technology, Vol. 5, No. 1, pp. 22-28, 2009.
[18] T. N. Lukas. "Linear system reduction by the modified factor
division method" IEEE Proceedings Vol. 133 Part D No. 6,
nov.-1986, pp-293-295.
[19] Gamperle R.,. Muller S. D. and Koumoutsakos P., "A Parameter
Study for Differential Evolution," Advances in Intelligent
Systems, Fuzzy Systems, Evolutionary Computation, pp. 293-
298, 2002.
[20] Zaharie D., "Critical values for the control parameters of
differential evolution algorithms," Proc. of the8th International
Conference on SoftComputing, pp. 62-67, 2002.
[1] M. J. Bosley and F. P. Lees, "A survey of simple transfer
function derivations from high order state variable models",
Automatica, Vol. 8, pp. 765-775, !978.
[2] M. F. Hutton and B. Fried land, "Routh approximations for
reducing order of linear time- invariant systems", IEEE Trans.
Auto. Control, Vol. 20, pp 329-337, 1975.
[3] R. K. Appiah, "Linear model reduction using Hurwitz
polynomial approximation", Int. J. Control, Vol. 28, no. 3, pp
477-488, 1978.
[4] T. C. Chen, C. Y. Chang and K. W. Han, "Reduction of transfer
functions by the stability equation method", Journal of Franklin
Institute, Vol. 308, pp 389-404, 1979.
[5] Y. Shamash, "Truncation method of reduction: a viable
alternative", Electronics Letters, Vol. 17, pp 97-99, 1981.
[6] P. O. Gutman, C. F. Mannerfelt and P. Molander, "Contributions
to the model reduction problem", IEEE Trans. Auto. Control,
Vol. 27, pp 454-455, 1982.
[7] Y. Shamash, "Model reduction using the Routh stability
criterion and the Pade approximation technique", Int. J. Control,
Vol. 21, pp 475-484, 1975.
[8] T. C. Chen, C. Y. Chang and K. W. Han, "Model Reduction
using the stability-equation method and the Pade approximation
method", Journal of Franklin Institute, Vol. 309, pp 473-490,
1980.
[9] Bai-Wu Wan, "Linear model reduction using Mihailov criterion
and Pade approximation technique", Int. J. Control, Vol. 33, pp
1073-1089, 1981.
[10] V. Singh, D. Chandra and H. Kar, "Improved Routh-Pade
Approximants: A Computer-Aided Approach", IEEE Trans.
Auto. Control, Vol. 49. No. 2, pp292-296, 2004.
[11] Stron Rainer and Price Kennth, Differential Evolution - "A
simple and efficient adaptive scheme forGlobal Optimization
over continuous spaces", Journal of Global Optimization,
Vol.11, pp. 341-359, 1997.
[12] Storn Rainer, Differential Evolution for Continuous Function
Optimization," http://www.icsi.berkeley.edu/-
storn/code.html,2005.
[13] DE bibliography, http://www.lut.fi/~jlampine/debiblio.htm
[14] S. Panda, S. K. Tomar, R. Prasad, C. Ardil, "Model Reduction
of Linear Systems by Conventional and Evolutionary
Techniques", International Journal of Computational and
Mathematical Sciences, Vol. 3, No. 1, pp. 28-34, 2009.
[15] S. Panda, S.C.Swain and A.K.Baliarsingh, "Power System
Stability Improvement by Differential Evolution Optimized
TCSC-Based Controller", Proceedings of International
Conference on Computing, (CIC 2008), Mexico City, Mexico,
Held on Dec., 3-5, 2008.
[16] S. Panda, S. K. Tomar, R. Prasad, C. Ardil, "Reduction of
Linear Time-Invariant Systems Using Routh-Approximation and
PSO", International Journal of Applied Mathematics and
Computer Sciences, Vol. 5, No. 2, pp. 82-89, 2009.
[17] S. Panda, J. S. Yadav, N. P. Patidar and C. Ardil, "Evolutionary
Techniques for Model Order Reduction of Large Scale Linear
Systems", International Journal of Applied Science, Engineering
and Technology, Vol. 5, No. 1, pp. 22-28, 2009.
[18] T. N. Lukas. "Linear system reduction by the modified factor
division method" IEEE Proceedings Vol. 133 Part D No. 6,
nov.-1986, pp-293-295.
[19] Gamperle R.,. Muller S. D. and Koumoutsakos P., "A Parameter
Study for Differential Evolution," Advances in Intelligent
Systems, Fuzzy Systems, Evolutionary Computation, pp. 293-
298, 2002.
[20] Zaharie D., "Critical values for the control parameters of
differential evolution algorithms," Proc. of the8th International
Conference on SoftComputing, pp. 62-67, 2002.
@article{"International Journal of Electrical, Electronic and Communication Sciences:59097", author = "J. S. Yadav and N. P. Patidar and J. Singhai and S. Panda and C. Ardil", title = "A Combined Conventional and Differential Evolution Method for Model Order Reduction", abstract = "In this paper a mixed method by combining an evolutionary and a conventional technique is proposed for reduction of Single Input Single Output (SISO) continuous systems into Reduced Order Model (ROM). In the conventional technique, the mixed advantages of Mihailov stability criterion and continued Fraction Expansions (CFE) technique is employed where the reduced denominator polynomial is derived using Mihailov stability criterion and the numerator is obtained by matching the quotients of the Cauer second form of Continued fraction expansions. Then, retaining the numerator polynomial, the denominator polynomial is recalculated by an evolutionary technique. In the evolutionary method, the recently proposed Differential Evolution (DE) optimization technique is employed. DE method is based on the minimization of the Integral Squared Error (ISE) between the transient responses of original higher order model and the reduced order model pertaining to a unit step input. The proposed method is illustrated through a numerical example and compared with ROM where both numerator and denominator polynomials are obtained by conventional method to show its superiority.
", keywords = "Reduced Order Modeling, Stability, Mihailov
Stability Criterion, Continued Fraction Expansions, Differential
Evolution, Integral Squared Error.", volume = "5", number = "9", pages = "1222-8", }
