Model Reduction of Linear Systems by Conventional and Evolutionary Techniques

Authors: S. Panda, S. K. Tomar, R. Prasad, C. Ardil

Abstract:

Reduction of Single Input Single Output (SISO) continuous systems into Reduced Order Model (ROM), using a conventional and an evolutionary technique is presented in this paper. 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. In the evolutionary technique method Particle Swarm Optimization (PSO) is employed to reduce the higher order model. PSO 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. Both the methods are illustrated through numerical example.

Keywords: Reduced Order Modeling, Stability, Continued Fraction Expansions, Mihailov Stability Criterion, Particle Swarm Optimization, Integral Squared Error.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1072874

References:

[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] J. Kennedy and R. C. Eberhart, "Particle swarm optimization", IEEE Int.Conf. on Neural Networks, IV, 1942-1948, Piscataway, NJ, 1995.
[12] S. Panda, and N. P. Padhy "Comparison of Particle Swarm Optimization and Genetic Algorithm for FACTS-based Controller Design", Applied Soft Computing. Vol. 8, pp. 1418-1427, 2008.
[13] Wan Bai-Wu, "Linear model reduction using Mihailov criterion and Pade approximation technique." Int. J. Control, 1981, No 33, pp 1073- 1089.
[14] Chen, C.F. and Sheih, L.S "A Novel approach to linear model simplification." Int. J. Control, 1968, No 8, pp 561-570.
[15] 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.
[16] M. Lal and H. Singh, "On the determination of a transfer function matrix from the given state equations." Int. J. Control, Vol. 15, pp 333-335, 1972.
[17] Sidhartha Panda, N.P.Padhy, R.N.Patel, "Power System Stability Improvement by PSO Optimized SSSC-based Damping Controller", Electric Power Components & Systems, Vol. 36, No. 5, pp. 468-490, 2008.
[18] Sidhartha Panda and N.P.Padhy, "Optimal location and controller design of STATCOM using particle swarm optimization", Journal of the Franklin Institute, Vol.345, pp. 166-181, 2008.