Reduction of Linear Time-Invariant Systems Using Routh-Approximation and PSO
Order reduction of linear-time invariant systems employing two methods; one using the advantages of Routh approximation and other by an evolutionary technique is presented in this paper. In Routh approximation method the denominator of the reduced order model is obtained using Routh approximation while the numerator of the reduced order model is determined using the indirect approach of retaining the time moments and/or Markov parameters of original system. By this method the reduced order model guarantees stability if the original high order model is stable. In the second 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 examples.
[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] S. John, R. Parthasarathy, "System Reduction by Routh Approximation
and Modified Cauer Continued Fraction", Electronics Letters, Vol. 15,
pp 691-692, 1979.
[14] 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.
[15] 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.
[16] 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.
[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] S. John, R. Parthasarathy, "System Reduction by Routh Approximation
and Modified Cauer Continued Fraction", Electronics Letters, Vol. 15,
pp 691-692, 1979.
[14] 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.
[15] 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.
[16] 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.
@article{"International Journal of Electrical, Electronic and Communication Sciences:62055", author = "S. Panda and S. K. Tomar and R. Prasad and C. Ardil", title = "Reduction of Linear Time-Invariant Systems Using Routh-Approximation and PSO", abstract = "Order reduction of linear-time invariant systems employing two methods; one using the advantages of Routh approximation and other by an evolutionary technique is presented in this paper. In Routh approximation method the denominator of the reduced order model is obtained using Routh approximation while the numerator of the reduced order model is determined using the indirect approach of retaining the time moments and/or Markov parameters of original system. By this method the reduced order model guarantees stability if the original high order model is stable. In the second 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 examples.
", keywords = "Model Order Reduction, Markov Parameters, Routh Approximation, Particle Swarm Optimization, Integral Squared Error, Steady State Stability.", volume = "3", number = "9", pages = "1752-8", }
