Conventional and PSO Based Approaches for Model Reduction of SISO Discrete Systems
Reduction of Single Input Single Output (SISO) discrete systems into lower order model, using a conventional and an evolutionary technique is presented in this paper. In the conventional technique, the mixed advantages of Modified Cauer Form (MCF) and differentiation are used. In this method the original discrete system is, first, converted into equivalent continuous system by applying bilinear transformation. The denominator of the equivalent continuous system and its reciprocal are differentiated successively, the reduced denominator of the desired order is obtained by combining the differentiated polynomials. The numerator is obtained by matching the quotients of MCF. The reduced continuous system is converted back into discrete system using inverse bilinear transformation. 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.
[1] R. Genesio and M. Milanese, "A note on the derivation and use of
reduced-order models", IEEE Transactions on Automatic Control, Vol.
21, pages 118-122, 1976.
[2] S. Panda, S. K. Tomar, R. Prasad and 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.
[3] S. Panda, S. K. Tomar, R. Prasad and 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.
[4] 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.
[5] J. S. Yadav, N. P. Patidar, J. Singhai, S. Panda and C. Ardil "A
Combined Conventional and Differential Evolution Method for Model
Order Reduction", International Journal of Computational Intelligence,
Vol. 5, No. 2, pp. 111-118, 2009.
[6] Y. Shamash, "Continued fraction methods for the reduction of discrete
time dynamic systems", Int. Journal of Control, Vol. 20, pages 267-268,
1974.
[7] C.P. Therapos, "A direct method for model reduction of discrete
system", Journal of Franklin Institute, Vol. 318, pp. 243-251, 1984.
[8] J.P. Tiwari, and S.K. Bhagat, "Simplification of discrete time systems by
improved Routh stability criterion via p-domain", Journal of IE (India),
Vol. 85, pp. 89-192, 2004.
[9] J. Kennedy and R. C. Eberhart, "Particle swarm optimization",
IEEEInt.Conf. on Neural Networks, IV, 1942-1948, Piscataway, NJ,
1995.
[10] Sidhartha Panda and N. P. Padhy, "Comparison of Particle Swarm
Optimization and Genetic Algorithm for FACTS-based Controller
Design", Applied Soft Computing, Vol. 8, Issue 4, pp. 1418-1427, 2008.
[11] A.C. Davies, "Bilinear transformation of polynomials," IEEE
Transaction on Circuits and systems, CAS-21, pp 792-794, 974.
[12] R. Parthasarthy and K.N. Jaysimha, "Bilinear Transformations by
Synthetic Division," IEEE Transaction on Automatic Control. Vol. 29,
No. 6, pp. 575-576, 1984.
[13] P.Gutman, C.F.Mannerfelt and P.Molandar, "Contribution to the model
reduction problem," IEEE Transaction on.Automatic Control, Vol. 27,
pp 454-455,1982.
[14] R. Parthasarthy and Sarasu John, "System reduction by Routh
approximation and modified Cauer continued fraction," Electronics
Letters, Vol. 5, No. 21, pp 691-692. 1979.
[15] Sunita Devi and Rajendra Prasad, "Reduction of Discrete time systems
by Routh approximation, National System Conference," IIT Kharagpur,
NSC 2003, pp 30-33, Dec 17-19, 2003.
[16] R. Parthasarthy and Sarasu John, "System Reduction using Cauer
Continued Fraction Expansion about s=0 and s= ∞," Electronics Letters,
Vol. 14, No. 8, pp .261-262, 1978.
[17] Ching-Shieh Hsieh and Chyi Hwang,"Model reduction of linear
discrete-time systems using bilinear Schwartz approximation,"
International Journal of System & Science, Vol .21, No 1, pp. 33-49,
1990.
[1] R. Genesio and M. Milanese, "A note on the derivation and use of
reduced-order models", IEEE Transactions on Automatic Control, Vol.
21, pages 118-122, 1976.
[2] S. Panda, S. K. Tomar, R. Prasad and 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.
[3] S. Panda, S. K. Tomar, R. Prasad and 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.
[4] 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.
[5] J. S. Yadav, N. P. Patidar, J. Singhai, S. Panda and C. Ardil "A
Combined Conventional and Differential Evolution Method for Model
Order Reduction", International Journal of Computational Intelligence,
Vol. 5, No. 2, pp. 111-118, 2009.
[6] Y. Shamash, "Continued fraction methods for the reduction of discrete
time dynamic systems", Int. Journal of Control, Vol. 20, pages 267-268,
1974.
[7] C.P. Therapos, "A direct method for model reduction of discrete
system", Journal of Franklin Institute, Vol. 318, pp. 243-251, 1984.
[8] J.P. Tiwari, and S.K. Bhagat, "Simplification of discrete time systems by
improved Routh stability criterion via p-domain", Journal of IE (India),
Vol. 85, pp. 89-192, 2004.
[9] J. Kennedy and R. C. Eberhart, "Particle swarm optimization",
IEEEInt.Conf. on Neural Networks, IV, 1942-1948, Piscataway, NJ,
1995.
[10] Sidhartha Panda and N. P. Padhy, "Comparison of Particle Swarm
Optimization and Genetic Algorithm for FACTS-based Controller
Design", Applied Soft Computing, Vol. 8, Issue 4, pp. 1418-1427, 2008.
[11] A.C. Davies, "Bilinear transformation of polynomials," IEEE
Transaction on Circuits and systems, CAS-21, pp 792-794, 974.
[12] R. Parthasarthy and K.N. Jaysimha, "Bilinear Transformations by
Synthetic Division," IEEE Transaction on Automatic Control. Vol. 29,
No. 6, pp. 575-576, 1984.
[13] P.Gutman, C.F.Mannerfelt and P.Molandar, "Contribution to the model
reduction problem," IEEE Transaction on.Automatic Control, Vol. 27,
pp 454-455,1982.
[14] R. Parthasarthy and Sarasu John, "System reduction by Routh
approximation and modified Cauer continued fraction," Electronics
Letters, Vol. 5, No. 21, pp 691-692. 1979.
[15] Sunita Devi and Rajendra Prasad, "Reduction of Discrete time systems
by Routh approximation, National System Conference," IIT Kharagpur,
NSC 2003, pp 30-33, Dec 17-19, 2003.
[16] R. Parthasarthy and Sarasu John, "System Reduction using Cauer
Continued Fraction Expansion about s=0 and s= ∞," Electronics Letters,
Vol. 14, No. 8, pp .261-262, 1978.
[17] Ching-Shieh Hsieh and Chyi Hwang,"Model reduction of linear
discrete-time systems using bilinear Schwartz approximation,"
International Journal of System & Science, Vol .21, No 1, pp. 33-49,
1990.
@article{"International Journal of Electrical, Electronic and Communication Sciences:53926", author = "S. K. Tomar and R. Prasad and S. Panda and C. Ardil", title = "Conventional and PSO Based Approaches for Model Reduction of SISO Discrete Systems", abstract = "Reduction of Single Input Single Output (SISO) discrete systems into lower order model, using a conventional and an evolutionary technique is presented in this paper. In the conventional technique, the mixed advantages of Modified Cauer Form (MCF) and differentiation are used. In this method the original discrete system is, first, converted into equivalent continuous system by applying bilinear transformation. The denominator of the equivalent continuous system and its reciprocal are differentiated successively, the reduced denominator of the desired order is obtained by combining the differentiated polynomials. The numerator is obtained by matching the quotients of MCF. The reduced continuous system is converted back into discrete system using inverse bilinear transformation. 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 = "Discrete System, Single Input Single Output (SISO),Bilinear Transformation, Reduced Order Model, Modified CauerForm, Polynomial Differentiation, Particle Swarm Optimization,Integral Squared Error.", volume = "7", number = "6", pages = "651-6", }
