Order Reduction of Linear Dynamic Systems using Stability Equation Method and GA

Authors: G. Parmar, R. Prasad, S. Mukherjee

Abstract:

The authors present an algorithm for order reduction of linear dynamic systems using the combined advantages of stability equation method and the error minimization by Genetic algorithm. The denominator of the reduced order model is obtained by the stability equation method and the numerator terms of the lower order transfer function are determined by minimizing the integral square error between the transient responses of original and reduced order models using Genetic algorithm. The reduction procedure is simple and computer oriented. It is shown that the algorithm has several advantages, e.g. the reduced order models retain the steady-state value and stability of the original system. The proposed algorithm has also been extended for the order reduction of linear multivariable systems. Two numerical examples are solved to illustrate the superiority of the algorithm over some existing ones including one example of multivariable system.

Keywords: Genetic algorithm, Integral square error, Orderreduction, Stability equation method.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1061595

References:

[1] R. Genesio and M. Milanese, ''A note on the derivation and use of reduced order models'', IEEE Trans. Automat. Control, Vol. AC-21, No. 1, pp. 118-122, February 1976.
[2] M. S. Mahmoud and M. G. Singh, Large Scale Systems Modelling, Pergamon Press, International Series on Systems and Control 1st ed., Vol. 3, 1981.
[3] M. Jamshidi, Large Scale Systems Modelling and Control Series, New York, Amsterdam, Oxford, North Holland, Vol. 9, 1983.
[4] S. K. Nagar and S. K. Singh, ''An algorithmic approach for system decomposition and balanced realized model reduction'', Journal of Franklin Inst., Vol. 341, pp. 615-630, 2004.
[5] V. Singh, D. Chandra and H. Kar, ''Improved Routh Pade approximants: A computer aided approach'', IEEE Trans. Automat. Control, Vol. 49, No.2, pp 292-296, February 2004.
[6] S. Mukherjee, Satakshi and R.C.Mittal, ''Model order reduction using response-matching technique'', Journal of Franklin Inst., Vol. 342 , pp. 503-519, 2005.
[7] B. Salimbahrami, and B. Lohmann, ''Order reduction of large scale second-order systems using Krylov subspace methods'', Linear Algebra Appl., Vol. 415, pp. 385-405, 2006.
[8] S. Mukherjee and R.N. Mishra, ''Reduced order modelling of linear multivariable systems using an error minimization technique'', Journal of Franklin Inst., Vol. 325, No. 2 , pp. 235-245, 1988.
[9] S. S. Lamba, R. Gorez and B. Bandyopadhyay, ''New reduction technique by step error minimization for multivariable systems'', Int. J. Systems Sci., Vol. 19, No. 6, pp. 999-1009, 1988.
[10] R. Prasad and J. Pal, ''Use of continued fraction expansion for stable reduction of linear multivariable systems'', Journal of Institution of Engineers, India, IE(I) Journal '' EL, Vol. 72, pp. 43-47, June 1991.
[11] R. Prasad, A. K. Mittal and S. P. Sharma, ''A mixed method for the reduction of multi-variable systems'', Journal of Institution of Engineers, India, IE(I) Journal - EL, Vol. 85, pp. 177-181, March 2005.
[12] T.C. Chen, C.Y. Chang and K.W. Han, ''Reduction of transfer functions by the stability equation method'', Journal of Franklin Inst., Vol. 308, No. 4, pp. 389-404, 1979.
[13] T.C. Chen, C.Y. Chang and K.W. Han, ''Model reduction using the stability equation method and the continued fraction method'', Int. J. Control, Vol. 32, No. 1, pp. 81-94, 1980.
[14] C.P. Therapos, ''Stability equation method to reduce the order of fast oscillating systems'', Electronic Letters, Vol. 19, No.5, pp.183-184, March 1983.
[15] T.C. Chen, C.Y. Chang and K.W. Han, ''Stable reduced order Pade approximants using stability equation method'', Electronic Letters, Vol. 16, No. 9, pp. 345-346, 1980.
[16] J. Pal, ''Improved Pade approximants using stability equation method'', Electronic Letters, Vol. 19, No.11, pp.426-427, May 1983.
[17] R. Parthasarathy and K.N. Jayasimha, ''System reduction using stability equation method and modified Cauer continued fraction'', Proceedings IEEE, Vol. 70, No. 10, pp.1234-1236, October 1982.
[18] C. Hwang, ''Mixed method of Routh and ISE criterion approaches for reduced order modelling of continuous time systems'', Trans. ASME, J. Dyn. Syst. Meas. Control, Vol. 106, pp. 353-356, 1984. World Academy of Science, Engineering and Technology 2 2007377
[19] S. Mukherjee, and R.N. Mishra, ''Order reduction of linear systems using an error minimization technique'', Journal of Franklin Inst., Vol. 323, No. 1, pp. 23-32, 1987.
[20] N.N. Puri, and D.P. Lan, ''Stable model reduction by impulse response error minimization using Mihailov criterion and Pade's approximation'', Trans. ASME, J. Dyn. Syst. Meas. Control, Vol. 110, pp. 389-394, 1988.
[21] P. Vilbe, and L.C. Calvez, ''On order reduction of linear systems using an error minimization technique'', Journal of Franklin Inst., Vol. 327, pp. 513-514, 1990.
[22] A.K. Mittal, R. Prasad, and S.P. Sharma, ''Reduction of linear dynamic systems using an error minimization technique'', Journal of Institution of Engineers IE(I) Journal - EL, Vol. 84, pp. 201-206, March 2004.
[23] G.D. Howitt, and R. Luus, ''Model reduction by minimization of integral square error performance indices'', Journal of Franklin Inst., Vol. 327, pp. 343-357, 1990.
[24] K. Deb, Optimization for Engineering Design: Algorithms and Examples, Prentice Hall, India, 1998
[25] D.E. Goldberg, Genetic Algorithms in Search, Optimization & Machine Learning, Pearson Education, India, 2003.
[26] E. J. Davison, ''A method for simplifying linear dynamic systems'', IEEE Trans. Automat. Control, Vol. AC-11, pp. 93-101, 1966.
[27] L.S. Shieh and Y.J. Wei, ''A mixed method for multivariable system reduction'', IEEE Trans. Automat. Control, Vol. AC-20, pp. 429-432, 1975.
[28] V. Krishnamurthy and V. Seshadri, ''Model reduction using the Routh stability criterion'', IEEE Trans. Automat. Control, Vol. AC-23, No.4, pp. 729-731, August 1978.
[29] J. Pal, ''Stable reduced order Pade approximants using the Routh Hurwitz array'', Electronic Letters, Vol. 15, No.8, pp.225-226, April 1979.
[30] P. O. Gutman, C.F. Mannerfelt and P. Molander, ''Contributions to the model reduction problem'', IEEE Trans. Automat. Control, Vol. AC-27, No.2, pp. 454-455, April 1982.
[31] R. Prasad and J. Pal, ''Stable reduction of linear systems by continued fractions'', J. Inst. Engrs. India, IE (I) Journal-EL, Vol. 72, pp. 113-116, October 1991.
[32] M. G. Safonov and R. Y. Chiang, ''Model reduction for robust control: a Schur relative error method'', Int. J. Adaptive Cont. and Signal Proc., Vol. 2, pp. 259-272, 1988.
[33] Y. Bistritz and U. Shaked, ''Minimal Pade model reduction for multivariable systems'', ASME Journal of Dynamic System Measurement and Control, Vol. 106, pp.293-299, 1984.
[34] R. Prasad, J. Pal and A. K. Pant, ''Multivariable system approximation using polynomial derivatives'', Journal of Institution of Engineers, India, IE(I) Journal - EL, Vol. 76, pp. 186-188, November 1995.