Relative Mapping Errors of Linear Time Invariant Systems Caused By Particle Swarm Optimized Reduced Order Model

The authors present an optimization algorithm for order reduction and its application for the determination of the relative mapping errors of linear time invariant dynamic systems by the simplified models. These relative mapping errors are expressed by means of the relative integral square error criterion, which are determined for both unit step and impulse inputs. The reduction algorithm is based on minimization of the integral square error by particle swarm optimization technique pertaining to a unit step input. The algorithm 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. Two numerical examples are solved to illustrate the superiority of the algorithm over some existing methods.

Authors:

• G. Parmar
• S. Mukherjee
• R. Prasad

Keywords:

• Order reduction
• Particle swarm optimization
• Relative mapping error
• Stability.

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. Jamshidi, Large Scale Systems Modelling and Control Series, New
York, Amsterdam, Oxford, North Holland, Vol. 9, 1983.
[3] 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.
[4] 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.
[5] S. Mukherjee, Satakshi and R.C.Mittal, "Model order reduction using
response-matching technique", Journal of Franklin Inst., Vol. 342 , pp.
503-519, 2005.
[6] 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.
[7] E. Layer, "Mapping error of simplified dynamic models in electrical
metrology", Proc. 16th IEEE Inst. and Meas. Tech. Conf., Vol. 3, pp.
1704-1709, May 24-26, 1999.
[8] E. Layer, "Mapping error of linear dynamic systems caused by reduced
order model", IEEE Trans. Inst. and Meas., Vol. 50, No. 3, pp. 792-799,
June 2001.
[9] 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.
[10] S. Mukherjee and R. N. Mishra, "Order reduction of linear systems using
an error minimization technique", Journal of Franklin Institute, Vol.
323, No. 1, pp. 23-32, 1987.
[11] S. S. Lamba, R. Gorez an 1987.d B. Bandyopadhyay, "New reduction
technique by step error minimization for multivariable systems", Int. J.
Systems Sci., Vol. 19, No. 6, pp. 999-1009, 1988.
[12] Mukherjee and R.N. Mishra, "Reduced order modeling of linear
multivariable systems using an error minimization technique", Journal
of Franklin Inst., Vol. 325, No. 2, pp. 235-245, 1988.
[13] 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.
[14] 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.
[15] 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.
[16] 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.
[17] J. Kennedy and R. C. Eberhart, "Particle swarm optimization", IEEE Int.
Conf. on Neural Networks, IV, 1942-1948, Piscataway, NJ, 1995.
[18] J. Kennedy and R. C. Eberhart, Swarm intelligence, 2001, Morgan
Kaufmann Publishers, San Francisco.
[19] R. C. Eberhart and Y. Shi, "Particle swarm optimization: developments,
applications and resources", Congress on evolutionary computation,
Seoul Korea, pp. 81-86, 2001.
[20] T.N. Lucas, "Further discussion on impulse energy approximation",
IEEE Trans. Automat. Control, Vol. AC-32, No. 2, pp. 189-190,
February 1987.
[21] Y. Shamash, "Linear system reduction using Pade approximation to
allow retention of dominant modes", Int. J. Control, Vol. 21, No. 2, pp.
257-272, 1975.
[22] M. F. Hutton and B. Friedland, "Routh approximation for reducing order
of linear, time invariant systems", IEEE Trans. Automat Control, Vol.
AC-20, No. 3, pp. 329-337, June 1975.
[23] 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.
[24] J. Pal, "Stable reduced order Pade approximants using the Routh
Hurwitz array", Electronic Letters, Vol. 15, No. 8, pp.225-226, April
1979.
[25] 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.
[26] 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.
[27] T.N. Lucas, "Factor division; a useful algorithm in model
reduction", IEE Proceedings, Vol. 130, No. 6, pp. 362-364, November
1983.
[28] R. Prasad and J. Pal, "Stable reduction of linear systems by continued
fractions", Journal of Institution of Engineers IE(I) Journal - EL, Vol.
72, pp. 113-116, October 1991.
[29] M. G. Safonov, R. Y. Chiang and D. J. N. Limebeer, "Optimal Hankel
model reduction for nonminimal systems", IEEE Trans. Automat
Control, Vol. 35, No.4, pp 496-502, April 1990.