Reduced Order Modelling of Linear Dynamic Systems using Particle Swarm Optimized Eigen Spectrum Analysis
The authors present an algorithm for order reduction of linear time invariant dynamic systems using the combined advantages of the eigen spectrum analysis and the error minimization by particle swarm optimization technique. Pole centroid and system stiffness of both original and reduced order systems remain same in this method to determine the poles, whereas zeros are synthesized by minimizing the integral square error in between the transient responses of original and reduced order models using particle swarm optimization technique, pertaining to a unit step input. 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 algorithm is illustrated with the help of two numerical examples and the results are compared with the other existing techniques.
[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, 1st Ed., International Series on Systems and Control, Vol. 3, Pergamon Press, 1981. [3] M. Jamshidi, Large Scale Systems Modelling and Control Series, Vol. 9, North Holland, Amsterdam, Oxford, 1983. [4] K. Jhou, J. C. Doyle, and K. Glover, Robust and Optimal Control, Prentice Hall, Upper Saddle River, New Jersey, 1996. [5] A. Varga, Model Reduction Software in the SLICOT Library, in: B. Datta (Ed.), Applied and Computational Control, Signals and Circuits, Vol. 629, Kluwer Academic Publishers, Boston, 2001. http://www.robotic.dlr.de/~varga/. [6] 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. [7] S. Mukherjee, Satakshi and R.C.Mittal, ''Model order reduction using response-matching technique'', Journal of Franklin Inst., Vol. 342, pp. 503-519, 2005. [8] G. Parmar, S. Mukherjee, and R. Prasad, ''System reduction using factor division algorithm and eigen spectrum analysis (Accepted for publication)'', Applied Mathematical Modelling, to be published. http://www.elsevier.com/locate/apm. [9] 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. [10] J. Pal, ''Stable reduced order Pade approximants using the Routh Hurwitz array'', Electronic Letters, Vol. 15, No. 8, pp.225-226, April 1979. [11] T.C. Chen, C.Y. Chang, and K.W. Han, ''Model reduction using the stability equation method and continued fraction method'', Int. J. Control, Vol. 32, No. 1, pp. 81-94, 1980. [12] R. Prasad, and J. Pal, ''Stable reduction of linear systems by continued fractions'', Journal of Institution of Engineers, India, IE(I) Journal - EL, Vol. 72, pp. 113-116, October 1991. [13] 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. [14] 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. [15] 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. [16] 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. [17] 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. [18] 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. [19] 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. [20] 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. [21] 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. [22] 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. [23] J. Kennedy, and R. C. Eberhart, ''Particle swarm optimization'', IEEE Int. Conf. on Neural Networks, IV, 1942-1948, Piscataway, NJ, 1995. [24] J. Kennedy, and R. C. Eberhart, Swarm intelligence, San Francisco: Morgan Kaufmann Publishers, 2001. [25] R. C. Eberhart, and Y. Shi, ''Particle swarm optimization: developments, applications and resources'', Congress on evolutionary computation, pp.81-86, 2001, Seoul Korea. [26] T.N. Lucas, ''Further discussion on impulse energy approximation'', IEEE Trans. Automat. Control, Vol. AC-32, No. 2, pp. 189-190, February 1987. [27] 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. [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] 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. [30] T. N. Lucas, ''Factor division: a useful algorithm in model reduction'', IEE Proceedings, Vol. 130, No. 6, pp. 362-364, November 1983. [31] B. C. Moore, ''Principal component analysis in linear systems: controllability, observability and model reduction'', IEEE Trans. Automat. Control, Vol. AC-26, No. 1, pp. 17-32, February 1981. [32] M. G. Safonov, and R. Y. Chiang, ''A Schur method for balanced-truncation model reduction'', IEEE Trans. Automat. Control, Vol. 34, No. 7, pp. 729-733, 1989. [33] 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, 1990.
[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, 1st Ed., International Series on Systems and Control, Vol. 3, Pergamon Press, 1981. [3] M. Jamshidi, Large Scale Systems Modelling and Control Series, Vol. 9, North Holland, Amsterdam, Oxford, 1983. [4] K. Jhou, J. C. Doyle, and K. Glover, Robust and Optimal Control, Prentice Hall, Upper Saddle River, New Jersey, 1996. [5] A. Varga, Model Reduction Software in the SLICOT Library, in: B. Datta (Ed.), Applied and Computational Control, Signals and Circuits, Vol. 629, Kluwer Academic Publishers, Boston, 2001. http://www.robotic.dlr.de/~varga/. [6] 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. [7] S. Mukherjee, Satakshi and R.C.Mittal, ''Model order reduction using response-matching technique'', Journal of Franklin Inst., Vol. 342, pp. 503-519, 2005. [8] G. Parmar, S. Mukherjee, and R. Prasad, ''System reduction using factor division algorithm and eigen spectrum analysis (Accepted for publication)'', Applied Mathematical Modelling, to be published. http://www.elsevier.com/locate/apm. [9] 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. [10] J. Pal, ''Stable reduced order Pade approximants using the Routh Hurwitz array'', Electronic Letters, Vol. 15, No. 8, pp.225-226, April 1979. [11] T.C. Chen, C.Y. Chang, and K.W. Han, ''Model reduction using the stability equation method and continued fraction method'', Int. J. Control, Vol. 32, No. 1, pp. 81-94, 1980. [12] R. Prasad, and J. Pal, ''Stable reduction of linear systems by continued fractions'', Journal of Institution of Engineers, India, IE(I) Journal - EL, Vol. 72, pp. 113-116, October 1991. [13] 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. [14] 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. [15] 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. [16] 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. [17] 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. [18] 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. [19] 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. [20] 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. [21] 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. [22] 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. [23] J. Kennedy, and R. C. Eberhart, ''Particle swarm optimization'', IEEE Int. Conf. on Neural Networks, IV, 1942-1948, Piscataway, NJ, 1995. [24] J. Kennedy, and R. C. Eberhart, Swarm intelligence, San Francisco: Morgan Kaufmann Publishers, 2001. [25] R. C. Eberhart, and Y. Shi, ''Particle swarm optimization: developments, applications and resources'', Congress on evolutionary computation, pp.81-86, 2001, Seoul Korea. [26] T.N. Lucas, ''Further discussion on impulse energy approximation'', IEEE Trans. Automat. Control, Vol. AC-32, No. 2, pp. 189-190, February 1987. [27] 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. [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] 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. [30] T. N. Lucas, ''Factor division: a useful algorithm in model reduction'', IEE Proceedings, Vol. 130, No. 6, pp. 362-364, November 1983. [31] B. C. Moore, ''Principal component analysis in linear systems: controllability, observability and model reduction'', IEEE Trans. Automat. Control, Vol. AC-26, No. 1, pp. 17-32, February 1981. [32] M. G. Safonov, and R. Y. Chiang, ''A Schur method for balanced-truncation model reduction'', IEEE Trans. Automat. Control, Vol. 34, No. 7, pp. 729-733, 1989. [33] 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, 1990.
@article{"International Journal of Electrical, Electronic and Communication Sciences:63693", author = "G. Parmar and S. Mukherjee and R. Prasad", title = "Reduced Order Modelling of Linear Dynamic Systems using Particle Swarm Optimized Eigen Spectrum Analysis ", abstract = "The authors present an algorithm for order reduction of linear time invariant dynamic systems using the combined advantages of the eigen spectrum analysis and the error minimization by particle swarm optimization technique. Pole centroid and system stiffness of both original and reduced order systems remain same in this method to determine the poles, whereas zeros are synthesized by minimizing the integral square error in between the transient responses of original and reduced order models using particle swarm optimization technique, pertaining to a unit step input. 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 algorithm is illustrated with the help of two numerical examples and the results are compared with the other existing techniques. ", keywords = "Eigen spectrum, Integral square error, Orderreduction, Particle swarm optimization, Stability.", volume = "1", number = "1", pages = "96-8", }
