Order Reduction by Least-Squares Methods about General Point ''a''
The concept of order reduction by least-squares moment matching and generalised least-squares methods has been extended about a general point ?a?, to obtain the reduced order models for linear, time-invariant dynamic systems. Some heuristic criteria have been employed for selecting the linear shift point ?a?, based upon the means (arithmetic, harmonic and geometric) of real parts of the poles of high order system. It is shown that the resultant model depends critically on the choice of linear shift point ?a?. The validity of the criteria is illustrated by solving a numerical example and the results are compared with the other existing techniques.
<p>[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 and its Applications, Vol. 415, pp. 385-405, 2006.
[7] G. Parmar, S. Mukherjee and R. Prasad, "System reduction using factor division algorithm and eigen spectrum analysis (Accepted for publication)", Applied Mathematical Modelling, In Press. http://www.elsevier.com/locate/apm.
[8] G. Parmar, R. Prasad and S. Mukherjee, "Order reduction of linear dynamic systems using stability equation method and GA", Int. Journal of Computer, Information, and Systems Science, and Engineering, Vol. 1, No. 1, pp. 26-32, 2007.
[9] Y. Shamash, ""Approximations of linear time invariant systems", Proc. Conf. on Pade approximants and their applications, P. R. Graves-Morris (Ed.), London Academic, 1973.
[10] G. A. Baker, Essentials of Pade approximation, Academic Press, New York, 1975.
[11] A. Bultheel and M. V. Barel, "Pade techniques for model reduction in linear system theory, A survey", Journal of Computational and Applied Mathematics, Vol. 14, pp. 401-438, 1986.
[12] F. F. Shoji, K. Abe and H. Takeda, "Model reduction for a class of linear dynamic systems", Journal of Franklin Inst., Vol. 319, pp. 549-558, 1985.
[13] T. N. Lucas and I. F. Beat, "Model reduction by least-squares moment matching", Electronics Letters, Vol. 26, No. 15, pp. 1213-1215, July 1990.
[14] T. N. Lucas and A. R. Munro, "Model reduction by generalised leastsquares method", Electronics Letters, Vol. 27, No. 15, pp. 1383-1384, July 1991.
[15] L. A. Aguirre, "The least-squares Padé method for model reduction", Int. Journal of Systems Science, Vol. 23, No. 10, pp. 1559-1570, 1992.
[16] L. A. Aguirre, "Model reduction via least-squares Padé simplification of squared-magnitude functions", Int. Journal of Systems Science, Vol. 25, No. 7, pp. 1191-1204, 1994.
[17] L. A. Aguirre, "Partial least-squares Padé reduction with exact retention of poles and zeros", Int. Journal of Systems Science, Vol. 25, No. 12, pp. 2377-2391, 1994.
[18] L. A. Aguirre, "Algorithm for extended least-squares model reduction", Electronics Letters, Vol. 31, No. 22, pp. 1957-1959, October 1995.
[19] R. Prasad, J. Pal and A. K. Pant, "Linear multivariable system reduction by continued fraction expansion about a general point ÔÇÿA- ", Advances in Modelling and Simulation, AMSE Press, Vol. 19, No. 4, pp. 47-58, 1990.
[20] S. C. Chuang, "Application of continued-fraction method for modelling transfer functions to give more accurate initial transient response", Electronics Letters, Vol. 6, No. 6, pp. 861-863, 1970.
[21] T.N. Lucas, "Further discussion on impulse energy approximation", IEEE Trans. Automat. Control, Vol. AC-32, No. 2, pp. 189-190, February 1987.
[22] R. Parthasarathy and S. John, "System reduction using Cauer continued fraction expansion about s = 0 and s = ∞ alternately", Electronics Letters, Vol. 14, No. 8, pp. 261-262, April 1978.
[23] S. A. Marshall, "Comments on ÔÇÿStable reduced order Pade approximants using the Routh Hurwitz array-", Electronics Letters, Vol. 15, No. 16, pp. 501-502, August 1974.
[24] T. C. Chen, C. Y. Chang and K. W. Han, "Model reduction using the stability equation method and the continued fraction method", Int. Journal of Control, Vol. 32, No. 1, pp. 81-94, 1980.
[25] J. Pal, "Stable reduced order Pade approximants using the Routh Hurwitz array", Electronic Letters, Vol. 15, No. 8, pp.225-226, April 1979.
[26] A. Lepschy and U. Viaro, "An improvement in Routh Pade approximation technique", Int. Journal of Control, Vol. 36, No. 4, pp. 643-661, 1982.
[27] J. Pal, "Improved Pade approximants using stability equation method", Electronic Letters, Vol. 19, No. 11, pp.426-427, May 1983.</p>
<p>[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 and its Applications, Vol. 415, pp. 385-405, 2006.
[7] G. Parmar, S. Mukherjee and R. Prasad, "System reduction using factor division algorithm and eigen spectrum analysis (Accepted for publication)", Applied Mathematical Modelling, In Press. http://www.elsevier.com/locate/apm.
[8] G. Parmar, R. Prasad and S. Mukherjee, "Order reduction of linear dynamic systems using stability equation method and GA", Int. Journal of Computer, Information, and Systems Science, and Engineering, Vol. 1, No. 1, pp. 26-32, 2007.
[9] Y. Shamash, ""Approximations of linear time invariant systems", Proc. Conf. on Pade approximants and their applications, P. R. Graves-Morris (Ed.), London Academic, 1973.
[10] G. A. Baker, Essentials of Pade approximation, Academic Press, New York, 1975.
[11] A. Bultheel and M. V. Barel, "Pade techniques for model reduction in linear system theory, A survey", Journal of Computational and Applied Mathematics, Vol. 14, pp. 401-438, 1986.
[12] F. F. Shoji, K. Abe and H. Takeda, "Model reduction for a class of linear dynamic systems", Journal of Franklin Inst., Vol. 319, pp. 549-558, 1985.
[13] T. N. Lucas and I. F. Beat, "Model reduction by least-squares moment matching", Electronics Letters, Vol. 26, No. 15, pp. 1213-1215, July 1990.
[14] T. N. Lucas and A. R. Munro, "Model reduction by generalised leastsquares method", Electronics Letters, Vol. 27, No. 15, pp. 1383-1384, July 1991.
[15] L. A. Aguirre, "The least-squares Padé method for model reduction", Int. Journal of Systems Science, Vol. 23, No. 10, pp. 1559-1570, 1992.
[16] L. A. Aguirre, "Model reduction via least-squares Padé simplification of squared-magnitude functions", Int. Journal of Systems Science, Vol. 25, No. 7, pp. 1191-1204, 1994.
[17] L. A. Aguirre, "Partial least-squares Padé reduction with exact retention of poles and zeros", Int. Journal of Systems Science, Vol. 25, No. 12, pp. 2377-2391, 1994.
[18] L. A. Aguirre, "Algorithm for extended least-squares model reduction", Electronics Letters, Vol. 31, No. 22, pp. 1957-1959, October 1995.
[19] R. Prasad, J. Pal and A. K. Pant, "Linear multivariable system reduction by continued fraction expansion about a general point ÔÇÿA- ", Advances in Modelling and Simulation, AMSE Press, Vol. 19, No. 4, pp. 47-58, 1990.
[20] S. C. Chuang, "Application of continued-fraction method for modelling transfer functions to give more accurate initial transient response", Electronics Letters, Vol. 6, No. 6, pp. 861-863, 1970.
[21] T.N. Lucas, "Further discussion on impulse energy approximation", IEEE Trans. Automat. Control, Vol. AC-32, No. 2, pp. 189-190, February 1987.
[22] R. Parthasarathy and S. John, "System reduction using Cauer continued fraction expansion about s = 0 and s = ∞ alternately", Electronics Letters, Vol. 14, No. 8, pp. 261-262, April 1978.
[23] S. A. Marshall, "Comments on ÔÇÿStable reduced order Pade approximants using the Routh Hurwitz array-", Electronics Letters, Vol. 15, No. 16, pp. 501-502, August 1974.
[24] T. C. Chen, C. Y. Chang and K. W. Han, "Model reduction using the stability equation method and the continued fraction method", Int. Journal of Control, Vol. 32, No. 1, pp. 81-94, 1980.
[25] J. Pal, "Stable reduced order Pade approximants using the Routh Hurwitz array", Electronic Letters, Vol. 15, No. 8, pp.225-226, April 1979.
[26] A. Lepschy and U. Viaro, "An improvement in Routh Pade approximation technique", Int. Journal of Control, Vol. 36, No. 4, pp. 643-661, 1982.
[27] J. Pal, "Improved Pade approximants using stability equation method", Electronic Letters, Vol. 19, No. 11, pp.426-427, May 1983.</p>
@article{"International Journal of Electrical, Electronic and Communication Sciences:61900", author = "Integral square error and Least-squares and Markovparameters and Moment matching and Order reduction.", title = " Order Reduction by Least-Squares Methods about General Point ''a''", abstract = "The concept of order reduction by least-squares moment matching and generalised least-squares methods has been extended about a general point ?a?, to obtain the reduced order models for linear, time-invariant dynamic systems. Some heuristic criteria have been employed for selecting the linear shift point ?a?, based upon the means (arithmetic, harmonic and geometric) of real parts of the poles of high order system. It is shown that the resultant model depends critically on the choice of linear shift point ?a?. The validity of the criteria is illustrated by solving a numerical example and the results are compared with the other existing techniques.
", keywords = "Integral square error, Least-squares, Markovparameters, Moment matching, Order reduction.", volume = "1", number = "2", pages = "340-8", }
