Chebyshev rational functions approximation for model order reduction using harmony search

A new method for model reduction of linear systems is presented, based on Chebyshev rational functions, using the Harmony Search (HS) algorithm. First, the full order system is expanded and then a set of parameters in a fixed structure are determined, whose values define the reduced order system. The values are obtained by minimizing the errors between the l first coefficients of the Chebyshev rational function expansion of full and reduced systems, using the HS algorithm. To assure stability, the Routh criterion is used as constraints in the optimization problem. To present the ability of the proposed method, three test systems are reduced. The results obtained are compared with other existing techniques. The results obtained show the accuracy and efficiency of the proposed method.

[1]  P. Parks.  A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov , 1962, Mathematical Proceedings of the Cambridge Philosophical Society.

[2]  Z. Elrazaz,et al.  On the selection of the dominant poles of a system to be retained in a low-order model , 1979 .

[3]  D. J. Wright,et al.  The continued fraction representation of transfer functions and model simplification , 1973 .

[4]  D. Wilson,et al.  Model reduction for multivariable systems , 1974 .

[5]  K. Han,et al.  Reduction of Transfer Functions by the Stability-Equation Method , 1979 .

[6]  S. Mukherjee,et al.  Order Reduction of Linear Dynamic Systems using Stability Equation Method and GA , 2007 .

[7]  C. F. Chen,et al.  A novel approach to linear model simplification , 1968 .

[8]  Maamar Bettayeb,et al.  H2 and H∞ optimal model reduction using genetic algorithms , 2011, J. Frankl. Inst..

[9]  M. Bettayeb,et al.  Characterization and computation of the solution to the optimal L ∞ approximation problem , 1994, IEEE Trans. Autom. Control..

[10]  Ian Postlethwaite,et al.  Multivariable Feedback Control: Analysis and Design , 1996 .

[11]  C. F. Chen.  Model reduction of multivariable control systems by means of matrix continued fractions , 1974 .

[12]  Maamar Bettayeb,et al.  Characterization of the solution to the optimal H∞ model reduction problem , 1993 .

[13]  M. Chidambara,et al.  Further comments on "A method for simplifying linear dynamic systems" , 1967 .

[14]  V. Singh,et al.  Obtaining Routh-Pade approximants using the Luus-Jaakola algorithm , 2005 .

[15]  Sidhartha Panda,et al.  Evolutionary Techniques for Model Order Reduction of Large Scale Linear Systems , 2009 .

[16]  R. K. Appiah,et al.  Padé methods of Hurwitz polynomial approximation with application to linear system reduction , 1979 .

[17]  James Lam,et al.  On H2 model reduction of bilinear systems , 2002, Autom..

[18]  G. Obinata,et al.  Authors' reply to "Comments on model reduction by minimizing the equation error" , 1983 .

[19]  Gian Antonio Mian,et al.  Optimality conditions in multivariable L2 model reduction , 1993 .

[20]  Dia I. Abu-Al-Nadi,et al.  Reduced Order Modeling of Linear MIMO Systems Using Particle Swarm Optimization , 2011 .

[21]  Z. Geem.  Music-Inspired Harmony Search Algorithm: Theory and Applications , 2009 .

[22]  Eduard Eitelberg.  Model reduction by minimizing the weighted equation error , 1981 .

[23]  B. Friedland,et al.  Routh approximations for reducing order of linear, time-invariant systems , 1975 .

[24]  E. Davison,et al.  On "A method for simplifying linear dynamic systems" , 1966 .

[25]  R. C. Mittal,et al.  Model order reduction using response-matching technique , 2005, J. Frankl. Inst..

[26]  R. Parthasarathy,et al.  System reduction by Routh approximation and modified Cauer continued fraction , 1979 .

[27]  K. Han,et al.  Model reduction using the stability-equation method and the continued-fraction method , 1980 .

[28]  Y. Shih,et al.  Model reduction of continuous and discrete multivariable systems by moments matching , 1978 .

[29]  J Lam,et al.  ON H2 MODEL REDUCTION OF BILINEAR SYSTEM , 2002 .

[30]  F. Lees.  The derivation of simple transfer function models of oscillating and inverting processes from the basic transformed equations using the method of moments , 1971 .

[31]  M. K. Pandey,et al.  System order reduction using GA for unit impulse input and a comparative study using ISE & IRE , 2009, ICAC3 '09.

[32]  F. P. Lees,et al.  The reduction of complex transfer function models to simple models using the method of moments , 1969 .

[33]  K. Han,et al.  Stability margins and model reduction , 1993 .

[34]  B. Moore.  Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

[35]  L. Silverman,et al.  Model reduction via balanced state space representations , 1982 .

[36]  Y. Shamash.  Stable reduced-order models using Padé-type approximations , 1974 .

[37]  R. Appiah.  Linear model reduction using Hurwitz polynomial approximation , 1978 .

[38]  A. S. I. Zinober,et al.  Adaptive relay control of second-order systems , 1975 .

[39]  V. Zakian,et al.  Simplification of linear time-invariant systems by moment approximants † , 1973 .

[40]  Zong Woo Geem,et al.  A New Heuristic Optimization Algorithm: Harmony Search , 2001, Simul..

[41]  M. Chidambara,et al.  Two simple techniques for the simplification of large dynamic systems , 1969 .

[42]  Mathukumalli Vidyasagar,et al.  Order reduction by ℓ 1- and ℓ ∞ -norm minimization , 1978 .

[43]  G. Obinata,et al.  A method for modeling linear time-invariant systems by linear systems of low order , 1976 .