AUTHORS: Sharad Kumar Tiwari, Gagandeep Kaur

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ABSTRACT: In this paper, we introduce a new clustering algorithm based on Lehmer measure for simplifying the order of large scale dynamical system. This algorithm is applied with the combined advantages of retaining the dominant poles and the Pade approximation. The dominant pole algorithm adapted to select poles to obtain the cluster centre. The selection of poles for the cluster is based on from the viewpoint of important poles contribution to the system is preserved by dominant pole algorithm. Having obtained the denominator polynomial of the reduced model, the numerator terms are obtained using the Pade approximation method. The reduction algorithm is fully computer oriented. The reduced model is stable if the original model is stable. Moreover, this method gives a better approximation in both the transient and the steady-state responses of the original system.

KEYWORDS: Order Reduction; Clustering; Lehmer measure; Dominant pole; Pade approximation; Integral Square Error.

REFERENCES:

[1]. S. C. Chuang, “Homographic transformation for the simplification of continuous-time transfer functions by Padé approximation,” Int. J. Control, vol. 23, no. 6, pp. 821–826, Jun. 1976.

[2]. A. Bultheel and M. V. Barel, “Padé techniques for model reduction in linear system theory: a survey,” J. Comput. Appl. Math., vol. 14, pp. 401–438, 1986.

[3]. C. F. Chen and L. S. Shieh, “An algebraic method for control systems design,” Int. J. Control, vol. 11, no. 5, pp. 717–739, May 1970.

[4]. R. Parthasarathy and S. John, “System reduction using Cauer continued fraction expansion about s= 0 and s=∞ alternately,” Electron. Lett., vol. 18, no. 23, pp. 9–10, 1978.

[5]. R. Prasad and J. Pal, “Use of continued fraction expansion for stable reduction of linear multivariable systems,” Inst. Eng. India Part Electr. Eng. Div., vol. 72, pp. 43–43, 1991.

[6]. M. Hutton and B. Friedland, “Routh approximations for reducing order of linear, timeinvariant systems,” IEEE Trans. Automat. Contr., vol. 20, no. 3, pp. 329–337, Jun. 1975.

[7]. V.Krishnamuthy and V. Seshadri, “Model Reduction Using the Routh Stability Criterion,” IEEE Trans. Automat. Contr., vol. 23, pp. 729– 731, 1978.

[8]. V. Krishnamurthi and V. Seshadri, “A simple and direct method of reducing order of linear systems using routh approximations in the frequency domain,” Autom. Control. IEEE …, vol. 21, no. 5, pp. 797–799, 1976.

[9]. R. K. Appiah, “Pade methods of Hurwitz polynomial approximation with application to linear system reduction,” Int. J. Control, vol. 29, no. 1, pp. 39–48, Jan. 1979.

[10].Y. Shamash, “Failure of the Routh-Hurwitz method of reduction,” IEEE Trans. Automat. Contr., vol. 25, no. 2, pp. 313–314, 1980.

[11].T. C. Chen, C. Y. Chang, and K. W. Han, “Reduction of transfer functions by the stabilityequation method,” Journal of the Franklin Institute, vol. 308, no. 4. pp. 389–404, 1979.

[12].T. C. Chen, C. Y. Chang, and K. W. Han, “Model reduction using the stability-equation method and the Padé approximation method,” J. Franklin Inst., vol. 309, no. 6, pp. 473–490, 1980.

[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, Jul. 1980.

[14].Y. Shamash, “Stable biased reduced order models using the Routh method of reduction,” Int. J. Syst. Sci., vol. 11, no. 5, pp. 641–654, Jan. 1980.

[15].G. Langhoz and Y. Bistrltz, “Model reduction of dynamic systems over a frequency interval†,” Int. J. Control, vol. 31, no. 1, pp. 51–62, Jan. 1980.

[16].Y. Bistritz and G. Langholz, “Model reduction by Chebyshev polynomial techniques,” IEEE Trans. Automat. Contr., vol. 24, no. 5, pp. 741–747, 1979.

[17].H. N. Soloklo, O. Nali, and M. M. Farsangi, “Model Reduction by Hermite Polynomials and Genetic Algorithm,” J. Math. Comput. Sci., vol. 9, pp. 188–202, 2014.

[18].B.W. Wan, “Linear model reduction using Mihailov criterion and Padé approximation technique,” Int. J. Control, vol. 33, no. 6, pp. 1073–1089, Jun. 1981.

[19].A. K. Sinha and J. Pal, “Simulation based reduced order modelling using a clustering technique,” Comput. Electr. Eng., vol. 16, no. 3, pp. 159–169, Jan. 1990.

[20].C.B. Vishwakarma and R. Prasad, “Clustering Method for Reducing Order of Linear System using Pade Approximation,” IETE Journal of Research, vol. 54, no. 5. p. 326, 2008.

[21].R. Komarasamy, N. Albhonso, and G. Gurusamy, “Order reduction of linear systems with an improved pole clustering,” J. Vib. Control, vol. 18, no. 12, pp. 1876–1885, Dec. 2011.

[22].C. B. Vishwakarma and R. Prasad, “MIMO System Reduction Using Modified Pole Clustering and Genetic Algorithm,” Model. Simul. Eng., vol. 2009, pp. 1–5, 2009.

[23].G. Parmar, S. Mukherjee, and R. Prasad, “System reduction using factor division algorithm and eigen spectrum analysis,” Appl. Math. Model., vol. 31, no. 11, pp. 2542–2552, Nov. 2007.

[24].K. B. Stolarsky, “Holder Means, Lehmer Means, and,” J. Math. Anal. Appl., vol. 202, pp. 810–818, 1996.

[25].M. Saadvandi, K. Meerbergen, and E. Jarlebring, “On dominant poles and model reduction of second order time-delay systems,” Appl. Numer. Math., vol. 62, no. 1, pp. 21–34, Jan. 2012.

[26].N. Martins and P. E. M. Quintão, “Computing dominant poles of power system multlvariable transfer functions,” IEEE Trans. Power Syst., vol. 18, no. 1, pp. 152–159, 2003.

[27].A. K. Mittal, R. Prasad, and S. P. Sharama, “Reduction of linear dynamic system using an error minimization technique,” J. Inst. Eng., vol. 84, pp. 201–204, 2004.

[28].T. Lucas, “Factor division: a useful algorithm in model reduction,” Control Theory Appl. 130, vol. 130, no. 6, pp. 1981–1983, 1983.

[29].S. R. Desai and R. Prasad, “A new approach to order reduction using stability equation and big bang big crunch optimization,” Syst. Sci. Control Eng., vol. 1, no. 1, pp. 20–27, Dec. 2013.

[30].B. Philip and J. Pal, “An evolutionary computation based approach for reduced order modelling of linear systems,” 2010 IEEE Int. Conf. Comput. Intell. Comput. Res., pp. 1–8, Dec. 2010.

[31].J. Pal, “Improved Pade approximants using stability equation method,” Electron. Lett., vol. 19, no. 11, pp. 3–4, 1983.

[32].H. N. Soloklo and M. M. Farsangi, “Multiobjective weighted sum approach model reduction by Routh-Pade approximation using harmony search,” Turkish J. Electr. Eng. Comput. Sci., vol. 21, pp. 2283–2293, 2013.

[33].M. Jamshidi, Large-Scale Systems: Modeling, Control and Fuzzy Logic. New York, NorthHolland, 1983.

[34].H. N. Soloklo, R. Hajmohammadi, and M. M. Farsangi, “Model order reduction based on moment matching using Legendre wavelet and harmony search algorithm,” Iran. J. Sci. Technol. , Trans. Electr. Eng., vol. 39, no. E1, pp. 39–54, 2015.

[35].J. Singh, C. B. Vishwakarma, and K. Chattterjee, “Biased Reduction Method by Combining Improved Modified Pole Clustering and Improved Pade Approximations,” Appl. Math. Model., 2015.