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On dominant poles and model reduction of second order time-delay systems. (English) Zbl 1236.93080
Summary: The method known as the Dominant Pole Algorithm (DPA) has previously been successfully used in combination with model order reduction techniques to approximate standard linear time-invariant dynamical systems and second order dynamical systems. In this paper, we show how this approach can be adapted to a class of second order delay systems, which are large scale nonlinear problems whose transfer functions have an infinite number of simple poles. Deflation is a very important ingredient for this type of methods. Because of the nonlinearity, many deflation approaches for linear systems are not applicable. We therefore propose an alternative technique that essentially removes computed poles from the system’s input and output vectors. In general, this technique changes the residues, and hence, modifies the order of dominance of the poles, but we prove that, under certain conditions, the residues stay near the original residues. The new algorithm is illustrated by numerical examples.

MSC:
93B60 Eigenvalue problems
93B55 Pole and zero placement problems
93C15 Control/observation systems governed by ordinary differential equations
Software:
JDQZ; DPA_TDEFL; QDPA; UMFPACK
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[1] Antoulas, A., Approximation of large-scale dynamical systems, (2005), SIAM Philadelphia, PA · Zbl 1112.93002
[2] Bellman, R.; Cooke, K.L., Differential-difference equations, (1963), Academic Press New York · Zbl 0118.08201
[3] ()
[4] Bezerra, L.H., IEEE trans. power syst., 30, 4, 2137-2157, (2008), written discussion to [15]
[5] Curtain, R.F.; Zwart, H., An introduction to infinite-dimensional linear systems theory, (1995), Springer-Verlag New York · Zbl 0839.93001
[6] T. Davis, Usersʼ guide for the Unsymmetric-pattern MultiFrontal Package (UMFPACK), Technical report TR-95-004, Computer and Information Sciences Department, University of Florida, Gainesville, FL, 1995.
[7] Denman, E.D.; Leyva-Ramos, J.; Jeon, G., The algebraic theory of latent projectors in lambda matrices, Appl. math. comput., 9, 173-300, (1981) · Zbl 0496.15012
[8] Gohberg, I.; Lerer, L., Resultants of matrix polynomials, Bull. amer. math. soc., 82, 565-567, (1976) · Zbl 0343.15011
[9] Hale, J.; Verduyn Lunel, S.M., Introduction to functional differential equations, (1993), Springer-Verlag · Zbl 0787.34002
[10] Harkort, C.; Deutscher, J., Krylov subspace methods for linear infinite-dimensional systems, IEEE trans. automatic control, 56, 2, 441-447, (2011) · Zbl 1368.93071
[11] Hu, H.Y.; Dowell, E.H.; Virgin, L.N., Stability estimation of high dimensional vibrating systems under state delay feedback control, J. sound vib., 214, 3, 497-511, (1998) · Zbl 1235.93188
[12] Jarlebring, E.; Meerbergen, K.; Michiels, W., A Krylov method for the delay eigenvalue problem, SIAM J. sci. comput., 32, 6, 3278-3300, (2010) · Zbl 1226.65069
[13] Lancaster, P., Lambda-matrices and vibrating systems, (2002), Dover Publications Mineola, NY · Zbl 1048.34002
[14] Mäkilä, P.; Partington, J., Laguerre and kautz shift approximations of delay systems, Internat. J. control, 72, 10, 932-946, (1999) · Zbl 0963.93042
[15] Martins, N.; Lima, L.T.G.; Pinto, H.J.C.P., Computing dominant poles of power system transfer functions, IEEE trans. power syst., 11, 162-170, (1996)
[16] Michiels, W.; Niculescu, S.-I., Stability and stabilization of time-delay systems. an eigenvalue based approach, (2007), SIAM Philadelphia, PA, USA
[17] W. Michiels, E. Jarlebring, K. Meerbergen, Krylov based model order reduction of time-delay systems, Technical report TW568, Dept. Comp. Sci., KU Leuven; SIAM J. Matrix Anal. Appl. (June 2010), in press, https://lirias.kuleuven.be/handle/123456789/269795. · Zbl 1247.65086
[18] MUMPS, mulfrontal massively parallel solver, (2009)
[19] Niculescu, S.-I., Delay effects on stability. A robust control approach, (2001), Springer-Verlag London
[20] Odabasioglu, A.; Celik, M.; Paganini, L., Prima: passive reduced-order interconnect macromodeling algorithm, IEEE TCAD integ. circuits syst., 17, 8, 645-654, (1998)
[21] Rommes, J.; Martins, N., Efficient computation of multivariable transfer function dominant poles using subspace acceleration, IEEE trans. power syst., 21, 1471-1483, (2006)
[22] Rommes, J.; Martins, N., Efficient computation of transfer function dominant poles using subspace acceleration, IEEE trans. power syst., 21, 1218-1226, (2006)
[23] Rommes, J.; Martins, N., Computing transfer function dominant poles of large-scale second-order dynamical systems, SIAM J. sci. comput., 30, 4, 2137-2157, (2008) · Zbl 1168.65338
[24] Rommes, J.; Sleijpen, G.L.G., Convergence of the dominant pole algorithm and Rayleigh quotient iteration, SIAM J. matrix anal. appl., 1, 346-363, (2008) · Zbl 1165.65016
[25] Sleijpen, G.; van der Vorst, H., A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM J. matrix anal. appl., 17, 401-425, (1996) · Zbl 0860.65023
[26] Smith, J.; Hauer, J.; Trudnowski, D., Transfer function identification in power system applications, IEEE trans. power syst., 8, 3, 1282-1290, (1993)
[27] Y. Yue, K. Meerbergen, Using model order reduction for design optimization of structures and vibrations, TW reports TW566, Department of Computer Science, K.U. Leuven, May 2010, URL https://lirias.kuleuven.be/handle/123456789/268881.
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