Van den Hof, Paul M. J.; Heuberger, Peter S. C.; Bokor, József System identification with generalized orthonormal basis functions. (English) Zbl 0848.93013 Automatica 31, No. 12, 1821-1834 (1995). Linear system identification is considered by using a set of flexible basis functions for the space of stable systems that generalize the classical Laguerre and Kautz bases. A least-squares identification of a finite number of expansion coefficients in the orthogonal series expansion of a transfer function is studied, and explicit bounds for the asymptotic bias and variance errors of the parameter estimates and the resulting transfer function estimates are derived and analyzed. The important advantage of the method is that a proper choice of basis functions (reflecting the dominant dynamics of the process to be modeled) can substantially diminish the number of expansion coefficients that should be estimated in a finite-length series expansion to obtain the approximate model up to a satisfactory accuracy. An illustrative simulation example is included. Reviewer: Z.Hasiewicz (Wrocław) Cited in 38 Documents MSC: 93B30 System identification 42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) Keywords:linear system identification; basis functions; least-squares identification; orthogonal series PDFBibTeX XMLCite \textit{P. M. J. Van den Hof} et al., Automatica 31, No. 12, 1821--1834 (1995; Zbl 0848.93013) Full Text: DOI References: [1] Clowes, G. J., Choice of the time scaling factor for linear system approximations using orthonormal Laguerre functions, IEEE Trans. Autom. Control, AC-10, 487-489 (1965) [2] De Callafon, R. A.; Van den Hof, P. M.J.; Steinbuch, M., Control relevant identification of a compact disc pick-up mechanism, (Proc. 32nd IEEE Conf. on Decision and Control. Proc. 32nd IEEE Conf. on Decision and Control, San Antonio, TX (1993)), 2050-2055 [3] De Vries, D. K., Identification of model uncertainty for control design, (Doctoral dissertation (1994), Mechanical Engineering Systems and Control Group, Delft University of Technology) [4] Fu, Y.; Dumont, G. A., An optimum time scale for discrete Laguerre network, IEEE Trans. Autom. Control, AC-38, 934-938 (1993) · Zbl 0800.93033 [5] Gottlieb, M. J., Concerning some polynomials orthogonal on finite or enumerable set of points, Am. J. Math., 60, 453-458 (1938) · JFM 64.0329.01 [6] Grenander, U.; Szegö, G., (Toeplitz Forms and Their Applications (1958), University of California Press: University of California Press Berkeley) · Zbl 0080.09501 [7] Hakvoort, R. G.; Van den Hof, P. M.J., An instrumental variable procedure for identification of probabilistic frequency response uncertainty regions, (Proc. 33rd IEEE Conf. on Decision and Control. Proc. 33rd IEEE Conf. on Decision and Control, Lake Buena Vista, FL (1994)), 3596-3601 [8] Hannan, E. J.; Wahlberg, B., Convergence rates for inverse Toeplitz matrix forms, J. Multiv. Anal., 31, 127-135 (1989) · Zbl 0689.62078 [9] Heuberger, P. S.C., On approximate system identification with system based orthonormal functions, (Doctoral dissertation (1991), Delft University of Technology: Delft University of Technology The Netherlands) [10] Heuberger, P. S.C.; Bosgra, O. H., Approximate system identification using system based orthonormal functions, (Proc. 29th IEEE Conf. on Decision and Control. Proc. 29th IEEE Conf. on Decision and Control, Honolulu, HI (1990)), 1086-1092 [11] Heuberger, P. S.C.; Van den Hof, P. M.J., A new signals and systems transform induced by generalized orthonormal basis functions, (Report N-487 (1995), Mechanical Engineering Systems and Control Group, Delft University of Technology), June 1995 [12] Heuberger, P. S.C.; Van den Hof, P. M.J.; Bosgra, O. H., A generalized orthonormal basis for linear dynamical systems, IEEE Trans. Autom. Control, AC-40, 451-465 (1995) · Zbl 0835.93011 [13] Heuberger, P. S.C.; Van den Hof, P. M.J.; Bosgra, O. H., A generalized orthonormal basis for linear dynamical systems, (Proc. 32nd IEEE Conf. on Decision and Control. Proc. 32nd IEEE Conf. on Decision and Control, San Antonio, TX (1993)), 2850-2855 [14] Kautz, W. H., Transient syythesis in the time domain, IRE Trans. Cire. Theory, CT-1, 29-39 (1954) [15] King, R. E.; Paraskevopoulos, P. N., Parametric identification of discrete time SISO systems, Int. J. Control, 30, 1023-1029 (1979) · Zbl 0418.93026 [16] Lee, Y. W., Synthesis of electrical networks by means of the Fourier transforms of Laguerre functions, J. Maths and Phys., 11, 83-113 (1933) · Zbl 0005.04509 [17] Lee, Y. W., (Statistical Theory of Communication (1960), Wiley: Wiley New York) · Zbl 0091.14002 [18] Ljung, L., (System Identification—Theory for the User (1987), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ) · Zbl 0615.93004 [19] Ljung, L.; Yuan, Z. D., Asymptotic properties of black-box identification of transfer functions, IEEE Trans. Autom. Control, AC-30, 514-530 (1985) · Zbl 0632.93075 [20] Nurges, Y., Laguerre models in problems of approximation and identification of discrete systems, Autom. Rem. Control, 48, 346-352 (1987) · Zbl 0625.93020 [21] Nurges, Y.; Yaaksoo, Y., Laguerre state equations for a multivariable discrete system, Autom. Rem. Control, 42, 1601-1603 (1981) · Zbl 0501.93040 [22] Szegö, G., (Orthogonal Polynomials (1975), American Mathematical Society: American Mathematical Society Providence, RI) · JFM 61.0386.03 [23] Van den Hof, P. M.J.; Heuberger, P. S.C.; Bokor, J., System identification with generalized orthonormal basis functions, (Proc. 33rd IEEE Conf. on Decision and Control. Proc. 33rd IEEE Conf. on Decision and Control, Lake Buena Vista, FL (1994)), 3382-3387 [24] Wahlberg, B., On the use of orthogonalized exponentials in system identification, (Report LiTH-ISY-1099 (1990), Department of Electrical Engineering, Linköping University: Department of Electrical Engineering, Linköping University Sweden) [25] Wahlberg, B., System identification using Laguerre models, IEEE Trans. Autom. Control, AC-36, 551-562 (1991) · Zbl 0738.93078 [26] Wahlberg, B., System identification using Kautz models, IEEE Trans. Autom. Control, AC-39, 1276-1282 (1994) · Zbl 0807.93065 [27] Wahlberg, B., Laguerre and Kautz models, (Preprints 10th IFAC Symp. on System Identification, Copenhagen, Vol. 3 (1994)), 1-12 [28] Wiener, N., (Extrapolation, Interpolation and Smoothing of Stationary Time Series (1949), MIT Press: MIT Press Cambridge, MA) · Zbl 0036.09705 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.