On approximation of stable linear dynamical systems using Laguerre and Kautz functions. (English) Zbl 0856.93017
The paper deals with approximation of stable linear dynamic systems by means of finite length orthogonal series expansions. The Laguerre and Kautz families of orthogonal functions are discussed, and it is shown that these sets of basis functions are optimal in a worst-case sense for systems with bounded transfer functions analytic outside a given disc. This fact is established in the Laplace domain by using conformal mapping techniques and reducing the Kautz approximation problem with two free parameters to two theoretically developed “one-parameter” Laguerre approximation problems. Then, the results concerning \(H_2\) and \(H_\infty\) Kautz approximation are given.
Reviewer: Z.Hasiewicz (Wrocław)
MSC:
| 93B11 | System structure simplification |
| 93C05 | Linear systems in control theory |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
References:
| [1] | Ahlfors, L. V., (Complex Analysis (1979), McGraw-Hill: McGraw-Hill New York) · Zbl 0395.30001 |
| [2] | Beckmann, P., (Orthogonal Polynomials for Engineers and Physicists (1973), The Golden Press) · Zbl 0253.42013 |
| [3] | Bodin, P.; Wahlberg, B., Thresholding in high order transfer function estimation, (Proc. 33th CDC (1994)), 3400-3405, Orlando, FL |
| [4] | Broome, P. W., Discrete orthonormal sequences, J. of the Association for Computing Machinary, 12, 2, 151-168 (1965) · Zbl 0139.11302 |
| [5] | Chihara, T. S., (An Introduction to Orthogonal Polynomials. Mathematics and its Applications, Vol. 13 (1978), Gordon and Breach: Gordon and Breach Reading, U.K) · Zbl 0389.33008 |
| [6] | Clement, P. R., Laguerre functions in signal analysis and parameter identification, J. Franklin Inst., 313, 2, 85-95 (1982) · Zbl 0481.93025 |
| [7] | Clement, P. R., Application of generalized Laguerre functions, Mathematics and Computers in Simulation, 27, 541-550 (1985) |
| [8] | Clowes, G. J., Choice of the time-scaling factor for linear system approximations using orthonormal Laguerre functions, IEEE Trans. Automat. Control, AC-10, 487-489 (1965) |
| [9] | Cluett, W. R.; Wang, L., Frequency smoothing using Laguerre model, (IEE Proceedings-D, 139 (1992)), 88-96, (1) · Zbl 0777.93016 |
| [10] | den Brinker, A. C., Adaptive orthonormal filters, (Preprints 12th IFAC World Congress, Vol. 5 (1993)), 287-290, Sydney, Australia · Zbl 0769.93077 |
| [11] | Dumont, G. A.; Zervos, C.; Pageau, G., Laguerre-based adaptive control of PH in an industrial bleach plant extraction stage, Automatica, 26, 781-787 (1990) · Zbl 0719.93056 |
| [12] | Dumont, G. A.; Zervos, C. C., Adaptive controllers based on orthonormal series representations, (2nd IFAC Workshop on Adaptive System in Control and Signal Processing (1986)), 371-376, Lund, Sweden |
| [13] | Finn, C.; Ydstie, E.; Wahlberg, B., Contrained predictive control using orthogonal expansions, AIChE J., 39, 1810-1826 (1993) |
| [14] | Fu, Y.; Dumont, G. A., An optimal time scale for discrete Laguerre network, IEEE Trans. Automat. Control, AC-38, 6, 934-938 (1993) · Zbl 0800.93033 |
| [15] | Fu, Y.; Dumont, G. A., On determination of Laguerre filter pole through step or impulse data, (Preprints 12th IFAC World Congress, Vol. 5 (1993)), 303-306, Sydney, Australia |
| [16] | Glader, C.; Högnäs, G.; Mäkilä, P. M.; Toivonen, H., Approximation of delay systems — a case study, Int. J. Control, 53, 369-390 (1991) · Zbl 0745.93016 |
| [17] | Glover, K.; Lam, J.; Partington, J. R., Rational approximations of a class of infinite-order systems II: Optimal convergence rates for \(L_∞\) approximates, Math. Control Sig. Sys., 4, 233-246 (1991) · Zbl 0733.41023 |
| [18] | Goodwin, G. C.; Gevers, M.; Ninness, B., Quantifying the error in estimated transfer function with application to model order selection, IEEE Trans. Automat. Control, AC-37, 7, 913-928 (1992) · Zbl 0767.93022 |
| [19] | Gottlieb, M. J., Polynomials orthogonal on finite or enumerable set of points, Amer. J. Mathematics, 60, 453-458 (1938) · JFM 64.0329.01 |
| [20] | Gu, G.; Khargonekar, P. P.; Lee, E. B., Approximation of infinite-dimensional systems, IEEE Trans. Automat. Control, AC-34, 610-618 (1989) · Zbl 0682.93035 |
| [21] | Gunnarsson, S.; Wahlberg, B., Some Asymptotic Results in Recursive Identification using Laguerre Models, Int. J. of Adaptive Control and Signal Processing, 5, 313-333 (1991) · Zbl 0742.93013 |
| [22] | Head, J. W., Approximation to transient by means of Laguerre Series, (Proc. Cambridge Phil. Soc., 52 (1956)), 64-651 · Zbl 0072.31804 |
| [23] | Henrici, P., Fast Fourier methods in computational complex analysis, SIAM Rev., 21, 481-527 (1979) · Zbl 0416.65022 |
| [24] | Heuberger, P., On Approximate System Identification with System Based Orthonormal Functions, (PhD thesis (1991), Delft University of Technology: Delft University of Technology Delft, The Netherlands) |
| [25] | Heuberger, P.; Van Den Hof, P.; Bosgra, O., Modelling linear dynamical systems through generalized orthonormal basis functions, (Preprints 12th IFAC World Congress, Vol. 5 (1993)), 283-286, Sydney, Australia |
| [26] | Horowitz, I. M., (Synthesis of Feedback Systems (1963), Academic Press: Academic Press New York) · Zbl 0121.07704 |
| [27] | Huggins, W. H., Signal theory, IRE Trans. on Circuit Theory, CT-3, 3, 210-216 (1956) |
| [28] | Kammler, D. W.; McGlinn, R. J., A bibliography for approximation with exponential sums, J. Comput. Appl. Math., 4, 2, 167-173 (1978) |
| [29] | Kautz, W. H., Network synthesis for specified transient response, (Technical report, 209 (1952), M.I.T. Research Lab., Electronics: M.I.T. Research Lab., Electronics Cambridge, MA) · Zbl 0143.41305 |
| [30] | Kautz, W. H., Transient synthesis in time domain, IRE Transactions on Circuit Theory, CT-1, 3, 29-39 (1954) |
| [31] | King, R. E.; Paraskevopoulos, P. N., Parametric identification of discrete-time SISO systems, Int. J. Control, 30, 1023-1029 (1979) · Zbl 0418.93026 |
| [32] | Lai, D. C., Signal processing with orthonormalized exponentials, Mathematics and Computers in Simulation, 27, 409-420 (1985) |
| [33] | Lampard, D. G.; Levan, N., A new construction of multiple Laguerre shifts, (Technical report (1992), Electrical Engineering, University of California: Electrical Engineering, University of California Los Angeles, CA) |
| [34] | Lee, Y. W., Synthesis of electrical networks by means of Fourier transforms of Laguerre’s functions, J. Math. Phy., 11, 83-113 (1931) · Zbl 0005.04509 |
| [35] | Lee, Y. W., (Statistical Theory of Communications (1961), J. Wiley & Sons: J. Wiley & Sons New York) |
| [36] | Lindskog, P.; Wahlberg, B., Application of Kautz models in system identification, (Preprints 12th IFAC World Congress, Vol. 5 (1993)), 308-312, Sydney, Australia |
| [37] | Lorentz, G. G., (Approximation of Functions (1986), Chelsea: Chelsea New York) · Zbl 0643.41001 |
| [38] | Mäkilä, P. M., Approximation of stable systems by Laguerre filters, Automatica, 26, 2, 333-345 (1990) · Zbl 0708.93007 |
| [39] | Mäkilä, P. M., Laguerre series approximation of infinite dimensional systems, Automatica, 26, 6, 985-996 (1990) · Zbl 0717.93028 |
| [40] | Mäkilä, P. M., Laguerre methods and \(H^∞\) identification of continuos-time systems, Int. J. Control, 53, 689-707 (1991) · Zbl 0744.93024 |
| [41] | Mäkilä, P. M.; Partington, J. R., Robust approximations and identification in \(H^∞\), International Journal of Control, 58, 3, 665-683 (1993) · Zbl 0782.93006 |
| [42] | Masani, P. R., (Norbert Wiener. Vita Mathematica, Vol. 5 (1990), Birkhäuser: Birkhäuser Basel, Boston, Berin) |
| [43] | Van Den Hof, P.; Heuberger, P.; Bokor, J., Identification with generalized orthonormal basis functions — statistical analysis and error bounds, (SYSID’94, Vol. 3 (1994)), 207-212, Copenhagen, Denmark |
| [44] | Mendel, J. M., A unified approach to the synthesis of orthonormal exponential functions useful in systems analysis, IEEE Trans. Systems Science and Cybernetics, 2, 54-62 (1966) |
| [45] | Neuman, J. V., Zur theorie der unbeschrakten matrizen, J. fur Math., 161, 208-236 (1929) · JFM 55.0825.01 |
| [46] | Nurges, Y., Laguerre models in problems of approximation and identification of discrete systems, Automat. and Remote Control, 48, 346-352 (1987) · Zbl 0625.93020 |
| [47] | Nurges, Y.; Yaaksoo, Y., Laguerre state equations for multivariable discrete systems, Automat. and Remote Control, 42, 1601-1603 (1981) · Zbl 0501.93040 |
| [48] | Partington, J. R., Approximation of delay systems by Fburier-Laguerre series, Automatica, 27, 569-572 (1991) · Zbl 0754.93019 |
| [49] | Pati, Y. C., Wavelets and time-frequency methods in linear systems and neural networks, (PhD thesis (1992), University of Maryland: University of Maryland College Park, MD) |
| [50] | Pati, Y. C.; Khrishnaprasad, P. S., Rational waveelets in approximation and identification of stable linear systems, (Proc. 31th CDC (1992)), 1502-1507, New York |
| [51] | Pinkus, A., (n-Widths in Approximation Theory (1985), Springer: Springer Berlin) · Zbl 0551.41001 |
| [52] | Rosenblum, M.; Rovnyak, J., (Hardy Classes and Operator Theory (1985), Oxford University Press: Oxford University Press New York) · Zbl 0586.47020 |
| [53] | Ross, D. C., Orthonormal exponentials, IEEE Trans. Comm. Electronics, CE-12, 173-176 (1964) |
| [54] | Schetzen, M., Power-series equivalence of some functional series with applications, IEEE Trans. on Circuit Theory, CT-17, 3, 305-313 (1970) |
| [55] | Söderström, T.; Stoica, P. G., (System Identification (1989), Prentice-Hall: Prentice-Hall Hemel Hempstead, U.K) · Zbl 0695.93108 |
| [56] | Steiglitz, K., Rational transform approximation via the Laguerre spectrum, J. Franklin Inst., 280, 387-394 (1965) · Zbl 0229.65096 |
| [57] | Szegö, G., Orthogonal Polynomials, (American Mathematical Society Colloqium Publication, Vol. XXII (1939), American Mathematical Society: American Mathematical Society Rhode Island, NY) · JFM 65.0278.03 |
| [58] | Tricomi, F., Trasformazione di Laplace e polinomi di Laguerre, I.R.C. Accad. Lincei, 21, 235 (1935) · JFM 61.0452.01 |
| [59] | Wahlberg, B., System identification using Laguerre models, IEEE Trans. Automat. Control, AC-36, 5, 551-562 (1991) · Zbl 0738.93078 |
| [60] | Wahlberg, B., System identification using Kautz models, IEEE Trans. Automat. Control, AC-39, 6, 1276-1281 (1994) · Zbl 0807.93065 |
| [61] | Wahlberg, B.; Hannan, E. J., Parametric signal modelling using Laguerre filters, Annals of Applied Probability, 3, 2, 467-496 (1993) · Zbl 0784.62080 |
| [62] | Wahlberg, B.; Lindskog, P., Approximate modelling by means of orthonormal functions, (De Masi, G.; Gombani, A.; Kurzhanski, A., Progress in Systems and Control Theory (1991), Birkhäuser: Birkhäuser Basel, Boston and Berlin) |
| [63] | Wahlberg, B.; Ljung, L., Hard frequency domain model error bounds from least-squares like identification techniques, IEEE Trans. Automat. Control, AC-37, 7, 900-912 (1992) · Zbl 0767.93021 |
| [64] | Wiener, N., The Theory of Prediction, (Modern Mathematics for Engineers (1956), McGraw-Hill: McGraw-Hill New York), Bechenbach · JFM 50.0195.03 |
| [65] | Young, T. Y.; Huggins, W. H., Complementary signals and orthogonalized exponentials, IRE Transactions on Circuit Theory, CT-9, 362-370 (1962) |
| [66] | Young, T. Y.; Huggins, W. H., Discrete orthonormal exponentials, (Proc. Nat. Elec. Conf. (1962)), 10-18 |
| [67] | Zervos, C.; Belanger, P. R.; Dumont, G.a., Controller tuning using orthonormal series identification, Automatica, 24, 165-175 (1988) · Zbl 0638.93047 |
| [68] | Zervos, C.; Dumont, G.a., Deterministic adaptive control based on Laguerre series representation, Int. J. Control, 48, 2333-2359 (1988) · Zbl 0656.93045 |
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