Abstract
Time-domain identification of nonlinear systems represented by functional expansions is considered. A general framework is defined for the analysis of three identification methods: the widely used cross-correlation method, Korenberg's method, and a suboptimal least-squares method based on a stochastic approximation algorithm. First, the major characteristics of the underlying estimation problem are pointed out. Then, the identification methods are interpreted as approximations to an optimal estimator, which helps gain insight into their internal functioning and to the investigation of their connections and differences. Examination of results previously published and of the simulations reported in this article indicate that stochastic approximation is an interesting alternative to other existing methods. Identification of a biological system stimulated by a non-Gaussian input confirms the practicality of this approach.
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References
Albert, A.E.; Gardner, L.A. Stochastic approximation and linear regression. Cambridge, MA: MIT Press: 1967.
Amorocho, J.; Brandstetter, A. Determination of nonlinear functional response functions in rainfallrunoff processes. Water Resources Research 7:1087–1108; 1971.
Aracil, J. Measurement of Wiener kernels with binary random signals. IEEE Trans. Automatic Control AC-15:123–125; 1970.
Boyd, S.; Tang, Y.S.; Chua, L.O. Measuring Volterra kernels. IEEE Trans. Circuits & Systems CAS-30:571–577; 1983.
Cameron, R.H.; Martin, W.T. The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals. Annals of Mathematics 48:385–392; 1947.
Domoment, G.; Goussard, Y., Bul-Mong-Hung. Peut-on étendre la notion d'élastance ventriculaire à des contractions non-périodiques? In: Biologie théorique. Paris: Editions CNRS; 1985: pp. 153–164.
Demoment, G.; Hinglais, J. Global parametric search and left ventricular identification: Evaluation of cardiac contractility through computed maximal isovolumic elastance. Annals of Biomedical Engineering 9:59–74; 1981.
Dvoretzky, A.. On stochastic approximation. Proc. 3rd Berkeley Symposium on Mathematical Statistics and Probability 1:39–55; 1956.
Fréchet, M. Sur les fonctionnelles continues. Annales de l'École Normale Supérieure 27:193–219; 1910.
French, A.S.; Butz, E.G. Measuring the Wiener kernels of a nonlinear system using the fast Fourier transform. International Journal of Control 17: 529–539; 1973.
Golub, G.H.; Van Loan, C.F. Matrix computations (2d ed.) Baltimore, MD: The Johns Hopkins University Press; 1989.
Goussard, Y. Identification de systèmes non-linéaires, représentés par développement fonctionnels et soumis à diverses entrées aléatoires, à l'aide d'une méthode d'approximation stochastique. Thèse de Docteur-Ingénieur, Université de Paris-Sud, Centre d'Orsay; 1983.
Goussard, Y.. Wiener kernel estimation: A comparison of cross-correlation and stochastic approximation methods. In: Advanced methods of physiological system modeling, Vol. 1. V.Z. Marmrelis, ed. Los Angeles: USC Biomedical Simulations Resource; 1987: pp. 289–302.
Goussard, Y; Bui-Mong-Hung; Demoment, G. Functional analysis of ventricular mechanics. How to extend the elastance concept to non-periodic contractions. In: Proc. 6th Int. Conf. CSDS. Philadephia, PA; 1984: pp. 308–311.
Goussard, Y.; Krenz, W. C.; Stark, L. An improvement of the Lee and Schetzen cross-correlation method. IEEE Trans. Automatic Control AC-30:895–898; 1985.
Hung, G.H.; Stark, L.: The kernel identification method (1910–1977)—Review of theory, calculation, application and interpretation. Mathematical Biosciences 37: 135–190; 1977.
Korenberg, M. J. Functional expansions, parallel cascades and nonlinear difference equations. In: Advanced methods of physiological system modeling. Vol. 1. V. Z. Marmarelis, ed. Los Angeles: USC Biomedical Simulations Resource; 1987: pp. 221–240.
Korenberg, M. J. Identifying nonlinear difference equation and functional expansion: The fast orthogonal algorithm. Annals of Biomedical Engineering 16: 123–142; 1988.
Korenberg, M.J.; Bruder, S.B.; McIlroy, P.J.. Exact orthogonal kernel estimation from finite data records: Extending Wiener's identification of nonlinear systems. Annals of Biomedical Engineering 16:201–214; 1988.
Korenberg, M.J.; Hunter, I.W.. The identification of nonlinear biological systems: Wiener kernel approaches. Annals of Biomedical Engineering 18:629–654; 1990.
Krausz, H.I. Identification of nonlinear system using random input train inputs. Biological Cybernetics 19:217–230; 1975.
Lee, Y.W.; Schetzen, M.; Measurement of the Wiener kernels of a nonlinear system by cross-correlation. International Journal of Control 2:237–254; 1965.
Ljung, L. Recursive identification methods for off-line identification problems. Proc. 6th Symp. Identification of Systems and Parameter Estimation. pp. 475–480; 1982.
Marmarelis, V.Z. A family of quasi-white noise signals and its optimal use in biological system identification. Part I: Theory. Biological Cybernetics 27:49–56; 1977.
Marmarelis, P.Z.; Marmarelis, V.Z. Analysis of physiological systems. New York: Plenum; 1978.
Palm, G.; Poggio, T. Stochastic identification methods for nonlinear systems: An extension of the Wiener theory. SIAM J. Applied Mathematics 34:524–534; 1978.
Rader, C.M.; Steinhardt, A.O. Hyperbolic Householder transformations. IEEE Trans. Acoustics, Speech & Signal Processing ASSP-34:1589–1602; 1986.
Saridis, G.N. Stochastic approximation methods for identification and control—A survey. IEEE Trans. Automatic Control AC-19:798–809; 1974.
Schetzen, M. A theory of nonlinear system identification. International Journal of Control 20:577–592; 1974.
Schetzen, M. Nonlinear system modeling based on the Wiener theory. Proc. IEEE 69:1557–1573; 1981.
Stankovic, S.S. On asymptotic properties of a real-time identification algorithm based on dynamic stochastic approximation. IEEE Trans. Automatic Control AC-23:58–61; 1978.
Stark, L. Neurological control systems: Studies in bioengineering. New York: Plenum; 1968.
Sunagawa, K.; Sagawa, K. Models of ventricular contraction based on time-varying elastance. CRC Crit. Review of Biomedical Engineering 7:193–228; 1982.
Usui, S.; Murase, K.; Watanabe, A. An optimization technique for Volterra series expansion in biological system identification. In: Proc. 6th IEEE/EMBS Conf., Los Angeles, CA; 1984: pp. 1–10.
Volterra, V. Theory of functionals. Glasgow: Blackie & Sons; 1930.
Watanabe, A.; Stark, L. Kernel method for nonlinear analysis: Identification of a biological control system. Mathematical Biosciences 27:99–108; 1975
Wiener, N. Nonlinear problems in random theory. New York: Wiley; 1958.
Yasui, S. Stochastic functional Fourier series, Volterra series, and nonlinear systems analysis. IEEE Trans. Automatic Control AC-24:230–242; 1979.
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Goussard, Y., Krenz, W.C., Stark, L. et al. Practical identification of functional expansions of nonlinear systems submitted to non-Gaussian inputs. Ann Biomed Eng 19, 401–427 (1991). https://doi.org/10.1007/BF02584318
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DOI: https://doi.org/10.1007/BF02584318