Abstract
A technique is described for exact estimation of kernels in functional expansions for nonlinear systems. The technique operates by orthogonalizing over the data record and in so doing permits a wide variety of input excitation. In particular, the excitation is not limited to inputs that are white, Gaussian, or lengthy. Diagonal kernel values can be estimated, without modification, as accurately as off-diagonal values. Simulations are provided to demonstrate that the technique is more accurate than the Lee-Schetzen method with a white Gaussian input of limited duration, retaining its superiority when the system output is corrupted by noise.
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References
Barrett, J.F. The use of functionals in the analysis of nonlinear physical systems. J. Elect. Control 15:567–615; 1963.
Barrett, J.F. Functional series representation of nonlinear systems—Some theoretical comments. 6th. IFAC Symp. Ident. Sys. Param. Est. 1:251–256; 1982.
Frechet, M. Sur les fonctionnelles continues. Ann. Sci. Ecole Normal Sup. 27:193–219; 1910.
Goussard, Y., Krenz, W.C., Stark, L. An improvement of the Lee and Schetzen cross-correlation method. IEEE Trans. Automat. Contr. AC-30:895–898; 1985
Hunter, I.W., Korenberg, M.J. The identification of nonlinear biological systems: Wiener and Hammerstein cascade models. Biol. Cybern. 55:135–144; 1986.
Korenberg, M.J. Identification of biological cascades of linear and static nonlinear systems. Proc. 16th Midwest Symp. Circuit Theory 18.2:1–9; 1973.
Korenberg, M.J. Crosscorrelation analysis of neural cascades. Proc. 10th Ann. Rocky Mountain Bioeng. Symp. 1:47–52; 1973.
Korenberg, M.J.. Statistical identification of parallel cascades of linear and nonlinear systems. 6th IFAC Symp. Ident. Sys. Param. Est. 1:580–585; 1982.
Korenberg, M. J. Identifying noisy cascades of linear and static nonlinear systems. 7th IFAC Symp. Ident. Sys. Param. Est. 1:421–426; 1985.
Korenberg, M.J. Orthogonal identification of nonlinear difference equation models. Proc. 28th Midwest Symp. Circuit Sys. 1:90–95; 1985.
Korenberg, M.J., Hunter, I.W. The identification of nonlinear biological systems:LNL cascade models. Biol. Cybern. 55:125–134; 1986.
Lee, Y.W., Schetzen, M. Measurement of the Wiener kernels of a nonlinear system by crosscorrelation. Int. J. Contr. 2:237–254; 1965.
Marmarelis, P.Z., Marmarelis, V.Z. Analysis of Physiological Systems. New York: Plenum; 1978.
Marmarelis, V.Z. Identification of nonlinear systems through quasi-white test signals. Ph.D. Thesis, California Institute of Technology, Pasadena, CA, 1976.
Melton, R.B., Kroeker, J.P. Wiener functional for an N-level uniformly distributed discrete random process. 6th IFAC Symp. Ident. Sys. Param. Est. 2:1169–1174; 1982.
Palm, G., Poggio, T. The Volterra representation and the Wiener expansion: Validity and pitfalls. Siam J. Appl. Math. 33:195–216; 1977.
Palm, G., Poggio, T. Stochastic identification methods for nonlinear systems: An extension of the Wiener theory. Siam J. Appl. Math. 34:524–534; 1978.
Sakuranaga, M., Sato, S., Hida, E., Naka, K.-I. Nonlinear analysis: Mathematical theory and biological applications. CRC Crit. Rev. Biomed. Eng. 14:127–184; 1986.
Schetzen, M. A theory of nonlinear system identification. Int. J. Contr. 20:557–592; 1974.
Swerup, C. On the choice of noise for the analysis of the peripheral auditory system. Biol. Cybern. 29:97–104; 1978.
Weiss, T.F. A model of the peripheral auditory system. Kybernetik 3:153–175; 1966.
Wiener, N. Nonlinear Problems in Random Theory. New York: Wiley; 1958.
Zadeh, L. On the representation of nonlinear operators. IRE Wescon. Conv. Rec. 2:105–113; 1957.
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Korenberg, M.J., Bruder, S.B. & McLlroy, P.J. Exact orthogonal kernel estimation from finite data records: Extending Wiener's identification of nonlinear systems. Ann Biomed Eng 16, 201–214 (1988). https://doi.org/10.1007/BF02364581
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DOI: https://doi.org/10.1007/BF02364581