Abstract
We consider the representation and identification of nonlinear systems through the use of parallel cascades of alternating dynamic linear and static nonlinear elements. Building on the work of Palm and others, we show that any discrete-time finite-memory nonlinear system having a finite-order Volterra series representation can be exactly represented by a finite number of parallel LN cascade paths. Each LN path consists of a dynamic linear system followed by a static nonlinearity (which can be a polynomial). In particular, we provide an upper bound for the number of parallel LN paths required to represent exactly a discrete-time finite-memory Volterra functional of a given order. Next, we show how to obtain a parallel cascade representation of a nonlinear system from a single input-output record. The input is not required to be Gaussian or white, nor to have special autocorrelation properties. Next, our parallel cascade identification is applied to measure accurately the kernels of nonlinear systems (even those with lengthy memory), and to discover the significant terms to include in a nonlinear difference equation model for a system. In addition, the kernel estimation is used as a means of studying individual signals to distinguish deterministic from random behaviour, in an alternative to the use of chaotic dynamics. Finally, an alternate kernel estimation scheme is presented.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Banyasz, C.S.; Haber, R.; Keviczky, L. Some estimation methods for nonlinear discrete time identification. IFAC Symp. Ident. Sys. Param. Est. 3:793–802; 1973.
Dieudonné, J. Foundations of modern analysis. Berlin, Heidelberg, New York: Springer; 1976.
Frechet, M. Sur les fonctionnelles continues. Annales Scientifiques de l'Ecole Normal Superieure 27:193–219; 1910.
Glass, L.; Mackey, M.C. From clocks to chaos. Princeton: Princeton University; 1988.
Golub, G.H.; Van Loan, C.F. Matrix computations (2nd ed.). Baltimore: Johns Hopkins Univ. Press; 1989.
Hunter, I.W.; Korenberg, M.J. The identification of nonlinear biological systems: Wiener and Hammerstein cascade models. Biol. Cybern. 55:135–144; 1986.
Kolmogoroff, A.N. On the representation of continuous functions of several variables by superposition of continuous functions of one variable and addition (Russian). Dokl. Akad. Nauk. SSSR 114:953–956; 1957; AMS Transl. 2:55–59; 1963.
Korenberg, M.J. Statistical identification of parallel cascades of linear and nonlinear systems. IFAC Symp. Ident. Sys. Param. Est. 1:580–585; 1982.
Korenberg, M.J. Functional expansions, parallel cascades and nonlinear difference equations. In: Marmarelis, V.Z. ed. Advanced methods of physiological system modeling. Los Angeles: USC Biomedical simulations Resource, Vol. 1; 1987: pp. 221–240.
Korenberg, M.J. Exact orthogonal estimation of kernels with biological applications. IEEE Montech Conference on Biomedical Technologies, pp. 27–32; 1987.
Korenberg, M.J. Identifying nonlinear difference equation and functional expansion representations: The fast orthogonal algorithm. Ann. Biomed. Eng. 16:123–142; 1988.
Korenberg, M.J. A robust orthogonal algorithm for system identification and time-series analysis. Biol. Cybern. 60:267–276; 1989.
Korenberg, M.J. A rapid and accurate method for estimating the kernels of a nonlinear system with lengthy memory. 15th Biennial Symp. Communications. June 1990; Queen's University, Kingston, Canada, pp. 57–60.
Korenberg, M.J. Some new approaches to nonlinear system identification and time-series analysis. Proc. 12th Annual International Conference of the IEEE Engineering in Medicine and Biology Society 12(1):20–21; 1990.
Korenberg, M.J.; Bruder, S.B.; McIlroy, P.J. Exact orthogonal kernel estimation from finite data records: Extending Wiener's identification of nonlinear systems. Ann. Biomed. Eng. 16:201–214; 1988.
Lee, Y.W.; Schetzen, M. measurement of the Wiener kernels of a non-linear system by cross-correlation. Int. J. Control 2:237–254; 1965.
Liebovitch, L.S. Introduction to the properties and analysis of fractal objects, processes, and data. In: Marmarelis, V.Z., ed. Advanced methods of physiological system modeling. New York: Plenum Press, Vol. 2; 1989: pp. 225–239.
Marmarelis, P.Z.; Marmarelis, V.Z. Analysis of physiological systems. The white-noise approach. New York: Plenum Press; 1978.
McIlroy, P.J.H. Applications of nonlinear systems identification. Kingston, Ontario, Canada; Queen's University; 1986. M.Sc. thesis.
Mo, L.; Elkasabgy, N. Elec-841 Report, Dept. Elect. Eng., Queen's Univ., Kingston, Ontario, Canada; 1984.
Palm, G. On the representation and approximation of nonlinear systems. Part II: Discrete time. Biol. Cybern. 34:49–52; 1979.
Shi, J.; Sun, H.H. Nonlinear system identification via parallel cascaded structure. Proc. 12th Annual International Conference of the IEEE Engineering in Medicine and Biology Society 12(4):1897–1898; 1990.
Wiener, N. Nonlinear problems in random theory. New York: Wiley; 1958.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Korenberg, M.J. Parallel cascade identification and kernel estimation for nonlinear systems. Ann Biomed Eng 19, 429–455 (1991). https://doi.org/10.1007/BF02584319
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02584319