Skip to main content
SpringerLink
Log in

Structure theorems for nonlinear systems

  • Published:
Multidimensional Systems and Signal Processing Aims and scope Submit manuscript

Abstract

One of the main results is a proposition to the effect that under some typically mild conditions finite sums of the form

$$\sum\limits_\ell {K_\ell \sigma } \left[ {\sum\limits_m {\eta _{\ell m} Q_m (\cdot) + \rho _\ell } } \right]$$

are dense in an important sense in the set of shift-invariant approximately-finite-memory mapsG(·) that take a certain type of subsetU ofR intoR, whereR is the set of real-valued functions defined onR n orZ n. Here theQ m (·) are linear, σ is any element of a certain set of nonlinear maps fromR toR, and the κℓ, ρℓ, and ηℓm are real constants. Approximate representations comprising only affine elements and lattice nonlinearities are also presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • I.W. Sandberg, “Nonlinear Input-Output Maps and Approximate Representations,”AT&T Technical Journal, vol. 64, No. 8, pp. 1967–1983, 1985.

    Google Scholar 

  • I.W. Sandberg, “The Mathematical Foundations of Associated Expansions for Mildly Nonlinear Systems,”IEEE Transactions on Circuits and Systems, vol. CAS-30, No. 7, pp. 441–455, 1983.

    Article  Google Scholar 

  • I.W. Sandberg, “Existence and Evaluation of Almost Periodic Steady-State Responses of Mildly Nonlinear Systems,”IEEE Transactions on Circuits and Systems, vol. CAS-31, No. 8, pp. 689–701, 1984.

    Article  Google Scholar 

  • S. Boyd, “Volterra Series: Engineering Fundamentals,” dissertation, University of California, Berkeley, 1985.

    Google Scholar 

  • M.H. Stone, “A Generalized Weierstrauss Approximation Theorem,” inStudies in Modern Analysis, Vol. 1, (R.C. Buck, ed.), Prentice-Hall: Englewood Cliffs, 1962.

    Google Scholar 

  • G. Cybenko, “Approximation by Superpositions of a Sigmoidal Function,”Mathematics of Control, Signals, and Systems, vol. 2, pp. 304–314, 1989. (See also C.K. Chui and L. Xin, “Approximation by Ridge Functions and Neural Networks with One Hidden Layer,” Center for Approximation Theory Report 222, Department of Mathematics, Texas A&M University, July 1990.)

    Google Scholar 

  • K. Hornik, M. Stinchcombe, and H. White, “Multilayer Feedforward Networks are Universal Approximators,”Neural Networks, vol. 2, pp. 359–366, 1989.

    Article  Google Scholar 

  • J. Park and I.W. Sandberg, “Universal Approximation Using Radial-Basis-Function Networks,”Proceedings of the Twenty-Eighth Annual Allerton Conference on Communication, Control, and Computing, Monticello, IL, October 3–5, 1990. (Also, to appear inNeural Computation, 1991.)

  • G. Bachman and L. Narici,Functional Analysis, New York: Academic Press, 1966.

    Google Scholar 

  • K. Yosida,Functional Analysis, New York: Academic Press, 1965.

    Google Scholar 

  • L.A. Liusternik and V.J. Sobolev,Elements of Functional Analysis, New York: Frederick Ungar, 1961.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Texas at Austin, 78712, Austin, TX

    Irwin W. Sandberg

Authors
  1. Irwin W. Sandberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sandberg, I.W. Structure theorems for nonlinear systems. Multidim Syst Sign Process 2, 267–286 (1991). https://doi.org/10.1007/BF01952236

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01952236

Keywords

Search

Navigation