Mathematical systems theory

, Volume 14, Issue 1, pp 83–94 | Cite as

Abstract bilinear systems: The forward shift approach

  • Arthur E. Frazho


A bilinear realization theory for a Volterra series input-output map is given. The approach involves a special transform representation for a Volterra series and certain shift operators on a Fock space. The approach yields in a very simple manner a theory of span reachability, observability and minimality for bilinear systems.


Computational Mathematic Realization Theory Shift Operator Simple Manner Bilinear System 
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  1. 1.
    P. Alper, A consideration of the discrete Volterra series,IEEE Trans. Automatic Control AC-10, 322–327 (1965).Google Scholar
  2. 2.
    A. V. Balakrishnan, On the state space theory of nonlinear systems,Functional Analysis and Optimization, Ed. E. R. Caianiello, Acad. Press, New York, 15–36, 1966.Google Scholar
  3. 3.
    A. V. Balakrishnan,Applied Functional Analysis, Springer-Verlag, New York, 1976.Google Scholar
  4. 4.
    J. S. Baras and R. W. Brockett, H2-functions and infinite-dimensional realization theory,SIAM J. Control 13, 221–231 (1975).Google Scholar
  5. 5.
    R. W. Brockett, On the algebraic structure of bilinear systems,Theory and Applications of Variable Structure Systems, Ed. R. Mohler and A. Ruberti, Academic Press, New York, 153–168, 1972.Google Scholar
  6. 6.
    R. W. Brockett, Finite and infinite dimensional bilinear realizations,J. Franklin Inst. 301, 509–520 (1976).Google Scholar
  7. 7.
    C. Bruni, G. DiPillo, and G. Koch, Bilinear systems: an appealing class of nearly linear systems in theory and applications,IEEE Trans. Automatic Control AC-19, 334–348 (1974).Google Scholar
  8. 8.
    C. T. Chen,Introduction to Linear System Theory, Holt, Rinehart and Winston, New York, 1970.Google Scholar
  9. 9.
    S. J. Clancy and W. J. Rugh, On the realization problem for stationary, homogeneous discrete-time systems,Automatica 14, 357–366 (1978).Google Scholar
  10. 10.
    S. J. Clancy, G. E. Mitzel, and W. J. Rugh, On transfer function representations for homogeneous nonlinear systems,IEEE Trans. on Automatic Control AC-24, 242–249 (1979).Google Scholar
  11. 11.
    P. D'Allessandro, A. Isidori, and A. Ruberti, Realization and structure theory of bilinear dynamical systems,SIAM J. Control 12, 517–535 (1974).Google Scholar
  12. 12.
    E. Fornasini and G. Marchesini, Algebraic realization theory of bilinear discrete-time inputoutput maps,J. Franklin Inst. 301, 143–159 (1976).Google Scholar
  13. 13.
    A. E. Frazho, Shift operators and bilinear system theory,Proc. of the 1978 Conference on Decision and Control, pp. 551–556.Google Scholar
  14. 14.
    A. E. Frazho, A shift operator approach to bilinear system theory,SIAM J. Control, to appear.Google Scholar
  15. 15.
    A. E. Frazho, Bilinear systems in Hilbert space, Submitted for publication.Google Scholar
  16. 16.
    P. A. Fuhrman, On realizations of linear systems and applications to some questions of stability,Math. Systems Theory 8, 132–141 (1974).Google Scholar
  17. 17.
    E. G. Gilbert, Functional expansions for the response of nonlinear differential systems,IEEE Trans. Automatic Control AC-22, 909–921 (1977).Google Scholar
  18. 18.
    E. G. Gilbert, Bilinear and 2-power input-output maps: finite dimensional realizations and the role of the functional series,IEEE Trans. Automatic Control AC-23, 418–425 (1978).Google Scholar
  19. 19.
    P. R. Halmos,Finite-Dimensional Vector Spaces, Springer-Verlag, New York,Google Scholar
  20. 20.
    H. Helson,Lectures on Invariant Subspaces, Acad. Press, New York, 1964.Google Scholar
  21. 21.
    J. W. Helton, Discrete time systems, operator models, and scattering theory,J. Functional Analysis 16, 15–38 (1974).Google Scholar
  22. 22.
    A. Isidori, Direct construction of minimal bilinear realizations from nonlinear input-output maps,IEEE Trans. Automatic Control AC-18, 626–631 (1973).Google Scholar
  23. 23.
    A. Isidori and A. Ruberti, Realization theory of bilinear systems,Geometric Methods in System Theory, Ed. D. Q. Mayne and R. W. Brockett, D. Reidel Publishing Co., Dordrecht, 1973.Google Scholar
  24. 24.
    R. E. Kalman, P. L. Falb, and M. A. Arbib,Topics in Mathematical System Theory, McGraw-Hill, New York, 1969.Google Scholar
  25. 25.
    G. Koch, A realization theorem for infinite dimensional bilinear systems.Richerche di Automatica 3 (1973).Google Scholar
  26. 26.
    Y. H. Ku and A. A. Wolf, Volterra-Wiener functionals for the analysis of nonlinear systems,J. Franklin Inst. 271, 9–26 (1966).Google Scholar
  27. 27.
    G. E. Mitzel and W. J. Rugh, Realization of stationary homogenous systems: the degree 2 case,Proceedings of the 1977 IEEE Conference on Decision and Control, New Orleans, LA, 783–787.Google Scholar
  28. 28.
    G. E. Mitzel, S. Clancy, and W. J. Rugh, On a multi-dimensional s-transform and the realization problems for homogeneous nonlinear systems,IEEE Trans. Automatic Control AC-22, 825–830 (1977).Google Scholar
  29. 29.
    A. W. Naylor and G. R. Sell,Linear Operator Theory in Engineering and Science, Holt, Rinehart and Winston, New York, 1971.Google Scholar
  30. 30.
    E. Nelson,Tensor Analysis, Princeton University Press, Princeton, 1967.Google Scholar
  31. 31.
    B. Sz.-Nagy and C. Foias,Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, 1970.Google Scholar

Copyright information

© Springer-Verlag New York Inc 1981

Authors and Affiliations

  • Arthur E. Frazho
    • 1
  1. 1.School of Aeronautics and AstronauticsPurdue UniversityWest LafayetteUSA

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