Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Alper, A consideration of the discrete Volterra series,IEEE Trans. Automatic Control AC-10, 322–327 (1965).
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.
A. V. Balakrishnan,Applied Functional Analysis, Springer-Verlag, New York, 1976.
J. S. Baras and R. W. Brockett, H2-functions and infinite-dimensional realization theory,SIAM J. Control 13, 221–231 (1975).
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.
R. W. Brockett, Finite and infinite dimensional bilinear realizations,J. Franklin Inst. 301, 509–520 (1976).
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).
C. T. Chen,Introduction to Linear System Theory, Holt, Rinehart and Winston, New York, 1970.
S. J. Clancy and W. J. Rugh, On the realization problem for stationary, homogeneous discrete-time systems,Automatica 14, 357–366 (1978).
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).
P. D'Allessandro, A. Isidori, and A. Ruberti, Realization and structure theory of bilinear dynamical systems,SIAM J. Control 12, 517–535 (1974).
E. Fornasini and G. Marchesini, Algebraic realization theory of bilinear discrete-time inputoutput maps,J. Franklin Inst. 301, 143–159 (1976).
A. E. Frazho, Shift operators and bilinear system theory,Proc. of the 1978 Conference on Decision and Control, pp. 551–556.
A. E. Frazho, A shift operator approach to bilinear system theory,SIAM J. Control, to appear.
A. E. Frazho, Bilinear systems in Hilbert space, Submitted for publication.
P. A. Fuhrman, On realizations of linear systems and applications to some questions of stability,Math. Systems Theory 8, 132–141 (1974).
E. G. Gilbert, Functional expansions for the response of nonlinear differential systems,IEEE Trans. Automatic Control AC-22, 909–921 (1977).
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).
P. R. Halmos,Finite-Dimensional Vector Spaces, Springer-Verlag, New York,
H. Helson,Lectures on Invariant Subspaces, Acad. Press, New York, 1964.
J. W. Helton, Discrete time systems, operator models, and scattering theory,J. Functional Analysis 16, 15–38 (1974).
A. Isidori, Direct construction of minimal bilinear realizations from nonlinear input-output maps,IEEE Trans. Automatic Control AC-18, 626–631 (1973).
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.
R. E. Kalman, P. L. Falb, and M. A. Arbib,Topics in Mathematical System Theory, McGraw-Hill, New York, 1969.
G. Koch, A realization theorem for infinite dimensional bilinear systems.Richerche di Automatica 3 (1973).
Y. H. Ku and A. A. Wolf, Volterra-Wiener functionals for the analysis of nonlinear systems,J. Franklin Inst. 271, 9–26 (1966).
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.
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).
A. W. Naylor and G. R. Sell,Linear Operator Theory in Engineering and Science, Holt, Rinehart and Winston, New York, 1971.
E. Nelson,Tensor Analysis, Princeton University Press, Princeton, 1967.
B. Sz.-Nagy and C. Foias,Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, 1970.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Frazho, A.E. Abstract bilinear systems: The forward shift approach. Math. Systems Theory 14, 83–94 (1981). https://doi.org/10.1007/BF01752391
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01752391