Abstract
In this paper we give necessary and sufficient conditions for a stationary finite Volterra series to have a linear (in the controls) analytic realization, which at the same time has a Hamiltonian structure. The result generalizes that for linear systems, where the condition is that the impulse response should be an odd function, and is expressed as a particular symmetry condition on the Volterra kernals. The relation between this problem and that of the inverse problem in Newtonian mechanics is explored. The finiteness of the Volterra series implies a nilpotence condition on a certain Lie algebra defined by the realization. The additional requirement that the system be Hamiltonian adds further structure to the classical representation of a nilpotent Lie algebra by lower triangular matrices.
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References
R. Abraham and J. E. Marsden,Foundations of Machines (2nd edition), Benjamin/Cummins, Reading, Mass, 1978.
P. d'Alessandro, A. Isidori and A. Ruberti, Realization and structure theory of bilinear systems,S.I.A.M. J. of Control and Opt., 12, 517–535 (1974).
L. Auslander and B. Kostant, Polarizations and unitary representations of solvable Lie groups,Inventions Math., 14. 255–354, (1977).
R. W. Brockett, On the algebraic structure of bilinear systems,Theory and Applications of Variable Structure Systems, (R. R. Mohler and A. Ruberti, Eds.), Academic Press, New York, pp. 153–168 (1972).
R. W. Brockett, Volterra series and geometric control theory,Automatica, 12, 167–176 (1976).
R. W. Brockett and A. Rahimi, Lie algebras and linear differential equations,ORD. DIFF. EQNS. (Ed. L. Weiss), Academic Press, New York (1972).
P. E. Crouch, Geometric structures in systems theory,Proc. I.E.E., 128, 242–252 (1981).
P. E. Crouch, Dynamical realizations of finite Volterra series,S.I.A.M. J. of Control and Opt., 19, 177–202, (1981).
P. E. Crouch and M. Irving, Dynamical realizations of homogeneous Hamiltonian systems,Control Theory Centre Report No. 122, University of Warwick, Coventry, (1984).
J. B. Goncalves, Nonlinear controllability and observability with applications to gradient systems, Ph.D. Thesis, University of Warwick, Coventry, England (1981).
J. E. Humphreys,Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, No. 9, Springer-Verlag (1972).
B. Kostant, Quantization and unitary representations,Lecture Notes in Mathematics, No. 170, Springer-Verlag (Ed. C. T. Team), pp. 87–208, (1970).
C. M. Lesiak and A. J. Krener, The existence and uniqueness of Volterra series for nonlinear systems,Trans. I.E.E.E., AC-23, 1090–1095 (1978).
R. M. Santilli,Foundations of Theoretical Mechanics I, Springer-Verlag, New York (1978).
A. J. Van der Schaft, System theoretic descriptions of physical systems, Doctorial Dissertation, University of Groningen, April 1983, to appear in:Mathematical Centre Tracts, Mathematical Centre, Amsterdam.
A. J. Van der Schaft, System theoretic properties of Hamiltonian systems with external forces, Memorandum No. 442, Department of Applied Mathematics, Technical Hogeschool, Twente (1983).
A. J. Van der Schaft, Controllability and observability for affine nonlinear Hamiltonian systems,I.E.E.E. Trans., AC-27, 490–492 (1982).
A. J. Van der Schaft, Observability and controllability for smooth nonlinear systems,S.I.A.M. J. of Control and Opt., 20, 338–354 (1982).
H. J. Sussmann, Existence and uniqueness of minimal realizations of nonlinear systems,Math. Systems Theory, 10, 263–284, (1977).
F. Takens, Variational and conservative systems, Report Z.W. 7603. University of Groningen, Groningen (1976).
P. E. Crouch, On the relation between internal and external symmetries of bilinear systems, Control Theory Centre Report, No. 80, University of Warwick, Coventry (1980).
M. Fliess, Fontionelles causales non lineaires et indeterminées Non Commutatives,Bull. Soc. Math., France, 109, 3–40, (1981).
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Work supported in part by the Office of Naval Research, Grant No. N00014-75-C-0648 while the author was on leave in the Division of Applied Sciences, Harvard University, Cambridge, Mass. U.S.A.
Work supported in part by S.E.R.C. Grant No. GR/B/9116.7.
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Crouch, P.E., Irving, M. On finite volterra series which admit hamiltonian realizations. Math. Systems Theory 17, 293–318 (1984). https://doi.org/10.1007/BF01744446
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DOI: https://doi.org/10.1007/BF01744446