Yaprak YALÇIN, Leyla Gören SÜMER, Salman KURTULAN

Discrete-time modeling of Hamiltonian systems

The problem of discrete-time modeling of the lumped-parameter Hamiltonian systems is considered for engineering applications. Hence, a novel gradient-based method is presented, exploiting the discrete gradient concept and the forward Euler discretization under the assumption of the continuous Hamiltonian model is known. It is proven that the proposed discrete-time model structure defines a symplectic difference system and has the energy-conserving property under some conditions. In order to provide alternate discrete-time models, 3 different discrete-gradient definitions are given. The proposed models are convenient for the design of sampled-data controllers. All of the models are considered for several well-known Hamiltonian systems and the simulation results are demonstrated comparatively.

Anahtar Kelimeler:

Hamiltonian systems, discrete-time control model, discrete gradient

Discrete-time modeling of Hamiltonian systems

Keywords:

Hamiltonian systems, discrete-time control model, discrete gradient,

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V.A. Arnold, Mathematical Methods of Classical Mechanics, New York, Springer-Verlag, 1983.
K.R. Meyer, G.R. Hall, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, New York, Springer-Verlag, 1992.
A. Van der Schaft, L2-Gain and Passivity Techniques in Nonlinear Control, London, Springer, 2000.
D.S. Laila, A. Astolfi, “Discrete-time IDA-PBC design for separable Hamiltonian systems”, 16th IFAC World Congress, Vol. 16, pp. 539–545, 2005.
D.S. Laila, A. Astolfi, “Direct discrete-time design for sampled-data Hamiltonian control systems”, 3rd IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Vol. 366, pp. 87–98, 2006.
D. Nesic, A.R. Teel, “A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models”, IEEE Transactions on Automatic Control, Vol. 49, pp. 1103–1122, 2004.
W. Kratz, “Discrete oscillation”, Journal of Difference Equations and Applications, Vol. 9, pp. 135–147, 2003.
R. Hilscher, V. Zeidan, “Symplectic difference systems: variable stepsize discretization and discrete quadratic functionals”, Linear Algebra and Its Applications, Vol. 367, pp. 67–104, 2003.
Y. Shi, “Weyl-Titchmarsh theory for a class of discrete linear Hamiltonian systems”, Linear Algebra and Its Applications, Vol. 416, pp. 452–519, 2006.
M. Bohner, “Linear Hamiltonian difference systems: disconjugacy and Jacobi-type conditions”, Journal of Mathe- matical Analysis and Applications, Vol. 199, pp. 804–826, 1996.
R. McLachlan, G. Quispel, N. Robidoux, “Geometric integration using discrete gradients”, Philosophical Transac- tions: Mathematical, Physical and Engineering Sciences, Vol. 357, pp. 1021–1045, 1999.
O. Gonzalez, J.C. Simo, “On the stability of symplectic and energy-momentum algorithms for non-linear Hamilto- nian systems with symmetry”, Computer Methods in Applied Mechanics and Engineering, Vol. 134, pp. 197–222, 1996.
V. Talasila, J. Clemente-Gallardo, A.J. Van der Schaft, “Discrete port-Hamiltonian systems”, Systems & Control Letters, Vol. 55, pp. 478–486, 2006.
L. G¨oren S¨umer, Y. Yal¸cın, “Gradient based discrete-time modeling and control of Hamiltonian systems”, IFAC World Congress, Vol. 17, pp. 212–217, 2008.
Y. Yal¸cın, L. G¨oren S¨umer, “Direct discrete-time control of port controlled Hamiltonian systems”, Turkish Journal of Electrical Engineering & Computer Sciences, Vol. 18, pp. 913–924, 2010.
L. G¨oren S¨umer, Y. Yal¸cın, “A direct discrete-time IDA-PBC design method for a class of underactuated Hamilto- nian systems”, IFAC World Congress, Vol. 18, pp. 13456–13461, 2011.
Y. Yal¸cın, S. Kurtulan, L. G¨oren S¨umer, “Discrete-time control of crane system via PBC technique”, International Review of Automatic Control - Theory and Applications, Vol. 4, pp. 581–587, 2011.
Y. Yal¸cın, L. G¨oren S¨umer, “Robust disturbance attenuation in Hamiltonian systems via direct digital control”, European Control Conference, Vol. 1, pp. 514–560, 2009.
W.E. Diewert, “Exact and superlative index numbers”, Journal of Econometrics, Vol. 4, pp. 115–145, 1976.
C.D. Ahlbrandt, A.C. Peterson, Discrete Hamiltonian Systems: Difference Equations, Continued Fractions and Riccati Equations, Boston, Kluwer Academic Publishers, 1996.