Abstract
Variational integrators is a a new discretization technique of the equations of motion of a mechanical system introduced by Veselov and further developed by J. Marsden an co-workers, which is now widely used by numerical analysts working in various applied fields. This discretization technique, unlike the usual discretization procedures familiar in control, e.g. zero-order-hold, can lead to simple and well-conditioned transformation formulas for the recovery of the continuous time parameters from the discretized model.We discuss variational integrators for linear second order mechanical systems and show that physically meaningful properties of the continuous-time model, like passivity, are preserved. Variational integrator discretization is also shown to provide well-conditioned models for the identification of continuous-time second-orders systems starting from measured data.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Austin, M.A., Krishnaprasad, P.S., Wang, L.S.: Almost poisson integration of rigid body dynamics. J. Comput. Phys. 52, 105–117 (1993)
Bruschetta, M., Picci, G., Saccon, A.: Discrete mechanical systems: Second order modelling and identification. In: Proceedings of the 15th IFAC System Identification Symposium (SYSID), [CD-ROM], Saint Malo, France, pp. 456–461 (2009)
Costa-Castello, R., Fossas, E.: On preserving passivity in sampled data linear systems. In: Proceedings of the 2006 American Control Conference, Minneapolis, MI, pp. 4373–4378 (2006)
De Angelis, M., Lus, H., Betti, R., Longman, R.W.: Extracting physiscal parameters of mechanical models from identified state-space representations. ASME Trans. Journal Appl. Mech. 69, 617–625 (2002)
Dieci, L., Papini, A.: Conditioning and Padè approximation of the logarithm of a matrix. SIAM J. Matr. Anal. Appl. 21, 913–930 (2000)
Faurre, P.: Réalisations Markoviennes de processus stationnaires. INRIA, Roquencourt, France, Rapport de Recherche no. 13 (1973)
Garnier, H., Wang, L. (eds.): Identification of Continuous-time Models from Sampled Data. Springer, London (2008)
Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer, Berlin (2005)
Jiang, J.: Preservation of passivity under discretization. Journal of the Franklin Institute 330, 721–734 (1993)
Laub, A., Arnold, W.F.: Controllability and observability criteria for linear second-order models. IEEE Trans. Automatic Control 29, 163–165 (1984)
Lin, W., Byrnes, C.I.: Passivity and absolute stabilization of a class of discrete-time nonlinear systems. Automatica 31(263-267) (1995)
Lus, H., De Angelis, M., Betti, R., Longman, R.W.: Constructing second-order models of mechanical systems from identified state-space realizations, part I: theoretical discussions. ASCE Journal of Engineering Mech. 129, 477–488 (2002)
Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numerica, 357–514 (2001)
Monaco, S., Normand-Cyrot, D., Tiefensee, F.: From passivity under sampling to a new discrete-time passivity concept. In: Proceedings of the 47th IEEE Conference on Decision and Control, Cancun Mexico, pp. 3157–3162 (2008)
Sinha, N.K., Rao, G.P.: Identification of Continuous-Time Systems. Kluwer Academic, Dordrecht (1991)
Stramigioli, S., Secchi, C., van der Schaft, A.: Sampled data system passivity and discrete port-Hamiltonian systems. IEEE Transactions on Robotics 21, 574–587 (2002)
van der Schaft, A.J.: L 2 Gain and Passivity Techniques in Nonlinear Control, 2nd edn. Springer, Heidelberg (2000)
Veselov, A.P.: Integrable discrete-time systems and difference operators. Funkts. Anal. Prilozhen (Funct. An. and Appl.) 22, 113 (1988)
Willems, J.C.: Dissipative dynamical systems, Part II: linear systems with quadratic supply rates. Arch. Ration. Mech. Anal. 45, 321–393 (1972)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Bruschetta, M., Picci, G., Saccon, A. (2010). How to Sample Linear Mechanical Systems. In: Willems, J.C., Hara, S., Ohta, Y., Fujioka, H. (eds) Perspectives in Mathematical System Theory, Control, and Signal Processing. Lecture Notes in Control and Information Sciences, vol 398. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-93918-4_31
Download citation
DOI: https://doi.org/10.1007/978-3-540-93918-4_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-93917-7
Online ISBN: 978-3-540-93918-4
eBook Packages: EngineeringEngineering (R0)