Implicit and explicit representations of continuous-time port-Hamiltonian systems
Authors
Fernando Castaños, Dmitry Gromov, Vincent Hayward, Hannah Michalska
Abstract
Implicit and explicit representations of smooth, finite-dimensional port-Hamiltonian systems are studied from the perspective of their use in numerical simulation and control design. Implicit representations arise when a system is modeled in Cartesian coordinates and when the system constraints are applied in the form of additional algebraic equations. Explicit representations are derived when generalized coordinates are used. A relationship between the phase spaces for both system representations is derived in this article, justifying the equivalence of the representations in the sense of preserving their Hamiltonian functions as well as their Hamiltonian symplectic forms, ultimately resulting in the same Hamiltonian flow.
Keywords
Port-Hamiltonian systems; Nonlinear implicit systems; Modeling of physical systems
Citation
- Journal: Systems & Control Letters
- Year: 2013
- Volume: 62
- Issue: 4
- Pages: 324–330
- Publisher: Elsevier BV
- DOI: 10.1016/j.sysconle.2013.01.007
BibTeX
@article{Casta_os_2013,
title={{Implicit and explicit representations of continuous-time port-Hamiltonian systems}},
volume={62},
ISSN={0167-6911},
DOI={10.1016/j.sysconle.2013.01.007},
number={4},
journal={Systems & Control Letters},
publisher={Elsevier BV},
author={Castaños, Fernando and Gromov, Dmitry and Hayward, Vincent and Michalska, Hannah},
year={2013},
pages={324--330}
}
References
- Singer, (2001)
- Schaft, A. J. Hamiltonian dynamics with external forces and observations. Mathematical Systems Theory vol. 15 145–168 (1981) – 10.1007/bf01786977
- Dalsmo, M. & van der Schaft, A. On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems. SIAM Journal on Control and Optimization vol. 37 54–91 (1998) – 10.1137/s0363012996312039
- Rheinboldt, W. C. Differential-algebraic systems as differential equations on manifolds. Mathematics of Computation vol. 43 473–482 (1984) – 10.1090/s0025-5718-1984-0758195-5
- Reich, S. On a geometrical interpretation of differential-algebraic equations. Circuits Systems and Signal Processing vol. 9 367–382 (1990) – 10.1007/bf01189332
- Arnold, (1989)
- Marsden, (1999)
- Smale, S. On the mathematical foundations of electrical circuit theory. Journal of Differential Geometry vol. 7 (1972) – 10.4310/jdg/1214430827
- Maschke, B. M., van der Schaft, A. J. & Breedveld, P. C. An intrinsic Hamiltonian formulation of the dynamics of LC-circuits. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications vol. 42 73–82 (1995) – 10.1109/81.372847
- van der Schaft, (2000)
- MaClamroch, Control of constrained Hamiltonian systems and applications to control of constrained robots. (1988)
- Hairer, (2006)
- Putting energy back in control. IEEE Control Systems vol. 21 18–33 (2001) – 10.1109/37.915398
- Leimkuhler, (2004)
- Lee, (2003)
- Giaquinta, (2004)
- Bullo, F. Stabilization of relative equilibria for underactuated systems on Riemannian manifolds. Automatica vol. 36 1819–1834 (2000) – 10.1016/s0005-1098(00)00115-1
- Giaquinta, (1996)
- Zia, R. K. P., Redish, E. F. & McKay, S. R. Making sense of the Legendre transform. American Journal of Physics vol. 77 614–622 (2009) – 10.1119/1.3119512
- Maschke, B. M., Van Der Schaft, A. J. & Breedveld, P. C. An intrinsic hamiltonian formulation of network dynamics: non-standard poisson structures and gyrators. Journal of the Franklin Institute vol. 329 923–966 (1992) – 10.1016/s0016-0032(92)90049-m
- Agrachev, (2004)
- Ortega, (1998)