On an intrinsic formulation of time-variant Port Hamiltonian systems
Authors
Markus Schöberl, Kurt Schlacher
Abstract
In this contribution we present an intrinsic description of time-variant Port Hamiltonian systems as they appear in modeling and control theory. This formulation is based on the splitting of the state bundle and the use of appropriate covariant derivatives, which guarantees that the structure of the equations is invariant with respect to time-variant coordinate transformations. In particular, we will interpret our covariant system representation in the context of control theoretic problems. Typical examples are time-variant error systems related to trajectory tracking problems which allow for a Hamiltonian formulation. Furthermore we will analyze the concept of collocation and the balancing/interaction of power flows in an intrinsic fashion.
Keywords
Nonlinear control systems; Differential geometric methods; Mathematical systems theory; Tracking applications; Mechanical systems
Citation
- Journal: Automatica
- Year: 2012
- Volume: 48
- Issue: 9
- Pages: 2194–2200
- Publisher: Elsevier BV
- DOI: 10.1016/j.automatica.2012.06.014
BibTeX
@article{Sch_berl_2012,
title={{On an intrinsic formulation of time-variant Port Hamiltonian systems}},
volume={48},
ISSN={0005-1098},
DOI={10.1016/j.automatica.2012.06.014},
number={9},
journal={Automatica},
publisher={Elsevier BV},
author={Schöberl, Markus and Schlacher, Kurt},
year={2012},
pages={2194--2200}
}
References
- Abraham, (1978)
- Cheng, D., Astolfi, A. & Ortega, R. On feedback equivalence to port controlled Hamiltonian systems. Systems & Control Letters vol. 54 911–917 (2005) – 10.1016/j.sysconle.2005.02.005
- Fujimoto, K. & Sugie, T. Canonical transformation and stabilization of generalized Hamiltonian systems. Systems & Control Letters vol. 42 217–227 (2001) – 10.1016/s0167-6911(00)00091-8
- Fujimoto, K., Sakurama, K. & Sugie, T. Trajectory tracking control of port-controlled Hamiltonian systems via generalized canonical transformations. Automatica vol. 39 2059–2069 (2003) – 10.1016/j.automatica.2003.07.005
- Giachetta, (1997)
- Gotay, M. J. A multisymplectic framework for classical field theory and the calculus of variations II: space + time decomposition. Differential Geometry and its Applications vol. 1 375–390 (1991) – 10.1016/0926-2245(91)90014-z
- Kanatchikov, I. V. Canonical structure of classical field theory in the polymomentum phase space. Reports on Mathematical Physics vol. 41 49–90 (1998) – 10.1016/s0034-4877(98)80182-1
- Maschke, B., Ortega, R. & Van Der Schaft, A. J. Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation. IEEE Transactions on Automatic Control vol. 45 1498–1502 (2000) – 10.1109/9.871758
- Nijmeijer, (1990)
- Putting energy back in control. IEEE Control Systems vol. 21 18–33 (2001) – 10.1109/37.915398
- Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica vol. 38 585–596 (2002) – 10.1016/s0005-1098(01)00278-3
- Saunders, (1989)
- Schöberl, M. & Schlacher, K. Geometric Analysis of Hamiltonian Mechanics using Connections. PAMM vol. 6 843–844 (2006) – 10.1002/pamm.200610401
- Schöberl, M. & Schlacher, K. Covariant formulation of the governing equations of continuum mechanics in an Eulerian description. Journal of Mathematical Physics vol. 48 (2007) – 10.1063/1.2735444
- Schöberl, M., Stadlmayr, R. & Schlacher, K. GEOMETRIC ANALYSIS OF TIME VARIANT HAMILTONIAN CONTROL SYSTEMS. IFAC Proceedings Volumes vol. 40 864–869 (2007) – 10.3182/20070822-3-za-2920.00143
- van der Schaft, (2000)