Finite rank distributed control for the resistive diffusion equation using damping assignment
Authors
Ngoc Minh Trang Vu, Laurent Lefèvre
Abstract
A first extension of the IDA-PBC control synthesis to infinite dimensional port Hamiltonian systems is investigated, using the same idea as for the finite dimensional case, that is transform the original model into a closed loop target Hamiltonian model using feedback control. To achieve this goal both finite rank distributed control and boundary control are used. The proposed class of considered port Hamiltonian distributed parameters systems is first defined. Then the matching equation is derived for this class before considering the particular case of damping assignment on the resistive diffusion example, for the radial diffusion of the poloidal magnetic flux in tokamak reactors.
Citation
- Journal: Evolution Equations & Control Theory
- Year: 2015
- Volume: 4
- Issue: 2
- Pages: 205–220
- Publisher: American Institute of Mathematical Sciences (AIMS)
- DOI: 10.3934/eect.2015.4.205
BibTeX
@article{Minh_Trang_Vu_2015,
title={{Finite rank distributed control for the resistive diffusion equation using damping assignment}},
volume={4},
ISSN={2163-2480},
DOI={10.3934/eect.2015.4.205},
number={2},
journal={Evolution Equations & Control Theory},
publisher={American Institute of Mathematical Sciences (AIMS)},
author={Minh Trang Vu, Ngoc and Lefèvre, Laurent},
year={2015},
pages={205--220}
}
References
- M. Becherif, Stability and robustness of disturbed-port controlled hamiltonian systems with dissipation,. in Proceedings of the 16th IFAC World Congress (Praha (2005)
- J. Blum, Numerical Simulation and Optimal Control in Plasma Physics,. Gauthier-Villars (1989)
- Bucalossi, J. et al. Feasibility study of an actively cooled tungsten divertor in Tore Supra for ITER technology testing. Fusion Engineering and Design vol. 86 684–688 (2011) – 10.1016/j.fusengdes.2011.01.114
- Jacob, B. & Zwart, H. J. Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces. (Springer Basel, 2012). doi:10.1007/978-3-0348-0399-1 – 10.1007/978-3-0348-0399-1
- Le Gorrec, Y., Zwart, H. & Maschke, B. Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM Journal on Control and Optimization vol. 44 1864–1892 (2005) – 10.1137/040611677
- Macchelli, A. Boundary energy shaping of linear distributed port-Hamiltonian systems. European Journal of Control vol. 19 521–528 (2013) – 10.1016/j.ejcon.2013.10.002
- Macchelli, A., van der Schaft, A. J. & Melchiorri, C. Port Hamiltonian formulation of infinite dimensional systems I. Modeling. 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601) 3762-3767 Vol.4 (2004) doi:10.1109/cdc.2004.1429324 – 10.1109/cdc.2004.1429324
- _____, Port Hamiltonian formulation of infinite dimensional systems. II. Boundary control by interconnection,. in 43rd IEEE Conference on Decisions and Control (CDC04) (2004)
- 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
- Ortega, R. & García-Canseco, E. Interconnection and Damping Assignment Passivity-Based Control: A Survey. European Journal of Control vol. 10 432–450 (2004) – 10.3166/ejc.10.432-450
- Schoberl, M. & Siuka, A. On Casimir functionals for field theories in Port-Hamiltonian description for control purposes. IEEE Conference on Decision and Control and European Control Conference 7759–7764 (2011) doi:10.1109/cdc.2011.6160430 – 10.1109/cdc.2011.6160430
- G. E. Swaters, Introduction to Hamiltonian Fluid Dynamics and Stability Theory,. Chapman & Hal/CRC (2000)
- van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics vol. 42 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
- Villegas, J. A., Zwart, H., Le Gorrec, Y. & Maschke, B. Exponential Stability of a Class of Boundary Control Systems. IEEE Transactions on Automatic Control vol. 54 142–147 (2009) – 10.1109/tac.2008.2007176
- Villegas, J. A., Zwart, H., Le Gorrec, Y., Maschke, B. & van der Schaft, A. J. Stability and Stabilization of a Class of Boundary Control Systems. Proceedings of the 44th IEEE Conference on Decision and Control 3850–3855 doi:10.1109/cdc.2005.1582762 – 10.1109/cdc.2005.1582762
- J. A. Villegas, Boundary control for a class of dissipative differential operators including diffusion systems,. in Proc. 7th International Symposium on Mathematical Theory of Networks and Systems (Kyoto (2006)
- T. N. M. Vu, Port-hamiltonian formulation for systems of conservation laws: application to plasma dynamics in tokamak reactors,. in 4th IFAC Workshop on Lagrangian and Hamiltonian Methods for Non Linear Control (2012)
- T. N. M. Vu, Geometric discretization for a plasma control model,. in IFAC Joint conference: 5th Symposium on System Structure and Control
- ____, An ida-pbc approach for the control of 1d plasma profile in tokamaks,. in 52nd IEEE Conference on Decision and Control (2013)
- T. N. M. Vu, Ida-pbc control for the coupled plasma poloidal magnetic flux and heat radial diffusion equations in tokamaks,. in 19th World Congress of the International Federation of Automatic Control (2014)
- J. Wesson, Tokamaks,. Third edition (2004)
- Witrant, E. et al. A control-oriented model of the current profile in tokamak plasma. Plasma Physics and Controlled Fusion vol. 49 1075–1105 (2007) – 10.1088/0741-3335/49/7/009