# American Institute of Mathematical Sciences

June  2015, 4(2): 205-220. doi: 10.3934/eect.2015.4.205

## Finite rank distributed control for the resistive diffusion equation using damping assignment

 1 LCIS Laboratoire de Conception et d'Intégration des Systèmes, Grenoble Alpes University, F-26902, France, France

Received  May 2014 Revised  October 2014 Published  May 2015

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: Ngoc Minh Trang Vu, Laurent Lefèvre. Finite rank distributed control for the resistive diffusion equation using damping assignment. Evolution Equations & Control Theory, 2015, 4 (2) : 205-220. doi: 10.3934/eect.2015.4.205
##### References:
 [1] M. Becherif and E. Mendes, Stability and robustness of disturbed-port controlled hamiltonian systems with dissipation,, in Proceedings of the 16th IFAC World Congress (Praha, (2005). [2] J. Blum, Numerical Simulation and Optimal Control in Plasma Physics,, Gauthier-Villars, (1989). [3] J. Bucalossi, A. Argouarch, V. Basiuk, et al., Feasibility study of an actively cooled tungsten divertor in tore supra for iter technology testing,, Fusion Engineering and Design, 86 (2011), 684. doi: 10.1016/j.fusengdes.2011.01.114. [4] B. Jacob and H. Zwart, Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces,, Operator Theory: Advances and Applications, (2012). doi: 10.1007/978-3-0348-0399-1. [5] Y. Le Gorrec, H. Zwart and B. M. Maschke, Dirac structures and boundary control systems associated with skew-symmetric differential operators,, SIAM J. of Control and Optimization, 44 (2005), 1864. doi: 10.1137/040611677. [6] A. Macchelli, Boundary energy shaping of linear distributed Port-Hamiltonian systems,, European Journal of Control, 19 (2013), 521. doi: 10.1016/j.ejcon.2013.10.002. [7] A. Macchelli, A. J. van der Schaft and C. Melchiorri, Port Hamiltonian formulation of infinite dimensional systems. I. Modeling,, Proc. 50th IEEE Conference on Decisions and Control (CDC04), (2004), 3762. doi: 10.1109/CDC.2004.1429324. [8] _________, Port Hamiltonian formulation of infinite dimensional systems. II. Boundary control by interconnection,, in 43rd IEEE Conference on Decisions and Control (CDC04), (2004). [9] R. Ortega, A. J. van der Schaft, B. Maschke and G. Escobar, Interconnection and damping assignment: Passivity-based control of port-controlled Hamiltonian systems,, Automatica, 38 (2002), 585. doi: 10.1016/S0005-1098(01)00278-3. [10] R. Ortega and E. Garcia-Canseco, Interconnection and damping assignement passivit-based control: A survey,, European Journal of Control, 10 (2004), 432. doi: 10.3166/ejc.10.432-450. [11] M. Schöberl and A. Siuka, On casimir functionals for field fheories in port-hamiltonian description for control purposes,, in 50nd IEEE Conference on Decision and Control, (2011). [12] G. E. Swaters, Introduction to Hamiltonian Fluid Dynamics and Stability Theory,, Chapman & Hal/CRC, (2000). [13] A. J. van der Schaft and B. M. Maschke, Hamiltonian formulation of distributed parameter systems with boundary energy flow,, J. of Geometry and Physics, 42 (2002), 166. doi: 10.1016/S0393-0440(01)00083-3. [14] J. Villegas, H. Zwart, Y. Le Gorrec and B. M. Maschke, Stability and stabilization of a class of boundary control systems,, IEEE Transaction On Automatic Control, 54 (2009), 142. doi: 10.1109/TAC.2008.2007176. [15] J. A. Villegas, H. Zwart, Y. Le Gorrec, B. Maschke and A. J. van der Schaft, Stability and stabilization of a class of boundary control systems,, in Proc. 44th IEEE Conference on Decision and Control and European Control Conference ECC 2005 (Seville, (2005), 3850. doi: 10.1109/CDC.2005.1582762. [16] J. A. Villegas, Y. Le Gorrec, H. Zwart and B. Maschke, 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), 297. [17] T. N. M. Vu, L. Lefèvre and B. Maschke, 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). [18] T. N. M. Vu, L. Lefèvre, R. Nouailletas and S. Brémond, Geometric discretization for a plasma control model,, in IFAC Joint conference: 5th Symposium on System Structure and Control, (). [19] ________, An ida-pbc approach for the control of 1d plasma profile in tokamaks,, in 52nd IEEE Conference on Decision and Control, (2013). [20] T. N. M. Vu, R. Nouailletas, L. Lefèvre, S. Brémond and F. Felici, 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). [21] J. Wesson, Tokamaks,, Third edition, (2004). [22] E. Witrant, E. Joffrin, S. Brémond, G. Giruzzi, D. Mazon, O. Barana and P. Moreau, A control-oriented model of the current profile on tokamak plasma,, Plasma Physics and Controlled Fusion, 49 (2007), 1075. doi: 10.1088/0741-3335/49/7/009.

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##### References:
 [1] M. Becherif and E. Mendes, Stability and robustness of disturbed-port controlled hamiltonian systems with dissipation,, in Proceedings of the 16th IFAC World Congress (Praha, (2005). [2] J. Blum, Numerical Simulation and Optimal Control in Plasma Physics,, Gauthier-Villars, (1989). [3] J. Bucalossi, A. Argouarch, V. Basiuk, et al., Feasibility study of an actively cooled tungsten divertor in tore supra for iter technology testing,, Fusion Engineering and Design, 86 (2011), 684. doi: 10.1016/j.fusengdes.2011.01.114. [4] B. Jacob and H. Zwart, Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces,, Operator Theory: Advances and Applications, (2012). doi: 10.1007/978-3-0348-0399-1. [5] Y. Le Gorrec, H. Zwart and B. M. Maschke, Dirac structures and boundary control systems associated with skew-symmetric differential operators,, SIAM J. of Control and Optimization, 44 (2005), 1864. doi: 10.1137/040611677. [6] A. Macchelli, Boundary energy shaping of linear distributed Port-Hamiltonian systems,, European Journal of Control, 19 (2013), 521. doi: 10.1016/j.ejcon.2013.10.002. [7] A. Macchelli, A. J. van der Schaft and C. Melchiorri, Port Hamiltonian formulation of infinite dimensional systems. I. Modeling,, Proc. 50th IEEE Conference on Decisions and Control (CDC04), (2004), 3762. doi: 10.1109/CDC.2004.1429324. [8] _________, Port Hamiltonian formulation of infinite dimensional systems. II. Boundary control by interconnection,, in 43rd IEEE Conference on Decisions and Control (CDC04), (2004). [9] R. Ortega, A. J. van der Schaft, B. Maschke and G. Escobar, Interconnection and damping assignment: Passivity-based control of port-controlled Hamiltonian systems,, Automatica, 38 (2002), 585. doi: 10.1016/S0005-1098(01)00278-3. [10] R. Ortega and E. Garcia-Canseco, Interconnection and damping assignement passivit-based control: A survey,, European Journal of Control, 10 (2004), 432. doi: 10.3166/ejc.10.432-450. [11] M. Schöberl and A. Siuka, On casimir functionals for field fheories in port-hamiltonian description for control purposes,, in 50nd IEEE Conference on Decision and Control, (2011). [12] G. E. Swaters, Introduction to Hamiltonian Fluid Dynamics and Stability Theory,, Chapman & Hal/CRC, (2000). [13] A. J. van der Schaft and B. M. Maschke, Hamiltonian formulation of distributed parameter systems with boundary energy flow,, J. of Geometry and Physics, 42 (2002), 166. doi: 10.1016/S0393-0440(01)00083-3. [14] J. Villegas, H. Zwart, Y. Le Gorrec and B. M. Maschke, Stability and stabilization of a class of boundary control systems,, IEEE Transaction On Automatic Control, 54 (2009), 142. doi: 10.1109/TAC.2008.2007176. [15] J. A. Villegas, H. Zwart, Y. Le Gorrec, B. Maschke and A. J. van der Schaft, Stability and stabilization of a class of boundary control systems,, in Proc. 44th IEEE Conference on Decision and Control and European Control Conference ECC 2005 (Seville, (2005), 3850. doi: 10.1109/CDC.2005.1582762. [16] J. A. Villegas, Y. Le Gorrec, H. Zwart and B. Maschke, 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), 297. [17] T. N. M. Vu, L. Lefèvre and B. Maschke, 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). [18] T. N. M. Vu, L. Lefèvre, R. Nouailletas and S. Brémond, Geometric discretization for a plasma control model,, in IFAC Joint conference: 5th Symposium on System Structure and Control, (). [19] ________, An ida-pbc approach for the control of 1d plasma profile in tokamaks,, in 52nd IEEE Conference on Decision and Control, (2013). [20] T. N. M. Vu, R. Nouailletas, L. Lefèvre, S. Brémond and F. Felici, 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). [21] J. Wesson, Tokamaks,, Third edition, (2004). [22] E. Witrant, E. Joffrin, S. Brémond, G. Giruzzi, D. Mazon, O. Barana and P. Moreau, A control-oriented model of the current profile on tokamak plasma,, Plasma Physics and Controlled Fusion, 49 (2007), 1075. doi: 10.1088/0741-3335/49/7/009.
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