A structured control model for the thermo-magneto-hydrodynamics of plasmas in tokamaks

Authors

Ngoc Minh Trang Vu, Laurent Lefèvre, Bernhard Maschke

Abstract

A thermo-magneto-hydrodynamics port-Hamiltonian model is derived for the plasmas in tokamaks. Electromagnetic field and material domain balance equations are expressed in covariant forms, together with the magneto-hydrodynamics interconnection structure connecting them together. The balance equations for the entropy, mass and momentum, as well as closure equations in the material domain, are derived from the Boltzmann equation (kinetic theory). The Gibbs–Duhem equation is used to compute the irreversible entropy source term and to define the interdomain ℜ-field of the model. All derived interdomain couplings in the material domain are represented using Dirac and Stokes–Dirac structures and the resistivity ℜ-field structure. The complete model is summarized in a Bond Graph.

Citation

• Journal: Mathematical and Computer Modelling of Dynamical Systems
• Year: 2016
• Volume: 22
• Issue: 3
• Pages: 181–206
• Publisher: Informa UK Limited
• DOI: 10.1080/13873954.2016.1154874

BibTeX

@article{Vu_2016,
  title={{A structured control model for the thermo-magneto-hydrodynamics of plasmas in tokamaks}},
  volume={22},
  ISSN={1744-5051},
  DOI={10.1080/13873954.2016.1154874},
  number={3},
  journal={Mathematical and Computer Modelling of Dynamical Systems},
  publisher={Informa UK Limited},
  author={Vu, Ngoc Minh Trang and Lefèvre, Laurent and Maschke, Bernhard},
  year={2016},
  pages={181--206}
}

Download the bib file

References

• Wesson J., Tokamaks (2004)
• Ariola M., Magnetic Control of Tokamak Plasmas (2008)
• Fusion, tokamaks, and plasma control: an introduction and tutorial. IEEE Control Systems vol. 25 30–43 (2005) – 10.1109/mcs.2005.1512794
• Emerging applications in tokamak plasma control. IEEE Control Systems vol. 26 35–63 (2006) – 10.1109/mcs.2006.1615272
• Blum J., Numerical Simulation and Optimal Control in Plasma Physics (1989)
• Marsden, J. E. & Weinstein, A. The Hamiltonian structure of the Maxwell-Vlasov equations. Physica D: Nonlinear Phenomena vol. 4 394–406 (1982) – 10.1016/0167-2789(82)90043-4
• Morrison, P. J. & Greene, J. M. Noncanonical Hamiltonian Density Formulation of Hydrodynamics and Ideal Magnetohydrodynamics. Physical Review Letters vol. 45 790–794 (1980) – 10.1103/physrevlett.45.790
• Tassi, E., Morrison, P. J., Waelbroeck, F. L. & Grasso, D. Hamiltonian formulation and analysis of a collisionless fluid reconnection model. Plasma Physics and Controlled Fusion vol. 50 085014 (2008) – 10.1088/0741-3335/50/8/085014
• 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
• Ou, Y. et al. Optimal Tracking Control of Current Profile in Tokamaks. IEEE Transactions on Control Systems Technology vol. 19 432–441 (2011) – 10.1109/tcst.2010.2046640
• Ou Y., IEEE Trans. Plasma Sci. (2010)
• Argomedo F.B., IEEE Trans. Automat. Contr. (2012)
• Schuster, E. & Krstić, M. Control of a non-linear PDE system arising from non-burning tokamak plasma transport dynamics. International Journal of Control vol. 76 1116–1124 (2003) – 10.1080/0020717031000135523
• Schuster, E., Krstić, M. & Tynan, G. Nonlinear Lyapunov-based burn control in fusion reactors. Fusion Engineering and Design vols 63–64 569–575 (2002) – 10.1016/s0920-3796(02)00272-7
• Schuster, E., Krstić, M. & Tynan, G. Burn Control in Fusion Reactors via Nonlinear Stabilization Techniques. Fusion Science and Technology vol. 43 18–37 (2003) – 10.13182/fst03-a246
• Morrison, P. J. Hamiltonian and action principle formulations of plasma physics. Physics of Plasmas vol. 12 (2005) – 10.1063/1.1882353
• 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
• Braginskii S.I., Reviews of Plasma Physics (1965)
• Duindam, V., Macchelli, A., Stramigioli, S. & Bruyninckx, H. Modeling and Control of Complex Physical Systems. (Springer Berlin Heidelberg, 2009). doi:10.1007/978-3-642-03196-0 – 10.1007/978-3-642-03196-0
• de Groot S.R., Non-Equilibrium Thermodynamics (1984)
• Karnopp D.C., System Dynamics: modeling and Simulation of Mechatronic Systems (2006)
• Kugi A., Non-linear Control Based on Physical Models: electrical, Hydraulic and Mechanical Systems (2000)
• Stramigioli S., Modeling and IPC Control of Interactive Mechanical Systems: a Coordinate-free Approach (2001)
• van der Schaft, A. L2 - Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering (Springer London, 2000). doi:10.1007/978-1-4471-0507-7 – 10.1007/978-1-4471-0507-7
• Golo, G., Talasila, V., van der Schaft, A. & Maschke, B. Hamiltonian discretization of boundary control systems. Automatica vol. 40 757–771 (2004) – 10.1016/j.automatica.2003.12.017
• Macchelli, A. & Melchiorri, C. Modeling and Control of the Timoshenko Beam. The Distributed Port Hamiltonian Approach. SIAM Journal on Control and Optimization vol. 43 743–767 (2004) – 10.1137/s0363012903429530
• Hamroun B., Trans. Fluid Mech. (2006)
• Baaiu, A. et al. Port-based modelling of mass transport phenomena. Mathematical and Computer Modelling of Dynamical Systems vol. 15 233–254 (2009) – 10.1080/13873950902808578
• Baaiu, A., Couenne, F., Lefevre, L., Le Gorrec, Y. & Tayakout, M. Structure-preserving infinite dimensional model reduction: Application to adsorption processes. Journal of Process Control vol. 19 394–404 (2009) – 10.1016/j.jprocont.2008.07.002
• Nishida, G., Takagi, K., Maschke, B. & Osada, T. Multi-scale distributed parameter modeling of ionic polymer-metal composite soft actuator. Control Engineering Practice vol. 19 321–334 (2011) – 10.1016/j.conengprac.2010.10.005
• Frankel T., The Geometry of Physics: An Introduction (2004)
• Choquet-Bruhat Y., Analysis Manifolds and Physics (1982)
• Olver, P. J. Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics (Springer New York, 1993). doi:10.1007/978-1-4612-4350-2 – 10.1007/978-1-4612-4350-2
• Maschke B., Advanced topics in control systems theory, in Chapter Compositional Modelling of Distributed Parameter Systems
• Courant, T. J. Dirac manifolds. Transactions of the American Mathematical Society vol. 319 631–661 (1990) – 10.1090/s0002-9947-1990-0998124-1
• 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
• van der Schaft A.J., Archiv für Elektronik und Übertragungstechnik (1995)
• Jamiolkowski A., Classical Electrodynamics (1985)
• Baaiu A., Math. Comput. Model. Dyn. Syst. (2009)
• Onsager L., Phys. Rev (1931)
• Boozer, A. H. Onsager symmetry of transport in toroidal plasmas. Physics of Fluids B: Plasma Physics vol. 4 2845–2853 (1992) – 10.1063/1.860159
• Garbet, X. et al. Thermodynamics of neoclassical and turbulent transport. Plasma Physics and Controlled Fusion vol. 54 055007 (2012) – 10.1088/0741-3335/54/5/055007
• Polner, M. & van der Vegt, J. J. W. A Hamiltonian vorticity–dilatation formulation of the compressible Euler equations. Nonlinear Analysis: Theory, Methods & Applications vol. 109 113–135 (2014) – 10.1016/j.na.2014.07.005