. Barton, This can significantly improve the plasma performance Felici and Sauter, 2012) The feed-forward commands can be computed off-line to find the optimal trajectory. The computation of the feedforward inputs can be done by substantial number of trial-and-error attempts and based on experience gained during operation of a particular device Another way is to use some model based techniques as for example extremum-seeking open-loop optimal control presented in, 2008.

J. Artaud, Predictive integrated modelling for ITER scenarios, EPS Conference on Plasma Physics, vol.29, 2005.

J. Artaud, The CRONOS suite of codes for integrated tokamak modelling, Nuclear Fusion, vol.50, issue.4, pp.43001-43041, 2010.
DOI : 10.1088/0029-5515/50/4/043001

K. Åström, T. Johan, and . Hägglund, Advanced PID control. ISA-The Instrumentation , Systems and Automation Society (cit, p.96, 2006.

J. Barton, Physics-model-based nonlinear actuator trajectory optimization and safety factor profile feedback control for advanced scenario development in DIII-D, Nuclear Fusion, vol.55, issue.9, pp.93005-93023, 2015.
DOI : 10.1088/0029-5515/55/9/093005

URL : https://doi.org/10.1088/0029-5515/55/9/093005

J. E. Barton, Robust control of the safety factor profile and stored energy evolutions in high performance burning plasma scenarios in the ITER tokamak, 52nd IEEE Conference on Decision and Control, pp.4194-4199, 2013.
DOI : 10.1109/CDC.2013.6760533

J. E. Barton, Simultaneous closed-loop control of the current profile and the electron temperature profile in the TCV tokamak, 2015 American Control Conference (ACC), pp.3316-3321, 2015.
DOI : 10.1109/ACC.2015.7171844

V. Basiuk, Simulations of steady-state scenarios for Tore Supra using the CRONOS code, Nuclear Fusion, vol.43, issue.9, p.822, 2003.
DOI : 10.1088/0029-5515/43/9/305

D. Bauer, On data preprocessing for subspace methods, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187), pp.2403-2408, 2000.
DOI : 10.1109/CDC.2000.914159

J. Blum, Numerical Simulation and Optimal Control in Plasma Physics: With Applications to Tokamaks, Gauthier-Villars Series in Modern Applied Mathemat) (cit, 1989.

C. Bohn and D. Atherton, An analysis package comparing PID anti-windup strategies, IEEE Control Systems 15.2, pp.34-40, 1995.
DOI : 10.1109/37.375281

B. Bondarenko and . Dmitrievich, Role played by OA Lavrent'ev in the formulation of the problem and the initiation of research into controlled nuclear fusion in the USSR, Physics-Uspekhi, vol.448, pp.844-851, 2001.

M. D. Boyer, First-principles-driven model-based current profile control for the DIII-D tokamak via LQI optimal control, Plasma Physics and Controlled Fusion, vol.55, issue.10, p.105007, 2013.
DOI : 10.1088/0741-3335/55/10/105007

M. D. Boyer, Backstepping Control of the Toroidal Plasma Current Profile in the DIII-D Tokamak, IEEE Transactions on Control Systems Technology, vol.22, issue.5, pp.1725-1739, 2014.
DOI : 10.1109/TCST.2013.2296493

B. Argomedo and F. , Model-based control of the magnetic flux profile in a tokamak plasma, 49th IEEE Conference on. IEEE, pp.6926-6931, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00541884

B. Argomedo and F. , A Strict Control Lyapunov Function for a Diffusion Equation With Time-Varying Distributed Coefficients, IEEE Transactions on Automatic Control, vol.58, issue.2, pp.290-303, 2013.
DOI : 10.1109/TAC.2012.2209260

URL : https://hal.archives-ouvertes.fr/hal-00719502

B. Argomedo and F. , Lyapunov-based distributed control of the safety-factor profile in a tokamak plasma, Nuclear Fusion, vol.53, issue.3, p.33005, 2013.
DOI : 10.1088/0029-5515/53/3/033005

URL : https://hal.archives-ouvertes.fr/hal-00788215

M. Chilali and P. Gahinet, H? design with pole placement constraints: an LMI approach, IEEE Transactions on Automatic Control, vol.413, pp.358-367, 1996.
DOI : 10.1109/9.486637

D. Ruscio and D. , Subspace System Identification: Theory and applications, In: Lecture notes, vol.37, p.45, 1997.

L. Duchesne, Subspace identification with multiple data sets, Guidance, Navigation, and Control Conference, 1996.
DOI : 10.2514/3.20031

M. Erba, Validation of a new mixed Bohm/gyro-Bohm model for electron and ion heat transport against the ITER, Tore Supra and START database discharges, Nuclear Fusion, vol.38, issue.7, p.1013, 1998.
DOI : 10.1088/0029-5515/38/7/305

F. Felici, M. De-baar, and M. Steinbuch, A dynamic state observer for real-time reconstruction of the tokamak plasma profile state and disturbances, 2014 American Control Conference, pp.4816-4823, 2014.
DOI : 10.1109/ACC.2014.6858781

F. Felici and O. Sauter, Non-linear model-based optimization of actuator trajectories for tokamak plasma profile control, Plasma Physics and Controlled Fusion, vol.54, issue.2, pp.25002-25020, 2012.
DOI : 10.1088/0741-3335/54/2/025002

F. Felici, Real-time physics-model-based simulation of the current density profile in tokamak plasmas, Nuclear Fusion, vol.51, issue.8, pp.83052-83077, 2011.
DOI : 10.1088/0029-5515/51/8/083052

F. Felici, Real-time control of tokamak plasmas: from control of physics to physics-based control, In: (cit. on pp, vol.5, issue.17, p.18, 2011.

A. Gahlawat, Bootstrap current optimization in tokamaks using sumof-squares polynomials, 51st Annual Conference on Decision and Control. IEEE, pp.4359-4365, 2012.
DOI : 10.1109/cdc.2012.6426638

Z. Gajic, Optimal control of singularly perturbed linear systems and applications, 2001.
DOI : 10.1201/9780203907900

O. Gaye, stabilization of the current profile in tokamak plasmas via an LMI approach, Automatica, vol.49, issue.9, pp.2795-2804, 2013.
DOI : 10.1016/j.automatica.2013.05.011

URL : https://hal.archives-ouvertes.fr/hal-00864243

O. Gaye, Robust stabilization of the current profile in tokamak plasmas using sliding mode approach in infinite dimension, Control Engineering Practice, vol.21, issue.10, pp.1350-1358, 2013.
DOI : 10.1016/j.conengprac.2013.05.003

URL : https://hal.archives-ouvertes.fr/hal-00845383

P. Geelen, Parameter estimation for a nonlinear control-oriented tokamak profile evolution model, Plasma Physics and Controlled Fusion, vol.57, issue.12, p.125008, 2015.
DOI : 10.1088/0741-3335/57/12/125008

URL : http://iopscience.iop.org/article/10.1088/0741-3335/57/12/125008/pdf

C. Gormezano, Steady state operation In: Nuclear Fusion 47, p.285, 2007.

C. Grimholt and S. Skogestad, Optimal PI-Control and Verification of the SIMC Tuning Rule, IFAC Proceedings Volumes 45.3, pp.11-22, 2012.
DOI : 10.3182/20120328-3-IT-3014.00003

C. Hall and J. Hung, Variable structure PID control to prevent integrator windup, 1999.

R. Hanus, M. Kinnaert, and J. Henrotte, Conditioning technique, a general anti-windup and bumpless transfer method, Automatica, vol.23, issue.6, pp.729-739, 1987.
DOI : 10.1016/0005-1098(87)90029-X

B. Haverkamp and T. Delft, Subspace method identification, theory and practice, 2000.

R. Hawryluk, AN EMPIRICAL APPROACH TO TOKAMAK TRANSPORT, pp.19-46, 1980.
DOI : 10.1016/B978-1-4832-8385-2.50009-1

F. Hinton and R. Hazeltine, Theory of plasma transport in toroidal confinement systems, Reviews of Modern Physics, vol.10, issue.2, pp.239-264, 1976.
DOI : 10.1103/PhysRevLett.25.916

S. Hirshman, B. Hawryluk, and . Birge, Neoclassical conductivity of a tokamak plasma, In: Nuclear Fusion, vol.173, p.611, 1977.

W. Kaplan, Advanced calculus, 2002.

T. Katayama, Subspace methods for system identification, pp.46-48, 2006.
DOI : 10.1007/1-84628-158-X

H. K. Khalil and J. Grizzle, Nonlinear systems, 1996.

D. Kim, Simple predictive electron transport models applied to sawtoothing plasmas, Plasma Physics and Controlled Fusion, vol.58, issue.5, pp.55002-55029, 2016.
DOI : 10.1088/0741-3335/58/5/055002

S. Kim and J. Lister, A potentially robust plasma profile control approach for ITER using real-time estimation of linearized profile response models, Nuclear Fusion, vol.52, issue.7, pp.74002-74019, 2012.
DOI : 10.1088/0029-5515/52/7/074002

P. Kokotovic, K. Hassan, J. Khalil, and . Reilly, Singular perturbation methods in control: analysis and design, 1999.
DOI : 10.1137/1.9781611971118

P. Kokotovic and R. Freeman, Robust nonlinear control design (cit, p.61, 1996.

M. Krstic and A. Smyshlyaev, Boundary control of PDEs: A course on backstepping designs, 2008.
DOI : 10.1137/1.9780898718607

L. Laborde, A model-based technique for integrated real-time profile control in the JET tokamak, Plasma Physics and Controlled Fusion, vol.47, issue.1, p.155, 2004.
DOI : 10.1088/0741-3335/47/1/010

J. D. Lawson, Some Criteria for a Power Producing Thermonuclear Reactor, Proceedings of the Physical Society. Section B, vol.70, issue.1, 1957.
DOI : 10.1088/0370-1301/70/1/303

D. Leith, R. Murray-smith, and . Bradley, Combination of data sets for system identification, IEE Proceedings D Control Theory and Applications, vol.140, issue.1, pp.11-18, 1993.
DOI : 10.1049/ip-d.1993.0002

L. Ljung, System identification, pp.43-50, 1998.
URL : https://hal.archives-ouvertes.fr/hal-00854858

J. Löfberg, YALMIP : a toolbox for modeling and optimization in MATLAB, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508), pp.284-289, 2004.
DOI : 10.1109/CACSD.2004.1393890

B. Maljaars, Model Predictive Control of the Current Density Distribution and Stored Energy in Tokamak Fusion Experiments using Trajectory Linearizations In: IFAC-PapersOnLine 48, pp.314-321, 2015.

D. Mazon, Active control of the current density profile in JET, Plasma Physics and Controlled Fusion, vol.45, issue.7, p.47, 2003.
DOI : 10.1088/0741-3335/45/7/102

D. N. Miller and R. Callafon, Subspace identification with eigenvalue constraints, Automatica, vol.49, issue.8, pp.2468-2473, 2013.
DOI : 10.1016/j.automatica.2013.04.028

D. Moreau, -profile in JET for steady state advanced tokamak operation, Nuclear Fusion, vol.43, issue.9, p.870, 2003.
DOI : 10.1088/0029-5515/43/9/311

D. Moreau, A two-time-scale dynamic-model approach for magnetic and kinetic profile control in advanced tokamak scenarios on JET, Nuclear Fusion, vol.48, issue.10, pp.106001-106040, 2008.
DOI : 10.1088/0029-5515/48/10/106001

D. Moreau, Plasma models for real-time control of advanced tokamak scenarios, Nuclear Fusion, vol.51, issue.6, pp.63009-63048, 2011.
DOI : 10.1088/0029-5515/51/6/063009

D. Moreau, Integrated magnetic and kinetic control of advanced tokamak plasmas on DIII-D based on data-driven models, Nuclear Fusion, vol.53, issue.6, pp.63020-63036, 2013.
DOI : 10.1088/0029-5515/53/6/063020

S. J. Moura, K. Hosam, and . Fathy, Optimal Boundary Control of Reaction???Diffusion Partial Differential Equations via Weak Variations, Journal of Dynamic Systems, Measurement, and Control, vol.135, issue.3, p.34501, 2013.
DOI : 10.1115/1.4023071

Y. Ou, Design and simulation of extremum-seeking open-loop optimal control of current profile in the DIII-D tokamak, Plasma Physics and Controlled Fusion, vol.50, issue.11, p.115001, 2008.
DOI : 10.1088/0741-3335/50/11/115001

A. (. Peeters, The bootstrap current and its consequences, Plasma Physics and Controlled Fusion, vol.42, issue.12B, p.231, 2000.
DOI : 10.1088/0741-3335/42/12B/318

G. Pereverzev and P. Yushmanov, ASTRA Automated system for transport analysis in a Tokamak Report IPP 5, 2002.

A. Polevoi, . Medvedev, and Y. Pustovitov, Possibility of Q> 5 stable, steadystate operation in ITER with moderate ?N and H-factor, In: Fusion Energy, 2002.

J. Ramos, Hermitian operator methods for reaction-diffusion equations, Numerical Methods for Partial Differential Equations, vol.29, issue.4, pp.241-287, 1987.
DOI : 10.1090/trans2/029/12

O. Sauter, C. Angioni, and Y. Lin-liu, Neoclassical conductivity and bootstrap current formulas for general axisymmetric equilibria and arbitrary collisionality regime, Physics of Plasmas, vol.6, issue.7, pp.2834-2839, 1994.
DOI : 10.1063/1.861356

O. Sautera, Y. Angioni, and . Lin-liub, Erratum, Canadian Journal of Anaesthesia, vol.39, issue.6, p.2834, 1999.
DOI : 10.1007/BF03008333

S. Skogestad, Probably the best simple PID tuning rules in the world, AIChE Annual Meeting, 2001.

L. Spitzer, Physics of Fully Ionized Gases, American Journal of Physics, vol.31, issue.11, 2013.
DOI : 10.1119/1.1969155

W. Suleiman and A. Monin, Linear multivariable system identification: multi-experiments case, Conference on Systems and Control. Marrakech, Morocco (cit, pp.41-45, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00155852

Y. Tang, C. Prieur, and A. Girard, Boundary control synthesis for hyperbolic systems: A singular perturbation approach, 53rd IEEE Conference on Decision and Control, pp.2840-2845, 2014.
DOI : 10.1109/CDC.2014.7039825

URL : https://hal.archives-ouvertes.fr/hal-01144135

S. Tarbouriech, Stability and stabilization of linear systems with saturating actuators, 2011.
DOI : 10.1007/978-0-85729-941-3

A. Taroni, Global and local energy confinement properties of simple transport coefficients of the Bohm type, Plasma Physics and Controlled Fusion, vol.36, issue.10, p.1629, 1994.
DOI : 10.1088/0741-3335/36/10/003

T. Taylor, Physics of advanced tokamaks, Plasma Physics and Controlled Fusion, vol.39, issue.12B, p.47, 1997.
DOI : 10.1088/0741-3335/39/12B/005

F. Troyon, MHD-Limits to Plasma Confinement, Plasma Physics and Controlled Fusion, vol.26, issue.1A, p.209, 1984.
DOI : 10.1088/0741-3335/26/1A/319

URL : https://infoscience.epfl.ch/record/120771/files/lrp_231_83_hq.pdf

R. Vazquez and M. Krstic, Control of turbulent and magnetohydrodynamic channel flows: boundary stabilization and state estimation, 2008.

M. Verhaegen and P. Dewilde, Subspace model identification Part 1. The output-error state-space model identification class of algorithms, International Journal of Control, vol.56, issue.5, pp.1187-1210, 1992.
DOI : 10.1109/TAC.1974.1100525

M. Verhaegen and V. Verdult, Filtering and system identification: a least squares approach, pp.41-49, 2007.
DOI : 10.1017/CBO9780511618888

A. Visioli, Modified anti-windup scheme for PID controllers, IEE Proceedings- Control Theory and Applications 150.1, pp.49-54, 2003.
DOI : 10.1049/ip-cta:20020769

N. M. Vu and . Trang, Plasma q -profile control in tokamaks using a damping assignment passivity-based approach, Control Engineering Practice, vol.54, issue.17, pp.34-45, 2016.
DOI : 10.1016/j.conengprac.2016.05.003

URL : https://hal.archives-ouvertes.fr/hal-01624840

E. Witrant and S. Brémond, Shape identification for distributed parameter systems and temperature profiles in tokamaks, IEEE Conference on Decision and Control and European Control Conference, pp.2626-2631, 2011.
DOI : 10.1109/CDC.2011.6160523

URL : https://hal.archives-ouvertes.fr/hal-00667227

E. Witrant, A control-oriented model of the current profile in tokamak plasma, Plasma Physics and Controlled Fusion, vol.49, issue.7, pp.1075-1099, 2007.
DOI : 10.1088/0741-3335/49/7/009

URL : https://hal.archives-ouvertes.fr/hal-00387534

J. G. Ziegler, B. Nathaniel, and . Nichols, Optimum Settings for Automatic Controllers, Journal of Dynamic Systems, Measurement, and Control, vol.115, issue.2B, 1942.
DOI : 10.1115/1.2899060