Model Predictive Control of Parabolic PDE Systems with Dirichlet Boundary Conditions via Galerkin Model Reduction
In this paper, we propose a framework to solve a closed-loop, optimal tracking control problem for a nonlinear parabolic partial differential equation (PDE) via diffusivity, interior, and boundary actuation. The approach is based on model reduction via proper orthogonal decomposition (POD) and Galerkin projection methods. A conventional integration-by-parts approach during the Galerkin projection fails to effectively incorporate the considered Dirichlet boundary control into the reduced-order model (ROM). To overcome this limitation we use a spatial discretization of the interior product during the Galerkin projection. The obtained low dimensional dynamical model is bilinear as the result of the presence of the diffusivity control term in the nonlinear parabolic PDE system. We propose a closed-loop optimal controller based on a nonlinear model predictive control (MPC) scheme aimed at bating the effect of disturbances with the ultimate goal of tracking a nominal trajectory. A quasi-linear approximation approach is used to solve on-line the quadratic optimal control problem subject to the bilinear reduced-order model associated with the MPC scheme. Based on the convergence properties of the quasi-linear approximation algorithm, the symptotical stability of the closed-loop nonlinear MPC scheme is discussed. Finally, the proposed approach is applied to the current profile control problem in tokamak plasmas and its effectiveness is demonstrated in simulations.
Control of the Current Profile Evolution During the Ramp-Up Phase at DIII-D
Setting up a suitable current profile has been demonstrated to be a key condition for advanced scenarios with improved confinement and possible steady-state operation. Experiments at DIII-D focus on creating the desired q profile during the plasma current ramp-up and early attop phases with the aim of maintaining this profile during the subsequent phases of the discharge. Active feedback control of the q profile evolution at DIII-D has already been demonstrated [1], and an open-loop control scheme has been proposed [2] based on a simplified control-oriented dynamic model [3]. The use of Corsica for both control testing and design is reported, and results of open-loop current profile control experiments are presented.
[1] J.R. Ferron, et al., Nucl. Fusion 46 (2006) L13.
[2] Y. Ou, et al., Proc. American Control Conf., New York (2007).
[3] Y. Ou, et al., Fusion Eng. & Design 82 (2007) 1153.
Model-Based Shape Control Design for the National Spherical Torus Experiment (NSTX)
Plasma shape and position control is a challenging problem due to the difficulties associated with real-time shape identification, plasma parameters measurement, and control method selection. The recent implementation of the real-time equilibrium reconstruction code rtEFIT on NSTX allows plasma shaping by controlling the magnetic flux at the plasma boundary. A non-model-based shape controller that exploits this capability has been recently proposed [1]. We describe current efforts to develop a robust model-based multi-input-multi-output (MIMO) H∞ controller to provide real-time shaping and position control in the presence of disturbances and uncertainties in the plasma parameters. The control design is based on linear plasma response models derived from fundamental physics assumptions. Computer simulation results illustrate the performance of the model-based shape control method.
[1] D.A. Gates, et al., Nucl. Fusion 46 (2006) 17–23.
*Supported by the Pennsylvania Infrastructure Technology Alliance (PITA), the NSF CAREER award program (ECCS-0645086), and US DOE DE-FG03-99ER54522.
Boundary Feedback Control for Mixing Enhancement of 2D Magnetohydrodynamic Channel Flow by Extremum Seeking
The interaction between electrically conducting fluids and magnetic fields in channel flows generates significant magnetohydrodynamics (MHD) effects, which often result in the need of higher pressure gradients to drive the fluid and lower heat transfer rates due to the laminarization of the flow. Active boundary control, either open-loop or closed-loop, can be employed to overcome this limitation. However, open-loop controllers generally have worse performance due to the uncertainties of the system. The extremum seeking scheme is a powerful tool to build feedback controllers based on existing open-loop controllers. In this work we demonstrate that by carefully tuning the extremum seeking the modified open-loop control scheme can be as good as the closed-loop control scheme presented in our earlier publications.