Real-time control of a magnetohydrodynamic flow

Abstract: We demonstrate the feedback control of a weakly conducting magnetohydrodynamic (MHD) flow via Lorentz forces generated by externally applied electric and magnetic fields. Specifically, we steer the flow of an electrolyte toward prescribed velocity or vorticity patterns using arrays of electrodes and electromagnets positioned around and beneath a fluid reservoir, with feedback provided by planar particle image velocimetry (PIV). Control is implemented using a model predictive control (MPC) framework, in which control signals are computed by minimizing a cost function over the predicted evolution of the flow. The predictor is constructed entirely from data using Koopman operator theory, which enables a linear representation of the underlying nonlinear fluid dynamics. This linearity allows the MPC problem to be solved by alternating between two small and efficiently solvable convex quadratic programs (QPs): one for the electrodes and one for the electromagnets. The resulting controller runs in a closed loop on a standard laptop, enabling real-time control of the flow. We demonstrate the functionality of the approach through experiments in which the flow is shaped to match a range of reference velocity fields and a time-varying vorticity field.