An adaptive compressed-sensing equation-free approach for closed-loop nonlinear control

We present a method which seeks to combine the efficiency of an optimal control command with the robustness of a closed-loop controller. While the performance of optimal control is excellent, it cannot account for perturbations to the system under control which make the resulting performance to drop. On the other hand, a closed-loop control makes use of the actual state of the system and is thus robust, at the price of limited performance. We here present a control method which is both robust (closed-loop) and achieves a near-optimal performance. The controller relies on an approximation of the optimal control command in the state space where the system lies upon reduction. The command can then be updated when a new observation of the system becomes available, hence achieving a closed-loop control. The approximation of the command in the state space is a computationally costly step but is achieved off-line and only once. To minimize its cost while achieving a good accuracy in the approximation, an adaptive multi-wavelets approach, combined with compressed-sensing acceleration exploiting compressibility, is here proposed. The control algorithm is demonstrated on the control of a cylinder wake in a 2-D flow.