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Feedback control synthesis for nonlinear, parameter-dependent fluid flowcontrol problems is considered. The optimal feedback law requires the solutionof the Hamilton-Jacobi-Bellman (HJB) PDE suffering the curse of dimensionality.This is mitigated by Model Order Reduction (MOR) techniques, where the systemis projected onto a lower-dimensional subspace, over which the feedbacksynthesis becomes feasible. However, existing MOR methods assume at least onerelaxation of generality, that is, the system should be linear, or stable, ordeterministic.  We propose a MOR method called Statistical POD (SPOD), which is inspired bythe Proper Orthogonal Decomposition (POD), but extends to more general systems.Random samples of the original dynamical system are drawn, treating time andinitial condition as random variables similarly to possible parameters in themodel, and employing a stabilizing closed-loop control. The reduced subspace ischosen to minimize the empirical risk, which is shown to estimate the expectedrisk of the MOR solution with respect to the distribution of all possibleoutcomes of the controlled system. This reduced model is then used to compute asurrogate of the feedback control function in the Tensor Train (TT) format thatis computationally fast to evaluate online. Using unstable Burgers' andNavier-Stokes equations, it is shown that the SPOD control is more accuratethan Linear Quadratic Regulator or optimal control derived from a model reducedonto the standard POD basis, and faster than the direct optimal control of theoriginal system.

Statistical Proper Orthogonal Decomposition for model reduction in feedback control