Solution of large optimal control problems bypassing open-loop model reduction
Our lab has developed a number of methods to apply control theory directly to the Navier-Stokes equation, bypassing the oft problematical step of open-loop model reduction, including:
Fourier transforming an entire linear H2/H∞ problem in a parallel (Orr-Sommerfeld/Squire) flow, decoupling a single large control/estimation problem into Nx ⨉ Nz smaller problems that may be solved separately and recombined, leading (when formulated correctly) to resolvable 3D control/estimation convolution kernels that ultimately decay exponentially with distance, and may be truncated and implemented in an overlapping decentralized manner that scales well to large 2D arrays of sensors and actuators,
considering a quasi-steady, parabolic-in-space formulation over a flat surface leveraging a boundary-layer formulation, and developing a corresponding controller which, interestingly, relaxes the constraint of causality, as the system considered is parabolic in space, not time,
determining a simplified formula for the controller which stabilizes a given system in the Minimum-Control-Energy (MCE) limit, which provides a computationally efficient approach for controlling high-dimensional problems with only a few unstable eigenvalues in this limit.
Our most recently method for this type of problem, which we are now further refining, is the
Oppositely-Shifted Subspace Iteration (OSSI) method, which is based on our new subspace iteration method for computing the Schur vectors corresponding, notably, to the m ≪ n central eigenvalues (nearest the imaginary axis) of the Hamiltonian matrix related to the Riccati equation of interest, thereby enabling efficient approximate computation of optimal control and estimation feedback rules more aggressive than those determined in the MCE limit.