A Noncausal Framework for Model-based Feedback Control of Spatially-developing Perturbations in Boundary Layers (Invited)

Abstract

We present a noncausal framework for model-based feedback stabilization of a large class of spatiallydeveloping boundary-layer flo w systems. The systems considered are (approximately) parabolic in the spatial coordinate x. This facilitates the application of a range of established feedback control theories which are based on the solution of differential Riccati equations which march over a finite horizon in x (rather than marching in t, as customary). However, unlike systems which are parabolic in time, there is no causality constraint for the feedback control of systems which are parabolic in space; that is, downstream information may be used to update the controls upstream. Thus, a particular actuator may be used to neutralize the effects of a disturbance which actually enters the system downstream of the actuator location. A numerically-tractable feedback control strategy is formulated which takes advantage of this special capability of feedback control rules in the spatially-parabolic setting in order to minimize a globally-defined cost function in an effort to maintain laminar boundary-layer flo w. We compute the state-feedback control gains at several spanwise wavenumbers β. We then inverse transform the result to obtain spatial convolution kernels for determining the control feedback. The effectiveness of the controls computed using these feedback kernels, which are well resolved on the computational grid and spatially localized in the spanwise direction, is tested using direct numerical simulation of the boundary-layer flo w system. A significant damping of the flo w perturbation is observed, which is of the same order as the damping that arises when applying significantly more expensive iterative adjoint-based control optimization schemes.