Riccati-less approach for optimal control and estimation: an application to two-dimensional boundary layers

The control of Tollmien-Schlichting (TS) in a 2D boundary layer is analysed by using numerical simulation. Full-dimensional optimal controllers are used in combination with a set-up of spatially localised inputs (actuators and disturbance) and outputs (sensors). The Adjoint of the Direct-Adjoint (ADA) algorithm, recently proposed by Pralits & Luchini (2010), is used to efficiently compute the Linear Quadratic Regulator (LQR) controller; the method is iterative and allows to by-pass the solution of the corresponding Riccati equation, unfeasible for high-dimensional systems. We show that an analogous iteration can be cast for the estimation problem; the dual algorithm is referred to as Adjoint of the Adjoint-Direct (AAD). By combining the solutions of the estimation and control problem, a full dimensional, model-free, Linear Gaussian Quadratic (LQG) controllers are obtained and used for the attenuation of the disturbances arising in the boundary layer flow. Model-free, full dimensional controllers turn out to be an excellent benchmark for evaluating the performance of the optimal control/estimation design based on open-loop model reduction. We show the conditions under which the two strategies are in perfect agreement by focusing on the issues arising when feedback configurations are considered. An analysis of the finite amplitude cases is also carried out addressing the limitations of the optimal controllers, the role of the estimation and the robustness to the nonlinearities arising in the flow of the control design.