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

The control of Tollmien-Schlichting waves in a two-dimensional boundary layer is analysed using numerical simulations. Full-dimensional optimal controllers are used in combination with a setup of spatially localized inputs (actuator and disturbance) and outputs (sensors). The adjoint of the direct-adjoint (ADA) algorithm, recently proposed by Pralits & Luchini (In Seventh IUTAM Symposium on Laminar-Turbulent Transition (ed. P. Schlatter & D. S. Henningson), vol. 18, 2010, Springer), is used to efficiently compute an optimal controller known as a linear quadratic regulator; the method is iterative and allows one to bypass the solution of the corresponding Riccati equation, which is infeasible for high-dimensional systems. We show that an analogous iteration can be made 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, full-dimensional linear quadratic Gaussian controllers are obtained and used for the attenuation of the disturbances arising in the boundary layer flow. The full-dimensional controllers turn out to be an excellent benchmark for evaluating the performance of the optimal control/estimation design based on reduced-order models. We show under which conditions the two strategies are in perfect agreement by focusing on the issues arising when feedback configurations are considered. An analysis of the finite-amplitude disturbances is also carried out by 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.