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

Abstract

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

Feedback control of vortex shedding using a full-order optimal compensator

“…In fact when a full-order approach is adopted, no approximations are introduced on the open-loop linearised dynamics, thus avoiding the so-called spill-over effects due to the undesired excitation of those stable dynamics which have not been retained in the open-loop ROM and which can affect the control action. With the same purpose, a more general and computationally demanding full-dimensional approach to the LQG design has been recently investigated by Semeraro et al (2013) in the control of a two-dimensional boundary layer. In particular, the authors have shown that for such a highly convective system, when sensors are located downstream of the actuators, only a full-order compensator can always guarantees the stability of the closed-loop plant while varying the design parameters.…”

On the wave-cancelling nature of boundary layer flow control

“…This is referred to in the literature as "reduce-then-design" process [44], and has seen several applications, both in numerical and experimental investigations [2,8,42,43]. An interesting reduced-order model for flow control is the Eigensystem realization algorithm (ERA), a technique proposed by Juang and Papa [24], that reproduces the input-output behaviour observed in a simulation or experiment without the need to solve adjoint equations.…”

Global stability and control of the confined turbulent flow past a thick flat plate

“…[1][2][3] Numerical modelling of fluid flows and mathematically rigorous theories for their control 4,5 have been usually first tested and verified in simplified conditions, at low or moderate Reynolds numbers. [6][7][8][9] Within this framework, the concept of structural sensitivity has gained interest with applications to a large variety of globally unstable flows. [10][11][12][13][14][15] Indeed this linearised approach allows one to predict, beforehand, the effective positioning of a flow disturbance, i.e., a passive device, able to shift the vortex-shedding frequency or even to completely suppress the global instability of the flow.…”