Mismatched Disturbance Attenuation of a Spatially Developing Free Shear Flow

This paper deals with the closed-loop control of a free shear flow subjected to mismatched disturbances. The goal is to maintain it in a desired state whatever the disturbances. Our approach consists of linearizing the Navier-Stokes equations about the desired state and next to perform a discretization step to obtain a linear state representation. The components of the state vectors are the stream function evaluated at different points of the space assumed to be estimated via image techniques. The proposed control law writes as the sum of two terms. The first one allows to impose a behavior in regulation while the second one describes a behavior in attenuation. We show that this second term writes simply as a feedback gain that has to apply to the perturbation vector. However, since we consider that the disturbance is unknown, we propose a disturbance observer to derive the control law. Next, we prove the input-to-state stability of our control scheme. Simulations results on the DNS (direct numerical simulation) solver Incompact3d validate our approach.