Control Input Separation by Actuation Mode Expansion for Flow Control Problems

Abstract

A control input separation method is proposed for reduced-order modelling in boundary control problems. The dynamics of flow systems are typically described by partial differential equations where the input affects the system through boundary conditions. From a control design perspective it is most desirable and natural to employ finite-dimensional representations in which the input enters the dynamics directly. The method proposed here to resolve the input from the boundary conditions is based on obtaining a proper orthogonal decomposition of the unforced flow of the system, and then augmenting this decomposition by optimally computed actuation modes, built using snapshots of the actuated flow. A reduced-order Galerkin model is then derived for this expansion, in which the input appears as an explicit term in the system dynamics. The model reduces exactly to the original baseline case under zero input conditions. The proposed method is then compared to an existing input separation technique, namely the sub-domain separation method. A boundary control example regarding the 2D incompressible Navier-Stokes equation is considered to illustrate the proposed method, where a controller is designed to achieve tracking of a desired 2D spatial profile for the flow velocity.