Reduced order models for closed-loop control of time-dependent flows

Controller development in the relatively young field of closed-loop (or feed back) flow control is constrained by the lack of proper models for the description of the dynamics and response of flows to a control input (actuation). If a set of ordinary differential equations could be derived that describes the unsteady flow with sufficient accuracy and that also models the effect of the actuation, control theory tools could be employed for controller design. In this paper reduced order models (ROMs) based on a Galerkin projection of the incompressible Navier-Stokes equations onto a proper orthogonal decomposition (POD) modal basis are described. The model coefficients can be modified or calibrated to make the model more accurate. An error-minimization technique is employed to obtain the coefficients that describe how the control enters the model equations. These models work well in the vicinity of the design operating point. Composite models are constructed by combining POD modes from different operating points. This approach results in more versatile ROMs that are valid for a larger range of operating conditions.