Using functional gains for effective sensor location in flow control: a reduced-order modelling approach

The problem of active feedback control of fluid flows falls into a class of problems in the area of distributed parameter control. Distributed parameter systems are typically defined by partial differential equations that model the time and spatial evolution of the process. We consider the problem of locating sensors for effective feedback control of a fluid flow problem described by the Navier–Stokes equations. In this setting, the state of the system is the velocity field , and hence all feedback laws are a function of this velocity field or, in most practical settings, a function of sensor outputs. In many designs, the feedback control law can be represented as a linear function of the state defined by an integral operator with a kernel function called the functional gain. In this paper we show that these functional gains can be used to determine effective sensor placement in complex flow control applications. The approach is to choose measurements of the state that would provide good quadrature points for the integral operator. We provide a computational validation of this approach by controlling the vortex shedding in a two-dimensional cylinder flow using a pair of fluid actuators on the cylinder surface. This model is linearized about the mean flow and a feedback control is designed by pole placement. Distributed parameter control theory yields the existence and form of the functional gains which are used to locate sensors. In particular, we use the location of the supports of the functional gains to determine two sets of four sensor locations in the wake. One of these measurement sets coincides with large magnitudes of the gain and the other set coincides with small magnitudes. Numerical experiments with a reduced-order model confirm superior performance of the closed-loop (CL) system using the former sensor set. We also show that choosing sensor locations associated with small magnitudes of the functional gains actually destabilizes the CL system.