Using Functional Gains for Optimal Sensor Placement in Fluid-Structure Interaction

Functional gains are integral kernels of the standard feedback operator and are useful in control of partial differential equations (PDEs). These functional gains provide physical insight into how the control mechanism is operating. In some cases, these functional gains can provide information about the optimal placement of actuators and sensors. The study is motivated by fluid flow control and focuses on the computation of these functions. However, for practical purposes, one must be able to compute these functions for a wide variety of PDEs. For higher dimensional systems, computing these gains is at least as challenging as the original simulation problem. To reduce the complexity of the governing equations, reduced-order models are often developed by reducing the PDEs to ordinary-differential equations (ODEs). In this study, we use proper orthogonal decomposition (POD)-Galerkin based approach and develop a reduced-order model of a bluff body wake. We solve the incompressible Navier-Stokes equations, simulate the flow past a circular cylinder, and record the snapshots of the flow field. We compute the POD eigenfunctions and project the Navier-Stokes equations onto these few of these eigenfunctions to develop a reduced-order model. Later, we modify the model by introducing a control function simulating suction actuation on the cylinder surface. We linearize the model about the mean flow and apply feedback control to suppress vortex shedding. We then compute the functional gains for the applied control. We identify these gains at various stations in the wake region and suggest optimum locations for the sensors.