Feedback control of transitional flows: A framework for controller verification using quadratic constraints

The dynamics of incompressible fluid flows are governed by a non-normal linear dynamical system in feedback with a static energy-conserving nonlinearity. These dynamics can be altered using feedback control but verifying performance of a given control law can be challenging. The conventional approach is to perform a campaign of high-fidelity direct numerical simulations to assess performance over a wide range of parameters and disturbance scenarios. In this paper, we propose an alternative simulation-free approach for controller verification. The incompressible Navier-Stokes equations are modeled as a linear system in feedback with a static and quadratic nonlinearity. The energy conserving property of this nonlinearity can be expressed as a set of quadratic constraints on the system, which allows us to perform a nonlinear stability analysis of the fluid dynamics with minimal complexity. In addition, the Reynolds number variations only influence the linear dynamics in the Navier-Stokes equations. Therefore, the fluid flow can be modeled as a parameter-varying linear system (with Reynolds number as the parameter) in feedback with a quadratic nonlinearity. The quadratic constraint framework is used to determine the range of Reynolds numbers over which a given flow will be stable, without resorting to numerical simulations. We demonstrate the framework on a reduced-order model of plane Couette flow. We show that our proposed method allows us to determine the critical Reynolds number, largest initial disturbance, and a range of parameter variations over which a given controller will stabilize the nonlinear dynamics.