Feedback control of unstable flows: a direct modelling approach using the Eigensystem Realisation Algorithm

Obtaining low-order models for unstable flows in a systematic and computationally tractable manner has been a long-standing challenge. In this study, we show that the Eigensystem Realisation Algorithm (ERA) can be applied directly to unstable flows, and that the resulting models can be used to design robust stabilising feedback controllers. We consider the unstable flow around a D-shaped body, equipped with body-mounted actuators, and sensors located either in the wake or on the base of the body. A linear model is first obtained using approximate balanced truncation. It is then shown that it is straightforward and justified to obtain models for unstable flows by directly applying the ERA to the open-loop impulse response. We show that such models can also be obtained from the response of the nonlinear flow to a small impulse. Using robust control tools, the models are used to design and implement both proportional and $\mathscr{H}_{\infty }$ loop-shaping controllers. The designed controllers were found to be robust enough to stabilise the wake, even from the nonlinear vortex shedding state and in some cases at off-design Reynolds numbers.