Abstract
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.
Highlights
Article structure In this paper, we build on the findings of Ma et al (2011) and Flinois et al (2015) to show that the Eigensystem Realisation Algorithm (ERA) is a practical and computationally cheap approach to obtain low-order models of unstable flows that are useful for feedback control
We show that ERA models can be readily obtained directly from the nonlinear code: if the impulse is sufficiently small, the early response of the system is approximately linear until disturbances grow enough for nonlinear effects to become significant
If projectionfree balanced proper orthogonal decomposition (BPOD) is used to obtain the model as in Flinois et al (2015), the analysis of actuator–sensor placement can be facilitated by extracting relevant information from the balanced modes
Summary
Over the last few decades, the aerospace and automotive industries among others have developed a keen interest in flow control. Article structure In this paper, we build on the findings of Ma et al (2011) and Flinois et al (2015) to show that the ERA is a practical and computationally cheap approach to obtain low-order models of unstable flows that are useful for feedback control We confirm that such models are in practice very similar to the ones obtained with BPOD and show that they can be obtained directly from the nonlinear system dynamics, without the need for a linearised or an adjoint solver. This is a key point, as it shows that it is possible to obtain ERA models for unstable systems with standard (nonlinear) computational fluid dynamics codes and even potentially with experiments
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