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

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 H ∞ 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.

Localised estimation and control of linear instabilities in two-dimensional wall-bounded shear flows

“…Another approach is the use of system identification methods in which low order models are obtained from a sample of input-output measurements. In particular the eigensystem realisation algorithm (ERA) (Juang & Pappa 1985) was recently used to construct reduced order models for fluid flows (Ma et al 2011;Illingworth et al 2012;Dadfar et al 2013;Semeraro et al 2013;Belson et al 2013;Flinois & Morgans 2016). ERA is based on the impulse response measurements and does not require prior knowledge of the high order system.…”

“…ERA is based on the impulse response measurements and does not require prior knowledge of the high order system. It is shown in Ma et al (2011) that ERA can theoretically obtain the same reduced-order models as BPOD and in Flinois & Morgans (2016) it is shown that ERA can also directly be applied to globally unstable flows.…”

“…The construction of an accurate linear state-space model describing the perturbation dynamics from all inputs to all outputs is the corner stone of linear model based control and is considered as a significant challenge . Limits related to unmodelled dynamics and nonlinearities are commonly assessed from case to case Fabbiane et al 2015) and/or addressed using robust design techniques such as H ∞ loop shaping (Jones et al 2015;Flinois & Morgans 2016). For example, in Jones et al (2015) the effect of nonlinearity is attenuated by a linear feedback controller that employs high loop gain over a selected frequency range.…”

“…Oscillator flows, such as bluff body flows and open cavity flows, are characterized by the presence of global instabilities that oscillate at a particular frequency and are rather insensitive to upstream perturbations. Modelling the external disturbance environment is thus less of an issue for suppressing global instabilities (Samimy et al 2007;Barbagallo et al 2009;Ma et al 2011;, but it raises different issues related to nonlinear saturation of global instabilities (Flinois & Morgans 2016). On the other hand amplifier flows, such as channel flows and boundary layer flows, are characterized by the presence of convective instabilities that amplify downstream (in space) in a broadband frequency spectrum in both space and time.…”

Reducing the pressure drag of a D-shaped bluff body using linear feedback control

“…Pastoor et al [24] used both slope-seeking control and a gain/phase-shift controller targeting synchronisation of the two shear layers, achieving a 15% drag reduction of the same experimental body. Stalnov et al [25,26] experimentally targeted reduced wake unsteadiness due to vortex shedding, using a phase-locked loop and reducing unsteadiness in the wake, while Flinois and Morgans [27] achieved full wake stabilisation at low Reynolds number, using balanced models derived from the unstable impulse response.…”