Feedback control of vortex shedding using data-driven modelling
This paper details the data-driven modelling and feedback control of vortex shedding past a circular cylinder at a Reynolds number of Re = 1000. We study the effect of varying the order of the reduced model for control design purposes and demonstrate that higher orders can lead to lower suppression of vortex shedding. We use the Bode integral theorem and a frequency-domain interpretation to show that this drop in performance is, in part, due to the classical ``waterbed effect'', which increases sensitivity in frequency bands of unmodelled dynamics. Training data from 2D unsteady simulation is used to obtain linear reduced-order state-space models of the system via dynamic mode decomposition with control. Using only lift measurement, we show that at least a 4th-order model is required for an LQG controller to suppress vortex shedding, with the best performance achieved with as few as 9 modes, whilst higher-order (>14) controllers show a significant decrease in performance. We study the influence of external disturbances, noise rejection, and parameter uncertainty on controller performance. A 28.6 dB reduction in lift coefficient variance is achieved, resulting in a 26% reduction in drag. We further show that, for control design purposes with practical actuation bandwidth, the closed-loop control delivers a significant 13.7% drag reduction within 3D DDES, despite having been trained with 2D URANS and therefore argue that 2D URANS simulation is sufficient for reduced-order model generation and control design.