Input–output measures for model reduction and closed-loop control: application to global modes

Abstract

Feedback control applications for flows with a large number of degrees of freedom require the reduction of the full flow model to a system with significantly fewer degrees of freedom. This model-reduction process is accomplished by Galerkin projections using a reduction basis composed of modal structures that ideally preserve the input-output behaviour between actuators and sensors and ultimately result in a stabilized compensated system. In this study, global modes are critically assessed as to their suitability as a reduction basis, and the globally unstable, two-dimensional flow over an open cavity is used as a test case. Four criteria are introduced to select from the global spectrum the modes that are included in the reduction basis. Based on these criteria, four reduced-order models are tested by computing open-loop (transfer function) and closed-loop (stability) characteristics. Even though weak global instabilities can be suppressed, the concept of reduced-order compensators based on global modes does not demonstrate sufficient robustness to be recommended as a suitable choice for model reduction in feedback control applications. The investigation also reveals a compelling link between frequency-restricted input-output measures of open-loop behaviour and closed-loop performance, which suggests the departure from mathematically motivated H ∞ -measures for model reduction toward more physically based norms; a particular frequency-restricted input-output measure is proposed in this study which more accurately predicts the closed-loop behaviour of the reduced-order model and yields a stable compensated system with a markedly reduced number of degrees of freedom.

Stochastic dynamics and model reduction of amplifier flows: the backward facing step flow

“…In the case where the most energetic dynamics are predominantly driven by some known instability at a given frequency, as in the present study, it is possible to build ROMs that focus in this particular frequency range by restricting the approximations of the Gramians to this range. For example, in view of closed-loop control, Barbagallo, Sipp & Schmid (2011) have shown that the ROM has to be accurate only in a small frequency band covering the dominant instability mechanisms to achieve efficient suppression of perturbations. Also, the ROMs designed in the present paper are not based on any specific input or output.…”

Eigenvalue sensitivity, singular values and discrete frequency selection mechanism in noise amplifiers: the case of flow induced by radial wall injection

“…In the case of boundary layers, which are also globally stable flows, the least-damped eigenvalues representing Tollmien-Schlichting waves were not found to be very sensitive (see Ehrenstein & Gallaire 2005;Alizard & Robinet 2007;Akervik et al 2008). In the case of an open cavity flow, Barbagallo, Sipp & Schmid (2009), Barbagallo et al (2011) showed that strongly damped global modes were difficult to compute and dependent on the shift value. This behaviour was well explained by the advection-diffusion model problem introduced by Trefethen & Embree (2005).…”

“…More complex configurations, in which the streamwise direction is solved for (global stability analysis), have also been considered: backward-facing step flow (Blackburn, Barkley & Sherwin 2008;Marquet et al 2008), boundary-layer flows (Alizard & Robinet 2007;Akervik et al 2008;Monokrousos et al 2010;Sipp et al 2010;Sipp & Marquet 2012), homogeneous jet flows (Garnaud et al 2013b), etc. Several studies pointed out the fact that stable eigenvalues are not the good quantities to focus on in the case of strongly convective flows (Sipp et al 2010;Sipp & Marquet 2012;Garnaud et al 2013b), either because they did not describe well the linear dynamics of the flow (Garnaud et al 2013a) or because these modes were extremely sensitive (Barbagallo, Sipp & Schmid 2011).…”

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

“…'Global modes' (e.g. Åkervik et al 2007;Henningson & Åkervik 2008;Barbagallo et al 2011;Ehrenstein, Passaggia & Gallaire 2011) and 'proper orthogonal modes' (e.g. Aubry et al 1988;Podvin & Lumley 1998;Graham, Peraire & Tang 1999;Ravindran 2000aRavindran ,b, 2002Ravindran , 2006Prabhu, Collis & Chang 2001;Ma & Karniadakis 2002;Noack et al 2003;Gloerfelt 2008;Siegel et al 2008;Barbagallo et al 2009Barbagallo et al , 2012Tadmor et al 2010) can yield successful ROMs, but 'balanced modes' have been shown to lead to superior performance in terms of robustness and required model order in a number of studies (e.g.…”