Closed-loop control of an open cavity flow using reduced-order models

Abstract

International audienceThe control of separated fluid flow by reduced-order models is studied using the two-dimensional incompressible flow over an open square cavity at Reynolds numbers where instabilities are present. Actuation and measurement locations are taken on the upstream and downstream edge of the cavity. A bi-orthogonal projection is introduced to arrive at reduced-order models for the compensated problem. Global modes, proper orthogonal decomposition (POD) modes and balanced modes are used as expansion bases for the model reduction. The open-loop behaviour of the full and the reduced systems is analysed by comparing the respective transfer functions. This analysis shows that global modes are inadequate to sufficiently represent the inputoutput behaviour whereas POD and balanced modes are capable of properly approximating the exact transfer function. Balanced modes are far more efficient in this process, but POD modes show superior robustness. The performance of the closed-loop system corroborates this finding: while reduced-order models based on POD are able to render the compensated system stable, balanced modes accomplish the same with far fewer degrees of freedom. © 2009 Cambridge University Press

BiGlobal linear stability analysis on low-Re flow past an airfoil at high angle of attack

“…As a precursor to devising control strategies, we believe it is important to understand and quantify the flow stability characteristics for the low-Re flow past airfoils. Computing the global spectrum of the flow has successfully led to reduced order models (e.g., the work of Berkooz et al 2 and Schmid 3 ) and control techniques (e.g., the work of Barbagallo et al 4 and Hervé et al 5 ). Linear stability analysis of the separated flow past an airfoil has been performed based on experimental (e.g., the work of Boutilier and Yarusevych 6 ) or direct numerical simulation (DNS) (e.g., the work of Jones et al 7 ) result.…”

Reduced order models for closed loop control: comparison between POD, BPOD, and global modes

“…As shown in [8], the stability of the compensated system using an arbitrary reduced model can be investigated replacing the linearized direct numerical simulation (plant) by a reduced model (thereafter called the reduced plant) which 510 FLOW CONTROL Figure 8 Performance of partial-state controller measured as the perturbation kinetic energy vs. time for di¨erent ROMs: (a) the number of included POD modes (in addition to the unstable global modes) is denoted by p; and (b) the number of included BPOD modes (in addition to the unstable global modes) is denoted by p Figure 9 σmax vs. the number of modes modeling the stable subspace for reduced models based on BPOD (1), POD (2), and global (3) modes captures accurately the input output relation. In this case, a reduced model composed of 8 unstable modes and 13 BPOD modes is considered.…”

“…In this case, the highly damped modes become very 508 FLOW CONTROL Figure 7 Comparison of transfer function of the ROM to the exact (1) transfer function for a projection basis of various order, using global modes (a) (2 ¡ p = 132; 3 ¡ 1201; 4 ¡ 3000; and 5 ¡ p = 4000), POD modes (b) (2 ¡ p = 10; 3 ¡ 28; and 4 ¡ p = 50), and BPOD modes (c) (2 ¡ p = 2; 3 ¡ 6; 4 ¡ 8; and 5 ¡ p = 10). Relative H∞-error norm as a function of the number p of included modes, using global (d ), POD (e), and BPOD modes (f ) sensitive to the action of the actuator when the ROM is constructed using a bi-orthogonal projection [8]. To overcome this feature, alternative projection techniques have been considered [9].…”

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

“…The conclusion was that the balanced POD modes produced nearly identical results with the balanced truncation modes, and both methods significantly outperformed the standard POD modes. Another comparison on a problem of designing closed-loop controllers for flow over a cavity was done in [13], where again the balanced POD modes achieved a stable closed-loop controller with fewer ROM degrees-of-freedom. A difficulty related to balanced truncation is that the linearized system must be stable.…”