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Input-output measures for model reduction and closed-loop control: application to global modes. (English) Zbl 1241.76163
Summary: 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 $${\mathcal{H}}_{\infty }$$-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.

##### MSC:
 76D55 Flow control and optimization for incompressible viscous fluids 76E09 Stability and instability of nonparallel flows in hydrodynamic stability 93B52 Feedback control
##### Keywords:
control theory; instability control
FreeFem++
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##### References:
 [1] DOI: 10.1146/annurev.fluid.39.050905.110153 · doi:10.1146/annurev.fluid.39.050905.110153 [2] DOI: 10.1137/S0895479899358595 · Zbl 1004.65044 · doi:10.1137/S0895479899358595 [3] DOI: 10.1063/1.2832773 · Zbl 1182.76313 · doi:10.1063/1.2832773 [4] DOI: 10.1146/annurev.fl.25.010193.002543 · doi:10.1146/annurev.fl.25.010193.002543 [5] Hecht, Freefem ++, The Book (2005) [6] DOI: 10.1017/S0022112009991418 · Zbl 1183.76701 · doi:10.1017/S0022112009991418 [7] Giannetti, J. Fluid Mech. 581 pp 167– (2007) · Zbl 1115.76028 · doi:10.1017/S0022112007005654 [8] DOI: 10.1115/1.3077635 · doi:10.1115/1.3077635 [9] DOI: 10.1007/s00162-010-0195-5 · Zbl 1272.76099 · doi:10.1007/s00162-010-0195-5 [10] DOI: 10.1063/1.3590732 · Zbl 06422410 · doi:10.1063/1.3590732 [11] DOI: 10.1017/S0022112008004394 · Zbl 1156.76374 · doi:10.1017/S0022112008004394 [12] Datta, Numerical Methods for Linear Control Systems (2003) [13] Antoulas, Contemp. Maths 280 pp 193– (2001) · doi:10.1090/conm/280/04630 [14] Burl, Linear Optimal Control. H (1999) [15] DOI: 10.1137/1.9780898718713 · doi:10.1137/1.9780898718713 [16] DOI: 10.1017/S0022112007005496 · Zbl 1175.76049 · doi:10.1017/S0022112007005496 [17] DOI: 10.1017/S0022112009992655 · Zbl 1189.76163 · doi:10.1017/S0022112009992655 [18] Zhou, Robust and Optimal Control (2002) [19] DOI: 10.2514/2.1570 · doi:10.2514/2.1570 [20] DOI: 10.1017/S0022112090000933 · doi:10.1017/S0022112090000933 [21] Sirovich, Q. Appl. Maths 45 pp 561– (1987) · Zbl 0676.76047 · doi:10.1090/qam/910462 [22] DOI: 10.1115/1.4001478 · doi:10.1115/1.4001478 [23] DOI: 10.1017/S0022112007008907 · Zbl 1172.76318 · doi:10.1017/S0022112007008907 [24] Schmid, Stability and Transition in Shear Flows (2000) [25] DOI: 10.1142/S0218127405012429 · Zbl 1140.76443 · doi:10.1142/S0218127405012429 [26] DOI: 10.1109/TAC.1981.1102568 · Zbl 0464.93022 · doi:10.1109/TAC.1981.1102568 [27] DOI: 10.1017/S0022112008003662 · Zbl 1165.76012 · doi:10.1017/S0022112008003662 [28] Lumley, Stochastic Tools in Turbulence (1970) · Zbl 0273.76035 [29] DOI: 10.1017/S0022112098001281 · Zbl 0924.76028 · doi:10.1017/S0022112098001281
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