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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
Software:
FreeFem++
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