Output Feedback Control for Transient Energy Growth Reduction and Transition Suppression in Shear Flows

Abstract

Transient energy growth~(TEG) of flow perturbations is an important mechanism for the laminar-to-turbulent transition that can be mitigated with feedback control. The suppression of TEG is formed as a common control objective for feedback flow control aimed at delaying transition to turbulence. In this context, a prevailing control approach is observer-based feedback, in which a full-state feedback controller is applied to state estimates from an observer. The present study identifies a fundamental performance limitation of observer-based feedback control designed using the separation principle: whenever the uncontrolled system exhibits TEG in response to optimal disturbances, control by observer-based feedback will necessarily lead to TEG in response to optimal disturbances for the closed-loop system as well. Alternative forms of output feedback could avoid such performance limitations. The dynamical compensator is then designed aiming at minimizing TEG. Control-oriented reduced-order models are presented to facilitate the computations required for such controller synthesis. A static output feedback~(SOF) gain is used as a prior-designed controller for the dynamical compensator to simplify the design approach. SOF TEG performance by consequence stands out as a worthy study. Thus, a static output feedback linear quadratic regulator~(SOF-LQR) is designed to reduce the worst-case TEG of flow perturbations. Optimal SOF gains are computed using a modified Anderson-Moore algorithm and accelerated by Armijo-type adaptations. To enhance performance, we propose two methods for sensor selection that enable sensor-based output feedback controllers to recover full-information control performance: one based on a sparse controller synthesis approach and one based on a balanced truncation procedure for model reduction. Results indicate that sensor configurations identified by both approaches allow sensor-based SOF- LQR controllers to recover full-information LQR control performance, both in reducing TEG and suppressing transition. The sensor selection methods and the resulting controllers exhibit robustness to Reynolds number variations. Non-linear direct numerical simulations also indicate that the designed SOF-LQR increases the transition thresholds with respect to the uncontrolled flows.