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We consider closed-loop control of a two-dimensional supersonic boundarylayer at M = 4.5 that aims at reducing the linear growth of second Mack modeinstabilities. These instabilities are first characterized with local spatialand global resolvent analyses, which allow to refine the control strategy andto select appropriate actuators and sensors. After linear input-output reducedorder models have been identified, multi-criteria structured mixed$H_{2}/H_{\infty}$ synthesis allows to fix beforehand the controller structureand to minimize appropriate norms of various transfer functions: the $H_{2}$norm to guarantee performance (reduction of perturbation amplification innominal condition) and the $H_{\infty}$ norm to maintain performance robustness(with respect to sensor noise) and stability robustness (with respect touncertain free-stream velocity/density variations). Both feedforward andfeedback setups, i.e. with estimation sensor placed respectivelyupstream/downstream of the actuator, allow to maintain the local perturbationenergy below a given threshold over a significant distance downstream of theactuator, even in the case of noisy estimation sensors or free-stream densityvariations. However, the feedforward setup becomes completely ineffective whenconvective time-delays are altered by free-stream velocity variations of $\pm5\%$, which highlights the strong relevance of the feedback setup forperformance robustness in convectively unstable flows.

Reactive control of 2 nd Mack mode in a supersonic boundary layer with freestream velocity/density variations