H2 optimal actuator and sensor placement in the linearised complex Ginzburg–Landau system

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Abstract
The linearised complex Ginzburg–Landau equation is a model for the evolution of small fluid perturbations, such as in a bluff body wake. By implementing actuators and sensors and designing an H2 optimal controller, we control a supercritical, infinite-domain formulation of this system. We seek the optimal actuator and sensor placement that minimises the H2 norm of the controlled system, from flow disturbances and sensor noise to a cost on the perturbation and input magnitudes. We formulate the gradient of the H2 squared norm with respect to the actuator and sensor placements and iterate towards the optimal placement. When stochastic flow disturbances are present everywhere in the spatial domain, it is optimal to place the actuator just upstream of the origin and the sensor just downstream. With pairs of actuators and sensors, it is optimal to place each actuator slightly upstream of each corresponding sensor, and scatter the pairs throughout the spatial domain. When disturbances are only introduced upstream, the optimal placement shifts upstream as well. Global mode and Gramian analyses fail to predict the optimal placement; they produce H2 norms about five times higher than at the true optimum. The wavemaker region is a better guess for the optimal placement.