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

K. K. Chen and C. W. Rowley

Journal of Fluid Mechanics 681 : 241260, June 2011.


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 to a cost on the perturbation and input magnitudes. We formulate the gradient of the H2 squared norm with respect to actuator and sensor positions, and iterate toward the optimal position. With a single actuator and sensor, 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 a corresponding sensor, and scatter the pairs throughout the spatial domain. Global mode and Gramian analyses fail to predict the optimal placement; they produce H2 norms about five times higher than at the true optimum. A wave maker formulation is better able to guess an initial condition for the iterator.

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