Actuator topology design using the controllability Gramian

This work develops a methodology for the optimal design of actuators for the vibration control of flexible structures. The objective is the maximization of a measure of the controllability Gramian. The test case is the embedding of piezoelectric inserts in elastic structures for vibration control in modal space. A topology optimization was formulated to distribute two material phases in the domain: a passive linear elastic material and an active linear piezoelectric material, with a volume constraint in the latter. The objective function is the trace of the controllability Gramian of a LQR control system. Analytical sensitivities for the finite element model are derived for the objective function and constraints. Results are shown for two dimensional vibration control of a short beam with varying number of electrodes (control inputs) and vibration modes.