Efficient Shape Control of Distributed Reflectors with Discrete Piezoelectric Actuators

In the current article, a novel approach to the shape control problem with a limited number of discrete actuators is presented. In optimizing the actuator locations, first the reference input was expanded with a representative set of eigenfunctions. From the expansion coefficients, the dominant modes in the reference input were identified. Next the distributed parameter system was rewritten in modal coordinates and the irrelevant modes were truncated. Finally, the actuator locations were selected such that the singular values of the controllability matrix of the truncated system were maximized. The steady state actuation forces were optimized such that a cost function based on the absolute value of the steady state error is minimized. To improve the transient response, a hybrid of dynamic modal control algorithm and the optimization method is developed. Four discrete linear actuators for the one-dimensionally curved reflector and eight actuators for the two-dimensionally curved reflector are utilized. The structures are controlled and assessments for experimental implementation are made.