Robust shape control of beams with load uncertainties by optimally placed piezo actuators

The shape of a laminated beam is controlled by an optimally placed piezo actuator so as to minimize its maximum deflection. The locations and magnitudes of the external loads are not known a priori and belong to a specified load uncertainty domain. The optimal actuator location is obtained for any load combinations and locations which are determined so as to produce the maximum deflection corresponding to the worst case of loading. In this sense, loading uncertainties lead to an anti-optimization problem which is coupled to the optimization problem via the design parameter and loading. The resulting coupled problems are solved simultaneously to compute the optimal actuator location and the least favourable loading condition. Multiple load cases are also considered. Numerical results are given to assess the effect of load uncertainty and actuator length on the actuator location and the design efficiency which is defined with respect to the corresponding uncontrolled beam.