Optimal Disturbance Suppression with Piezoelectric Patch Actuators - A Lagrangian Approach

The Lagrangian Approach is proposed for the modelling and analysis of piezoelectric structures. Considering the electric voltage as the input to the system, one remains with the open problem to determine the optimal spatial distribution of the piezoelectric actuator. It is well known that the problem can be solved exactly for certain loads acting on the structure. This exact compensation requires a spatial distribution of the piezoelectric material, which is practically unattainable. A more practical approach is to approximate the ideal distribution with a finite number of actuators such that the applied voltages become the new design parameters. From a mathematical point of view, there exists a natural product between the stress and the strain tensor and in the linear scenario, one derives from this product an inner product on a certain Hilbert space. Based on this inner product, on can approximate the desired distribution by the given actuator configuration such that the error is minimized in the induced norm.