Two considerations for the design of a robust optimal smart structure where control energy is expensive.

Large vibrations in structures are normally unwanted due to corresponding large displacements and accelerations. Other deleterious effects may be caused by the large stresses which are coexistent with these large vibrations; resulting in ultimate failure or at the very least a dramatic reduction in the fatigue life of the structure. A possible solution to this problem is to adopt a "smart structures" approach by using piezoceramic actuators in a feedback control loop to reduce these unwanted vibrations. Two distinct problems will be considered here (although distinct they are of equal importance to the whole control system). The first problem is the optimal placement of piezoceramics for a given amount of material so as to obtain "maximum modal controllability". A cantilevered plate is examined as an example. The second problem considered is the design of robust optimal controllers which maximize the damping of the lightest damped mode in the closed loop system. Restricting the position of the poles of the controller to lie within a certain domain effectively limits the bandwidth of the controller and ensures robustness. The non-linear problem of maximizing the minimum damping using only a certain amount of control energy whilst the controller is restricted in the aforementioned way is solved. In the case of expensive control energy which motivates our study it is seen that the results agree with our intuition via some simple examples.