Improving active eigenvector assignment through passive modifications

Specifications on the dynamic behavior of feedback-controlled vibrating systems are often expressed in terms of its eigenstructure, i.e. eigenvalues and eigenvectors. The notion of controllability establishes the possibility to assign eigenvalues through state feedback, but it is not adequate to assure the assignment of arbitrary eigenvectors. Indeed, assignable eigenvectors are just those belonging to the allowable vector subspace, which depends on the physical properties of the vibrating system (mass, damping and stiffness matrices) and of the actuators. To overcome this limitation, this paper proposes a hybrid approach that exploits passive modification of the system physical parameters to modify the allowable subspace in such a way that it spans (or closely approximates) the desired eigenvectors. Then, once that the system modifications have been computed, standard techniques for control synthesis can be employed to compute the gains assigning the desired poles and the eigenvectors. The modification of the allowable subspace is cast in this work as a rank minimization problem, which can be efficiently tackled through semi-definite programming. The proposed method is numerically validated on a lumped parameter system, by proving that the assignment of eigenvectors by hybrid control is significantly enhanced compared with sole active control.