LAUSR

Abstract Eigenvalues and eigenvectors play a fundamental role in determining the dynamic behavior of a vibrating system. Thus, an important goal in vibration control is to modify the eigenstructure to match the design specifications. Feedback control is a popular approach to this problem, which is however not always satisfactory. In fact, while system controllability suffices to assure that any requirement on the eigenvalues can be met by the controlled system, desired closed-loop eigenvectors cannot be attained in general, due to inherent limitations of active control. To overcome such a problem, this paper proposes a hybrid approach in which the mechanical system and the controller are concurrently designed to improve the attainability of the desired eigenstructure. Indeed, the suitable modification of the system inertial and elastic parameters modifies the set of eigenvectors that can be achieved through active control. In this work is demonstrated that such an objective can be effectively accomplished by minimizing the rank of a certain matrix which depends on the features of the original system and on the desired eigenpairs. Two algorithms for rank minimization are described and adjusted for the problem under consideration. The method is validated with three examples. In the first one, eigenstructure assignment of a 5 degrees of freedom system that previously appeared in the literature is performed, demonstrating the effectiveness of the proposed approach with respect to the state of the art. In the second example the same 5 degrees of freedom system is considered by also including damping, to evaluate the method capability to deal with damped systems. The third one is a 30 degrees of freedom system that enables the comparison of the two algorithms for different choices of the actuation.