An Optimization Approach for Minimum Norm and Robust Partial Quadratic Eigenvalue Assignment Problems for Vibrating Structures

The Partial Quadratic Eigenvalue Assignment Problem (PQEVAP) concerns the reassignment of a small number of undesirable eigenvalues of a quadratic matrix pencil, while leaving the remaining large number of eigenvalues and the corresponding eigenvectors unchanged. The problem arises in controlling undesirable resonance in vibrating structures and in stabilizing control systems. The solution of this problem requires computations of a pair of feedback matrices. For practical effectiveness, these feedback matrices must be computed in such a way that their norms and the Preprint submitted to Elsevier 23 October 2008 condition number of the closed-loop eigenvector matrix are as small as possible. These considerations give rise to Minimum Norm Partial Quadratic Eigenvalue Assignment Problem (MNPQEVAP) and Robust Partial Quadratic Eigenvalue Assignment Problem (RPQEVAP), respectively. In this paper we propose new optimization based algorithms for solving these problems. The problems are solved directly in a second-order setting without resorting to a standard first-order formulation so as to avoid the inversion of a possibly ill-conditioned matrix and the loss of exploitable structures of the original model. The invariance of the large number of eigenvalues and eigenvectors by application of feedback is guaranteed by a proven mathematical result. Furthermore, the gradient formulas needed to solve the problems by existing optimization techniques are computed using only the few eigenvalues and eigenvectors of the associated quadratic pencil that can be computed using the state-of-the-art computational techniques. Above all, the proposed methods do not require the reduction of the model order or the order of the controller, even when the underlying finite element model has a very large degree of freedom. These attractive features, coupled with minimal computational requirements, such as solutions of small diagonal Sylvester equations and small algebraic linear systems, make the proposed algorithms ideally suited for application to large real-life structures. Numerical results show significant improvement in feedback norms and in the condition number of the closed-loop system. Also, the closed-loop eigenvalues have acceptable accuracy. ∗ Corresponding author. Email addresses: brahma [email protected] (Sanjoy Brahma ),