A sylvester-equation based parametric approach for minimum norm and robust Partial Quadratic Eigenvalue Assignment Problems

The Partial Quadratic Eigenvalue Assignment Problem is the problem of reassigning a small number of undesirable eigenvalues of a quadratic matrix pencil using feedback. The problem arises in controlling resonance in vibrating structures and also in stabilizing control systems. The solution involves the computation of a pair of feedback matrices. For practical effectiveness, the magnitudes of the feedback norms need to be reduced and the conditioning of the closed-loop eigenvalues needs to be improved. In this paper, we propose new optimization methods for solving these problems. An important practical aspect of these methods is that the gradient formulas needed to solve the underlying unconstrained optimization problems are computed using only a small number of eigenvalues and eigenvectors of the quadratic pencil, which are all that can be computed using the state-of-the-art computational techniques.