A norm-minimizing parametric algorithm for quadratic partial eigenvalue assignment via Sylvester equation

Abstract

In this paper, we propose a Sylvester equation based parametric algorithm for generating a family of feedback matrices to solve the quadratic partial eigenvalue problem (QPEVAP) arising in controlling dangerous vibrations, such as resonance, in structures like bridges, highways, air and spacecrafts. It is then shown how the parametric matrix can be optimally chosen using a numerical optimization technique so that feedback matrices have minimum norms. Such minimum-norm feedback gains naturally lead to smaller control signals and are useful in reducing noises. The distinguished features of the algorithm making it applicable to even very large practical structures are: (i) the algorithm works directly in quadratic setting without any need to transformation to a standard state-space form, which might require ill-conditioned matrix inversion and destroys the exploitable structural properties, (ii) no model reduction is needed, no matter how large the problem is, (iii) knowledge of only a small amount of eigenvalues and eigenvectors of the associated quadratic eigenvalue problem that can be computed using the state-of-the-art matrix computational techniques, is sufficient for its implementation, and (iv) the algorithm is capable of exploiting the structural properties of the system, such as the sparsity, bandedness, symmetry and positive definiteness, etc., in a computational setting.