Robust Partial Pole Assignment for Vibrating Systems With Aerodynamic Effects

This note proposes a novel algorithm for robust partial eigenvalue assignment (RPEVA) problem for a cubic matrix pencil arising from modeling of vibrating systems with aerodynamic effects. The RPEVA problem for a cubic pencil is the one of choosing suitable feedback matrices to reassign a few (say k<3n) unwanted eigenvalues while leaving the remaining large number (3n-k) of them unchanged, in such a way that the the eigenvalues of the closed-loop matrix are as insensitive as possible to small perturbation of the data. The latter amounts to minimizing the condition number of the closed-loop eigenvector matrix. The problem is solved directly in the cubic matrix polynomial setting without making any transformation to a standard first-order state-space system. This allows us to take advantage of the exploitable structures such as the sparsity, definiteness, bandness, etc., very often offered by large practical problems. The major computational requirements are: i) solution of a small Sylvester equation, ii) QR factorizations, and iii) solution of a standard least squares problem. The least-squares problem result from matrix rank-two update techniques used in the algorithm for reassigning complex eigenvalues. The practical effectiveness of the method is demonstrated by implementational results on simulated data provided by the Boeing company