Partial Eigenvalue Assignment Problem for the Quadratic Matrix Pencil

This thesis is devoted to the study of comparisons of two types of methods for solving Quadratic Partial Eigenvalue Assignment Problem (QPEVAP). Numerical experiments performed by us confirm that the direct second-order methods are superior to the first-order methods, both in eigenvalue assignment and norm reduction of the feedback matrices. First, the definition of a Partial Eigenvalue Assignment Problem for the Quadratic Matrix Pencil is derived. Second, modeling of vibrations in an electric pole of a mass-spring system is described. Two standard approaches to compute eigenvalues and eigenvectors of a Quadratic Matrix Pencil are defined, one with finding the relation between standard eigenvalue problems and quadratic eigenvalue problems and the other with finding the relation between generalized and quadratic eigenvalue problems. Third, the existence and uniqueness results for both the problems, the matrix second order case and for the partial eigenvalue assignment problem for the matrix pencil, are defined. Orthogonality relations between the eigenvectors of the linear and quadratic matrix pencil are defined. The eigenvalue assignment problem for the quadratic matrix pencil via the first-order reformulation is solved. Last, the solutions are proposed for the partial eigenvalue assignment problems for the quadratic pencil where only the partial knowledge of eigenvalues and eigenvectors are required. The latter makes the approach completely viable for the practical applications because the real-life problems are capable of computing only a small part of the spectra of the associated quadratic pencil. Some real-life numerical examples are given.