Robust partial eigenvalue assignment problem for the second-order system

In this paper, we study the partial eigenvalue assignment problem for the second-order system, where only a small part of eigenvalues of the open-loop system is to be reassigned, and the rest are required to remain unchanged. It is desirable that the feedback controller not only assigns specific eigenvalues to the second-order closed-loop system but also that the system is robust, or insensitive to perturbations. We propose a numerical method such that the condition number of the matrix of the eigenvectors of the closed-loop system is minimized. In the method, we only need the knowledge of the eigenvalues to be altered and the corresponding eigenvectors, while we do not need the knowledge of the eigenstructures that are required to remain unchanged and are often unknown. Numerical examples show that the present method often leads to better conditioned closed-loop system.