A structural property of subspaces of assignable eigenvectors and generalized eigenvectors of closed-loop system

In this paper, we investigate the structural property of subspaces which consist of assignable eigenvectors and generalized eigenvectors of prescribed closed-loop system, which is rather a classical topic. It is shown that among the subspaces relative to different eigenvalues to be assigned there is a close relationship which only depends on and can be described by the structure of controllable subspace of system (A, B). This structural property determines the prescribed Jordan form of (A+BK) and the choices of assignable eigenvectors and generalized eigenvectors. As an example of applications of the result the paper also proved that for any assignable prescribed Jordan form with multiple eigenvalues almost for any choice of parameter vectors a state feedback K can be got by the method of parametric solution of eigenstructure assignment.