An algorithm to assign canonical forms by state feedback

Abstract

A procedure for assigning an arbitrary normalized Hessenberg matrix is presented. Specifically, given a controller-Hessenberg pair (A,b) and a normalized upper Hessenberg matrix B, the algorithm computes an upper triangular matrix L=(l/sub ij/) and a row vector f/sup T/ such that L(A-b/sup T/f/sup T/)L/sup -1/=B/sup T/. It is formulated in such a way that the columns of L can be scaled to have unit lengths. Since the class of normalized Hessenberg matrices contains important canonical forms, such as companion, Schwarz, Rough, a Jordan matrix associated with a single eigenvalue, or any bidiagonal matrix having its eigenvalues on the diagonal, the method can be used to assign all these important canonical forms and also an arbitrary set of eigenvalues. After the initial reduction to the Hessenberg form, the method requires only the evaluation of a simple recursion, which becomes extremely simplified in all of these cases. It is also easy to program on a computer. A theoretical operations count suggests that the method is more efficient as an eigenvalue-assignment procedure than the best-known previous procedures.<>