Ackermann’s Method: Revisited, Extended, and Generalized to Uncontrollabe Systems

The celebrated method of Ackermann for eigenvalue assignment of single-input controllable systems is revisited in this paper, contributing an elegant proof. The new proof facilitates a compact formula which consequently permits an extension of the method to what we call incomplete assignment of eigenvalues. The inability of Ackermann’s formula to deal with uncontrollable systems is considered a weakness inherent in the method. The notion of incomplete assignment leads to a straightforward generalization of our method to eigenvalue assignment of uncontrollable systems, thus mitigating such a drawback of a popular method. Further results concerning the incomplete assignment are stated, verified, and commented. Such results reveal the trace of the state matrix
$A$
a worthy feature pertinent to an open loop system. Finally, four numerical examples are worked out to demonstrate cases of incomplete, and uncontrollable eigenvalues assignment. The examples consider a case where the structure of the feedback matrix can be easily simplified. The paper ends with a commentary brief concerning some commonly used MATLAB commands for eigenvalue assignment.