Exact pole assignment using direct or dynamic output feedback

This note addresses the problem of the assignability of the eigenvalues of the matrix
A + BPC
by choice of the matrix
P
. This mathematical problem corresponds to pole assignment in the direct output feedback control problem, and by proper changes of variables it also represents the pole assignment problem with dynamic feedback controllers. The key to our solution is the introduction of the new concept of local complete assignability which in loose terms is the arbitrary perturbability, of the eigenvalues of
A + BPC
by perturbations of
P
. If n
x
is the order of the system, we show that if
A + BP_{0}C
has distinct eigenvalues, a necessary and sufficient condition for local complete assignability at P
0
is that the matrices
C[A + BP_{0}C]^{i-1}B
be linearly independent, for
1 \leq i \leq n_{x}
. In special cases, this condition reduces to known criteria for controllability and observability. Although these latter properties are necessary conditions for assignability, we also address the question of the assignability of uncontrollable or unobservable systems both by direct output feedback and dynamic compensation. The main result of this note yields an algorithm that assigns the closed-loop poles to arbitrarily chosen values in the direct and in the dynamic output feedback control problems.