Eigenvalues assignment in uncontrollable linear systems

Abstract

It is shown that in uncontrollable linear system ẋ = Ax + Bu it is possible to assign arbitrarily the eigenvalues of the closed-loop system with state feedbacks u = Kx, K ∈ ℜ^n⨉m if rank [A B] = n. The design procedure consists in two steps. In the step 1 a nonsingular matrix  M ∈ ℜ^n⨉m is chosen so that the pair (MA,MB) is controllable. In step 2 the feedback matrix K is chosen so that the closed-loop matrix A_c = A  − BK has the desired eigenvalues. The procedure is illustrated by simple example.