Repeated eigenstructure assignment for controlled invariant subspaces

This paper is concerned with the computation of basis matrices for the subspaces that lie at the core of the so-called geometric approach to control theory, namely the supremal output-nulling, reachability and stabilisability subspaces. Importantly, we also consider the problem of computing the feedback matrices that render these subspaces invariant with respect to the closed loop, while simultaneously assigning the assignable eigenstructure of the closed loop. Differently from the classical techniques presented in the literature so far on this topic, which are based on the standard pole assignment algorithms and are therefore applicable only in the non-defective case, the method presented in this paper can be applied in the case of closed-loop eigenvalues with arbitrary multiplicity.