Controlled invariance in systems over rings

The definition of controlled invariant (i.e. (A,B)-invariant) subspaces of a linear system is extended to systems over rings. It is observed that in this more general setting, the equivalence of the geometric and the feedback characterization is no longer true. Particular attention is paid to the weakly unobservable space V*, and conditions are given for this space to satisfy the feedback characterization. These conditions have the form of the existence of a factorization of the transfer function. An application to the disturbance rejection problem is given