Small (A, B)-invariant subspaces in disturbance decoupling problems

The problem of synthesizing a state feedback law which decouples the output of a system with respect to a nonmeasurable disturbance, or disturbance decoupling problem (DDP), is considered. The key geometric tool for expressing the solvability conditions is the maximum (A,B)-invariant or controlled invariant subspace contained in a given subspace. The authors describe the properties of the smallest (A,B)-invariant submodule in a particular lattice, whose existence is guaranteed under a mild hypothesis. They show the usefulness of such an object in the study of a DDP involving a system whose coefficients depend on a parameter. More precisely, they take into consideration the cases in which the system may be modeled as a parameter-dependent real linear system or as a system with coefficients in a ring. The results consists of necessary and sufficient conditions for the solvability of the DDP stated in terms of output of a recursive geometric procedure