Up to now the use of geometric methods in the study of disturbance decoupling problems (DDPs) for systems over a ring has provided only necessary conditions for the existence of solutions. In this paper we study such problems, considering separately the case in which only static feedback solutions are allowed, and the one in which dynamic feedback solutions are admitted. In the first case, we give a complete geometric characterization of the solvability conditions of such problems for injective systems with coefficients in a commutative ring. Practical procedures for testing the solvability conditions and for constructing solutions, if any exist, are given in the case of systems with coefficients in a principal ideal domain (PID). In the second case, we give a complete geometric characterization of the solvability conditions for systems with coefficients in a PID.

The model matching and the left factorization problems for nonlinear systems are investigated using an approach based on the structural algorithm. Sufficient conditions for the solvability of the first and necessary conditions, which in some cases and from a local point of view are also sufficient, for that of the second problem are found in terms of equalities between ranks or structures at infinity.