Detectability of linear systems over a principal ideal domain with unknown inputs

Observability and detectability conditions are obtained for a sort of linear time invariant systems affected by unknown inputs. It is assumed that the system can be described using matrices whose elements are within a principal ideal domain (PID). It is shown that, under suitable hypothesis, the obtained conditions are necessary and sufficient. The analysis is carried out by making use of the Smith normal form of matrices over a PID. The obtained conditions are a generalization of known conditions for particular sorts of systems which are included in those considered here.