A geometric approach to 2-D implicit systems

The existence of solutions with certain geometric properties for the implicit Roesser model (IRM) and the second implicit Fornasini-Marchesini model is considered. Invariant subspaces and some geometric notions which do not have one-dimensional counterparts are defined. Using these notions, a Fornasini-Marchesini-like model is analyzed which encompasses two alternative representations of the IRM. For the IRM, boundary conditions are considered on any side of the boundary to investigate existence of the solutions. Recursions to compute relevant subspaces for each model are provided. The 2-D output nulling (AEB)-invariant subspaces of a dual system are related to observer design. An asymptotic state-space observer is constructed for each model by using geometric techniques, and an illustrative example is included.