Algebraic and geometric indices for singular systems

Observability spaces for a minimal singular system are defined from two points of view. One method is algebraic in nature, depending on filtrations of the Wedderburn-Forney spaces and global pole spaces. The second method is geometric and proceeds by defining algorithmically a chain of subspaces. It is shown that the two methods coincide and that the observability indices defined in this way coincide with an appropriate set of row indices attached to a good left matrix fraction decomposition of the original (possibly improper) transfer function. The algebraic formulation for the controllability is given, and the corresponding identification of controllability indices with appropriate column degrees is described