Generalized Notions in Geometry and Duality

We show the dynamical significance of the subspaces generated by the steps of the recursions for the supremal (A, E, Image(B))-invariant subspace and the infimal almost controllability subspace for discrete-time singular systems. We also show the dynamical interpretation of these recursions when performed on the dual system. We thus give duality results for subspaces computed from the original system matrices. Our approach should be contrasted with previous results which give duality results in terms of (different) subspaces computed from a slow/fast decomposition of the system.