The pole placement map, its properties, and relationships to system invariants

A number of properties of the complex and real pole placement map (PPM) which relate to the dimensions of their images and relate them to known system invariants are derived. It is shown that the two dimensions are equal and that their computation is equivalent to determining the rank of the corresponding differential. A new expression for the differential of the PPM allows the derivation of relationships between the Markov parameters and the Plucker matrix invariant of the system. Conditions for pole assignability are derived, based on the relationships between the rank of the Plucker matrix and the rank of the differential of the PPM