Properties of the pole placement map

Some further properties of the complex and real pole placement map (PPM) are derived which relate to the dimension of their images. It is shown that these two invariants are equal, and that their computation is equivalent to a rank determination of the corresponding differential. A new expression of 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