Data Driven Local Coordinates: Some New Topological and Geometrical Results

Certain topological and geometrical properties of data driven local coordinates (DDLC) for state-space systems as introduced in (Wolodkin et al., 1997) and (McKelvey and Helmersson, 1999) are derived. First the special case of SISO systems with McMillan degree n 1 is discussed in order to provide some insights into the geometry of the DDLC construction. Then for the MIMO case with general n it is shown that the set of transfer functions corresponding to the parameter space contains a nonvoid open subset of the manifold of transfer functions of order n and that the estimation problem is locally well posed. Moreover, it is stated that the parameter space always contains points corresponding to non minimal systems and a result on the number of disconnected components of the equivalence classes in the space n 2 n m s (obtained by an embedding of the system matrices A B C ) concludes this contribution.