On parametrizations for the minimal partial realization problem

The paper considers parametrizations for minimal partial realizations of a finite sequence of Markov parameters. First it is shown that all minimal partial realizations have the same set of input and output Kronecker indices. A parametrization is obtained formulated in terms of a minimal set of extension entries. Then an I/O canonical form is used as a parametrization for the set of minimal partial realizations. It is shown that this allows to define the invariants of the problem. Moreover, the I/O canonical form is shown to contain a set of parameters that can be chosen such that any minimal partial realization can be represented.