Distribution of invariant indices and static output feedback pole placement

This paper deals with the static output feedback pole placement problem, examined via the eigenstructure assignment method. We transform the so-called Sylvester equations, occurring in the static output feedback (SOF) eigenstructure assignment problem, to a block Hankel matrix equation. This way we get rid of the redundant data and we treat only the significant ones. As a result, important properties of SOF-assignability appear to have an expression in terms of the distributions of the controllability and observability indices. First of all we present a necessary condition for the solvability of the Sylvester equations and we show of its relations with Kimura´s condition and Plucker´s matrix rank deficiency. According the distributions of invariant indices, the set of systems is partitioned into categories. For some of them SOF-assignability is impossible, for others is a linear and for the rest a non linear problem. For the linear case we present a parameterization of the eigenvectors related to a specific pole placement. For the non-linear one, the equations are weakly coupled and allow an analytic solution in relatively simple cases.