System Assignment and Pole Placement for Symmetric Realisations

In this paper we consider the problem of system assignment for the class of linear output feedback systems having symmetric state space realisations. For such systems the classical pole placement task can be considered as a degenerate case of a more general system assignment problem. It is shown that a symmetric state space realisation can be assigned arbitrary (real) poles via output feedback if and only if there are at least as many system inputs as states. The task of computing feedback gains for system assignment is approached by deriving gradient ows which minimize suitable least squares distance functions on smooth manifolds of output feedback equivalent realisations. These ordinary di erential equations provide insight into the complex structure of the systems assignment and pole placement problems. Computing the limiting values of the ows provides a method of determining optimal feedback gains for the system assignment (pole placement) problem even when exact solutions to the problem does not exist. The methods are also generalised to a simultaneous multiple system assignment problem.