Robust Stabilization of a Family of Plants with Varying Number of Right Half Plane Poles

We consider the problem of robustly stabilizing frequency domain families of plants by removing the assumption that all the plants in the family have the same number of unstable poles. We discover that this assumption which is made in much of the existing works on robust stabilization is in some sense necessary. We show that it is necessary for us to restrict the plants to have no more unstable poles than the nominal plant and under this restriction the family of plants can be robustly stabilized if and only if a Nevanlinna-Pick matrix related to the size of the family is nonnegative definite. When this condition holds then there are in fact no plants in the family with fewer unstable poles than the nominal plant. We also characterize the largest achievable stability margins and obtain a parameterization of the set of robustly stabilizing compensators.