Minimal or maximal realizations?

Transfer function matrices arise in control systems from setting variable parameters to their nominal values. They are often the starting point of feedback system design calculations based on minimal order state space realizations. The McMillan degree and therefore the minimal order of realizations is, in general, a discontinuous function of the variable parameter. In view of this, we show that using minimal realizations of a given plant transfer matrix for feedback stabilization can lead to nominally stable systems that are destabilized by infinitesimal parameter perturbations. The remedy for such structural instability is to use maximal minimal realizations, that is, minimal realizations in which the orders of the antistable (all RHP poles) parts are maximized over the set of uncertain parameters. The antistable McMillan degree ν_max⁺ of such systems is invariant under small perturbations, except on an algebraic variety, and this leads to reliable stabilization. Although the controller designed by this means stabilizes a ball of plants around the perturbed nominal it cannot, in general, simultaneously stabilize the plants lying on the algebraic variety where ν⁺ is less than ν_max⁺.