Loss of structurally stable regulation implies loss of stability

Consider the problem of selecting a single compensator capable of regulating any member of a family of linear systems in the presence of a specified exogenous system. Necessary and sufficient conditions for the existence of a structurally stable, regulating and stabilizing compensator at a particular plant are well known. We refer to systems for which these conditions fail as critical. Clearly any choice of compensator must fail in at least one respect at a critical point. Either it is not structurally stable, or it does not asymptotically reject the exogenous system, or it does not stabilize the closed-loop system. However, existing theory does not reveal which of these occurs. In fact, we show that a compensator designed to regulate, with structural stability, a specified exogenous system, is necessarily destabilizing at a critical point. As an application of this result we derive an upper bound on structurally stable regulation under unstructured perturbation for any choice of system metric.