Identification for control: can the optimal restricted complexity model always be identified?

We consider model based certainty equivalence optimal control design for a high order plant on the basis of a model that is identified within a restricted complexity model set. Each model in this model set yields a “restricted complexity” controller. This defines a controller set which contains a “best” controller for the high order plant, i.e. the controller that achieves the lowest cost on the high order plant. The following question is addressed: can the restricted complexity model (or models) that produces this “best” restricted complexity controller always be obtained by identification from plant data, given adequate experimental conditions? The answer is `not always´