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'.<<ETX>>