Analysis of the issue of consistency in identification for robust control

Given measured data generated by a discrete-time linear system we propose a model consisting of a linear, time-invariant system affected by norm-bounded perturbation. Under mild assumptions, the plants belonging to the resulting uncertain family form a convex set. The approach depends on two key parameters: an a priori given bound of the perturbation, and the input used to generate the data. It turns out that the size of the uncertain family can be reduced by intersecting the model families obtained by making use of different inputs. Two model validation problems in this identification scheme are analyzed, namely the worst and the best invalidation problems. It turns out that while the former is a max-min optimization problem subject to a spherical constraint, the latter is a quadratic optimization problem with a quadratic and a convex constraint. For a given energy level, the worst invalidation problem computes the family of models which can be invalidated for all possible noise sequences; i.e. the union of all possible uncertain families is obtained. The best invalidation problem, on the other hand, yields the family of those models which can never be invalidated for any possible input of fixed energy and perturbation; in other words the intersection of all uncertain families is computed. An important consequence of the best invalidation problem is that for finite length measurements not all models can be invalidated, using fixed-energy inputs.