Identification of an LFT uncertainty model by minimizing the ν-gap metric

An uncertainty model in the form of a linear fractional transformation (LFT) is composed of a nominal model augmented by an uncertainty description. The size of the uncertainty required to cover a given set of models depends not only on the set of models, but also on the nominal model. Thus, the size of the uncertainty can be minimized by choosing the nominal model optimally according to some metric. If the uncertainty model is to be used for control design, a suitable metric is the ?-gap metric. It is shown that the optimal solution in terms of the ?-gap metric has to satisfy a bilinear matrix inequality (BMI) for every model in the model set. To solve this nonconvex optimization problem, the BMIs are linearized to enable an iterative solution constrained by linear matrix inequalities (LMIs), where each iteration is a convex optimization problem. It is proved that the iteration converges to the optimal solution satisfying the BMIs. Because the solution is obtained as the frequency response at selected frequencies, the final model is determined by fitting a model to the frequency responses. A state-space model is used because the fitting can then easily be done subject to the same BMIs/LMIs to guarantee an optimal model. The procedure is illustrated by an application to uncertainty modeling of the product composition dynamics of a distillation column.