Model validation in closed loop

In Ljung (1998), a model validation method was proposed that is based on the estimation of an unbiased model of the model error which relates the inputs to the simulation errors. In this paper, we extend this methodology to closed-loop validation, i.e. the validation of an open-loop model on the basis of closed-loop validation data. We show that, perhaps surprisingly, the same model may fail to be validated with open-loop data, while it is validated by data collected in closed loop. In addition, we show that the uncertainty sets generated by models validated in closed loop are better tuned towards control design, i.e. the controllers that stabilize all members of such a set are less conservative than those that stabilize all members of a set validated in open loop. The comparison between both methods is illustrated by means of a numerical example