Identification for control: closing the loop gives more accurate controllers

We compare open loop versus closed loop identification when the identified model is used for control design, and when the system itself belongs to the model class, so that only variance errors are relevant. For three different control design criteria (minimum variance, LQG and model reference control) we show that, under those conditions, a better performance is achieved by closing the loop during the identification. The measure of performance is the variance of the error between the output of the ideal closed loop system (with the ideal controller) and that of the actual closed loop system (with the controller computed from the identified model)