On closed-loop identification: error distributions in the ν-gap metric

The paper is a result of the observation that direct identification in closed-loop, using prediction error methods, with reference inputs which are chosen to excite the plant input and output equally tends to provide "good" models (in terms of small δ_ν(P,Pˆ)). We show, by an analysis of asymptotic parameters and frequency response covariances, why this is so (it is because the expected identification errors are uniformly distributed across frequency). For a more practical reference injection scheme, based on a normalized coprime factorization of the controller, the expected identification errors in the ν-gap are inversely proportional to the H_∞ loop-shaping cost function, on a frequency by frequency basis. Thus, the identified model is likely to be good at frequencies where the existing controller is poor and vice-versa. These results suggest a scheme for iterative identification for control, based on successive prediction error identification and a control design step which trades off the H_∞ loop-shaping cost function of the next controller and the model against the ν-gap distance between the next and the current controller. We analyse some properties of this scheme