Closed-loop identification of multivariable processes with part of the inputs controlled

In many multivariable industrial processes a subset of the available input signals is being controlled. In this paper it is analysed in which sense the resulting partial closed-loop identification problem is actually a full closed-loop problem, or whether one can benefit from the presence of non-controlled inputs to simplify the identification problem. The analysis focuses on the bias properties of the plant estimate when applying the direct method of prediction error identification, and the possibilities to identify (parts of) the plant model without the need of simultaneously estimating full-order noise models.