On the uniqueness of prediction error models for systems with noisy input-output data

This paper addresses the uniqueness problem of the prediction error (PE) identification for a class of linear systems with noisy input and output data. Necessary and sufficient conditions are derived for the corresponding PE loss function to have (asymptotically) a unique global minimum. The results indicate that a PE algorithm may give very bad parameter estimates for systems not satisfying these conditions. Such a possibility is illustrated by a numerical example. While the PE method is used as a vehicle for illustration, the derived conditions for global uniqueness (or identifiability) apply to any consistent estimation method based on second-order data.