On the Optimal Estimation of Errors in Variables Models for Robust Control

There exists a substantial literature dealing with the problem of errors-in-variables identification. It is known, for example, that there is an equivalence class of models that give compatible descriptions of the input-output data. In the current paper, we impose a mild restriction so as to avoid certain singular possibilities. This leads to a parameterization of the equivalence class of models via a single real parameter. We then use this result to show that there exists a model which is optimal in the sense that minimizes the maximal weighted infinity norm of the error between the chosen model and all members of the equivalence class. This model is unique and is independent of the weighting function used in the infinity norm. It is thus the natural choice to be used in applications such as robust control. The result is also compared with more conventional estimates provided by prediction error methods.