L1 - L2 robust estimation in prediction error system identification

The problem of robust system identification with corrupted data remains a difficulty. In this paper we shall put ourselves in the prediction error framework. We shall present a mixed L₁-L₂ estimator based on a parameterized objective function leading to an alternative solution fighting against the outliers, based on the well-known Huber´s M-estimate. A simple physical insight on the main noise characteristics of the data leads to a convenient choice of the scaling factor which automatically determines the balance between L₁ and L₂ contributions in the estimation procedure. Moreover, a general formalism leads to concise expressions of the gradient and the Hessian of the objective function which facilitate the estimation algorithm´s synthesis. These expressions are established in the classical case of linear models. A new decision tool, namely the L₁ - contribution function of the residuals is proposed, which helps the user to determine the convenient model. Finally, some results on the frequency response of the best estimate are given for a semi finite acoustic duct used as an experimental set-up.