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 L
₁
-L
₂
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 L
₁
and L
₂
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 L
₁
- 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.