Robust nonlinear parameter estimation in the bounded noise case

When modeling a system, it is of importance to assess the uncertainty on the estimated values of the parameters. This is usually done by taking advantage of the asymptotic properties of maximum likelihood estimators. However, the significance of the results obtained can be questionned when little information is available on the noise statistical properties and when the number of data points is limited. Membership set theory then appears as a promising tool to overcome some of these difficulties. Its purpose is to characterize the set of all the parameter vectors that are consistent with the assumed knowledge of bounds on the acceptable errors between the data and the model outputs. However, the membership set estimators presented in the literature so far are restricted to model linear in the parameters. The approach described here has been designed for handling nonlinear models as well as linear ones. It involves the maximisation of the number of data points that do not have to be considered as outliers and the characterization of the boundary of the domain of the parametric space where this number is maximum. The method is applied to two examples. It is shown to be extremely robust to outliers and to be able to handle even models that are not uniquely Identifiable.