A new approach to optimal filter design for nonlinear systems

Optimal filters for nonlinear systems are in general difficult to derive/implement. The common approach is to obtain approximate solutions, e.g. based on linearizations along the performed path, as done in extended Kalman filters. However, no optimality properties can be guaranteed using these approximations, not even the stability of the estimation error is ensured. In this paper, a new method is presented, able to overcome this problem. The method is based on the direct identification of the filter from a set of data. Under the standard assumptions of the filtering literature, i.e. known system and noise properties, the data can be generated by simulations of system equations. A further feature of the method arises from the fact that, in most practical situations, the system to filter is not known, but it is possible to perform measurements on it. In these situations, a two-step approach is typically adopted: 1) a model of the system is identified from the available measurements; 2) a filter is designed from the identified model. A relevant problem of the two-step design is that, in presence of modeling errors, the two-step filter may display large performance deteriorations. The direct method allows the direct design of the filter from the measurements, avoiding this problem. The method is developed within a set membership framework. Optimal filters for nonlinear systems are obtained, where optimality refers to the minimization of the induced norm from the noise to the estimation error.