Inversion of non-linear stochastic models for the purpose of parameter estimation

Prediction error and maximum likelihood estimation of non-linear stochastic models requires inversion of the model, a step which may require substantial efforts, either in terms of manual calculations or through the use of software capable of symbolic computations. In this paper we show that model inversion can be easily implemented in numerical software such as, e.g. Simulink and Matrix X, by means of a feedback connection based on the model. It is further shown how the gradients, used for the optimization of the cost function, can be generated by a linear time-varying feedback system associated with the non-linear model. In addition, we derive sufficient conditions for the existence of a stable causal inverse as well as sufficient conditions for the initial transient to decay. These conditions are given in terms of properties for a linear time-varying system associated with the non-linear model. The method is illustrated on numerical examples.