Extended recursive least squares algorithm for nonlinear stochastic systems

The strong consistency of parameter estimation has always been one of the main problems in system identification theory especially for the nonlinear systems. Although there are several approaches and algorithms set up for the nonlinear stochastical system, the strong consistency of the parameter estimation has not been discussed and proved for most of them. In this paper, for a class of discrete-time nonlinear stochastic systems, an extended recursive least squares (ERLS) algorithm is developed. It has been proved that the proposed ERLS algorithm has strong consistency without the strict restrictions on the system, i.e. (1) the persistent excitation condition has been replaced by a less restrictive condition; (2) the system noises don´t have to be white noise; and (3) the variance functions of the system noises don´t have to be bounded. The convergence rate of the parameter estimation can also be worked out. The results presented in this paper can be applied in more general classes of nonlinear systems such as some of the variable parameter and state dependent parameter nonlinear stochastic systems as well.