Nonparametric algorithms for identification of nonlinear autoregressive systems with exogenous inputs

This paper is concerned with nonparametric identification of nonlinear autoregressive systems with exogenous inputs (NARX), i.e., $y_{k+1}=f(y_k,cdots,y_{k+1 n_0},u_k,cdots,u_{k+1-n_0})+varepsilon_{k+1}$. Kernel functions based stochastic approximation algorithms with expanding truncations are designed for recursively estimating the value of $f(cdot)$ at any given $[y^{(1)},cdots,y^{(n_0)},u^{(1)},cdots,u^{(n_0)}]^{tau}in mathbf{R}^{2n_0}$. The estimates are shown to be strongly consistent. The NARX systems considered in this paper include the one in cite{ZhaoChen} as a special case. A numerical example is given to justify the theoretical analysis.