The nonparametric identification for nonlinear autoregressive systems with exogenous inputs (NARX) described by  yk +1= f ( yk ,..., yk +1- n  0 , uk ,..., uk +1- n  0 )+e k +1 is considered. First, a condition on  f (·) is introduced to guarantee ergodicity and stationarity of { yk } . Then the kernel function based stochastic approximation algorithm with expanding truncations (SAAWET) is proposed to recursively estimate the value of  f (φ*) at any given φ* Δ/= [ y  (1) ,..., y  (  n  0 ), u  (1) ,..., u  (  n  0 )] τ  ∈  R  2  n  0 . It is shown that the estimate converges to the true value with probability one. In establishing the strong consistency of the estimate, the properties of the Markov chain associated with the NARX system play an important role. Numerical examples are given, which show that the simulation results are consistent with the theoretical analysis. The intention of the paper is not only to present a concrete solution to the problem under consideration but also to profile a new analysis method for nonlinear systems. The proposed method consisting in combining the Markov chain properties with stochastic approximation algorithms may be of future potential, although a restrictive condition has to be imposed on  f (·), that is, the growth rate of  f ( x ) should not be faster than linear with coefficient less than 1 as || x || tends to infinity.
['Wenxiao Zhao', 'Han-Fu Chen', 'Wei Xing Zheng']
Recursive Identification for Nonlinear ARX Systems Based on Stochastic Approximation Algorithm