Identification for Hammerstein nonlinear ARMAX systems based on multi-innovation fractional order stochastic gradient

•The fractional order gradient method is improved.•Muti-innovation fractional order stochastic gradient algorithm is proposed.•Variable gradient order and forgetting factor for step size and is developed to improve the convergence performance.

A multi-innovation fractional order stochastic gradient (MIFOSG) algorithm, which involves a variable initial value scheme, is investigated to identify the Hammerstein nonlinear ARMAX systems in this paper. Firstly, according to an improved fractional order gradient method, the MIFOSG algorithm is proposed. Furthermore, according to the martingale convergence theorem, the convergence analysis of the proposed algorithm is developed. In addition, for the purpose of improving the convergence performance, a forgetting factor on step size and a variable gradient order are introduced. Given a sufficiently large number of independent runs, the effectiveness of the proposed algorithm is demonstrated in two numerical examples finally.