Design of momentum fractional LMS for Hammerstein nonlinear system identification with application to electrically stimulated muscle model

. Fractional calculus extends the scope of adaptive algorithms supporting the design of novel fractional methods that outperform standard strategies in various applications arising in applied physics and engineering. In this study, a momentum fractional least-mean-square (M-FLMS) algorithm for nonlinear system identification using a first and fractional-order gradient information is proposed. The M-FLMS avoids being trapped in local minima and provides faster convergence than the standard FLMS. The convergence and complexity analysis of the M-FLMS are given along with simulation results of a benchmark nonlinear system identification problem. The M-FLMS accuracy is verified through a parameter estimation problem for a nonlinear Hammerstein structure, modeling an electrically stimulated muscle (ESM) for rehabilitation of paralyzed muscles. The proposed method is studied in detail for different levels of noise variance, fractional orders and proportion of gradients used in the current update.