Design of modified fractional adaptive strategies for Hammerstein nonlinear control autoregressive systems

Abstract

In this study, modified fractional least mean square (FrLMS) algorithms are formulated for parameter estimation of Hammerstein nonlinear control autoregressive system (HNCAR) by exploiting the fractional calculus concepts in weight adaptation mechanism of the algorithm. In modified FrLMS (MFrLMS) of first kind, forgetting factor is applied to exploit the strength of both standard LMS and FrLMS algorithms. The MFrLMS algorithm of second kind is based on single fractional weight adaptation term in cost function for finding the optimal values instead of considering both fractional and first derivative terms as in FrLMS approach. Performance analysis of the proposed methods are carried out on the basis of optimization ability to the true values of HNCAR system, and comparison is made in terms of accuracy, convergence and complexity. Reliability and effectiveness of the proposed scheme is validated through performance indices based on mean square error, variance account for and Nash Sutcliffe efficiency for sufficient large number of independent runs.

Design of auxiliary model and hierarchical normalized fractional adaptive algorithms for parameter estimation of bilinear‐in‐parameter systems

“…Minimizing the objective function () by taking first‐order derivative and fractional derivative with respect to $$$ {\hat{w}}_k $$$, 37,45 and combined with the negative gradient search, the conventional fractional least mean square (FLMS) algorithm is expressed as: $$$$ {\hat{w}}_k(n)={\hat{w}}_k\left(n-1\right)-\frac{\mu }{2}\frac{\partial J(n)}{\partial {\hat{w}}_k}-\frac{\mu_{fr}}{2}\frac{\partial^{fr}J(n)}{\partial {\hat{w}}_k^{fr}}, $$$$ where $$$ \mu $$$ and $$$ {\mu}_{fr} $$$ are the step size parameter of the filter corresponding to first‐order derivative and fractional derivative of objective function, $$$ k=0,1,\dots, M-1 $$$.…”

Modified Volterra LMS algorithm to fractional order for identification of Hammerstein non‐linear system

“…Ortiguierra and his team introduced the fundamental concept and theory of F-SP [30,31]. Afterwards, these concepts have been applied effectively to solve many problems including system identification [44][45][46][47], speech enhancement [48], active noise control [35] and channel equalisation [36]. The aim of present study is to step further in this domain by proposing fractional variant of VLMS algorithm for parameter estimation of H-BJ system.…”

Auxiliary Model-Based Multi-Innovation Fractional Stochastic Gradient Algorithm for Hammerstein Output-Error Systems

“…To the best of our knowledge, different fractional-order gradient methods have been produced [34][35][36]. For example, in [37], a fractional-order SG algorithm was designed to identify the Hammerstein nonlinear ARMAX systems by an improved fractional-order gradient method.…”