Design of normalized fractional adaptive algorithms for parameter estimation of control autoregressive autoregressive systems

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

“…For $$$ x>a, fr>-1 $$$, the fractional order derivative 49,50 for the function $$$ g(x)={x}^{\lambda } $$$ is given as: $$$$ {\kern0.01em }_a{D}_x^{fr}{x}^{\lambda }=\frac{\Gamma \left(\lambda +1\right)}{\Gamma \left(\lambda - fr+1\right)}{x}^{\lambda - fr}. $$$$ The objective function 46 for the BIP model is given as: $$$$ J(n)=E\left[{\left|e(n)\right|}^2\right]={\left|e(n)\right|}^2, $$$$ where $$$ E\left(\cdot \right) $$$ represents an expectation operator, the estimation error $$$ e(n) $$$ is the difference between desired response $$$ y(n) $$$ and the estimated filter response $$$ \hat{y}(n) $$$, and it is writ...…”

“…When the input vector $$$ \boldsymbol{x}(n) $$$ is divided with its norm in Equation (), the weight update equation of normalized AFLMS (NAFLMS) algorithm 46,47 is written as: $$$$ \hat{\boldsymbol{w}}(n)=\hat{\boldsymbol{w}}\left(n-1\right)+\left[\mu +\frac{\mu_{fr}}{\Gamma \left(2- fr\right)}{\left|\hat{\boldsymbol{w}}\right|}^{1- fr}\left(n-1\right)\right]\circ \frac{e(n)\boldsymbol{x}(n)}{\left\Vert \boldsymbol{x}(n)\right\Vert }. $$$$ Introducing the adjustable gain parameter $$$ \omega $$$ in the same way, the NMFLMS algorithm is given as: …”

“…And fractional identification algorithms are applied widely for parameter estimation of different system models such as, Chaudhary et al 40 and Cheng et al 41 addressed Hammerstein nonlinear ARMAX systems, Raja et al 42 estimated CARMA systems, Chaudhary et al 43,44 identified input nonlinear Box–Jenkins systems and nonlinear Hammerstein CAR systems. Some modified fractional LMS strategies have been applied to many research fields, Chaudhary et al 45 formulated the modified fractional least mean square algorithms for parameter estimation of Hammerstein nonlinear control autoregressive system by introducing an adjustable gain parameter in weight update equation, and then the authors used the proposed methods for parameter identification of CARAR systems 46 . In the following studies, the authors designed the normalized fractional adaptive strategies for automatic tuning of the step size parameter in nonlinear system identification of the Hammerstein nonlinear control autoregressive model 47 .…”

“…Inspired by the work in References 46,47, this article presents the auxiliary model based $$$ \varepsilon \hbox{-} \mathrm{normalized} $$$ modified fractional least mean square ($$$ \varepsilon \hbox{-} \mathrm{NMAFLMS} $$$) algorithm and the normalized modified hierarchical fractional least mean square (NMHFLMS) algorithm for BIP system by using the auxiliary model identification idea and the hierarchical identification principle. The main contributions of this article are as follows.…”

Three‐stage forgetting factor stochastic gradient parameter estimation methods for a class of nonlinear systems

“…The modeling of industrial systems plays a crucial role in system analysis and control 1,2 . System identification is essential for constructing the mathematical models of systems, 3,4 and parameter estimation is the foundation of system identification 5,6 . The study of new parameter estimation methods has become the eternal theme of system control 7,8 .…”