Design of sign fractional optimization paradigms for parameter estimation of nonlinear Hammerstein systems


Fractional calculus plays a fundamental role in understanding the physics of nonlinear systems due to its heritage of uncertainty, nonlocality and complexity. In this study, novel sign fractional least mean square (F-LMS) algorithms are designed for ease in hardware implementation by applying sign function to input data and estimation error corresponding to first and fractional-order derivative terms in weight update mechanism of the standard F-LMS method. Theoretical expressions are derived for proposed sign F-LMS and its variants; strength of methods for different fractional orders is evaluated numerically through computer simulations for parameter estimation problem based on nonlinear Hammerstein system for low and high signal–noise variations. Comparison of the results from true parameters of the model illustrates the worth of the scheme in terms of accuracy, convergence and robustness. The stability and viability of design methodologies are examined through statistical observations on sufficiently large number of independent runs through mean square deviation and Nash–Sutcliffe efficiency performance indices.

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J :

Cost function

w :

Weights of a filter

x :

Input signal

µ 1 :

Step size for first-order gradient

v :

Noise signal

w :

Weight vector

\({\varvec{\uptheta}}\) :

Parameter vector

\(\delta\) :

Fitness function

E :

Error signal

M :

Number of filter taps

fr :

Fractional order

µ 2 :

Step size for fractional-order gradient

y :

Output signal

φ :

Information vector

\({\hat{\boldsymbol{\uptheta}}}\) :

Estimated parameter vector

σ 2 :



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Correspondence to Muhammad Saeed Aslam.

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Chaudhary, N.I., Aslam, M.S., Baleanu, D. et al. Design of sign fractional optimization paradigms for parameter estimation of nonlinear Hammerstein systems. Neural Comput & Applic 32, 8381–8399 (2020).

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  • Nonlinear system identification
  • Sign regressors
  • Fractional adaptive signal processing
  • Hammerstein models
  • Control structures