CFEM-LMS¶

Implements CFEM-LMS, a convex combination of the LMS filter and a fractional-order error-modified LMS filter for identifying Van der Pol-Duffing oscillator systems.

Built on a functional-link (FLANN) structure, the method runs two adaptive filters in parallel. One is the ordinary least mean square (LMS) filter; the other, FEM-LMS, replaces the squared-error cost with an error-function (erf) cost that saturates large errors and then takes the \(v\)-order Caputo fractional gradient, yielding an extra fractional term whose nonlocality improves identification of the nonlinear dynamics. The two weight vectors are mixed by a sigmoid-controlled parameter \(\lambda_t\) so the overall filter inherits the fast convergence of LMS and the low steady-state error of FEM-LMS.

Each branch updates with a first-order gradient term plus a Caputo fractional term; the Caputo derivative of \(w^{1-v}\) contributes the \(|w_t|^{1-v}/\Gamma(2-v)\) factor. The combination weight follows from a sigmoid of an auxiliary scalar \(a_t\) adapted to minimize the overall squared error.

\[ \begin{aligned} e_t &= d_t - w_t^\top x_t, \\ w^{(1)}_{t+1} &= w^{(1)}_t + \gamma\, e_t\, x_t, \\ w^{(2)}_{t+1} &= w^{(2)}_t + \gamma\, \psi(e_t)\, x_t + \gamma_f\, \psi(e_t)\, x_t \odot \frac{|w^{(2)}_t|^{\,1-v}}{\Gamma(2-v)}, \\ \lambda_t &= \frac{1}{1 + e^{-a_t}}, \\ w_{t+1} &= \lambda_t\, w^{(1)}_{t+1} + (1-\lambda_t)\, w^{(2)}_{t+1}. \end{aligned} \]

where \(w^{(1)}\) is the LMS branch and \(w^{(2)}\) the FEM-LMS branch, \(x_t\) is the FLANN-expanded input, \(d_t\) the desired output, \(e_t\) the error, \(\psi(\cdot)\) the erf-based saturation nonlinearity of the modified cost, \(\gamma\) and \(\gamma_f\) the integer- and fractional-order step sizes, \(v\in(0,1)\) the fractional order, \(\Gamma\) the gamma function, \(\lambda_t\in(0,1)\) the mixing parameter from sigmoid of auxiliary variable \(a_t\), and \(\odot\) elementwise product.

Reference: Kai-Li Yin, Yi-Fei Pu, Lu Lu, "Combination of fractional FLANN filters for solving the Van der Pol-Duffing oscillator", Neurocomputing 399 (2020) 183-192. https://doi.org/10.1016/j.neucom.2020.02.022

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