FGD-RBFNN (UAV)

Implements FGD-RBFNN, a fractional gradient descent rule for training the output weights of a radial basis function neural network.

The RBFNN is used as an online identifier inside the active fault-tolerant controller of a plant-protection UAV. Each output weight is updated by a convex combination (mixing parameter \(q\)) of the ordinary integer-order gradient and a fractional-order (\(\upsilon\)) gradient of the squared-error cost \(\varepsilon\). The fractional term, obtained from the Riemann–Liouville derivative of the cost with respect to the weight, contributes a \(w_i^{1-\upsilon}\) factor scaled by \(1/\Gamma(2-\upsilon)\), which lets the step adapt to the current weight magnitude.

\[ \begin{aligned} -\nabla_{w_i}\varepsilon(n) &= \phi_i(x,x_i)\, e_k(n), \\ -\nabla^{\upsilon}_{w_i}\varepsilon(n) &= \phi_i(x,x_i)\, e_k(n)\, \frac{w_i^{\,1-\upsilon}(n)}{\Gamma(2-\upsilon)}, \\ w_i(n+1) &= w_i(n) + e_k(n)\,\Big( q\,\hbar + (1-q)\,\hbar_{\upsilon}\, w_i^{\,1-\upsilon}(n) \Big)\,\phi_i(x,x_i) \end{aligned} \]

where \(w_i\) is the \(i\)-th output weight, \(n\) the iteration index, \(e_k(n)\) the instantaneous output error, \(\phi_i(x,x_i)\) the \(i\)-th Gaussian basis activation, \(\upsilon \in (0,1)\) the fractional order, \(q \in [0,1]\) the convex mixing parameter, \(\hbar\) and \(\hbar_{\upsilon}\) the integer- and fractional-order step sizes, and \(\Gamma(\cdot)\) the gamma function (folded into \(\hbar_{\upsilon}\) in the combined form).

Reference: Lianghao Hua, Jianfeng Zhang, Dejie Li, Xiaobo Xi, "Fractional Gradient Descent RBFNN for Active Fault-Tolerant Control of Plant Protection UAVs", Computer Modeling in Engineering & Sciences 2024. https://doi.org/10.32604/cmes.2023.030535

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