An optimized radial basis function neural network with modulation-window activation function

Abstract

It is a crucial basis to improve the performance of neural network by constructing an appropriate activation function. This paper proposes a novel modulation window radial basis function neural network (MW-RBFNN) with an adjustable activation function. In this MW-RBFNN, a raised cosine radial basis function (RC-RBF) is adaptively modulated by an exponential function, and served as a shape-tunable activation function of MW-RBFNN. Compared with the basic RC-RBF neural network, the approximating ability of MW-RBFNN is improved due to its shape-tunable activation function. Besides, the computation of MW-RBFNN is far less than that of Gaussian radial basis function neural network (GRBFNN) because the MW-RBFNN is compactly supported. The training algorithm of MW-RBFNN is provided and its approximating ability is proved. Moreover, the regulation mechanism of the modulation index for the NN’s performance is proved and the regulating algorithm of the modulation index in MW-RBFNN is given. The computational complexity of MW-RBFNN is also analyzed. Five typical application cases are presented to illustrate the effectiveness of this proposed MW-RBFNN.

Appendix I

MW-RBFNN-based Jacobian in PID control system.

The output of the MW-RBFNN shown in Fig. 8 can be rewritten as

$$ y_{m} = \sum\limits_{k = 1}^{r} {\left( {\omega_{k} + {\mathbf{V}}_{k} {\mathbf{x}}} \right)\phi_{Wk} ({\mathbf{x}})} = \sum\limits_{k = 1}^{r} {\left( {\omega_{k} + \sum\limits_{i = 1}^{3} {v_{k,i} x_{i} } } \right)\phi_{Wk} ({\mathbf{x}})} , $$

(43)

where x = [Δu(k), y(k), y(k-1)]^T, and $\phi_{Wk} ({\mathbf{x}})$ is descried by (10). Therefore,

$$ \frac{\partial y}{{\partial \Delta u}} \approx \frac{{\partial y_{m} }}{\partial \Delta u} = \sum\limits_{k = 1}^{r} {\left\{ {v_{k,1} \phi_{Wk} ({\mathbf{x}}) + \left( {\omega_{k} + \sum\limits_{i = 1}^{3} {v_{k,i} x_{i} } } \right)\frac{{\partial \phi_{Wk} ({\mathbf{x}})}}{\partial \Delta u}} \right\}} , $$

(44)

where,

$$ \frac{{\partial \phi_{Wk} ({\mathbf{x}})}}{\partial \Delta u} = \prod\limits_{i = 2}^{3} {e^{{ - \alpha \left[ {p_{i} (x_{i} ) - p_{i} (x_{i}^{(k)} )} \right]^{2} }} \left\{ {0.5 + 0.5\cos \left[ {\pi \left( {p_{i} (x_{i} ) - p_{i} (x_{i}^{(k)} )} \right)} \right]} \right\}} \times \frac{{\partial \zeta_{c} \left[ {p_{1} (x_{1} ) - p_{1} (x_{1}^{(k)} )} \right]}}{\partial \Delta u}. $$

(45)

where x₁ = Δu(k), p_i(x_i) is described by (5), and

$$ \zeta_{c} \left[ {p_{1} (x_{1} ) - p_{1} (x_{1}^{(k)} )} \right] = e^{{ - \alpha \left[ {p_{1} (x_{1} ) - p_{1} (x_{1}^{(k)} )} \right]^{2} }} \left\{ {0.5 + 0.5\cos \left[ {\pi \left( {p_{1} (x_{1} ) - p_{1} (x_{1}^{(k)} )} \right)} \right]} \right\}. $$

(46)

Thus, in (45), $\frac{{\partial \zeta_{c} \left[ {p_{1} (x_{1} ) - p_{1} (x_{1}^{(k)} )} \right]}}{\partial \Delta u}$ is obtained by (47), i.e.,

$$ \begin{aligned}& \frac{{\partial \zeta_{c} \left[ {p_{1} (x_{1} ) - p_{1} (x_{1}^{(k)} )} \right]}}{\partial \Delta u} = e^{{ - \alpha \left[ {p_{1} (x_{1} ) - p_{1} (x_{1}^{(k)} )} \right]^{2} }} \left\{ { - \frac{{\pi \sin \left[ {\pi \left( {p_{1} (x_{1} ) - p_{1} (x_{1}^{(k)} )} \right)} \right]}}{{2\left( {x_{1}^{(2)} - x_{1}^{(1)} } \right)}}} \right\} \\&\quad - \left\{ {1 + \cos \left[ {\pi \left( {p_{1} (x_{1} ) - p_{1} (x_{1}^{(k)} )} \right)} \right]} \right\}\left[ {\frac{{\alpha \left[ {p_{1} (x_{1} ) - p_{1} (x_{1}^{(k)} )} \right]e^{{ - \alpha \left[ {p_{1} (x_{1} ) - p_{1} (x_{1}^{(k)} )} \right]^{2} }} }}{{\left( {x_{1}^{(2)} - x_{1}^{(1)} } \right)}}} \right] \hfill \\ \end{aligned} $$

(47)