Neural Network Modeling Based on Lead Elements Approximation to $\mathrm{s}^{\alpha}$ and Its Application

To obtain a dynamic model with good generalization and a high accuracy, a new neural network modeling method with fractional filters is proposed. The filter contains the operator
$\mathrm{s}^{\alpha}$
derived from the time domain, but the geometric and physical meaning of
$\mathrm{s}^{\alpha}$
is not clear and the calculation speed is slow. In the present study,
$\mathrm{s}^{\alpha}$
is deduced from the frequency domain (sine and cosine functions) to make it easier to understand, and the cascade leading elements (CL) are used to approximate
$\mathrm{s}^{\alpha}$
reasonably, which makes the calculation speed of
$\mathrm{s}^{\alpha}$
faster than other methods. A particle swarm optimization (PSO) algorithm is used in neural network modeling and its application (PID tuning). The simulation results show that the CL algorithm realizes good time savings, while the model has good generalization and high accuracy. It can be applied to a wide variety of projects.