## 4 Simulation  
The model description of a PMSM with friction disturbance is realized by Equation (10). In the simulation, the model parameters are set as \(\sigma_{0}=260\), \(\sigma_{1}=2.5\), \(\sigma_{2}=0.02\), \(f_{c}=0.28\), \(f_{s}=0.34\), \(k_{v}=0.01\), \(J=1\), \(b=1\). Firstly, to show the influence of friction on the control performance, the PD control algorithm is adopted to the controlled object. According to the bandwidth-based parametric tuning method in linear systems, the proportional integral differential (PID) parameters can be adjusted according to the closed-loop bandwidth. The relationship between PID parameters and control bandwidth is approximately expressed as follows:  
\[\left\{ \begin{array}{l}k_{p}=2\omega_{0}\\ k_{i}=\omega_{0}^{2}\\ k_{d}=\frac{1}{2}\omega_{0}\end{array} \right. \tag{27}\]  
Figure 5: Block diagram of speed controller for PMSM. where \(\omega_{0}\) is the closed-loop bandwidth. In the simulation, the closed-loop bandwidth of the speed control is set as \(\omega_{0}=10\), so the parameters of the PD controller in the speed control loop are set as \(k_{p}=20\) and \(k_{d}=5\). In order to reduce the influence of the current response delay on speed control, the bandwidth of the current control loop should be much larger than that of the speed control loop. Therefore, the closed-loop bandwidth of the current control is set as \(\omega_{0}=200\), so the parameters of two PI controllers in the current control loop are set as \(k_{p}=400\) and \(k_{i}=40,000\). Figure 6 shows the simulation results of position and speed tracking. Due to the interference of friction torque, the waveform in Figure 6 is distorted. When the speed passes through the zero point, the position tracking has a flat top, and the speed tracking has a dead zone. It indicates that, due to the nonlinear influence of friction torque, especially at the zero point of angular velocity, the control performance of the linear PD is utterly poor. To verify that the proposed algorithm can effectively restrain friction torque, comparative simulation experiments are carried out, in which three algorithms are designed as the speed controller, including the nonlinear proportional differential (NPD), typical ADRC and SRBF-MADRC. The parameters of the three control algorithms are shown in Table 1. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**SRBF-MADRC**} \\ \hline TD: & \(\delta=5\) \\ NLSEF: & \(k_{p}=20,k_{d}=5,\alpha=0.75,\lambda=0.02\) \\ MESO: & \(\beta_{11}=100\), \(\beta_{12}=300\), \(\hat{f}_{c}=0.22\), \(\hat{k}_{v}=0.008\), \(b_{0}=1\) \\ SRBF: & \(c=\begin{bmatrix}-2&-1&1&2\end{bmatrix}\), \(b_{j}=0.5\), \(\eta=0.3\), \(\gamma=0.05\) \\ \hline \hline \multicolumn{2}{c}{**ADRC**} \\ \hline TD: & \(\delta=5\) \\ NLSEF: & \(k_{p}=20,k_{d}=5,\alpha=0.75,\lambda=0.02\) \\ ESO: & \(\beta_{01}=1000\), \(\beta_{02}=3000\), \(\beta_{03}=10,000\), \(b_{0}=1\) \\ \hline \hline \multicolumn{2}{c}{**NPD**} \\ \hline TD: & \(\delta=5\) \\ NLSEF: & \(k_{p}=20,k_{d}=5,\alpha=0.75,\lambda=0.02\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of three control algorithms. Figure 6: Simulation results of position and speed tracking. Notes: (**a**) Position tracking; (**b**) Speed tracking. Before conducting comparative experiments, it is necessary to verify the convergence of the SRBF neural network. The main principle of an RBF based supervisory control system is that the NPD control plays a leading role and the SRBF plays a regulating role when control errors occur. The SRBF employs the gradient descent method to realize the learning of neural network weights, and finally makes the output errors converge to the threshold. According to the authors in [32], the learning rate factor \(\eta\) is adjusted to coordinate the learning speed and stability of the SRBF. When \(\eta=0.3\), the SRBF obtains a fine learning speed and stability.