## 5 Compensation for Frictional Disturbance Observer
### Design of the Friction Disturbance Observer  
The schematic diagram of the classical disturbance observer is shown in Figure 5. The equivalent disturbance \(d\) caused by the disturbance is generated between the output of the  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline \(T_{c}/N\!\cdot\!m\) & 0.459 \\ \(T_{s}/N\!\cdot\!m\) & 0.459 \\ \(v/rad\!\cdot\!s^{-1}\) & 0.3236 \\ \(B/N\!\cdot\!m\!\cdot\!s\!\cdot\!rad^{-1}\) & 0.005735 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Static parameters of the Stribeck friction force.  
Figure 4: Stribeck friction model.  
controlled object and the output of the nominal model. Then, the equivalent compensation is added to the control input, achieving complete suppression of the disturbance.  
\(G(s)\) is the PID controller as the main controller in the speed loop; the inverse of the nominal model of the actual object \(G_{p}(s)\) is denoted as \(G_{p}^{-1}(s)\); \(Q(s)\) is the low-pass filter; and \(\hat{d}\) is the estimated value of the external disturbance \(d\).  
The low-pass filter \(Q(s)\) is designed to have different characteristics in different frequency bands: in the low-frequency band, \(Q(s)\approx 1\), which allows the system to effectively suppress external disturbances with lower frequencies; in the high-frequency band, \(Q(s)\approx 0\), which significantly reduces the impact of measurement noise on the system.  
Next, the low-pass filter \(Q(s)\) is designed in the following form [30]:  
\[Q(s)=\frac{g_{0}}{s^{n}+g_{n-1}s^{n-1}+\cdots+g_{1}s+g_{0}} \tag{37}\]  
where the coefficients \(g_{0}\), \(g_{1}\),..., \(g_{n}\) represent the parameters of the filter and are associated with the dynamic characteristics of the control system.  
In the system studied in this paper, \(Q(s)\) is represented as follows:  
\[Q(s)=\frac{1}{\tau s+1} \tag{38}\]  
In summary, the model of the frequency-domain-based DOB designed in this paper is as follows:  
\[Q(s)=\frac{s}{0.002s+1} \tag{39}\]  
After introducing the disturbance observer into the dual-motor servo system, its structure is shown in Figure 6. The expression of the disturbance observer is given by the following equation:  
\[\hat{d}=Q(s)(J_{n}s\omega_{\mathrm{out}}\ -K_{\mathrm{i}}Ni) \tag{40}\]  
Using Equation (38), the following can be derived:  
\[\hat{d}=\frac{1}{\tau s+1}\big{(}J_{s}s\omega_{\mathrm{out}}\ -k_{t}i_{q}\big{)} \tag{41}\]  
From the above equation, it can be seen that the disturbance observer contains a first-order differential term. Even if the noise amplitude is small, as long as the frequency is sufficiently high, the rate of change of the deviation may still become large, leading to a larger output from the differential part, which ultimately affects the system performance  
Figure 5: Control structure of the classical DOB.  
and makes it sensitive to high-frequency noise. To avoid this issue, the above equation is modified as follows:  
\[\dot{d}=\frac{I_{n}\omega_{\text{out}}}{\tau}-\frac{1}{\tau s+1}\bigg{(}\frac{I_ {n}\omega_{\text{out}}}{\tau}+K_{\text{i}}Ni\bigg{)} \tag{42}\]  
The improved disturbance observer is shown in Figure 7.