## 3 Adaptive Fractional-Order Anti-Saturation Synchronous Control Method  
### FDC Unification Module (FDC-Ulm)  
According to the frequency-domain control theory, the dynamic response characteristics of the LTI system are determined by the FDC of the control system. In literature [28], the relationship between the input and output signals of the control system was analyzed in detail, and the output signal corresponding to the input signal \(r=\sin(\omega_{r}t)\) was obtained as follows:  
\[y=K\sin(\omega_{r}(t-\Delta t)). \tag{7}\]  
The signals have the same frequency (\(f=\omega_{r}/2\pi\)), but there is a certain time delay \(\Delta t_{i}\) and an amplitude ratio \(K_{i}\) between them. Therefore, the synchronization error expression of the dual-motor system can be defined as follows:  
\[e_{syn}=K_{1}\sin(\omega_{r}(t-\Delta t_{1}))-K_{2}\sin(\omega_{r}(t-\Delta t _{2})), \tag{8}\]  
\[\Delta t_{i}=\varphi_{i}/\omega_{r}, \tag{9}\]  
\[K_{i}=10^{A_{i}/20}, \tag{10}\]  
where \(\varphi_{i}(\omega)=\arctan\left(\text{Im }[G_{i}(\omega)]/\text{Re }[G_{i}(\omega)]\right)\) and \(A_{i}(\omega)=20\text{log}_{10}\left|\right.G_{i}(\omega)\left|\right.\) are the phase and amplitude characteristics of the transfer function shown in Equation (5), respectively. According to Equations (8)-(10), to achieve high-synchronous motion performance of the dual-motor system, both feedback control systems must have consistent FDC. That is, the dual-motor system must satisfy the following condition:  
\[C(s)F(s)G_{Y}(s)K_{F}=0, \tag{11}\]  
where \(C(s)=[c_{1}(s),c_{2}(s)]\), \(G_{Y}(s)=\text{diag}(g_{y1}(s),g_{y2}(s))\). \(F(s)=\text{diag}(f_{1}(s),f_{2}(s))\), in which \(f_{1}(s)\) is the transfer function of the FDC-UM. \(K_{F}=[k_{f1},\,-k_{f2}]^{T}\) is the gain matrix used to adjust the  
Figure 2: The transfer block diagram of the proposed adaptive fractional-order anti-saturation synchronous control method. control performance of the system. According to Equations (5) and (11), we can derive the following when \(\varphi_{1}\geq\varphi_{2}\) in the main frequency domain. \[\left\{ \begin{array}{l}f_{1}(s)=\Big{(}k_{f1}c_{1}(s)g_{y1}(s)\Big{)}^{-1}k_{f2 }c_{2}(s)g_{y2}(s)\\ f_{2}(s)=1\end{array} \right., \tag{12}\]  
When \(\varphi_{1}<\varphi_{2}\) in the main frequency domain,  
\[\left\{ \begin{array}{l}f_{1}(s)=1\\ f_{2}(s)=\Big{(}k_{f2}c_{2}(s)g_{y2}(s)\Big{)}^{-1}k_{f1}c_{1}(s)g_{y1}(s)\\ \end{array} \right.. \tag{13}\]  
After the compensation, the transfer function of the feedback control system is as follows:  
\[G_{fi}(s)=\frac{y_{i}(s)}{r(s)}=\frac{k_{fi}f_{i}(s)c_{i}(s)g_{yi}(s)}{1+k_{fi} f_{i}(s)c_{i}(s)g_{yi}(s)}. \tag{14}\]  
Based on the analysis performed above, we know that the dynamic response characteristics of the control system are related to the frequency of the control signal. Through compensation, the dual-motor system meets the requirements of \(A_{f1}(\omega)=A_{f2}(\omega)\) and \(\varphi_{f1}(\omega)=\varphi_{f2}(\omega)\), so the two motor systems possess a unified dynamic response characteristic and thus can achieve a synchronous motion. To verify the effectiveness of the FDC-UM, we used the chirp signal for the experiments. Figure 3 shows the working principle of the FDC-UM and the synchronization error comparison by the control methods of the PID feedback control method (Method 1 in Figure 3a), the force compensation cross-coupling synchronous control method (Method 2 in Figure 3b), and the FDC-UM based synchronous control method (Method 3 in Figure 3c). Figure 3d shows that both Method 2 and Method 3 can ensure excellent synchronous-motion performance of the dual-motor system, and the FDC-UM-based synchronous control method achieves the best performance, which can avoid the effects of control signal variation in frequency. Figure 3: Working principle of the FDC-UM and comparison of synchronization errors with three different control methods: PID feedback method, force compensation method, and the proposed FDC-UM method.