## 3 Adaptive Fractional-Order Anti-Saturation Synchronous Control Method
### Adaptive Fractional-Order Anti-Saturation Module (AFOAM)
#### 3.2.2 AFOAM Design and Implementation  
A. AFOAM design  
Combined with the above theoretical peak voltage calculation, we can obtain the saturation situation of the control voltage by analyzing the input signal when the frequency domain characteristics of the dual-motor system are determined. On this basis, we developed an AFOAM based on fractional-order frequency-domain control theory to realize the specific amplitude characteristic adjustment in which the saturation status of the control voltage is determined by analyzing the input signal when the frequency-domain characteristics of the dual-motor system are determined. The implementation block diagram of AFOAM is shown in Figure 5. When the input signal is determined, the controller calculates the peak voltage according to Equation (17) to adjust the control system to avoid the saturation effect, i.e., not adjust the control parameters in real time during motion. As shown in Equation (20), we only need to adjust the amplitude characteristic at \(\omega_{k}\). Since the adjustment at a specific frequency will affect the amplitude and phase characteristics of the surrounding frequency and will further influence the dynamic response performance of the feedback control system, we propose the AFOAM in this paper to adjust the specific FDC accurately and flexibly by introducing the fractional-order lead-lag controller (FOLLC) as the fractional-order FDC planning target. The expression for the FOLLC is described as follows:  
\[C_{f}(s)=K_{\mathrm{C}}\bigg{(}\frac{1+\lambda xs^{\alpha}}{1+\lambda s^{ \alpha}}\bigg{)}, \tag{23}\]  
where \(K_{\mathrm{C}}\), \(\lambda\), and \(x\) are coefficients. The order \(\alpha\) can be any real number in the range (0, 2), which enables the FOLLC to accurately describe the specified FDC at the specified frequency through accurate parameter selection. The transfer function from the input signal \(r\) to the feedback control voltage \(u_{FBCi}\) with the AFOAM is as follows:  
\[\overline{G}_{ui}(s)=\frac{k_{fi}f_{i}(s)c_{i}(s)T_{a}(s)}{1+k_{fi}f_{i}(s)c_{ i}(s)g_{yi}(s)T_{a}(s)}, \tag{24}\]  
where \(T_{a}(s)\) represents the transfer function of the AFOAM. With the AFOAM, the voltage amplitude characteristic of the feedback control system is changed as  
\[\overline{A}_{ui}(\omega)=A_{ui}(\omega)+A_{f}(\omega), \tag{25}\]  
where \(A_{f}\) (\(\omega\)) is the amplitude characteristic of the FOLLC. According to the transfer block diagram of the proposed adaptive fractional-order anti-saturation synchronous control method, the transfer function of AFOAM can be expressed as follows:  
\[T_{a}(s)=\frac{K_{\mathrm{C}}(1+\lambda xs^{\alpha})}{(1+\lambda s^{\alpha}-K _{\mathrm{C}}(1+\lambda xs^{\alpha}))k_{fi}f_{i}(s)c_{i}(s)g_{yi}(s)+\lambda s ^{\alpha}+1}. \tag{26}\]  
B. Parameter determination  
It can be seen from Equation (26) that the parameters of the AFOAM can be determined by selecting the parameters of the FOLLC. Therefore, we first need to establish the relationship between the parameters of the FOLLC and the FDC adjustment. Equation (23) can be rewritten as  
\[C_{f}(j\omega)=K_{\mathrm{C}}\bigg{(}\frac{1+x\lambda(j\omega)^{\alpha}}{1+ \lambda(j\omega)^{\alpha}}\bigg{)}. \tag{27}\]  
The amplitude and phase characteristics of the FOLLC can be calculated as follows:  
\[A_{f}(\omega)=20\log_{10}\bigg{(}K_{\mathrm{C}}\sqrt{\frac{1+x^{2}\lambda^{2 }\omega^{2\alpha}+2x\lambda\omega^{\alpha}\cos(\alpha\pi/2)}{1+\lambda^{2} \omega^{2\alpha}+2\lambda\omega^{\alpha}\cos(\alpha\pi/2)}}\bigg{)}, \tag{28}\]  
Figure 5: Implementation block diagram of AFOAM. \[\varphi_{f}(\omega)=\tan^{-1}\biggl{[}\frac{(x-1)\lambda\omega^{\alpha}\sin(\pi\pi/ 2)}{1+(x+1)x\lambda^{2}\omega^{2\alpha}\cos(\pi\pi/2)}\biggr{]}, \tag{29}\]  
Let  
\[g(\omega)=K_{\rm C}\sqrt{\frac{1+x^{2}\lambda^{2}\omega^{2\alpha}+2x\lambda \omega^{\alpha}\cos(\alpha\pi/2)}{1+\lambda^{2}\omega^{2\alpha}+2\lambda\omega^ {\alpha}\cos(\pi\pi/2)}}. \tag{30}\]  
The logarithmic function is monotonic, and thus, \(g(\omega)\) represents the trend of \(A_{f}\) (\(\omega\)).