## 3 Adaptive Fractional-Order Anti-Saturation Synchronous Control Method
### Adaptive Fractional-Order Anti-Saturation Module (AFOAM)  
#### 3.2.1 Peak Voltage Calculation and Amplitude Characteristic Adjustment  
Value Determination  
To avoid the nonlinear problem caused by the saturation effect, it is necessary to recognize the determinants of the control voltage in the feedback control loop and calculate the required suppression value of the control voltage. According to the dynamic response  
Figure 4: The saturation effect of the control voltage on the synchronization error of the dual-motor system: (**a**) under different control methods; (**b**) control voltage of the motor system in the range of [\(-u_{max}\), \(u_{max}\)]; (**c**) partial enlarged view of (**a**); (**d**) partial enlarged view of (**b**).  
characteristics of the control system and Equations (4) and (14), the transfer function from the input signal \(r\) to the feedback control voltage \(u_{FBCi}\) is as follows:  
\[G_{ui}(s)=\frac{u_{FBCi}(s)}{r(s)}=\frac{k_{fi}f_{i}(s)c_{i}(s)}{1+k_{fi}f_{i}(s )c_{i}(s)g_{yi}(s)}. \tag{16}\]  
In the frequency domain, \(s=j\omega\), so the amplitude characteristic \(A_{ui}(\omega)\) of Equation (16) is \(20\text{log}_{10}(\mid G_{ui}(\omega)\mid)\). Because the input signal amplitude in the frequency domain can be obtained by spectrum analysis, the calculation expression of the theoretical peak voltage of the control system can be expressed as  
\[u_{i}^{\text{max}}=\big{|}s_{p}(\omega_{k})\big{|}10^{A_{ui}(\omega_{k})/20}+u _{F\text{C}i}, \tag{17}\]  
where \(\mid s_{p}(\omega_{k})\mid\) is the amplitude spectrum of the input signal when \(\omega=\omega_{k}\), and \(u_{F\text{C}i}\) is the friction compensation voltage. Therefore, the amount by which the theoretical peak voltage exceeds the voltage limitation can be calculated as follows:  
\[\widetilde{u}_{i}=\text{sgn}_{i}(u)|u_{i}^{\text{max}}-u_{\text{max}}|, \tag{18}\]  
in which the symbolic function for determining whether the theoretical peak voltage exceeds the voltage limitation is defined as follows:  
\[\text{sgn}_{i}(u)=\left\{\begin{array}{ll}1,\ for\ u_{i}^{\text{max}}>u_{ \text{max}},\\ 0,\ for\ u_{i}^{\text{max}}\leq u_{\text{max}}.\end{array}\right.. \tag{19}\]  
According to Equation (17), we know that the control voltage can be changed by adjusting the spectrum of the input signal or the amplitude characteristic of the control system. Since the main work of this paper is the design of the synchronous control system, the adjustment of the input signal will directly affect the trajectory of the dual-motor system. Therefore, in order to avoid the saturation effect, the amplitude characteristic \(A_{ui}(\omega)\) at \(\omega_{k}\) must be adjusted to be the value as shown in Equation (20).  
\[\overline{A}_{ui}(\omega_{k})=20\text{log}_{10}\Big{|}10^{A_{ui}(\omega_{k})/ 20}-\widetilde{u}_{i}/\big{|}s_{p}(\omega_{k})\big{|}\Big{|}. \tag{20}\]  
The amplitude characteristic adjustment value of each motor system is  
\[\Delta A_{i}=\overline{A}_{ui}(\omega_{k})-A_{ui}(\omega_{k}). \tag{21}\]  
To ensure that the adjustment of the amplitude characteristic does not affect the unification of the dynamic response characteristics of the dual-motor system, it is necessary to unify the adjustment value to a smaller value \(\Delta A\), so that the anti-saturation module can satisfy the adjustment requirements of both feedback control loops simultaneously. The amplitude characteristic adjustment value can be determined as follows:  
\[\Delta A=-MAX\{|\Delta A_{1}|,|\Delta A_{2}|\}. \tag{22}\]