## 3 Adaptive Fractional-Order Anti-Saturation Synchronous Control Method
### Adaptive Fractional-Order Anti-Saturation Module (AFOAM)
#### 3.2.2 AFOAM Design and Implementation
Because the FOLLC has equal-order properties, that is, the orders of the differential operator in the numerator and denominator polynomials are the same, we can deduce that the FOLLC has the following properties: (1) \(A_{f}\) (\(\omega\)) \(\mid_{\omega\to 0}\) = 20 log\({}_{10}\)(\(K_{\rm C}\)) and (2) \(\varphi_{f}\) (\(\omega\)) \(\mid_{\omega\to 0}\) = \(\varphi_{f}\) (\(\omega\)) \(\mid_{\omega\to\infty}\) = 0; (3) when \(\omega=\omega_{m}\), the FOLLC reaches its extreme phase characteristic value. By deriving Equation (29), we obtain  
\[C_{f}(s)=K_{\mathrm{C}}\bigg{(}\frac{1+\lambda xs^{\alpha}}{1+\lambda s^{ \alpha}}\bigg{)}, \tag{23}\]0  
The parameter \(\lambda\) can be obtained by Equation (32). \[C_{f}(s)=K_{\mathrm{C}}\bigg{(}\frac{1+\lambda xs^{\alpha}}{1+\lambda s^{ \alpha}}\bigg{)}, \tag{23}\]1  
Because the middle- and low-frequency domains are the main control-frequency domains of the motor system, the FDC adjustment should avoid excessive influence on the amplitude and phase characteristics in these areas. However, owing to the continuity of the FDC adjustment, this influence cannot be completely avoided. To minimize the influence of the FDC adjustment caused by the AFOAM on the outer area of the specified angular frequency, especially the middle- and low-frequency domains, we need to let \(K_{\rm C}\) = 1. According to the above analysis, to minimize the influence range of the FDC adjustment, we determine \(A_{f}\) (\(\omega\)) \(\mid_{\omega\to+\infty}\) = \(\Delta A\) and \(\mid A_{f}\) (\(\omega_{k}\)) + \(\Delta A\mid\) \(\leq\)\(\epsilon_{1}\). Because the logarithmic function is a monotonic function, \(A_{f}\) (\(\omega_{k}\)) satisfies \(\left|\dot{g}(\omega_{k})\right|\) \(\leq\)\(\epsilon_{2}\) simultaneously, where \(\epsilon_{1}\) and \(\epsilon_{2}\) represent the minimum values. Thus, parameter \(x\) can be calculated by Equation (33). \[C_{f}(s)=K_{\mathrm{C}}\bigg{(}\frac{1+\lambda xs^{\alpha}}{1+\lambda s^{ \alpha}}\bigg{)}, \tag{23}\]2  
To simplify the calculation, we let \(A_{f}\) (\(\omega_{k}\)) \(\approx\)\(\Delta A\) and substitute it into Equation (28), and then we can obtain the relationship between \(\omega_{j}\) and \(\alpha\) by the following expression:  
\[C_{f}(s)=K_{\mathrm{C}}\bigg{(}\frac{1+\lambda xs^{\alpha}}{1+\lambda s^{ \alpha}}\bigg{)}, \tag{23}\]3  
where \(\omega_{j}\) satisfies the condition \(\omega_{m}\) = (\(\omega_{k}\omega_{j}\))\({}^{1/2}\), and (\(\omega_{k}\)-\(\omega_{j}\)) is the range of the FDC adjustment. Since the variables \(x\), \(\omega_{k}\), and \(\omega_{j}\) are not less than zero, the order \(\alpha\) should be in the range of [1, 2). We can see that when the order \(\alpha\) = 1, \(\omega_{j}\) = 0, which means that the FDC adjustment of the AFOAM will influence the amplitude and phase characteristics at all frequencies. Under this condition, because \(\omega_{m}\) = (\(\omega_{k}\omega_{j}\))\({}^{1/2}\), \(\omega_{m}\) will be zero by calculation, so the parameters of the AFOAM cannot be determined uniquely and adaptively. Therefore, to reduce the influence of the FDC adjustment, the order must be in the range of (1, 2). Through the derivation of Equation (30), we obtain  
\[C_{f}(s)=K_{\mathrm{C}}\bigg{(}\frac{1+\lambda xs^{\alpha}}{1+\lambda s^{ \alpha}}\bigg{)}, \tag{23}\]4  
For Equation (35), \(\left|\dot{g}(\omega_{k})\right|\) \(\leq\)\(\epsilon_{2}\), so there is a minimum value \(\epsilon\), making \(\left|\dot{g}(\omega_{k})\right|\) = 0. Combining Equations (34) and (35), and the above conditions, the optimal model for parameter \(\alpha\) can be deduced as follows:\[\left\{ \begin{array}{l}\min\ Z(\alpha)=\omega_{k}-\omega_{k}\big{(}-2\sqrt{x} \cos(\alpha\pi/2)/(x+1)\big{)}^{2/\alpha}\\ s.t.\quad 1<\alpha<2.\\ \left|A_{f}(\omega_{k})+\Delta A\right|\leq\varepsilon_{1}.\\ \left|\hat{g}(\omega_{k})-\varepsilon\right|=0.\\ \end{array} \right., \tag{36}\]  
Through Equation (36), parameter \(\alpha\) can be solved and determined.