## 3 ADRC Design and Parameterization
### Fuzzy-ADRC Design  
In practice, the system receives a small disturbance from the outside world, which will cause the motor speed to decrease and not reach the set speed value after ADRC closed-loop control and there is still a gap in the set speed. The motor's measured speed value and set value error are both tiny, which makes it challenging to employ closed-loop control to produce the necessary regulating effect. The speed error can be scaled up by introducing a proportionality factor and then passing it to the ADRC for further control. In order to achieve flexible adjustment of the proportionality coefficient, this paper introduces FC and utilizes the fuzzy rule of \"small error, large gain\". The obtained small error in rotational speed is amplified by the proportional coefficient and then transmitted to the ADRC system. The AC PMSM servo system's robustness and disturbance compensation are significantly enhanced by this approach.  
The FC used in this paper utilizes the ability of the adaptive method of fuzzy control to enable the controller to provide the best estimate of the desired parameters within a certain range. First, the system's fuzzy inference rules are formulated. Then, a fuzzy rule table is created, utilizing the fuzzy subset affiliation function for the velocity error, acceleration, and the fuzzy control model of each parameter. This fuzzy rule table is designed to effectively compensate for minor disturbances within the system. After defuzzification, the necessary proportional amplification factor for the speed error can be identified and paired with the regulation of the ADRC and PI controllers to determine the motor's final speed output. Figure 4 is the overall schematic diagram of the Fuzzy-ADRC control system.  
Where err is the output variable and r and rc are the fuzzy input variables for the controller, respectively. In their domain S, seven subsets of languages are specified as \"negative large (n3)\", \"minus medium (n2)\", \"negative small (n1)\", \"zero (O)\", \"positive small (p1)\", \"plus medium (p2)\", and \"positive large (p3)\". The fundamental domains of the input quantities r and rc are [\(-\)1000, 1000] and [\(-\)100, 100], respectively, and the fundamental domain of the output quantity err is [\(-\)100, 100]. The affiliation function of each variable has seven grades corresponding to the fuzzy linguistic variables, and the affiliation function curves are in the form of a combination of affiliation functions with the Gaussian normal distribution type (gaussmf) on both sides and the triangular type (trimf) in the middle.  
The defuzzification processing methodology uses the average weighting method, and the fuzzy inference is of the Mamdani type. Table 2 displays a list of fuzzy control rules for determining parameter rectification.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline r/rc & n3 & n2 & n1 & O & p1 & p2 & p3 \\ \hline n1 & O & O & O & n1 & n2 & n2 & n3 \\ n2 & O & n1 & n1 & n1 & n2 & n3 & n3 \\ n1 & n1 & n1 & n2 & n2 & n3 & n3 & p2 \\ O & n2 & n2 & p3 & p3 & p3 & p2 & p2 \\ p1 & n2 & p3 & p3 & p2 & p2 & p1 & p1 \\ p2 & p3 & p3 & p2 & p1 & p1 & p1 & O \\ p3 & p3 & p2 & p1 & p1 & O & O & O \\ \hline \end{tabular}
\end{table}
Table 2: Table of fuzzy control rules.  
Figure 4: PMSM speed control system with Fuzzy-ADRC.  
After system debugging, it was found that the proportional coefficient value output is too large to cause the system oscillation caused by speed overshooting, so in this paper, r and rc were multiplied by coefficients of 0.001. The following equation is constructed to represent the motor speed of the fuzzy control output:  
\[Nref=(30/pi)*1.18\omega_{m}+0.1 err \tag{36}\]  
The values of the final parameters of the simulation model constructed in this paper are listed in Table 3.