## 3 Design of the Master and Auxiliary Compound Control
### Variable Universe Fuzzy PID Controller
#### 3.1.2 Contraction-Expansion Factor Adjustment Module  
The idea of a variable domain can be understood as the domain of adjusted input and output variables according to certain control rules according to actual requirements. Let us set [\(-\)E, E] and [\(-\)EC, EC] as the initial universe of input variables \(e\) and \(ec\), respectively. The initial output domains of the proportional coefficient increment, \(\Delta K_{p}\), the integral coefficient increment \(\Delta K_{i}\) and the differential coefficient increment \(\Delta K_{d}\) are [\(-\)Up, Up], [\(-\)Ui, Ui] and [\(-\)Ud, Ud], respectively. The initial domains are transformed into [\(-a(e)\)E, \(a(e)\)E], [\(-a(ec)\)EC, \(a(ec)\)EC], [\(-\beta(kp)\)Up, \(\beta(kp)\)Up], [\(-\beta(ki)\)Ui, \(\beta(ki)\)Ui] and [\(-\beta(kd)\)Ud, \(\beta(kd)\)Ud] through the contraction-expansion factor, that is, the domain of work changes with the change in the contraction-expansion factor, ensuring the number of actual work fuzzy values is always in the highest position, and a relatively satisfactory control effect is obtained. The factors \(a(x)\) and \(\beta(x)\) are the contraction-expansion factors of the input and output domains, respectively. There are currently three design methods: the functional contraction-expansion factor, the fuzzy reasoning contraction-expansion factor, and the error classification contraction-expansion factor. This paper chooses the calculation method based on the fuzzy inference type contraction-expansion factor, because the contraction-expansion factor calculation model based on fuzzy rules satisfies the monotonicity, duality, coordination, normality and zero avoidance of the contraction-expansion factor, and avoids the functional contraction-expansion factor's calculation model parameter selection [15; 25]. At the same time, fuzzy rules are used to express the change law of the contraction-expansion factor to realize the online automatic adjustment of the contraction-expansion factor. The contraction-expansion factor model based on fuzzy rules is based on the actual control deviation and deviation change rate, and uses easy-to-understand language to describe the change law of the universe. That is, the universe shrinks when the deviation becomes smaller, and the universe expands when the deviation becomes larger. The size of the quantization factor and the scale factor actually reflects the expansion change of the corresponding domain. The fuzzy reasoning expansion factor is used to establish another fuzzy controller on the basis of the basic fuzzy PID controller, in order to modify the parameters of the quantization factor and the scale factor. (1) Input universe contraction-expansion factor fuzzy control  
The contraction-expansion factors \(a(e)\) and \(a(ec)\) of the input universe are divided into five fuzzy language variables, which are extra-small (VS), small (S), medium (M), large (B), extra-large (VB). The uniform triangular membership distribution function is used, as shown in Figure 12a. The adjustment rules are as follows: when \(e\) and \(ec\) are large, the input domain should remain large; when \(e\) and \(ec\) are small, reduce the input domain to improve the pressure control accuracy, as shown in Table 5.  
Figure 12: The contraction–expansion factors’ membership functions: (**a**) Input variables \(a(e)\) and \(a(ec)\); (**b**) Output variables \(\beta(kp)\), \(\beta(ki)\) and \(\beta(kd)\). 2. Output universe contraction-expansion factor fuzzy control  
The contraction-expansion factors \(\beta\)(_kp_), \(\beta\)(_ki_) and \(\beta\)(_kd_) of the output universe are divided into seven fuzzy language variables, which are extremely-small (_Z_), very small (VS), pretty small (S), small (SB), medium (M), large (B), extra-large (VB). A triangular membership function is used, as shown in Figure 12b. The adjustment rule of the output universe contraction-expansion factor: when \(e\) and \(ec\) are large, and the signs of the two are the same, this indicates that the target value and the process variable are very different.