# Research on the Autonomous Control Technology Used in the Slurry Mixing System of Cementing Units
## 3 Design of the Slurry Mixing System Controller  
### Design of the Smith Fuzzy PID Controller  
The conventional PID control and the typical Smith predictive compensation algorithm for the time-delay system still have limitations in terms of the control of the slurry density time-delay system. Therefore, a composite control strategy combining PID control, Smith predictive compensation control, and fuzzy control is proposed, namely Smith fuzzy PID control.  
In practical applications, the principle of the Smith fuzzy PID control scheme is shown in Figure 3, in which the design of the fuzzy controller mainly includes three processes: fuzzification, fuzzy inference, and defuzzification [23].  
(1)  
Fuzzification  
The input of the fuzzy controller is the difference \(e\) between the measured density of the densitometer and the set density, and the change rate \(ec\) of the difference between the measured density and the set density. The output is the correction values \(\Delta K_{p}\), \(\Delta K_{i}\), and \(\Delta K_{d}\) of the PID parameters. In this paper, the domain of discourse of input and output variables is described as the set {NB,NM,NS,ZO,PS,PM,PB}. The domain of discourse range of each input and output variable is shown in Table 1.  
According to the characteristics of the slurry mixing control system, the input and output of the system using a combination of linear trigonometric functions and Gaussian functions to describe the membership function, in which the Gaussian function is selected on both sides and the trigonometric function is selected in the middle. Through the fuzzy control design toolbox of MATLAB R2023a, the types and parameters of input and output membership functions of the system can be designed, and the input membership function, as  
(3) Defuzzification  
What is obtained through fuzzy inference is a fuzzy quantity, and defuzzification is the process of converting the abstract results obtained by fuzzy inference into precise values that can be recognized by the actuator through the use of various methods. There are a number of defuzzification methods, and the applicable calculation method should be reasonably selected according to the control requirements of the density control system. Commonly used methods include the maximum membership degree method, the center of gravity method, and the weighted fuzzy mean method. The center of gravity method can respond to the input signal even if the input signal changes ever so slightly and has  
Figure 4: Input membership function. (**a**) \(\varepsilon\) membership function plots. (**b**) \(\alpha\) membership function plots. (**c**) \(\alpha\) membership function plots.  
Figure 5: Output membership function. (**a**) \(K_{p}\) membership function plots. (**b**) \(K_{i}\) membership function plots. (**c**) \(K_{d}\) membership function plots.  
the characteristics of precise values and high accuracy. In light of the above, the center of gravity method was used for the defuzzification process described herein.  
After the precise \(\Delta K_{p}\), \(\Delta K_{i}\), and \(\Delta K_{d}\) are obtained, tuning should also be combined with the actual PID parameters. The tuning expression is as follows:  
\[K_{\rm X}=K_{\rm xo}+k_{\rm X}\cdot\Delta K_{X} \tag{7}\]  
where \(K_{\rm xo}\) is the initial value of each PID parameter; \(k_{x}\) is the quantization factor; and \(\Delta K_{x}\) is the correction value after defuzzification.  
Relative parameters of the system are modified by fuzzy control, which successfully overcomes the shortcomings of the system, such as low control accuracy and poor stability.