# Research on Variable Universe Fuzzy Multi-Parameter Self-Tuning PID Control of Bridge Crane
## 3 Variable Domain Fuzzy Multi-Parameter Self-Tuning PID Controller
### Fuzzy Control Rules Determination  
The foundation for implementing fuzzy control is the fuzzy control rule table. In practice, the fuzzy control rules are usually designed based on expert experience. Considering the importance of parameters like proportional, integral, and derivative in PID control, the following adjustment principles have been formulated:  
1. For large differences \(\left|e\right|\) between the desired and actual values, a larger \(K_{P}\) parameter is selected to quickly reach the predetermined position. To avoid issues such as sudden changes in the rate of change and integral saturation, smaller \(K_{I}\) and \(K_{D}\) parameters are used. 2. When the difference \(\left|e\right|\) and the rate of change of the difference \(\left|ec\right|\) are moderate, smaller \(K_{P}\) and \(K_{I}\) parameters are selected to prevent excessive overshoot and other problems. At the same time, a moderate \(K_{D}\) parameter is selected to balance the response speed of the crane. By doing so, the crane's motion can be controlled smoothly and accurately. 3. When the error \(\left|e\right|\) is small, the focus is on maintaining a stable operating state, so smaller \(K_{P}\) and \(K_{I}\) parameters are selected. However, in order to prevent load oscillation and improve the crane's anti-interference ability, a larger \(K_{D}\) parameter is selected when the rate of change of the error \(\left|ec\right|\) is small. This is because a larger \(K_{D}\) can help to dampen any oscillations that may occur. Conversely, when the rate of change of the error \(\left|ec\right|\) is large, a smaller \(K_{D}\) parameter is selected to prevent overshoot and instability. Based on the above three parameter adjustment rules, the input and output variables of the fuzzy PID controller are partitioned into 7 levels. Each represents a fuzzy subset: negative big (NB), negative medium (NM), negative small (NS), zero (Z), positive small (PS), positive medium (PM), and positive big (PB). These fuzzy subsets are utilized to denote the linguistic values of the input and output variables, and the fuzzy control rules are formulated based on them. The \(\Delta K_{P}\), \(\Delta K_{I}\) and \(\Delta K_{D}\) values obtained through fuzzy rules reasoning are represented in the following Tables 1-3 where the fuzzy control rules for \(\Delta K_{P}\), \(\Delta K_{I}\) and \(\Delta K_{D}\) are shown [31]. From Figures 3-5, we can more intuitively see the quantitative relationship between input \(e\), \(ec\) and output \(\Delta K_{P}\), \(\Delta K_{I}\) and \(\Delta K_{D}\). To tune the PID and PD controllers of this system, we use the same fuzzy control rules to adjust the gain tuning quantities \(\Delta K_{P}\), \(\Delta K_{I}\) and \(\Delta K_{D}\). Triangular membership functions are used for the fuzzy input and output membership functions, and the centroid method is used to defuzzify the fuzzy quantities, which provides us with the PID parameter tuning quantities \(\Delta K_{P}\), \(\Delta K_{I}\) and \(\Delta K_{D}\). The initial \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\Delta K_{P}\)} & \multicolumn{8}{c}{_ec_} \\ \cline{2-10}  & & **NB** & **NM** & **NS** & **Z** & **PS** & **PM** & **PB** \\ \hline \multirow{8}{*}{\(e\)} & NB & PB & PB & PM & PM & PS & PS & Z \\ \cline{2-10}  & NM & PB & PB & PM & PM & PS & Z & Z \\ \cline{2-10}  & NS & PM & PM & PM & PS & Z & NS & NM \\ \cline{2-10}  & Z & PM & PS & PS & Z & NS & NM & NM \\ \cline{2-10}  & PS & PM & PS & Z & NS & NM & NM & NM \\ \cline{2-10}  & PM & Z & Z & NS & NM & NM & NM & NB \\ \cline{2-10}  & PB & Z & NS & NS & NM & NM & NB & NB \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fuzzy control rules for \(\Delta K_{P}\).