## 4 Controller Design
Consequently, fixed factor parameters will render the controller incapable of adapting to complex and variable real-world operating conditions, significantly limiting its versatility and adaptability. To ensure that the control performance consistently maintains excellent robustness, as well as dynamic and static performance, an intelligent optimization algorithm is employed to dynamically adjust the quantization factors, _Ke_ and _Kex_; and the proportional factors, _Kup_, _Kui_, and _Kud_, of the fuzzy controller in real time. PSO is an algorithm that iterates the individual positions of a group of particles to finally obtain the best solution [31; 32; 33]. However, when addressing complex high \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(e\) & _ec_ = NB & _ec_ = NM & _ec_ = NS & _ec_ = ZO & _ec_ = PS & _ec_ = PM & _ec_ = PB \\ \hline NB & PB/PB/PB & PB/PB/PB & PB/PB/PB & PB/PB/NB & PS/NS/NB & ZO/NM/NM & NS/NB/NS \\ NM & PB/PB/PB & PB/PB/PB & PM/PM/PM & PM/PM/PM & PM/PM/NM & ZO/PM/ZO & NS/NB/PS & NM/NB/PM \\ NS & PB/PB/PB & PM/PM/PM & PM/PS/PM & PS/PS/NS & NS/NB/PM & NM/NB/PB & NB/NB/PB \\ ZO & ZO/ZO/ZO/ZO & ZO/ZO/ZO & ZO/ZO/ZO & ZO/ZO/ZO/ZO & ZO/ZO/ZO & ZO/ZO/ZO/ZO \\ PS & NB/NB/PB & NM/NB/PB & NS/NB/PM & PS/PS/NS & PM/PS/PM & PM/PM/PM & PB/PB/PB \\ PM & NM/NB/PM & NS/NB/PS & ZO/NM/ZO & PM/PM/NM & PB/PM/PM & PB/PB/PB & PB/PB/PB \\ PB & NS/NB/NS & ZO/NM/NM & PS/NS/NB & PB/PB/NB & PB/PB/PB & PB/PB/PB & PB/PB/PB \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fuzzy rules for _dKp_, _dKi_, and _dkd_. Figure 5: Schematic diagram of fuzzy PID control. -dimensional optimization problems, the issue of PSO being prone to falling into local optima becomes particularly prominent. During the later stages of the search process, the diversity of the particle swarm diminishes rapidly, leading to premature convergence and preventing the algorithm from locating the global optimal solution [34; 35]. Additionally, the selection of PSO-related parameters heavily relies on extensive experimentation and empirical knowledge. Inappropriate parameter choices may result in either excessively slow convergence or a complete failure to converge [36; 37]. Therefore, this paper introduces an enhanced Circle chaotic mapping into the PSO, which utilizes the high complexity and randomness of the chaotic particle swarm to enhance the global search capability of the particle swarm algorithm, avoid the problems of premature convergence and falling into the local optimal solution of the algorithm, and enhance the optimization accuracy of the algorithm [38]. To enhance the equilibrium between the algorithm's global exploration and local exploitation, a formula for inertia weight that decreases linearly is incorporated. Circle's formula for chaotic mapping:  
\[x_{i+1}=mod\bigg{(}x_{i}+0.2-\frac{1}{4\pi}\cdot\sin(2\pi x_{i}),1\bigg{)} \tag{21}\]  
Linearly decreasing inertia weight formula:  
\[\omega=\omega_{\mathrm{s}}-(\omega_{\mathrm{s}}-\omega_{e})\cdot\frac{d}{K} \tag{22}\]  
where \(\omega_{\mathrm{s}}\) is initial inertia weights, \(\omega_{e}\) is final inertia weights, \(d\) is current number of iterations, and \(K\) is total number of iterations. The enhanced CPSO's flowchart is presented in Figure 6. The principle of the enhanced CPSO fuzzy PID control is presented in Figure 7.  
Figure 6: Flowchart of the enhanced CPSO.