## 3 Methodology
### Ant Colony Optimization
The parameters used in the ACO algorithm are summarized in Table 5. This table lists the symbols and their corresponding descriptions, including pheromone intensity (\(\tau_{i,j}(t)\)), control parameters (\(\alpha\), \(\beta\), \(\rho\)), and the number of ants (\(k\)). Additionally, it includes the probability of ant movement (\(P_{i,j}^{k}(t)\)), visibility value (\(\eta_{i,j}\)), change in pheromone intensity (\(\Delta\tau_{i,j}(t)\)), and other related constants (\(\zeta\), \(T\)). These parameters are crucial for understanding the behavior and optimization process of the ACO algorithm. The proposed Algorithm 1, titled \"FOPID Tuning Ant Colony Optimization\" systematically tunes Fractional Order PID controllers using ACO. Initially, the search space and algorithm parameters are declared as outlined in Table 5. Allowable discrete values are initialized with equal pheromone levels (\(\tau\)). The algorithm iterates through several steps: computing the selection probability of design variables (Equation (28)), setting probability ranges, and generating random positions for each ant based on cumulative probability ranges (roulette-wheel design). The control signal values are assigned and the objective function is evaluated. The algorithm updates the best and worst objective functions, strengthens the pheromone of the best path, and evaporates pheromones from other paths. The tuning values are printed after completing the iterations, ensuring optimal \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Description** \\ \hline \(\tau_{i,j}(t)\) & Pheromone intensity at time \(t\) between points \(i\) and \(j\) \\ \(\alpha\) & Parameter controlling the influence of pheromone intensity \\ \(\beta\) & Parameter controlling the influence of visibility \\ \(\rho\) & Pheromone evaporation rate \\ \(k\) & Number of ants \\ \(P_{i,j}^{k}(t)\) & Probability of ant \(k\) moving from point \(i\) to \(j\) \\ \(\eta_{i,j}\) & Visibility value between points \(i\) and \(j\) \\ \(\Delta\tau_{i,j}(t)\) & Change in pheromone intensity \\ \(S_{k}\) & Set of objects to be scheduled \\ \(\zeta\) & Constant used in Global Pheromone Update \\ \(T\) & Total unit of the path taken by an ant \\ \hline \hline \end{tabular}
\end{table} 
Table 5: ACO algorithm parameters. controller parameters. This structured approach leverages the robustness and efficiency of ACO in optimizing FOPID controllers for complex systems. ```
1:Step 1: Declare search space and algorithm parameters (Table 5)
2:Step 2: Initialize allowable discrete values \(m\) with equal pheromone \(\tau\)
3:for\(i=1\):\(T\)do
4:for\(j=1\):\(ls\)do
5:Step 3: Compute selection probability of design variables (Equation (28))
6:for\(j=1\):\(ls\)do
7:Set probability ranges \(r\vec{w}s\), \(r\vec{w}e\) with steps of \(p_{j}\)
8:for\(k=1\):\(N\)do
9:Step 4: Compute cumulative probability ranges (roulette-wheel design)
10:Step 5: Generate random positions \(\vec{r}_{k}=rand(1)\) for each ant
11:for\(m=1\):\(ls\)do
12:if\(rws(m)<r_{k}<rwe(m)\), assign \(\vec{f}_{m}=\vec{x}_{m}\)
13:for\(k=1\):\(N\)do
14:Step 6: Assign \(\vec{f}_{m}\) to the control signal
15:Step 7: Evaluate the objective function
16:Step 8: Update best \(f_{best}\) and worst \(f_{worst}\) objective functions
17:for\(j=1\):\(ls\)do
18:Step 9: Strengthen pheromone of the best path (Equations (29) and (30))
19:Step 10: Evaporate pheromone from other paths (\(\Delta\tau_{i,j}(t)=0\))
20:Step 11: Print tuning values
```  
**Algorithm 1** FOPID tuning Ant Colony Optimization