## 2 Differential Step-Scaling Antilion Algorithm
### Algorithm Idea and Specific Steps
The controlled plant is a second-order system with time delay:

\[G_{p}(s)=\frac{e^{-6s}}{s^{2}+3s+1} \tag{21}\]

The observer is designed as

\[F_{1}=\frac{\omega_{o}^{2}}{\left(s+\omega_{o}\right)^{2}},\ \ F_{2}=\frac{ \omega_{o}^{2}s^{2}}{\left(s+\omega_{o}\right)^{2}} \tag{22}\]

The controller takes the proportional controller \(\omega_{c}\). Thus, we obtain a parameter set \(\{\omega_{o},\omega_{c}\}\) to be optimized. There are two parameters optimization objectives: one is to minimize the IAE index of the system; the other is to reduce the overshoot of the system as much as possible while considering the IAE index. Since in this paper the performance of DSALO and OEALO has been verified in Section 3 that they are both better than the original ALO and PSO, only DSLAO and OEALO are considered for comparison in this section. Given the randomness of the algorithms, each algorithm is run 25 times, and only the best results are recorded. The authors run the simulation of ADRC parameters optimization in the following hardware environment: Intel(r) Core(tm) I7-8700K, Nvidia GeForce GTX 1060, and 16 GB memory. The software environment is a Windows 10 operating system and Matlab R2020b. According to statistics, when the number of populations is set to 50 and the algorithm iterates 100 times, the average time of running DSALO to optimize the active disturbance rejection controller is 32 min, and the average time of running OEALO is 34 min. Given the complexity of the algorithm and the long control cycle in the process industry, such operation time is acceptable. It is worth noting that due to the existence of uncertainty, parameter optimization usually needs to be run at regular intervals. A large number of iterations can obtain better parameters but is also time-consuming, which may lead to a phenomenon that the optimization time takes up too much ratio of running interval or even is longer than the running interval. This phenomenon may cause parameter updates to be delayed, which in turn has a negative impact on the effectiveness of control. So, in the problem of ADRC parameter optimization, it needs a compromise between the optimization effect and computation time. That is the reason why we choose to set the number of iterations to 100. The search results of parameters under the two targets are listed in Table 3. Figures 8 and 9 show the corresponding output. Figure 7: Control structure. Figure 4:0 Using IAE as an indicator. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Performance** & **Methods** & **Index Number** & \(\omega_{c}\) & \(\omega_{o}\) \\ \hline IAE & DSALO & \(2.6116\times 10^{3}\) & \(1.1899\) & \(0.6133\) \\ OEALO & \(2.6776\times 10^{3}\) & \(1.18026\) & \(0.57996\) \\ IAE + overshoot & DSALO & \(3.4860\times 10^{3}\) & \(0.99298\) & \(0.54947\) \\ OEALO & OEALO & \(3.5029\times 10^{3}\) & \(0.9577\) & \(0.5967\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optimization results. Figure 9: Using IAE+ overshoot as an indicator. There is also one thing worth noting. The DSALO algorithm can be applied in industrial ADRC systems in this way: first collect input and output data of controlled plant to establish current model of the plant, then integrate the model in simulation software and take the control performance as objective function/fitness, finally run the simulation and DSALO simultaneously and the solutions of DSALO are the optimized parameters of ADRC. It can be seen from the diagram, whichever kind of index or which kinds of optimization algorithm are used, the optimized performances of disturbance rejection (200 to 250 s) are very good. The difference mainly exists in the procedure of setpoint tracking (0 to 50 s). Using the parameters obtained by the DASLO algorithm can obtain smaller indexes. Combined with the meaning of IAE index, which is the corresponding energy consumption (coal, fuel, natural gas, etc. ), the parameters obtained by DSALO have smaller performance metrics, which means better economy of the optimized system and better for the environment.