## 2 Materials and Methods
### Uncertain Parameter Range Corresponds to a Random Number Range  
This section uses the sufficient condition proved in Appendix A to analyze the uncertain linear system of the fractional-order particle swarm algorithm for finding the range of uncertain parameters corresponding to the range of random numbers. Herein, we selected the maximum inertia weight \(\omega_{max}\) of the fractional-order particle swarm algorithm to be 0.9, the minimum inertia weight \(\omega_{min}\) to be 0.4, the individual learning parameter \(c_{1}\) to be 2, and the group learning parameter \(c_{2}\) also to be 2. Moreover, we set the individual learning parameter constant value \(c_{10}\) to 1 and the group learning parameter constant value \(c_{20}\) to 1. Therefore, the following Equation (20) can be obtained:  
\[x(k+1)=\Bigg{(}A_{10}+\sum_{j=1}^{2}\varepsilon_{j}A_{j}\Bigg{)}\cdot x(k)+ \Bigg{(}B_{I0}+\sum_{j=1}^{2}\varepsilon_{j}B_{j}\Bigg{)}\cdot u(k) \tag{20}\]  
where  
\[A_{10}=\left[ \begin{array}{cccc}-\lambda I&-(\lambda/2)(1-\lambda)I&-( \lambda/6)(1-\lambda)(2-\lambda)I&-(\lambda/24)(1-\lambda)(2-\lambda)(3- \lambda)&\omega(k)I\\ I&0&0&0&0\\ 0&I&0&0&0\\ 0&0&I&0&0\\ (-\lambda-1)I&-(\lambda/2)(1-\lambda)I&-(\lambda/6)(1-\lambda)(2-\lambda)I&-( \lambda/24)(1-\lambda)(2-\lambda)(3-\lambda)&\omega(k)I\end{array} \right]\]  
\[A_{1}=\left[ \begin{array}{cccc}\lambda I&(\lambda/2)(1-\lambda)I&(\lambda/6 )(1-\lambda)(2-\lambda)I&(\lambda/24)(1-\lambda)(2-\lambda)(3-\lambda)I&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ \lambda I&(\lambda/2)(1-\lambda)I&(\lambda/6)(1-\lambda)(2-\lambda)I&( \lambda/24)(1-\lambda)(2-\lambda)(3-\lambda)I&0\end{array} \right]\]  
\[A_{2}=\left[ \begin{array}{cccc}I&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ I&0&0&0&0\end{array} \right]\]  
\[B_{1}=\left[ \begin{array}{cccc}I&0\\ 0&0\\ 0&0\\ 0&0\\ 0&0\\ I&0\end{array} \right]B_{2}=\left[ \begin{array}{cccc}0&I\\ 0&0\\ 0&0\\ 0&0\\ 0&I\end{array} \right]\]  
\[\varepsilon_{1}=\left[ \begin{array}{cccc}-1&1\end{array} \right]\]  
\[\varepsilon_{2}=\left[ \begin{array}{cccc}-1&1\end{array} \right]\]  
and \(\omega(k)=\omega_{min}+((iter_{max}-k)/iter_{max})\times(\omega_{max}-\omega_ {min})\). In this system, the inertia weight parameter of each generation is a constant value, which does not affect the robustness and controllability of Equation (20). As the fractional-order particle swarm algorithm generates different values in Equation (20) based on different values of \(\lambda\), the ranges of random numbers \(r_{1}\) and \(r_{2}\) corresponding to \(\varepsilon_{1}\) and \(\varepsilon_{2}\), respectively, are also different. As the value of \(\lambda\) is usually between 0 and 2, this study breaks \(\lambda\) into 20 values and deduces them one by one at the intervals of 0.1. Table 1 shows the \(r_{1}\) and \(r_{2}\) for different \(\lambda\) values according to Equation (20), where the random number range of \(r_{2}\) is between 0 and 1, and the random number range \(r_{1}\) will change as per different \(\lambda\) values. Among them, the range of \(r_{1}\) obtained with \(\lambda\) = 0.3 is the widest and least conservative. Thus, the final range of \(r_{1}\) and \(r_{2}\) is  
\[\left\{ \begin{array}{c}r_{1}\in(0.20874\;1]\\ r_{2}\in[0\;1]\end{array} \right.. \tag{21}\]  
The fractional-order particle swarm algorithm using the random number of Equation (21) is called the controllability fractional-order particle swarm optimizer (CFPSO) algorithm. This study quotes [PERSON] [21] when it is time to implement the controllable fractional-order particle swarm algorithm. If it meets the conditions, it will be executed. The conditions are as follows:  
\[\left\{ \begin{array}{c}\|pbest_{i}(k+1)-pbest_{i}(k)\|<e_{pbest_{i}}\\ \|gbest(k+1)-gbest(k)\|<e_{gbet}\end{array} \right. \tag{22}\]  
Among them, \(e_{pbest_{i}}\) and \(e_{gbet}\) are set to \(10^{-4}\).