## 2 Materials and Methods
### Robust and Controllable  
Combine the \"linearly decreasing inertia weight\" of Equation (7) with Equation (5) and expand Equation (6) into Equations (9) and (10). \[ \begin{array}{rl}&V_{i}(k+1)=\omega(k)\cdot V_{i}(k)-(c_{1}\cdot r_{1}\cdot \lambda+c_{2}\cdot r_{2})p_{i}(k)-c_{1}\cdot r_{1}\cdot\frac{\lambda}{2}(1- \lambda)p_{i}(k-1)\\ &-c_{1}\cdot r_{1}\cdot\frac{\lambda}{6}(1-\lambda)(2-\lambda)p_{i}(k-2)-c_{1 }\cdot r_{1}\cdot\frac{\lambda}{24}(1-\lambda)(2-\lambda)(3-\lambda)p_{i}(k-3) \\ &+c_{1}\cdot r_{1}P_{i}^{l}(k)+c_{2}\cdot r_{2}\cdot P^{g}(k)\end{array}  \tag{9}\]  
\[ \begin{array}{rl}&p_{i}(k+1)=\omega(k)\cdot V_{i}(k)+(1-c_{1}\cdot r_{1} \cdot\lambda-c_{2}\cdot r_{2})p_{i}(k)\\ &-c_{1}\cdot r_{1}\cdot\frac{\lambda}{2}(1-\lambda)p_{i}(k-1)-c_{1}\cdot r_{1} \cdot\frac{\lambda}{6}(1-\lambda)(2-\lambda)p_{i}(k-2)\\ &-c_{1}\cdot r_{1}\cdot\frac{\lambda}{24}(1-\lambda)(2-\lambda)(3-\lambda)p_{ i}(k-3)+c_{1}\cdot r_{1}P_{i}^{l}(k)+c_{2}\cdot r_{2}\cdot P^{g}(k)\end{array}  \tag{10}\]  
Subsequently, rewrite these equations into the state Equation (11):  
\[x(k+1)=A_{I}\cdot x(k)+B_{I}\cdot u(k) \tag{11}\]  
where \(x(k)=\left[p_{i}^{T}(k),p_{i}^{T}(k-1),\,p_{i}^{T}(k-2),\,p_{i}^{T}(k-3),\,V_ {i}^{T}(k)\right]^{T}\in R^{n}\) denotes the system state vector; \(u(k)=\left[p_{i}^{T}(k),p^{gT}(k)\right]^{T}\in R^{n}\) represents the control input vector; \(i=1,2,\ldots,m\) (\(m\) denotes the number of particles) and \(n=5\theta\), while \(\theta\) represents the di mension for the search space, and \(k\) is the iteration index. \(A_{I}\) denotes the system matrix, and \(B_{I}\) represents the input matrix, as shown in the following Equation (12):  
\[A_{I}=\left[ \begin{array}{cccccc}A_{0}&A_{1}&A_{2}&A_{3}&A_{4}\\ I&0&0&0&0\\ 0&I&0&0&0\\ 0&0&I&0&0\\ A_{5}&A_{6}&A_{7}&A_{8}&A_{9}\end{array} \right]\qquad B_{I}=\left[ \begin{array}{ ccc}c_{1}\cdot r_{1}\cdot I&c_{2}\cdot r_{2}\cdot I\\ 0&0\\ 0&0\\ 0&0\\ c_{1}\cdot r_{1}\cdot I&c_{2}\cdot r_{2}\cdot I\end{array} \right] \tag{12}\]  
where  
\[A_{0}=(1-c_{1}\cdot r_{1}\cdot\lambda-c_{2}\cdot r_{2})\cdot I\]  
\[A_{1}=-c_{1}\cdot r_{1}\cdot\frac{\lambda}{2}(1-\lambda)\cdot I\]  
\[A_{2}=-c_{1}\cdot r_{1}\cdot\frac{\lambda}{6}(1-\lambda)(2-\lambda)\cdot I\]  
\[A_{3}=-c_{1}\cdot r_{1}\cdot\frac{\lambda}{24}(1-\lambda)(2-\lambda)(3- \lambda)\cdot I\]  
\[A_{4}=\omega(k)\cdot I\]  
\[A_{5}=(-c_{1}\cdot r_{1}\cdot\lambda-c_{2}\cdot r_{2})\cdot I\]  
\[A_{6}=-c_{1}\cdot r_{1}\cdot\frac{\lambda}{2}(1-\lambda)\cdot I\]  
\[A_{7}=-c_{1}\cdot r_{1}\cdot\frac{\lambda}{6}(1-\lambda)(2-\lambda)\cdot I\]  
\[A_{8}=-c_{1}\cdot r_{1}\cdot\frac{\lambda}{24}(1-\lambda)(2-\lambda)(3- \lambda)\cdot I\]  
\[A_{9}=\omega(k)\cdot I\]  
and \(I\) denotes the \(\theta\times\theta\) unit matrix. \(r_{1}\) and \(r_{2}\) denote random numbers between 0 and 1. Equations (9) and (10) are equivalent to Equation (11) based on Equation (12) and matrix multiplication rules. If each pair \((A_{I},B_{I})\) is controllable, then the state Equation (11) is said to be robust and controllable [27]. Suppose that a fixed fractional value \(\lambda\), an inertial weight constant reference value \(\omega_{0}\), a volume learning parameter constant reference value \(c_{10}\), and a group learning parameter constant reference value \(c_{20}\) are selected as the nominal values of the fractional-order particle swarm algorithm. In that case, the equation of state (11) can be rewritten as an uncertain linear system, i.e., the system will be transformed into a nominal fractional-order particle swarm optimization (FPSO) combined with uncertain matrices:  
\[A_{1}=-c_{1}\cdot r_{1}\cdot\frac{\lambda}{2}(1-\lambda)\cdot I\]0  
where  
\[A_{I0}=\left[ \begin{array}{cccc}a_{0}&a_{1}&a_{2}&a_{3}&a_{4}\\ I&0&0&0&0\\ 0&I&0&0&0\\ 0&0&I&0&0\\ a_{5}&a_{6}&a_{7}&a_{8}&a_{9}\end{array} \right]\qquad B_{I0}=\left[ \begin{array}[] {cccc}c_{10}\cdot I&c_{20}\cdot I\\ 0&0\\ 0&0\\ c_{10}\cdot I&c_{20}\cdot I\end{array}