# Improving Particle Swarm Optimization Analysis Using Differential Models
## 4 Differential Evolution PSO
### Selection of Parameters  
In order to verify the performance of the three different differential evolution PSO algorithms proposed above, it is necessary to give the selection methods of inertia weight \(\omega\), cognitive coefficient \(c_{1}\), social coefficient \(c_{2}\), and step size \(h\). Since the step size \(h\) is generated after the differential model (Equation (38)) is differenced, first consider the conditions that need to be satisfied when the differential equation system (Equation (38)) is stable.  
To further simplify the model, let:  
\[y_{jk}(t)=x_{jk}(t)-\frac{\varphi_{1}p_{jk}+\varphi_{2}p_{gk}}{\varphi} \tag{49}\]  
Thus, transforming the differential model (Equation (38)) into:  
\[\left\{\begin{array}{l}\frac{dv_{jk}(t)}{dt}=(\omega-1)v_{jk}(t)-\varphi y_ {jk}(t)\\ \frac{dy_{jk}(t)}{dt}=\omega v_{jk}(t)-\varphi y_{jk}(t)\end{array}\right. \tag{50}\]  
Its singularity is:  
\[(\omega-1)v_{jk}(t)-\varphi y_{jk}(t)=0,\ \omega v_{jk}(t)-\varphi y_{jk}(t)=0 \tag{51}\]  
Calculated:  
\[v_{jk}(t)=y_{jk}(t)=0 \tag{52}\]  
The coefficient determinant is:  
\[\left|\begin{matrix}\omega-1&-\varphi\\ \omega&-\varphi\end{matrix}\right|=\varphi\]  
Since random numbers \(r_{1}\) and \(r_{2}\) can only be selected in (0, 1), in the previous formula, \(\varphi=c_{1}r_{1}+c_{2}r_{2}\). Calculate the characteristic equation:  
\[\left[\begin{matrix}\lambda-(\omega-1)&\varphi\\ -\omega&\lambda+\varphi\end{matrix}\right]=0\]  
Arrangement available:  
\[\lambda^{2}+(1+\varphi-\omega)\lambda+\varphi=0 \tag{53}\]  
The characteristic root \(\lambda\) is:  
\[\lambda_{1,2}=\frac{-(1+\varphi-\omega)\pm\sqrt{\left(1+\varphi-\omega\right) ^{2}-4\varphi}}{2} \tag{54}\]  
Ordinary differential equation stability theory states that if the real part of the eigenvalue is negative, its solution is zero,  
\(\left(v_{jk}(t),\ y_{jk}(t)\right)=(0,\ 0)\) is asymptotically stable.  
According to the usual parameter selection method, the inertia weight \(0\)\(<\omega\leq 1\), \(\varphi=c_{1}r_{1}+c_{2}r_{2}>0\), that is, \(1+\varphi-\omega>0\), so:If \(\left(1+\varphi-\omega\right)^{2}-4\varphi\geq 0\), then since \(1+\varphi-\omega>\sqrt{\left(1+\varphi-\omega\right)^{2}-4\varphi}\), this equation has \(Re(\lambda_{1,2})<0\);  
If \(\left(1+\varphi-\omega\right)^{2}-4\varphi<0\), then \(Re(\lambda_{1,2})=\frac{-\left(1+\varphi-\omega\right)}{2}<0\).  
Therefore, its solution is zero \(\left(v_{jk}(t),y_{jk}(t)\right)=\left(0,\ 0\right)\) is asymptotically stable, and:  
\[\begin{array}{ll}\underset{t\rightarrow+\infty}{\lim}y_{jk}(t)&=\underset{ t\rightarrow+\infty}{\lim}\left(x_{jk}(t)-\frac{\varphi_{1}p_{k}+\varphi_{2}p_{jk}}{ \varphi}\right)=0\\ &\Rightarrow\underset{t\rightarrow+\infty}{\lim}x_{jk}(t)=\frac{\varphi_{1}p_ {k}+\varphi_{2}p_{jk}}{\varphi}\end{array} \tag{55}\]  
This is the same as the existing theoretical results. The results show that the existing parameter selection method in the standard PSO algorithm can ensure the stability of the differential model (Equation (38)). Therefore, it is only necessary to discuss the selection method of the step size, and the selection methods of other parameters refer to the parameter settings of the standard PSO algorithm.