# Improving Particle Swarm Optimization Analysis Using Differential Models
## Unified Model for PSO
### Evolutionary Behavior Analysis of PSO Algorithm  
Taking \(v_{jk}(t+1)\) as a first-order approximation, that is, \(v_{jk}(t+1)=v_{jk}(t)+\frac{dv_{jk}(t)}{dt}\), we substitute Equation (2) and obtain Equation (3) as:  
\[\left\{ \begin{array}{l}\frac{dv_{jk}(t)}{dt}=(\omega\eta-1)v_{jk}(t)-\eta \sigma x_{jk}(t)+\eta\varphi_{p}\\ \frac{dx_{jk}(t)}{dt}=\omega\eta v_{jk}(t)-\eta\sigma x_{jk}(t)+\eta\varphi_{p} \end{array} \right. \tag{3}\]  
with the definition:  
\[\mathbf{y}(t)= \begin{bmatrix}v_{jk}(t)\\ x_{jk}(t)\end{bmatrix} \!\!,\ \mathbf{u}(t)= \begin{bmatrix}p_{jk}(t)\\ p_{gk}(t)\end{bmatrix} \!\!,\ \mathbf{A}= \begin{bmatrix}\omega\eta-1&-\eta\varphi\\ \omega\eta&-\eta\varphi\end{bmatrix} \!\!,\ \mathbf{B}= \begin{bmatrix}\eta\varphi_{1}&\eta \varphi_{2}\\ \eta\varphi_{1}&\eta\varphi_{2}\end{bmatrix} \]  
Then, we have Equation (4):  
\[\mathbf{y}^{\prime}(t)=\mathbf{A}\mathbf{y}(t)+\mathbf{B}\mathbf{u}(t) \tag{4}\]  
Solve to get Equation (5):  
\[\mathbf{y}(t)=e^{\mathbf{A}(t-t_{0})}\mathbf{y}(t_{0})+\int_{t_{0}}^{t}e^{\mathbf{A}(t-\tau)} \mathbf{B}\mathbf{u}(\tau)d\tau \tag{5}\]It can be seen from Equation (5) that when the eigenvalue of the matrix has a negative real part, Equation (4) converges, and its characteristic equation is:  
\[|\lambda\mathbf{I}-\mathbf{A}|=\lambda^{2}+(1-\omega\eta+\eta\varphi)\lambda+\eta\varphi=0 \tag{6}\]  
Find the characteristic root as:  
\[\lambda_{1,2}=\frac{\omega\eta-\eta\varphi-1\pm\sqrt{\left(1-\omega\eta+\eta \varphi\right)^{2}-4\eta\varphi}}{2} \tag{7}\]  
Since \(\eta\), \(\varphi>0\), as long as \(\omega\eta-\eta\varphi-1<0\), the eigenvalue of matrix \(\mathbf{A}\) can be guaranteed to be a negative real part, that is, as long as \(\omega\eta-\eta\varphi-1<0\), described by Equation (3). The PSO evolution equation converges, while:  
\[\lim_{t\rightarrow+\infty}\mathbf{y}(t)=\lim_{t\rightarrow+\infty}e^{A(t-t_{0}) }\mathbf{y}(t_{0})+\int_{t_{0}}^{t}e^{A(t-\tau)}\mathbf{B}\mathbf{u}(\tau)d\tau= \begin{bmatrix} 0\\ \frac{\varphi_{p}}{\varphi}\end{bmatrix}  \tag{8}\]  
That is:  
\[\lim_{t\rightarrow+\infty}v_{jk}(t)=0,\ \lim_{t\rightarrow+\infty}x_{jk}(t)= \frac{\varphi_{p}}{\varphi} \tag{9}\]  
That is:  
\[\lim_{t\rightarrow+\infty}(\varphi_{1}+\varphi_{2})x_{jk}(t)=\varphi_{1}p_{ jk}+\varphi_{2}p_{gk} \tag{10}\]  
Since \(\varphi_{1}\) and \(\varphi_{2}\) are random variables, the above formula must be established if and only if \(\lim_{t\rightarrow+\infty}x_{jk}(t)=p_{jk}=p_{gk}\). It can be seen from the above analysis that when \(\omega\eta-\eta\varphi-1<0\), all particles will converge to the global optimal position \(\mathbf{p}_{g}\). That is, when \(\mathbf{p}_{g}\) is fixed, \(\mathbf{x}_{j}(t)\) converges to \(\mathbf{p}_{g}\). Therefore, in the evolution of the PSO algorithm, as long as the global optimal position \(\mathbf{p}_{g}=\mathbf{x}^{*}\) can be explored, it can be guaranteed that all particles will eventually converge at \(\mathbf{p}_{g}=\mathbf{x}^{*}\). In addition, from Equation (4) and \(\mathbf{u}(t)= \begin{bmatrix}p_{jk}(t)\\ p_{gk}(t)\end{bmatrix} \), the PSO is a linear system under the step signal input and the amplitude of its input step signal changes with evolution [22]. The upper bound estimate for the convergence of Equation (4) is given below.