# PSO Optimized Active Disturbance Rejection Control for Aircraft Anti-Skid Braking System
## 4 Controller Design
### Optimization of Control Parameters Based on Improved Particle Swarm Optimization
\begin{array}{ll}\omega_{\text{min}}-(\omega_{\text{max}}- \omega_{\text{min}})\frac{f-f_{\text{min}}}{f-f_{\text{min}}}&f\leq\bar{f}\\ \omega_{\text{max}}&f>\bar{f}\end{array} \right. \tag{25}\]  
In the above formula, \(f\) is the current fitness value, \(\bar{f}\) is the average fitness value, and \(f_{\text{min}}\) is the minimum fitness value, while \(\omega_{\text{max}}\) and \(\omega_{\text{min}}\) are the maximum and minimum weights, respectively. This paper improves the standard particle swarm. The concept behind the improvement is to adjust the inertia weight and the learning factor of the particle swarm, and to introduce the natural selection mechanism. The steps for obtaining the minimum value problem of the improved particle swarm are as follows:  
A. Set the size of the particle swarm, that is, how many particles there are in the swarm, and set the starting position and speed of the particles. Set the maximum and minimum values of the learning factor. Specify the maximum and minimum weights. B. Calculate the fitness of each particle. Store the particle position and fitness value at the moment in the individual extreme value in \(P_{best}\). Among all the individual extreme values, the particle with the lowest fitness is selected by comparison. Store the particle's position and fitness value in the global extremum \(g_{best}\). C. Use the synchronous learning factor to update the speed of the particle:  
\[\left\{ \begin{array}{ll}c_{1}=c_{2}=c_{\text{max}}-(c_{\text{max}}-c_{\text{ min}})\frac{f}{f}\\ V_{ij}(t+1)=\omega V_{ij}(t)+c_{1}r_{1}\big{(}P_{ij}-X_{ij}\big{)}+c_{2}r_{2} \big{(}P_{gj}-X_{ij}\big{)}\\ X_{ij}(t+1)=X_{ij}(t)+V_{ij}(t+1)\end{array} \right. \tag{26}\]  
D. Perform adaptive weights adjustment:  
\[\omega=\left\{ \begin{array}{ll}\omega_{\text{min}}-(\omega_{\text{max}}- \omega_{\text{min}})\frac{f-f_{\text{min}}}{f-f_{\text{min}}}&f\leq\bar{f}\\ \omega_{\text{max}}&f>\bar{f}\end{array} \right. \tag{27}\]In the above formula, \(f\) is the current fitness value, \(\tilde{f}\) is the average fitness value, and \(f_{\text{min}}\) is the minimum fitness value, while \(\omega_{\text{max}}\)  and \(\omega_{\text{min}}\)  are the maximum and minimum weights, respectively. It can be seen from Figure 4 that the optimization process is as follows: First, a particle swarm is generated, and the swarm particles are assigned to the parameters \(k_{1}\), \(k_{2}\), \(\beta_{1}\), \(\beta_{2}\), and \(\beta_{3}\). Then, the model of the control system is run to obtain the performance index corresponding to the set of controller parameters. Then, the performance index is transferred to the particle swarm as the fitness value of the current particle. Finally, it is judged whether the termination condition is met. The error performance index adopts the ITAE criterion, so the fitness function form of the particle swarm algorithm is as follows:  
\[\text{fitness}\ =\int_{0}^{\infty}t|e(t)|dt \tag{28}\]  
The tuned parameters are the observer gains \(\beta_{1}\), \(\beta_{2}\), and \(\beta_{3}\), and the gain coefficients \(k_{1}\) and \(k_{2}\) in the control law. Therefore, the particle swarm algorithm is initialized to 5 dimensions, the initial scale is 50 particles, the termination condition is 50 iterations, the inertia weights are \(\omega_{\text{max}}=0.9\) and \(\omega_{\text{min}}=0.36\), and the learning factors are \(c_{\text{max}}=2.2\) and \(c_{\text{min}}=0.8\). The initialization parameters are shown in Table 3. The fitness curves of the standard particle swarm and the improved particle swarm algorithm are shown in Figure 8. We can see that the improved particle swarm algorithm has a faster convergence speed and better optimization effect (the fitness value is smaller).