# PSO Optimized Active Disturbance Rejection Control for Aircraft Anti-Skid Braking System
## 4 Controller Design  
The main purpose of this work is to design a control law, so that the anti-skid braking system of the aircraft can be stabilized near the optimal value of the slip rate to improve the efficiency and safety of the braking process.  
Figure 3: Relationship between adhesion coefficient \(\mu\) and slip rate \(\lambda\).  
Design and Parameter Optimization of Aircraft Anti-Skid Braking Active Disturbance Rejection Controller  
The performance of aircraft braking systems is mainly dependent on the tire/road characteristics and actuator dynamics. Accordingly, the main purpose of the control algorithm will be to maintain the system slip rate near optimal values, thus improving braking efficiency. To solve this problem, this work designs an active disturbance rejection controller, starting from the analysis of tire/road friction characteristics and system equilibrium.  
Figure 4 is a schematic diagram of the balance point of the system when the control input of the controller is constant (i.e., \(T_{b}=T_{b}^{\rm ss}\)). The system variable \(\lambda\) exhibits equilibrium points, which can be seen in Figure 4:  
(1) If \(T_{b}^{\rm ss}>\lambda_{max}\psi(\lambda)\), that is, if the brake torque is greater than the maximum friction value the tire can generate at the time, then the slip rate has a tendency of increasing, so the unique equilibrium point is \(\lambda_{ss}=1\).  
(2) If \(T_{b}^{\rm ss}<\lambda_{max}\psi(\lambda)\), then the system has at most three equilibria, namely \(\lambda_{1}^{\rm ss}=1\), \(\lambda_{2}^{\rm ss}=\lambda_{2}\), and \(\lambda_{3}^{\rm ss}=\lambda_{3}\), where \(\lambda_{2}\) and \(\lambda_{3}\) are the two possibly coincident solutions of \(T_{b}^{\rm ss}=\psi(\lambda)\), as shown in Figure 4. To facilitate the analysis of system dynamics with control action, in this section we assume the braking torque of the actuator as \(T_{b}=Ku\). The system trajectories under control can be seen in Figure 5.  
Where \(K=-r,0,r\). Therefore, the slip rate change rate concerning actuator dynamics yields the following second-order equation:  
\[\left\{\begin{array}{l}\dot{\lambda}=-((1-\lambda)/J\omega)(\Psi(\lambda)-T _{b})\\ \hat{T}_{b}=u\end{array}\right. \tag{16}\]  
Accordingly, the control system will maintain the wheel slip ratio around the acceptable values by generating a suitable value of variable \(r\). To achieve this goal, this work designs an active disturbance rejection controller Figure 6.  
Figure 4: Equilibrium points for aircraft anti-skid braking model.  
(1) According to system dynamics, the input \(v\) of the controller is the desired slip rate, so we arrange the transient process of the controller as follows:  
\[\left\{\begin{array}{l}e_{\lambda}=v_{1}-v\\ fh=fhan(e_{\lambda},v_{2},r_{0},h)\\ v_{1}=v_{1}+Hv_{2}\\ v_{2}=v_{2}+Hfh\end{array}\right. \tag{17}\]  
(2) Then, we track and estimate the states and disturbance of the system by output \(y\) and input \(v\).  
\[\left\{\begin{array}{l}e=z_{1}-y\\ fe=fal(e,0.5,\delta)\\ fe_{1}=fal(e,0.25,\delta)\\ z_{1}=z_{1}+h(z_{2}-\beta_{01}e)\\ z_{2}=z_{2}+h(z_{3}-\beta_{02}fe+b_{0}u)\\ z_{3}=z_{3}+h(-\beta_{03}fe_{1})\end{array}\right. \tag{18}\]  
Figure 5: System trajectories of states \(\lambda\) and \(T_{b}\) when \(K=r\) (**a**), \(K=0\) (**b**), and \(K=-r\) (**c**).  
Figure 6: The structure of the ADRC.  
(3) The nonlinear state error feedback is as follows:  
\[\left\{\begin{array}{l}e_{1}=v_{1}-z_{1}\\ e_{2}=v_{2}-z_{2}\\ u_{0}=\beta_{1}fal(e_{1},\alpha_{1},\delta)+\beta_{2}fal(e_{2},\alpha_{2}, \delta)\end{array}\right. \tag{19}\]  
(4) The disturbance compensation process is as follows:  
\[u=u_{0}-(z_{3}(t))/b_{0} \tag{20}\]