# PSO Optimized Active Disturbance Rejection Control for Aircraft Anti-Skid Braking System
## 4 Controller Design
### Stability Analysis  
Stability may be the most important indicator of a control system, especially for aircraft anti-skid braking systems. The stability of ADRC is also one of the key points of academic attention, and many landmark achievements have emerged [18; 19; 20; 21]. Stability analysis of an ADRC system was performed via the describing function method by [PERSON] et al. [22; 23]. In addition, time domain convergences of the ADRC system were proven [22; 24]. However, for typical nonlinear ADRC systems, there is still no suitable Lyapunov function, and there are many restrictions on the parameters, which makes the method difficult to apply. In this work, we adapt the absolute stability method to prove the stability of the ADRC designed  
Figure 8: Particle swarm algorithm fitness value change curve. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Parameter** & \(k_{1}\) & \(k_{2}\) & \(\beta_{1}\) & \(\beta_{2}\) & \(\beta_{3}\) \\ \hline Position upper limit & 300 & 300 & 200 & 1000 & 4000 \\ Position lower limit & 0.5 & 0.5 & 50 & 50 & 50 \\ Velocity upper limit & 1 & 1 & 2 & 5 & 5 \\ Velocity lower limit & \(-1\) & \(-1\) & \(-2\) & \(-5\) & \(-5\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Initialization parameter of particle swarms. for aircraft anti-skid braking system. Moving forward, we work under the following assumptions:  
(A1) The input of TD is 0 (i.e., \(v=0\)); hence, the output value of TD \(v_{1}\) is also zero. (A2) Nonlinear state error feedback is converted to linear state error feedback by first-order approximation. \[u=\sum_{i=1}^{n}k_{i}z_{i}-z(n+1)/b \tag{29}\]  
(A3) The nonlinear function \(\varphi(x)\) is chosen in the design of ESO, such that for all \(x_{0},x\varphi(x)>0\), and \(\varphi F(\mu_{1},\mu_{2})\) The following two requirements are introduced: \(\varphi(0)=0\); \(\exists 0<\mu_{1}<\mu_{2}\) for \(\forall x0,\mu_{1}x^{2}<x\varphi(x)<\mu_{2}x^{2}\) (i.e., \(\varphi(x)F(0,+]\)). Then, the output of ESO can be rewritten as:  
\[\left\{ \begin{array}{c}\dot{z}_{1}=z_{2}-\beta_{1}\varphi(z_{1}-y)\\ \vdots\\ z_{n}=z_{(n+1)}-\beta_{n}\varphi(z_{1}-y)+bu\\ \dot{z}(n+1)=-(n+1)\varphi(z_{1}-y))\end{array} \right. \tag{30}\]  
(A4) Let the object system under control be a linear time-invariant system. Then, the state space form of the controlled object can be expressed as:  
\[\left\{ \begin{array}{c}\dot{x}_{1}=x_{2}\\ \vdots\\ \dot{x}_{n}=-a_{n}x_{1}-a_{(n-1)}x_{2}-\ldots-a_{1}x_{n}+bu\\ \dot{z}(n+1)=-(n+1)\varphi(z_{1}-y)\,)\end{array} \right. \tag{31}\]  
Substituting Equation (29) into Equation (31) yields:  
\[\left\{ \begin{array}{c}\dot{X}=A_{11}X+A_{12}Z+a_{13}z(n+1)\\ y=x_{1}\end{array} \right. \tag{32}\]  
where \(X=[x_{1},x_{2},\ldots,x_{n}]^{T}\), \(Z=[z_{1},z_{2},\ldots,Z_{n}]^{T}\), \(a_{13}=[0,\ldots,0,-1]^{T}\in R^{n}\)  
\[A_{11}=\left[ \begin{array}{ccccc}0&1&0&\cdots&0\\ 0&0&1&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&1\\ -a_{n}&-a_{n-1}&-a_{n-2}&\cdots&a_{1}\end{array} \right]\quad A_{12}=\left[ \begin{array}{ccccc}0&\cdots&0\\ 0&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&0\\ -bk_{1}&\cdots&-bk_{n}\end{array} \right]\]  
Substituting Equation (29) into Equation (30), yields:  
\[\left\{ \begin{array}{c}\dot{X}=A_{11}X+A_{12}Z+a_{13}z(n+1)\\ \dot{Z}=A_{22}Z+b_{2}u\\ z(n+1)=\beta(n+1)u\\ u=-\varphi(y)\\ y=c_{1}^{T}X+c_{2}^{T}Z\end{array} \right. \tag{33}\]  
Let \(Y=A_{11}X+a_{13}z(n+1)\), \(z(n+1)^{\prime}=z(n+1)/\beta_{(n+1)}\), and we get:  
\[\left\{ \begin{array}{c}\dot{Y}=A_{11}Y+A_{11}A_{12}Z+a_{13}\beta(n+1)u\\ \dot{Z}=A_{22}Z+b_{2}u\\ z(n+1)=u\\ u=-\varphi(y)\\ y=c_{1}^{T}A_{11}{}^{-1}Y+c_{2}^{T}Z-c_{1}^{T}A_{11}{}^{-1}a_{13}\beta(n+1)z(n+1 )^{\prime}\end{array} \right. \tag{34}\]Equation (34) conforms to one standard form of the first critical situation of absolute stability [25], that is,  
\[\left\{ \begin{array}{l}\dot{x}=Ax+bu\\ \varepsilon=u\\ u=-\varphi(y)\\ y=c^{T}x=p\varepsilon\end{array} \right. \tag{35}\]  
where \(A=\left[ \begin{array}{cc}A_{11}&A_{12}A_{12}\\ 0&A_{22}\end{array} \right]\)  
\(b=\left[