# PSO Optimized Active Disturbance Rejection Control for Aircraft Anti-Skid Braking System
## 4 Controller Design
### Stability Analysis
\begin{array}{cc}a_{13}\beta_{n+1}\\ b_{2}\end{array} \right]\)  
\(C^{T}=\left[ \begin{array}{cc}c_{1}^{T}A_{11}{}^{-1}&c_{2}^{T}\end{array} \right]\)  
\(\rho=-c_{1}^{T}A_{11}^{-1}a_{13}\beta(n+1)=\beta(n+1)/a_{n}\)  
Theorem 2.1: _The necessary condition of absolute stability of Equation (35) in regard to \(F(0;+1]\) is \(Re\lambda(A)<0\); \(\rho>0\)[26]. If the linear time-invariant object mentioned above is asymptotically stable (i.e., all the eigenvalues of \(A_{11}\) have a negative real part), at the same time we can ensure all the eigenvalues of \(A_{12}\) have a negative real part by choosing an appropriate value of parameter ki. Hence, all the eigenvalues of \(A\) have a negative real part; thus, we get the following theorem._  
Theorem 2.2: _The necessary conditions of absolute stability of the ADRC system equation (i.e., Equation (34)) are that the linear time-invariant object is asymptotic and the control parameters are positive, while at the same time ensuring that \(Re\lambda(A_{22})<0\). Consider now the candidate Lyapunov function: \(U(x,y)=x^{T}Px+\alpha\left(y-c^{T}x\right)^{2}+\beta\int_{0}^{y}\varphi(y)dy\). If \(P\) is chosen as positive definite matrix, then function \(U(x,y)\) is a zero solution. Take the time derivative of Equation (36) and substitute Equation (35) into the derivative equation \(\dot{U}(x,y)=-x^{T}Qx-2d^{T}x\varphi(y)+\gamma\varphi^{2}(y)+2\alpha\varphi y \varphi(y)\), where \(Q=-PA-A^{T}P,d=Pb-\left(\alpha\rho c+1/2\beta A^{T}c\right)\), \(\gamma=\beta\left(\rho+c^{T}b\right)\) as \(y\varphi(y)>0,\rho=\beta_{(n+1)}/a_{n}>0\), provided \(\alpha\geq 0\), and the former three terms of Equation 30 are negative quadratic forms of \(x\) and \(\varphi\). Then, \(\dot{U}\) is negative, thus guaranteeing the global asymptotic stability of Equation (35)._  
Lemma 1: _Let \(Q=Q^{T}\in R^{n},\gamma\in R,d\in R^{n},C^{T}=\left[ \begin{array}{cc}Q&d\\ d^{T}&\gamma\end{array} \right]\) be positive definite if and only one of the following two conditions hold: (1) \(Q>0,\gamma-d^{T}Q^{-1}d>0\); (2) \(\gamma>0,Q-\frac{1}{\gamma}dd^{T}>0\);_  
Theorem 2.3: _Suppose that \(\lambda\) and \(\beta\) are non-negative real numbers that are not equal to zero at the same time, and \(P\) is a positive definite matrix that can make \(Q\) and \(d\) satisfy the conditions of Lemma 1. If this is the case, then the zero solution, according to \(F(0;+1]\) of Equation (35), is absolute stable ([PERSON], [PERSON] 1990)._  
Theorem 2.4: _Suppose that \(\alpha\) and \(\beta\) are non-negative real numbers that are not equal to zero at the same time. This makes \(T(s)=(2\alpha\rho+\beta S)W(s)\) a positive real function, where \(W(s)=cT(s1-A)-1b+\rho s\) is the transfer function of the system represented in Equation (35). Moreover, two additional factors need to be met: (1) \(T(s)\) has at least one negative real part pole; (2) when \(\alpha=0\) for all \(0<e<+1\), the linearized system \(\phi(y)=ey\) is asymptotically stable. Then, the zero solution of system Equation (30), according to \(F(0;+1]\), is absolute stable. As for the aircraft anti-skid braking system, according to the first approximation, Equation (16) can be approximated as:_  
\[\left\{ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=-a_{1}x_{1}-a_{2}x_{2}+g(x_{1},x_{2})+bu\\ y=x_{1}\end{array} \right. \tag{36}\]  
_where \(x_{1}=\lambda\), \(x_{2}=-((1-\lambda)/J_{w}\omega)(\Psi(\lambda)-T_{b})\), and term \(g(x_{1},x_{2})\) only contains terms higher than one degree of \(x_{i}\)._Through transformation we can get the following:  
\[\left\{ \begin{array}{l}\dot{X}=A_{11}X+A_{12}Z+a_{13}z_{n+1}+a_{14}g(Y)\\ \dot{Z}=A_{22}Z+b_{2}u\\ z_{n+1}=\beta_{n+1}u\\ u=-\varphi(y)\\ y=c_{1}^{T}X+c_{2}^{T}Z\end{array} \right. \tag{37}\]  
Comparing this with Equation (35), the only difference is that Equation (37) has a nonlinear function item \(g(Y)\), and \(g(Y)\) only contains items higher than one degree. Thus, we can draw a conclusion that is similar to that of the first approximation theorem [27] (i.e., Theorem 5).