# Backstepping Adaptive Neural Network Control for Electric Braking Systems of Aircrafts
## 3 Control Strategy
### Controller Design  
Based on the backstepping theory, the high order system is divided into three sub systems, and the Lyapunov function of the system is built [33; 34]. The error variables of system are defined as:  
\[\left\{\begin{array}{l}e_{1}=x_{1}-p^{*}\\ e_{2}=x_{2}-\alpha_{1}\\ e_{3}=x_{3}-\alpha_{2}\end{array}\right., \tag{25}\]  
where \(e_{1}\), \(e_{2}\), \(e_{3}\) are the errors,\(\alpha_{1}\) and \(\alpha_{2}\) are the virtual control variables of the sub system.  
Step 1:  
For the first sub system, the Lyapunov function is selected as:  
\[V_{1}=\frac{e_{1}^{2}}{2}. \tag{26}\]  
Obtain the derivative of Equation (26) as:  
\[\dot{V}_{1}=e_{1}\dot{e}_{1}=e_{1}\Big{(}a_{1}e_{2}+a_{1}\alpha_{1}+f_{1}( \mathbf{x})-\dot{p}^{*}\Big{)}. \tag{27}\]  
The virtual control variable \(\alpha_{1}\) is designed as:  
\[\alpha_{1}=\frac{1}{a_{1}}\Big{(}-\mu_{1}e_{1}-\dot{f}_{1}(\mathbf{x})+\dot{p }^{*}\Big{)}. \tag{28}\]  
Take Equation (28) into Equation (27) as:  
\[\dot{V}_{1}=a_{1}e_{1}e_{2}+e_{1}\Delta_{1}(\mathbf{x})-\mu_{1}e_{1}^{2}, \tag{29}\]  
where \(\dot{f}_{1}(\mathbf{x})\) is the designed neural network function, which will be used to approximate the unknown continuous function \(f_{1}(\mathbf{x})\), and \(\Delta_{1}(\mathbf{x})=f_{1}(\mathbf{x})-\dot{f}_{1}(\mathbf{x})\) is the estimated error.  
Step 2:For the second sub system, the item \(e_{1}e_{2}\) in (29) needs to be offset, and the Lyapunov function is selected as:  
\[V_{2}=V_{1}+\frac{e_{2}^{2}}{2}. \tag{30}\]  
Obtain the derivative of Equation (30) as:  
\[\dot{V}_{2}=\dot{V}_{1}+e_{2}\dot{e}_{2}=\dot{V}_{1}+e_{2}\Big{(}a_{2}e_{3}+a_{ 2}\alpha_{2}+f_{2}(\mathbf{x})-\dot{\alpha}_{1}\Big{)}. \tag{31}\]  
The virtual control variable \(\alpha_{2}\) is designed as:  
\[\alpha_{2}=\frac{1}{a_{2}}\Big{(}-a_{1}e_{1}-\mu_{2}e_{2}-\hat{f_{2}}(\mathbf{ x})+\dot{\alpha}_{1}\Big{)}. \tag{32}\]  
Take Equation (32) into Equation (31) as:  
\[\dot{V}_{2}=\sum_{i=1}^{2}\Big{(}-\mu_{i}e_{i}^{2}+e_{i}\Delta_{i}(\mathbf{x}) \Big{)}+a_{2}e_{2}e_{3}, \tag{33}\]  
where \(\hat{f_{2}}(\mathbf{x})\) is the designed neural network function, which will be used to approximate the unknown continuous function \(f_{2}(\mathbf{x})\), and \(\Delta_{2}(\mathbf{x})=f_{2}(\mathbf{x})-\hat{f_{2}}(\mathbf{x})\) is the estimated error.  
Step 3:  
For the third sub system, the Lyapunov function is selected as:  
\[V_{3}=V_{2}+\frac{e_{3}^{2}}{2}. \tag{34}\]  
Obtain the derivative of Equation (34) as:  
\[\dot{V}_{3}=\dot{V}_{2}+e_{3}\dot{e}_{3}=\dot{V}_{2}+e_{3}\Big{(}a_{3}U_{a}+f_ {3}(\mathbf{x})-\dot{\alpha}_{2}\Big{)}. \tag{35}\]  
The control law of \(U_{a}\) is designed as:  
\[U_{a}=\frac{1}{a_{3}}\Big{(}-a_{2}e_{2}-\mu_{3}e_{3}-\hat{f_{3}}(\mathbf{x})+ \dot{\alpha}_{2}\Big{)}. \tag{36}\]  
Take Equation (36) into Equation (35) as:  
\[\dot{V}_{3}=\sum_{i=1}^{3}\Bigg{(}-\mu_{i}e_{i}^{2}+\frac{e_{i}}{a_{i}}\Delta_ {i}(\mathbf{x})\Bigg{)}, \tag{37}\]  
where \(\hat{f_{3}}(\mathbf{x})\) is the designed neural network function, which will be used to approximate the unknown continuous function \(f_{3}(\mathbf{x})\), and \(\Delta_{3}(\mathbf{x})=f_{3}(\mathbf{x})-\hat{f_{3}}(\mathbf{x})\) is the estimated error.  
After the control variable \(U_{a}\) is designed, we need to design the neural network function \(\hat{f_{i}}(\mathbf{x})\)\(i=1,2,3\) and keep the control system stable.