## 3 Control Law and Stability
### The Design of the Backstepping Controller  
The design process of the backstepping method [26; 27; 28; 29] is completed by constructing the intermediate quantity \(e_{i}=x_{i}-\alpha_{i-1}\), where \(\alpha_{i}\) is the virtual control quantity of step \(i\). The final virtual control quantity \(\alpha_{n}\) is part of the actual control quantity \(u(t)\) applied to the system. The backstepping method is adopted to reconstruct the system Formula (10) as follows:  
**Step 1**. For the first subsystem \(\dot{x}_{1}(t)=b_{1}x_{2}(t)+f_{1}(x_{1})+w_{1}(t)\), defining \(e_{1}=y-y_{d}\), where \(y_{d}\) is the expected tracking trajectory, and virtual control signal \(\alpha_{1}\) is introduced, an equation can be obtained as follows:  
\[\dot{e}_{1}=\dot{x}_{1}-\dot{y}_{d}=b_{1}e_{2}-b_{1}\alpha_{1}+f_{1}(x_{1})+w_{ 1}(t)-\dot{y}_{d} \tag{12}\]  
where \(e_{2}=x_{2}-\alpha_{1}\). **Step 2**. By introducing the virtual control quantity \(\alpha_{2}\) into the second subsystem, the following equation can be obtained:  
\[\dot{e}_{2}=\dot{x}_{2}-\dot{\alpha}_{1}=b_{2}e_{3}+b_{2}\alpha_{2}-\dot{ \alpha}_{1}+f_{2}(\overline{x}_{2})+w_{2} \tag{13}\]  
where \(e_{3}=x_{3}-\alpha_{2}\). **Step \(k\). ** By analogy, introducing a virtual control quantity \(\alpha_{k}\) is introduced into the \(k\) subsystem, defining \(e_{k+1}=x_{k+1}-\alpha_{k}\), the following equation can be obtained:  
\[\dot{e}_{k}=\dot{x}_{k}-\dot{\alpha}_{k-1}=b_{k}e_{k+1}+b_{k}\alpha_{k}-\dot{ \alpha}_{k-1}+f_{k}(\overline{x}_{k})+w_{k} \tag{14}\]  
let \(e_{n}=x_{n}-\alpha_{n-1}\); then, for the last subsystem:  
\[\dot{e}_{n}=\dot{x}_{n}-\dot{\alpha}_{n-1}=b_{n}u+f_{n}(\overline{x}_{n})+w_{ n}-\dot{\alpha}_{n-1} \tag{15}\]  
therefore, the system Equation (10) can be converted to these equations:  
\[\left\{ \begin{array}{l}\dot{e}_{k}=b_{k}e_{k+1}+b_{k}\alpha_{k}-\dot{\alpha }_{k-1}+f_{k}(\overline{x}_{k})+w_{k},\;1\leq k\leq n-1\\ \dot{e}_{n}=b_{n}u-\alpha_{n-1}+f_{n}(\overline{x}_{n})+w_{n}\end{array} \right. \tag{16}\]  
where \(\alpha_{0}=y_{d}\). By determining the virtual control quantity \(\alpha_{k}\) and \(u\), the system is stable and can reach the desired tracking trajectory, and the vibration error is eliminated. When \(k=1\), choosing the Lyapunov function gives the following equation:  
\[V_{1}=\frac{1}{2b_{1}}e_{1}^{2} \tag{17}\]  
taking the derivative of \(V_{1}\) gives the following equation:  
\[\dot{V}_{1}=e_{1}(e_{2}+\alpha_{1}+\hat{f}_{1})+\frac{1}{b_{1}}e_{1}w_{1} \tag{18}\]  
where \(\hat{f}_{1}=\frac{f_{1}(\overline{x}_{1})-\dot{y}_{d}}{b_{1}}\). Defining \(\alpha_{1}=-\lambda_{1}e_{1}-\varphi_{1}\), \(\lambda_{1}>0\), \(\varphi_{1}\) is used to approximate a fuzzy system with nonlinear function \(\hat{f}_{1}\). An equation can be obtained as follows:  
\[\dot{V}_{1}=-\lambda_{1}e_{1}^{2}+e_{1}e_{2}+e_{1}(\hat{f}_{1}-\varphi_{1})+ \frac{1}{b_{1}}e_{1}w_{1} \tag{19}\]  
When \(k=2\), designing the Lyapunov function gives the following equation:  
\[V_{2}=V_{1}+\frac{1}{2b_{2}}e_{2}^{2} \tag{20}\]  
defining \(\alpha_{2}=-\lambda_{2}e_{2}-e_{1}-\varphi_{2}\), \(\lambda_{2}>0\), \(\varphi_{2}\) is used to approximate a fuzzy system with nonlinear function \(\hat{f}_{2}\). An equation can be obtained as follows:  
\[\dot{V}_{2}=-\sum_{i=1}^{2}\lambda_{i}e_{i}^{2}+\sum_{i=1}^{2}e_{i}(\hat{f}_{i }-\varphi_{i})+e_{2}e_{3}+\sum_{i=1}^{2}e_{i}w_{i} \tag{21}\]where \(\hat{f}_{2}=\frac{\hat{f}_{2}-\hat{a}_{1}}{\hat{b}_{2}}\).