## 3 Control Law and Stability
### The Design of Adaptive Fuzzy Control Based on the Backstepping Method
The signals of all closed-loop systems are bounded, and for a given attenuation coefficient \(\rho>0\), the tracking performance index satisfies the following expression:  
\[\sum_{i=1}^{n}\int_{0}^{T}e_{i}^{2}(s)ds\leq\frac{1}{d_{0}}V(0)+\frac{1}{d_{0} }Tb_{0}+\sum_{i=1}^{n}\frac{1}{2d_{0}}\int_{0}^{T}\rho^{2}w_{i}^{2}dt,\,T\in[0,\infty  
by reducing the above equation, the following inequality can be obtained:  
\[\dot{V}\leq-\sum\limits_{i=1}^{n}{{\lambda_{i}}{e_{i}^{2}}}+\sum\limits_{i=1}^{n}{ \overset{\sim}{\theta}}_{i}^{T}\Bigg{(}{{e_{i}}{\xi_{i}^{*}}(\overline{x}_{i})- \frac{1}{{r_{i}}}\overset{\cdot}{\theta}}_{i}\Bigg{)}+\sum\limits_{i=1}^{n}{{| {e_{i}}{\varepsilon_{i}}|}}+\sum\limits_{i=1}^{n}{\frac{1}{{b_{i}}}{e_{i}}{w_ {i}}} \tag{34}\]  
defining the equation \(S=-\sum\limits_{i=1}^{n}{{\lambda_{i}}{e_{i}^{2}}}+\sum\limits_{i=1}^{n}{ \overset{\sim}{\theta}}_{i}^{T}\Bigg{(}{{e_{i}}{\xi_{i}^{*}}(\overline{x}_{i})- \frac{1}{{r_{i}}}\overset{\cdot}{\theta}}_{i}\Bigg{)}+\sum\limits_{i=1}^{n}{{| {e_{i}}{\varepsilon_{i}}|}}+\sum\limits_{i=1}^{n}{\frac{1}{{b_{i}}}{e_{i}}{w_ {i}}}\) and, at the same time, defining \(a_{i}={\lambda_{i}}-\frac{1}{2}-\frac{1}{2\rho^{2}{b_{i}^{2}}}\), solving for \({\lambda_{i}}\), and putting it into \(S\) gives the following equation:  
\[S=-\sum\limits_{i=1}^{n}{{a_{i}}{e_{i}^{2}}}-\frac{1}{2}{\sum\limits_{i=1}^{n }{{e_{i}^{2}}}}-\sum\limits_{i=1}^{n}{\frac{1}{{2\rho^{2}{b_{i}^{2}}}}{e_{i}^ {2}}}+\sum\limits_{i=1}^{n}{\overset{\sim}{\theta}}_{i}^{T}\Bigg{(}{{e_{i}}{ \xi_{i}^{*}}(\overline{x}_{i})-\frac{1}{{r_{i}}}\overset{\cdot}{\theta}}_{i} \Bigg{)}+\sum\limits_{i=1}^{n}{{|{e_{i}}{\varepsilon_{i}}|}}+\sum\limits_{i=1} ^{n}{\frac{1}{{b_{i}}}{e_{i}}{w_{i}}} \tag{35}\]  
shrinking the above equation, because of inequalities \(-\frac{1}{2}\sum\limits_{i=1}^{n}{{e_{i}^{2}}}+\sum\limits_{i=1}^{n}{{|{e_{i} }{\varepsilon_{i}}|}}\leq\frac{1}{2}\sum\limits_{i=1}^{n}{{\varepsilon_{i}^{2}}}\) and \(-\sum\limits_{i=1}^{n}{\frac{1}{{2\rho^{2}{b_{i}^{2}}}}{\varepsilon_{i}^{2}}}+ \sum\limits_{i=1}^{n}{\frac{1}{{b_{i}}}{e_{i}}{w_{i}}}\leq\sum\limits_{i=1}^{n} {\frac{1}{{2\rho^{2}{w_{i}^{2}}}}}\), and considering the adaptive Equation (30), the following inequalities are obtained:  
\[S\leq-\sum\limits_{i=1}^{n}{{a_{i}}{e_{i}^{2}}}+\sum\limits_{i=1}^{n}{\frac{{ k_{i}}}{{r_{i}}}\Bigg{(}{{\overset{\to}{2\theta}}}_{i}^{*T}\overset{\cdot}{ \theta}}_{i}-2{{\overset{\to}{\theta}}}_{i}^{T}\overset{\cdot}{\theta}}_{i} \Bigg{)}+\frac{1}{2}\sum\limits_{i=1}^{n}{{\varepsilon_{i}^{2}}}+\sum\limits_ {i=1}^{n}{\frac{1}{{2\rho^{2}{w_{i}^{2}}}}} \tag{36}\]  
According to inequality \({{\overset{\to}{\theta}}}_{i}^{T*}\overset{\cdot}{\theta}}_{i}^{*}+{{ \overset{\to}{\theta}}}_{i}^{T}\overset{\cdot}{\theta}}_{i}\geq 2{{\overset{\to}{ \theta}}}_{i}^{*T}\overset{\cdot}{\theta}}_{i}\), the inequality \(2{{\overset{\to}{\theta}}}_{i}^{*T}\overset{\cdot}{\theta}}_{i}-2{{ \overset{\to}{\theta}}}_{i}^{T}\overset{\Defining the compact set \(\Omega_{0}=\{X|V(X)\leq C_{0}\}\), \(C_{0}=V(0)+\frac{b_{0}+c_{0}}{a_{0}}\), from the definition of Equation (31), it can be seen that all signals of the closed-loop system are bounded:  
\(\left(e_{1},\,e_{2},\,\ldots,e_{n},\,\widetilde{\theta}_{1},\,\widetilde{\theta }_{2},\,\ldots,\widetilde{\theta}_{n}\right)^{T}\in\Omega_{0}\). Let \(d_{0}=\min\{a_{i};\,1,2,\,\ldots,n\}\).