# Fuzzy Adaptive Asymptotic Control for a Class of Large-Scale High-Order Unknown Nonlinear Systems
## 2 Preliminaries and Problem Formulation
### Stability Analysis  
Differentiating the surface errors:  
\[\dot{z}_{k,1} = Q_{k,1}^{*}{}^{\rm T}\Psi_{k,1}(\bar{x}_{k,1})+\varepsilon_{k,1 }(\bar{x}_{k,1})+x_{k,1+1}-\dot{y}_{k,d}\] \[\dot{z}_{k,r_{k}} = Q_{k_{r_{k}}}^{*}{}^{\rm T}\Psi_{k_{r_{k}}}+\varepsilon_{k,r_{k }}+x_{k,r_{k}+1}+\frac{\hat{M}_{k,r_{k}-1}^{2}\varepsilon_{k_{r_{k}}-1}}{ \sqrt{\hat{M}_{k,r_{k}-1}^{2}\varepsilon_{k_{r_{k}}-1}^{2}+\sigma^{2}(t)}}+z_ {k,r_{k}-1}+\frac{\varepsilon_{k_{r_{k}}-1}}{\tau_{k,r_{k}-1}}\] \[\dot{z}_{k,n_{k}} = Q_{k,n_{k}}^{*}{}^{\rm T}\Psi_{k,n_{k}}+\varepsilon_{k,n_{k}}+u_{ k}+\frac{\hat{M}_{k,n_{k}-1}^{2}\varepsilon_{k,n_{k}-1}}{\sqrt{\hat{M}_{k,n_{k}-1}^ {2}\varepsilon_{k,n_{k}-1}^{2}+\sigma^{2}(t)}}+z_{k,n_{k}-1}+\frac{\varepsilon _{k,n_{k}-1}}{\tau_{k,n_{k}-1}} \tag{34}\]  
Differentiating the boundary layer errors \(e_{k,r_{k}}=s_{k,r_{k}}-\alpha_{k,r_{k}}\):  
\[\dot{e}_{k,r_{k}} = -\frac{\varepsilon_{k,r_{k}}}{\tau_{k,r_{k}}}-\frac{\hat{M}_{k,r _{k}}^{2}\varepsilon_{k,r_{k}}}{\sqrt{\hat{M}_{k,r_{k}}^{2}\varepsilon_{k,r_{ k}}^{2}+\sigma^{2}(t)}}-z_{k,r_{k}}+B_{k,r_{k}}(z_{k,1},\ldots,z_{k,(r_{k}+1)}, \varepsilon_{k,1},\ldots,\varepsilon_{k,r_{k}},\hat{Q}_{1}, \tag{35}\] \[\ldots,\hat{Q}_{k},\hat{\varepsilon}_{1},\ldots,\hat{z}_{k,r_{k} }y_{k,d},\dot{y}_{k,d},\dot{y}_{k,d},\sigma(t),\sigma(t),\hat{M}_{1},\ldots, \hat{M}_{i}),r_{k}=1,\ldots,n_{k}-1\] \[\text{where}\] \[B_{k,1}(\cdot) = -\frac{\partial\alpha_{k,1}}{\partial x_{k,1}}\dot{x}_{k,1}- \frac{\partial\alpha_{k,1}}{\partial y_{k,d}}\dot{y}_{k,d}-\frac{\partial \alpha_{k,1}}{\partial\dot{y}_{k,d}}\dot{y}_{k,d}-\frac{\partial\alpha_{k,1}} {\partial\dot{W}_{k}}\dot{\dot{W}}_{k}-\frac{\partial\alpha_{k,1}}{\partial \dot{\bar{\varepsilon}}_{k}}\dot{\dot{\varepsilon}}_{k}\] \[B_{k,r_{k}}(\cdot) = -\sum_{q=1}^{r_{k}}\frac{\partial\alpha_{k,r_{k}}}{\partial x_{k, d}}\dot{x}_{k,q}-\frac{\partial\alpha_{k,r_{k}}}{\partial\dot{Q}_{k}}\dot{ \dot{Q}}_{k}-\frac{\partial\alpha_{k,r_{k}}}{\partial\dot{\bar{\varepsilon}}_{ k}}\dot{\dot{\varepsilon}}_{k}-\frac{\partial\alpha_{k,1}}{\partial y_{k,d}} \dot{y}_{k,d}-\frac{\partial\alpha_{k,r_{k}}}{\partial\dot{e}_{k,(r_{k}-1)}} \dot{e}_{k,(r_{k}-1)}\] (36) \[-\frac{\partial\alpha_{k,1}}{\partial\dot{y}_{k,d}}\ddot{y}_{k,d}- \frac{\partial\alpha_{k,1}}{\partial\sigma(t)}\sigma(t)-\frac{\partial\alpha_{k,r _{k}}}{\partial\dot{\bar{M}}_{k,(r_{k}-1)}}\dot{M}_{k,(r_{k}-1)},r_{k}=2, \ldots,n_{k}-1\]  
Define Lyapunov function \(V_{k}\):  
\[V_{k}=V_{k,n_{k}}+\sum_{q=1}^{n_{k}-1}\frac{1}{2}\varepsilon_{k,q}^{2}+\sum_{q= 1}^{n_{k}-1}\frac{1}{2\beta_{k,q}}\bar{M}_{k,q}^{2} \tag{37}\]  
where \(\beta_{k,q},q=1,2,\ldots,n_{k}-1\) are constants that need to be designed. According to the above design steps of the controller, we have the following theorem. **Theorem 1**. : _Consider system (1), Assumption 1, and \(V_{k}(0)\leq p\) for any initial with \(p\) being a positive constant, virtual control laws (11), (20), actual control law (29), nonlinear filters (14), (23), and the adaptive laws (32) guarantee the following:_  
* _An output signal tracks the desired trajectory asymptotically, i.e.,_ \(\lim_{t\to\infty}z_{k,1}\to 0\)* _The boundedness of all signals in the closed-loop system._  
**Proof. ** According to Assumption 1 and the bounded initial states, \(\Omega_{k,0}=\{[y_{k,d},\dot{y}_{k,d},\dot{y}_{k,d}]^{T}:y_{k,d}^{2}+\dot{y}_{k,d}^ {2}+\dot{y}_{k,d}^{2}+\dot{y}_{k,d}^{2}\leq B_{k,0}\}\) is compact in \(\Re^{3}\), where \(B_{k,0}>0\), the set \(\Omega_{k,1}=\{V_{k}(t)\leq p\}\) is compact in \(\Re^{3n}\). Thus, \(\Omega_{k,0}\times\Omega_{k,1}\) are compact in \(\Re^{3n+3}\). In addition, \(\sigma(t)\) satisfies \(\lim_{t\to\infty}\int_{0}^{t}\sigma(q)\,dq\leq\sigma_{k,1}<+\infty\) and \(|\dot{\sigma}(t)|\leq\sigma_{k,2}<+\infty\) with \(\sigma_{k,1}\) and \(\sigma_{k,2}\) as constants. Thus, \(|B_{k,d}(\cdot)|\leq M_{k,q}\) on \(\Omega_{k,0}\times\Omega_{k,1}\), where \(M_{k,q}\) denotes an unknown positive constant that can be estimated by \(\tilde{M}_{k,q}\).