# Fuzzy Adaptive Asymptotic Control for a Class of Large-Scale High-Order Unknown Nonlinear Systems
## 2 Preliminaries and Problem Formulation
: _For any \(w\in\Re\), inequality \(0\leq|w|-\frac{w^{2}}{\sqrt{w^{2}+e^{2}}}\leq\varepsilon\) is satisfied, and \(\varepsilon>0\) is a constant._  
**Lemma 4** ([16]). : _For the uniform continuity function \(f(\cdot)\), if \(\lim_{t\to\infty}\int_{0}^{t}f(\tau)d\tau\) is bounded, we have \(lim_{t\to\infty}f(t)=0\)._  
**3. Main Results**  
_3.1. Fuzzy Adaptive DSC Design_  
Define the notions \(Q_{k}=\max\{\|Q^{*}_{k,1}\|^{2},\ldots,\|Q^{*}_{k,n_{k}}\|^{2}\},\varepsilon_{ k}=\max\{\tilde{\varepsilon}_{k,1},\ldots,\tilde{\varepsilon}_{k,n_{k}}\}\). Instead of \(Q^{*}_{k_{rR}}\) and \(\tilde{\varepsilon}_{k_{rR}}\), \(Q_{k}\) and \(\tilde{\varepsilon}_{k}\) will be used to update the parameters of FLSs such that the tedious analytic computation is avoided. Define the notions \(\tilde{Q}_{k},\tilde{Q}_{k},\tilde{\varepsilon}_{k},\tilde{\varepsilon}_{k} \). \(\tilde{Q}_{k}\) and \(\tilde{\varepsilon}_{k}\) denote the estimated values of \(Q_{k},\tilde{\varepsilon}_{k}\), respectively. \(\tilde{Q}_{k}=Q_{k}-\tilde{Q}_{k}\) and \(\tilde{\varepsilon}_{k}=\tilde{\varepsilon}_{k}-\tilde{\varepsilon}_{k}\) represent the estimated errors. _Step \((k,1)\)_: For the \(k\)th subsystem, the first surface error is defined:  
\[z_{k,1}=x_{k,1}-y_{k,d} \tag{5}\]  
Differentiating \(z_{k,1}\)  
\[\dot{z}_{k,1}=x_{k,1+1}+{Q^{*}_{k,1}}^{T}\Psi_{k,1}(\tilde{x}_{k,1})+ \varepsilon_{k,1}(\tilde{x}_{k,1})-\dot{y}_{k,d} \tag{6}\]  
For the \(k\)th subsystem, defining \(V_{k,1}\)  
\[V_{k,1}=\frac{1}{2}z^{2}_{k,1}+\frac{1}{2\gamma_{k,1}}\tilde{Q}^{2}_{k}+\frac{ 1}{2\gamma_{k,2}}\tilde{\varepsilon}^{2}_{k,1} \tag{7}\]  
\[\dot{V}_{k,1} \leq z_{k,1}\big{(}-K_{k,1}z_{k,1}-\tanh(\frac{z_{k,1}}{\omega(t)})\frac{1 }{2\eta_{k,1}}\|\Psi_{k,1}\|^{2}\hat{Q}_{k}-\tanh(\frac{z_{k,1}}{\omega(t)})\eta _{k,1}- \tag{13}\] \[\tanh(\frac{z_{k,1}}{\omega(t)})\hat{\varepsilon}_{k}+\dot{y}_{k, \hat{d}}\big{)}-z_{k,1}\dot{y}_{k,\hat{d}}+z_{k,1}\tanh(\frac{z_{k,1}}{\omega(t )})\frac{1}{2\eta_{k,1}}\|\Psi_{k,1}\|^{2}Q_{k}+\] \[z_{k,1}\tanh(\frac{z_{k,1}}{\omega(t)})\eta_{k,1}+z_{k,1}\tanh( \frac{z_{k,1}}{\omega(t)})\bar{\varepsilon}_{k}+\big{(}\frac{1}{2\eta_{k,1}} \|\Psi_{k,1}\|^{2}Q_{k}+\eta_{k,1}+\bar{\varepsilon}_{k}\big{)}\kappa\omega(t)-\] \[\frac{1}{\gamma_{k,1}}\tilde{Q}_{k}\dot{\hat{Q}}_{k}-\frac{1}{ \gamma_{k,2}}\bar{\varepsilon}_{k}\dot{\hat{\varepsilon}}_{k}+z_{k,1}(x_{k,2 }-\alpha_{k,1})\] \[= -K_{k,1}z_{k,1}^{2}+\frac{1}{\gamma_{k,1}}\tilde{Q}_{k}(\Xi_{1} ^{\mathcal{Q}_{k}}-\dot{\hat{Q}}_{k})+\frac{1}{\gamma_{k,2}}\bar{\varepsilon }_{k}(\Xi_{1}^{\bar{\varepsilon}_{k}}-\dot{\hat{\varepsilon}}_{k})+z_{k,1}(x_ {k,2}-\alpha_{k,1})+\] \[(\frac{1}{2\eta_{k,1}}\|\Psi_{k,1}\|^{2}Q_{k}+\eta_{k,1}+\bar{ \varepsilon}_{k})\kappa\omega(t)\]  
Design the novel nonlinear filter:  
\[\tau_{k,1}\dot{s}_{k,1} = -e_{k,1}-\frac{\tau_{k,1}\hat{M}_{k,1}^{2}e_{k,1}}{\sqrt{\hat{M}_ {k,1}^{2}e_{k,1}^{2}+\sigma^{2}(t)}}-\tau_{k,1}z_{k,1}\] \[\dot{z}_{k,1}=x_{k,1+1}+{Q^{*}_{k,1}}^{T}\Psi_{k,1}(\tilde{x}_{k,1})+ \varepsilon_{k,1}(\tilde{x}_{k,1})-\dot{y}_{k,d} \tag{6}\]0  
where \(\tau_{k,1}\) and \(e_{k,1}:=s_{k,1}-\alpha_{k,1}\) denote the filter time constant and the first boundary layer error, respectively; \(\hat{M}_{k,1}\) is illustrated later; \(\sigma(t)\) satisfies \(\lim_{t\to\infty}\int_{0}^{t}\sigma(q)\,dq\leq\sigma_{k,1}<+\infty\) and \(|\hat{\sigma}(t)|\leq\sigma_{k,2}<+\infty\) with \(\sigma_{k,1}\) and \(\sigma_{k,2}\) as constants. A nonlinear filter is designed to avoid the repeated differentiation of virtual control law such that the noise rejection capability is improved. The filter is characterized by that it is a hyperplane, which can make the system state slide along the hyperplane to the desired state point, and it can realize a mandatory system state switching on this plane, so that the system has the characteristics of robustness for noise. **Remark 1**. : _Nonlinear filter (14) releases the necessaryconditions for directly using \(x_{k,2}-\alpha_{k,1}\), thus avoiding the \"complexity explosion\" problem and improving the robustness of the system.