## 4 Adaptive Robust Model Free Control
### Adaptive Fuzzy Neural Network Compensator  
If disturbances, uncertainties, and abrupt external forces are too large, TDE estimation error will also change greatly. TDE error is obtained as follow:  
\[\xi(t)=\hat{F}(t)-F(t) \tag{26}\]  
where \(|\xi(t)|\leq Z\). To take advantages of the artificial neural network and fuzzy inference system, a fuzzy neural network is proposed for compensating the TDE error. It has been proved that the fuzzy neural network has a nonlinear function approachability with arbitrary precision [43]. Then, we obtain the new control law:  
\[u=u_{eq}+u_{cor}+u_{NN} \tag{27}\]  
where \(u_{eq}\) denotes the same designed as in Equation (16), and \(u_{NN}\) denotes compensation term. Define \(\xi(x,w)\) as the fuzzy approximation function of \(\xi(t)\), and \(\xi(x,w)\) can be described as  
\[\xi(x,w)=\frac{\sum_{i=1}^{h}w_{i}(\prod_{j=1}^{h}u_{A_{j}^{i}}(x_{j}))}{\sum_ {i=1}^{h}(\prod_{j=1}^{h}\mu_{A_{j}^{i}}(x_{j}))}=w^{T}\Phi(x) \tag{28}\]  
where \(h\) denotes the total number of fuzzy rules, \(n\) is the number of states, and \(w=[w_{1},w_{2},..,w_{h}]^{T}\) is a vector of design parameters. \(\mu_{A_{j}^{i}}(x_{j})=exp(-\frac{\|x_{j}-c_{ji}\|}{h_{i}})\) is the Gaussian function as the membership function of the input variable \(x_{j}\). \(c_{ji}\) and \(b_{i}\) are the mean value and standard deviation of the Gaussian function, respectively. \(\Phi(x)=[\Phi_{1}(x),\Phi_{2}(x),...,\Phi_{h}(x)]^{T}\) is the fuzzy basis function vector, and defined as  
\[\Phi_{i}(x)=\frac{\prod_{j=1}^{n}\mu_{A_{j}^{i}}(x_{j})}{\sum_{i=1}^{h}(\prod_ {j=1}^{h}\mu_{A_{j}^{i}}(x_{j}))} \tag{29}\]  
Define \(w^{*}\) as the optimal weight and the approximation error is described as:  
\[\delta=\xi(x,w^{*})-\xi(x,\widehat{w}) \tag{30}\]and \(w^{*}\) should be satisfied with  
\[w^{*}=argmin_{w\in\Omega}\{sup|\hat{\xi}(x,w)-\xi(x,\hat{w})|\} \tag{31}\]  
where \(\Omega\) is the set of \(w\). Combining Equations (27), (16), (25), and (10), the system error function can be attained:  
\[ \begin{split}&\ddot{e}+\frac{1}{\gamma_{2}}\frac{n}{m}(\dot{e}^{ 2-m/n}|+\gamma_{1}ae^{a-1}\dot{e}^{[2-m/n]})+(\eta+\widehat{Z})\\ sat(s(t),\sigma)=\bar{\mathrm{M}}^{-1}(\dot{\Phi}(x,w^{*})-\delta +u_{FNN})\end{split}  \tag{32}\]  
To eliminate the TDE error, i.e., to let \(lim_{t\rightarrow\infty}e(t)=0\), the following equation should be satisfied:  
\[u_{FNN}=-\xi(x,\hat{w})=-\hat{w}^{T}\Phi(x) \tag{33}\]  
The adaptive rate of weight can be designed as follows:  
\[\hat{w}=\kappa\mathrm{M}s\gamma_{2}\frac{m}{n}|e|^{\frac{m}{n}-1}\Phi(x) \tag{34}\]  
where \(\kappa>0\). **Theorem 2**. : _Considering the nonlinear and unknown system in Equation (10), when the proposed control laws in Equations (27) and (33) with adaptive law in Equation (34) are employed in the whole process, the convergence and stability of the system are guaranteed._  
Proof. : Define the Lyapunov function as follows:  
\[V_{2}=V_{1}+\frac{1}{2\kappa}\widetilde{w}^{2} \tag{35}\]  
where \(\widetilde{w}=w^{*}-\widetilde{w}\). Differentiate the Lyapunov function \(V_{2}\), which leads to:  
\[ \begin{split}\dot{V}_{2}&=\dot{V}_{1}+\frac{1}{ \kappa}\widetilde{w}\dot{\psi}\\ &=s(\dot{e}+\gamma_{1}a|e|^{a-1}\dot{e}+\gamma_{2}\frac{m}{n}|e| ^{\frac{m}{n}-1}(\bar{\mathrm{M}}^{-1}u-\bar{\mathrm{M}}^{-1}\mathrm{F}(t)\\ &+\bar{\mathrm{M}}^{-1}\mathbf{e}_{Gk}-\bar{x}_{d}))+(\widehat{ Z}-Z)\hat{Z}+\frac{1}{\kappa}\widetilde{w}\dot{\psi}\end{split}  \tag{36}\]  
where  
\[ \begin{split} u&=u_{eq}+u_{cor}+u_{NN}\\ &=-\bar{\mathrm{M}}\frac{1}{\gamma_{2}}\frac{n}{m}((\dot{e}^{2-m /n})+\gamma_{1}a|e|^{a-1}e^{[2-m/n]})+\hat{\mathrm{F}}(t)\\ &-\mathbf{e}_{Gk}+\bar{\mathrm{M}}\dot{x}_{d}-\bar{\mathrm{M}}( \eta+\hat{Z})sign(s)-\hat{w}^{T}\xi(x)\end{split}  \tag{37}\]  
Then,  
\[