## 3 An Adaptive Robust Controller Using SMCs and RBFNNs
### Dead Zone Compensation Algorithm  
The dead zone inverse can be expressed in the equivalent form as [42]:  
\[D^{-1}(u)=u+u_{NN} \tag{6}\]  
Figure 4: Dead zone compensation scheme. Figure 3: Dead zone inverse. From Equations (4) and (6), one can get \(u_{NN}\) as follows:  
\[u_{NN}= \begin{cases}m^{-1}(u)-u&u_{C}<0\\ 0&u_{C}=0\\ n^{-1}(u)-u&u_{C}>0\end{cases}  \tag{7}\]  
According to the approximation theory, the ideal output values of the dead zone estimation and compensation are calculated as:  
\[\left\{ \begin{array}{c}D(u_{C})=\{W_{D}\}^{T}\cdot\zeta_{D}(u_{C})+\delta_{D }(u_{C})\\ u_{NN}=\{W_{C}\}^{T}\cdot\zeta_{C}(u)+\delta_{C}(u)\end{array} \right. \tag{8}\]  
where \(W_{D}\) and \(W_{C}\) are the values of the ideal weights, \(\zeta_{D}(u_{C})\) and \(\zeta_{C}(u)\) are outputs of the hidden layer, and \(\delta_{D}(u_{C})\) and \(\delta_{C}(u)\) are modeling errors. The actual output can be given as follows:  
\[\left\{ \begin{array}{c}\hat{D}(u_{C})=\left\{\hat{W}_{D}\right\}^{T}\cdot \left[\zeta_{D}(u_{C})\right]\\ \hat{u}_{NN}=\left\{\hat{W}_{C}\right\}^{T}\cdot\left[\zeta_{C}(u)\right] \end{array} \right. \tag{9}\]  
where \(\hat{W}_{D}\) and \(\hat{W}_{C}\) are estimators of the weight values. To define \(\widetilde{W}_{D}\) and \(\widetilde{W}_{C}\) as the error weights estimators:  
\[\left\{ \begin{array}{c}\widetilde{W}_{D}=W_{D}-\hat{W}_{D}\\ \widetilde{W}_{C}=W_{C}-\hat{W}_{C}\end{array} \right. \tag{10}\]  
The purpose of dead zone compensation is to generate an actual control signal with the dead zone \(u_{d}\), tracking the ideal control law \(u\). The relationship between \(u\) and \(u_{D}\) is given by  
\[ \begin{array}{rl}u_{D}=&u-\left\{\hat{W}_{D}\right\}^{T}\cdot\left\{\zeta_ {D}^{\prime}(u_{c})\right\}\cdot\left\{\widetilde{W}_{C}\right\}^{T}\cdot \left\{\zeta_{C}(u)\right\}+\\ &\left\{\widetilde{W}_{D}\right\}^{T}\cdot\left\{\zeta_{D}^{\prime}(u_{c}) \right\}\cdot\left\{\hat{W}_{C}\right\}^{T}\cdot\left\{\zeta_{C}(u)\right\}+d \end{array}  \tag{11}\]  
\(d\) _is the modeling mismatch, represented as:_  
\[d=-\left\{\widetilde{W}_{D}\right\}^{T}\cdot\left\{\zeta_{D}^{\prime}(u_{c}) \right\}\cdot\left\{W_{C}\right\}^{T}\cdot\left\{\zeta_{C}(u)\right\}-b+ \delta_{D}(u_{C}) \tag{12}\]  
_Here,_  
\[b=\{W_{D}\}^{T}\cdot\left\{\left(\zeta_{D}^{\prime}\left(u+\left\{\hat{W}_{C }\right\}^{T}\cdot\left[\zeta_{C}(u)\right]\right)\right)\cdot\left(\delta_{C }(u)\right)\right\}+\{W_{D}\}^{T}\cdot\left\{R\left(\widetilde{W}_{C},u\right) \right\}+\delta_{D}(u_{C}) \tag{13}\]  
_with\(R\left(\hat{W}_{C},u\right)\) defined in the proof._  
**Proof. ** From Equations (3) and (8), the dead zone output of the FFSGR is:  
\[ \begin{array}{rl}u_{D}&=D(u_{C})=\{W_{D}\}^{T}\cdot\zeta_{D}(u_{C})+\delta_ {D}(u_{C})\\ &=\{W_{D}\}^{T}\cdot\zeta_{D}(u+\hat{u}_{NN}(u))+\delta_{D}(u+\hat{u}_{NN}(u)) \end{array}  \tag{14}\]  
and from Equations (5) and (6), one has:  
\[u=D\Big{(}D^{-1}(u)\Big{)}=D(u+u_{NN}(u)) \tag{15}\]Substituting Equation (8) into (15):  
\[ \begin{array}{ll}u&=\{W_{D}\}^{T}\cdot\zeta_{D}\big{(}u+u_{NN}(u)\big{)}+\delta_ {D}(u+u_{NN}(u))\\ &=\{W_{D}\}^{T}\cdot\zeta_{D}\big{(}u+\{W_{C}\}^{T}\cdot\zeta_{C}(u)+\delta_{C} (u)\big{)}+\delta_{D}(u+u_{NN}(u))\\ &=\{W_{D}\}^{T}\cdot\zeta_{D}\bigg{(}u+\big{\{}\tilde{W}_{C}\big{\}}^{T}\cdot \zeta_{C}(u)+\big{\{}\widetilde{W}_{C}\big{\}}^{T}\cdot\zeta_{C}(u)+\delta_{C} (u)\bigg{)}+\delta_{D}(u+u_{NN}(u))\end{array}  \tag{16}\]  
Let \(x=u+\big{\{}\tilde{W}_{C}\big{\}}^{T}\cdot\zeta_{C}(u)+\big{\{}\widetilde{W}_ {C}\big{\}}^{T}\cdot\zeta_{C}(u)+\delta_{C}(u)\), \(x_{0}=u+\big{\{}\tilde{W}_{C}\big{\}}^{T}\cdot\zeta_{C}(u)\) Then, \(x-x_{0}=\big{\{}\widetilde{W}_{C}\big{\}}^{T}\cdot\zeta_{C}(u)+\delta_{C}(u)\).