## 3 An Adaptive Robust Controller Using SMCs and RBFNNs
### Dead Zone Compensation Algorithm
\begin{array}{ll}\dot{V}&=&-s^{T}(t)K_{g}s(t)+s^{T}(t)\big{\{}\hat{W}_{D} \big{\}}^{T}\cdot\left\{\zeta^{\prime}_{D}(u_{c})\right\}\cdot\left\{ \widetilde{W}_{C}\right\}^{T}\cdot\left\{\zeta_{C}(u)\right\}-\\ &s^{T}(t)\big{\{}\widetilde{W}_{D}\big{\}}^{T}\cdot\left\{\zeta^{\prime}_{D}(u_ {c})\right\}\cdot\left\{\hat{W}_{C}\right\}^{T}\cdot\left\{\zeta_{C}(u)\right\} +\\ &s^{T}(t)\widetilde{f}-s^{T}(t)d-s^{T}(t)K_{s}\text{sgn}(s(t))+\\ &{\sum\limits_{k=1}^{n}\left\{\widetilde{W}_{Dk}\right\}^{T}}\Big{(} \zeta^{\prime}_{D}(u_{Ck})\cdot\left\{\hat{W}_{Ck}\right\}^{T}\cdot\zeta_{C}(u _{k})\cdot s_{k}+K_{1}\cdot\|s_{k}\|\cdot\hat{W}_{Dk}\Big{)}+\\ &{\sum\limits_{k=1}^{n}\left\{\widetilde{W}_{Ck}\right\}^{T}}\Big{(}-\zeta_{C}(u _{k})\cdot s_{k}\cdot\hat{W}_{Dk}\cdot\zeta^{\prime}_{D}(u_{Ck})+K_{1}\cdot\|s_ {k}\|\cdot\hat{W}_{Ck}+K_{2}\cdot\|s_{k}\|\cdot\|\hat{W}_{C}\|_{F}\cdot\hat{W}_ {Ck}\Big{)}\end{array}  \tag{37}\]Since  
\[\left\{ \begin{array}{l}s^{T}(t)\Big{\{}\widetilde{W}_{D}\Big{\}}^{T}\cdot\big{\{} \zeta^{\prime}_{D}(u_{c})\big{\}}\cdot\big{\{}\hat{W}_{C}\big{\}}^{T}\cdot[ \zeta_{C}(u)]=\sum\limits_{k=1}^{n}\Big{\{}\widetilde{W}_{Dk}\Big{\}}^{T}\cdot \big{\{}\zeta^{\prime}_{Dk}(u_{c})\big{\}}\cdot\big{\{}\hat{W}_{Ck}\big{\}}^{T} \cdot\big{\{}\zeta_{Ck}(u)|s_{k}(t)\\ s^{T}(t)\Big{\{}\hat{W}_{D}\Big{\}}^{T}\cdot\big{\{}\zeta^{\prime}_{D}(u_{c}) \big{\}}\cdot\big{\{}\widetilde{W}_{C}\big{\}}^{T}\cdot[\zeta_{C}(u)]=\sum \limits_{k=1}^{n}\Big{\{}\widetilde{W}_{Ck}\Big{\}}^{T}\cdot\zeta_{Ck}(u) \cdot\big{\{}\hat{W}_{Dk}\Big{\}}^{T}\cdot\big{\{}\zeta^{\prime}_{Dk}(u_{C}) \Big{\}}\end{array} \right. \tag{38}\]  
then. \[\dot{V} = -s^{T}(t)K_{S}s(t)+\sum\limits_{k=1}^{n}\Big{\{}\widetilde{W}_{Dk }\Big{\}}^{T}K_{1}\|s\|\dot{W}_{Dk}+\sum\limits_{k=1}^{n}\Big{\{}\widetilde{W} _{Ck}\Big{\}}^{T}\Big{(}K_{1}\|s\|\dot{W}_{Ck}+K_{2}\|s\|\|\dot{W}_{C}\|_{F} \dot{W}_{Ck}\Big{)}+\] \[s^{T}(t)\widetilde{f}-s^{T}(t)d-s^{T}(t)K_{s}\mathrm{sgn}(s(t))\] \[= -s^{T}(t)K_{S}s(t)+K_{1}\|s\|\operatorname{tr}\Big{(}\widetilde{ W}_{D}^{T}\hat{W}_{D}\Big{)}+\|s\|\operatorname{tr}\Big{(}\widetilde{W}_{C}^{T}K_{1} \hat{W}_{C}+\widetilde{W}_{C}^{T}K_{2}\|\dot{W}_{C}\|_{F}\dot{W}_{C}\Big{)}-\] \[-s^{T}(t)d+s^{T}(t)\Big{(}\widetilde{f}-K_{s}\mathrm{sgn}(s(t)) \Big{)}\] \[= -s^{T}(t)K_{S}s(t)+K_{1}\|s\|\operatorname{tr}\Big{(}\widetilde{ W}_{D}^{T}\Big{(}W_{D}-\widetilde{W}_{D}\Big{)}\Big{)}-s^{T}(t)d+s^{T}(t)\Big{(} \widetilde{f}-K_{s}\mathrm{sgn}(s(t))\Big{)}+\] \[\|s\|\operatorname{tr}\Big{(}\widetilde{W}_{C}^{T}K_{1}\Big{(}W_{ C}-\widetilde{W}_{C}\Big{)}\Big{)}+\|s\|\operatorname{tr}\Big{(}\widetilde{W}_{C}^{T}K_{2 }\|\dot{W}_{C}\|_{F}\Big{(}W_{C}-\widetilde{W}_{C}\Big{)}\Big{)}\]  
According to the inequality \(\operatorname{tr}(\widetilde{x}(x-\widetilde{x}))\leq\|\widetilde{\alpha}\|_{F }\|x\|_{F}-\|\widetilde{\alpha}\|_{F}^{2}\), one has:  
\[\left\{ \begin{array}{l}\operatorname{tr}\Big{(}\widetilde{W}_{D}^{T}\Big{(}W _{D}-\widetilde{W}_{D}\Big{)}\Big{)}\leq\|\widetilde{W}_{D}\|_{F}\|W_{D}\|_{F }-\|\widetilde{W}_{D}\|_{F}^{2}\leq\|\widetilde{W}_{D}\|_{F}\Big{(}W_{DM}-\| \widetilde{W}_{D}\|_{F}\Big{)}\\ \operatorname{tr}\Big{(}\widetilde{W}_{C}^{T}K_{1}\Big{(}W_{C}-\widetilde{W}_ {C}\Big{)}\Big{)}\leq K_{1}\bigg{(}\|\widetilde{W}_{C}\|_{F}\|W_{C}\|_{F}-\| \widetilde{W}_{C}\|_{F}^{2}\Big{)}\leq K_{1}\|\widetilde{W}_{C}\|_{F}\Big{(}W _{CM}-\|\widetilde{W}_{C}\|_{F}\Big{)}\\ \operatorname{tr}\Big{(}\widetilde{W}_{C}^{T}K_{2}\|\dot{W}_{C}\|_{F}\Big{(}W _{C}-\widetilde{W}_{C}\Big{)}\Big{)}\leq K_{2}\|\widetilde{W}_{C}\|_{F}W_{CM}^ {2}+2K_{2}\|\widetilde{W}_{C}\|_{F}^{2}W_{CM}-K_{2}\|\widetilde{W}_{C}\|_{F}^{ 3}\end{array} \right. \tag{40}\]  
According to Theorem 2, combining Equations (40) and (39):  
\[\dot{V} \leq -\|s\|\] \[= -\|s\|\]  
Let \(m(x)=K_{2}x^{3}+(K_{1}-2K_{2}W_{CM}-a_{2})x^{2}-\big{(}K_{1}W_{CM}+K_{2}W_{CM} ^{2}+a_{3}\big{)}x\), \(C=\inf\{m(x),x>0\}\), and \(n(x)=m(x)+C\). Then  
\[\dot{V}\leq-\|s\|\left\{