## 3 An Adaptive Robust Controller Using SMCs and RBFNNs
### Dead Zone Compensation Algorithm
 \tag{30}\]  
where \(f=M(q)\ddot{q}_{s}+C\Big{(}q,\dot{q}\Big{)}\dot{q}_{s}+G(q)+F\Big{(}\dot{q} \Big{)}+\tau_{d}\), which represents the system dynamics. The ideal adaptive robust control law using RBFNNs and a sliding mode is:  
\[u=f_{0}+K_{g}s(t)-K_{s}\text{sgn}(s(t)) \tag{31}\]  
where \(f_{0}=M_{0}(q)\ddot{q}_{s}+C_{0}\Big{(}q,\dot{q}\Big{)}\dot{q}_{s}+G_{0}(q)+F_ {0}\Big{(}\dot{q}\Big{)}+\tau_{0}\), and the modeling error is  
\[ \begin{split}&\widetilde{f}=f-f_{0}\\ &\quad=\Delta M(q)\ddot{q}_{s}+\Delta C\Big{(}q,\dot{q}\Big{)} \dot{q}_{s}+\Delta G(q)+\Delta F\Big{(}\dot{q}\Big{)}+\Delta\tau_{d}\end{split}  \tag{32}\]  
Figure 5: Structure block diagram of the control system. \(\|\widetilde{f}\|\leq f_{M}\), \(f_{M}\) is the maximum value of \(\widetilde{f}\). \(K_{g}>0\) and \(K_{s}>0\) are often chosen as diagonal matrices. Substituting ideal control law (31) and the relationship with dead zone output Equations (11) into (30) yields the closed loop error dynamics:  
\[ \begin{array}{ll}M(q)\dot{s}(t)=&-C\Big{(}q,\dot{q}\Big{)}s(t)-K_{g}s(t)+\left\{ \hat{W}_{D}\right\}^{T}\cdot\left\{\zeta^{\prime}_{D}(u_{c})\right\}\cdot\left\{ \widetilde{W}_{C}\right\}^{T}\cdot\left\{\zeta_{C}(u)\right\}-\\ &\left\{\widetilde{W}_{D}\right\}^{T}\cdot\left\{\zeta^{\prime}_{D}(u_{c}) \right\}\cdot\left\{\hat{W}_{C}\right\}^{T}\cdot\left\{\zeta_{C}(u)\right\}+ \widetilde{f}-d-K_{s}\text{sgn}(s(t))\end{array}  \tag{33}\]  
**Theorem 3**. : _Given the system in (1), selecting the tracking control law (31) plus the dead zone estimation and compensation law (9) as well as the adaptive tuning law (25), the closed loop error of the control system is uniformly bounded. Moreover, the tracking error can be kept as small as desired by increasing the gain \(K_{g}\)._  
**Proof. ** To guarantee the stability of the total control system, the Lyapunov function was chosen as follows:  
\[V=\frac{1}{2}s^{T}(t)M(q)s(t)+\frac{1}{2}{\sum\limits_{k=1}^{n}\left\{ \widetilde{W}_{Dk}\right\}^{T}}S^{-1}\Big{\{}\widetilde{W}_{Dk}\Big{\}}+ \frac{1}{2}{\sum\limits_{k=1}^{n}\left\{\widetilde{W}_{Ck}\right\}^{T}}T^{-1} \Big{\{}\widetilde{W}_{Ck}\Big{\}} \tag{34}\]  
and the derivative was:  
\[\dot{V}=s^{T}(t)\Big{(}M(q)\dot{s}(t)+C\Big{(}q,\dot{q}\Big{)}s(t)\Big{)}+{ \sum\limits_{k=1}^{n}\left\{\widetilde{W}_{Dk}\right\}^{T}}S^{-1}\Big{\{} \dot{\widetilde{W}}_{Dk}\Big{\}}+{\sum\limits_{k=1}^{n}\left\{ \widetilde{W}_{Ck}\right\}^{T}}T^{-1}\Big{\{}\dot{\widetilde{W}}_{Ck}\Big{\}} \tag{35}\]  
Substituting Equations (33) into (35), one gets:  
\[ \begin{array}{ll}\dot{V}&=&s^{T}(t)\left(\begin{array}{c}-K_{g}s(t)+\left\{ \hat{W}_{D}\right\}^{T}\cdot\left\{\zeta^{\prime}_{D}(u_{c})\right\}\cdot \left\{\widetilde{W}_{C}\right\}^{T}\cdot\left\{\zeta_{C}(u)\right\}-\\ &\left\{\widetilde{W}_{D}\right\}^{T}\cdot\left\{\zeta^{\prime}_{D}(u_{c}) \right\}\cdot\left\{\hat{W}_{C}\right\}^{T}\cdot\left\{\zeta_{C}(u)\right\}+ \\ &\widetilde{f}-d-K_{s}\text{sgn}(s(t))\end{array}\right)+\\ &{\sum\limits_{k=1}^{n}\left\{\widetilde{W}_{Dk}\right\}^{T}}S^{-1}\Big{\{} \dot{\widetilde{W}}_{Dk}\Big{\}}+{\sum\limits_{k=1}^{n}\left\{ \widetilde{W}_{Ck}\right\}^{T}}T^{-1}\Big{\{}\dot{\widetilde{W}}_{Ck}\Big{\}} \\ &=&-s^{T}(t)K_{g}s(t)+s^{T}(t)\Big{\{}\hat{W}_{D}\right\}^{T}\cdot\left\{\zeta^{ \prime}_{D}(u_{c})\Big{\}}\cdot\left\{\widetilde{W}_{C}\right\}^{T}\cdot\left\{ \zeta_{C}(u)\right\}-\\ &s^{T}(t)\Big{\{}\widetilde{W}_{D}\Big{\}}^{T}\cdot\left\{\zeta^{\prime}_{D}(u _{c})\right\}\cdot\left\{\hat{W}_{C}\right\}^{T}\cdot\left\{\zeta_{C}(u)\right\} +\\ &s^{T}(t)\widetilde{f}-s^{T}(t)d-s^{T}(t)K_{s}\text{sgn}(s(t))+\\ &{\sum\limits_{k=1}^{n}\left\{\widetilde{W}_{Dk}\right\}^{T}}S^{-1}\Big{\{} \dot{\widetilde{W}}_{Dk}\Big{\}}+{\sum\limits_{k=1}^{n}\left\{\widetilde{W}_{Ck }\right\}^{T}}T^{-1}\Big{\{}\dot{\widetilde{W}}_{Ck}\Big{\}}\end{array}  \tag{36}\]  
Combining adaptive law (25) with (36):  
\[