# Wall-Climbing Mobile Robot for Inspecting DEMO Vacuum Vessel
## 5 Adsorption Control Analysis
### Sliding Mode Robust Controller  
Equation (12) shows that the WMR adsorption system is a second-order non-linear system. Due to the presence of parameter uncertainty, gas leakage, and unknown disturbances, we designed a robust sliding mode controller to control the adsorption system [36,37]. The control block diagram for the adsorption system is shown in Figure 16. The following is the derivation of the control system. Firstly, Equation (12) is written as a state equation:  
\[\left\{ \begin{aligned} \dot{w}_{1}=w_{2}\\ \dot{w}_{2}=u+f(w)\end{aligned} \right. \tag{13}\]  
where \(f(w)=-aw_{2}-bw_{1}+\Delta\) is the system unknown variable, \(u=cU\) is the control input. Assuming that the pressure command is \(w_{d}\), the error and its derivative are:  
\[e=w_{1}-w_{d},\ \dot{e}=w_{2}-\dot{w}_{d} \tag{14}\]  
Define the sliding mode function as  
\[s=ce+\dot{e},\ c>0 \tag{15}\]  
\[\dot{s}=c\dot{e}+\ddot{e}=c\dot{e}+\dot{w}_{2}-\ddot{w}_{d}=c\dot{e}+f(w)+u- \ddot{w}_{d} \tag{16}\]  
The RBF neural network structure proposed by [PERSON] and [PERSON] can approximate any continuous function with arbitrary accuracy [38, 39, 40]. Therefore, the RBF neural network can be used to approximate \(f(w)\):  
\[h_{j}=\exp\left(\frac{\left\|\mathbf{w}-d_{j}\right\|^{2}}{2b_{j}^{2}}\right) \tag{17}\]  
Figure 16: Robust sliding mode control of adsorption system. \[f(w)=\mathbf{W}^{*T}\mathbf{h}(\mathbf{w})+\epsilon \tag{18}\]  
where \(\mathbf{w}= \begin{bmatrix}w_{1}&w_{2}\end{bmatrix} ^{T}\) is the input to the network, \(j\) is the \(j\)-th node of the hidden layer of the network, \(\mathbf{h}= \begin{bmatrix}h_{j}\end{bmatrix} ^{T}\) is the output of the Gaussian basis function of the network, \(\mathbf{W}^{*}\) is the ideal weight of the network, and \(\epsilon\) is the approximation error of the network \(\epsilon\leq\epsilon_{N}\). The network input is obtained as \(\mathbf{w}= \begin{bmatrix}w_{1}&w_{2}\end{bmatrix} ^{T}\), then the network output is:  
\[\hat{f}(w)=\mathbf{\Phi}^{T}\mathbf{h}(\mathbf{w}) \tag{19}\]  
where \(\mathbf{\Phi}\) is the actual network weight, due to  
\[f(w)-\hat{f}(w)=\mathbf{W}^{*T}\mathbf{h}(\mathbf{w})+\epsilon-\mathbf{\Phi}^{T}\mathbf{h}(\mathbf{w}) =-\widetilde{\mathbf{W}}^{T}\mathbf{h}(\mathbf{w})+\epsilon \tag{20}\]  
Define the Lyapunov function as:  
\[V=\frac{1}{2}s^{2}+\frac{1}{2\gamma}\widetilde{\mathbf{W}}^{T}\widetilde{\mathbf{W}} \tag{21}\]  
where \(\gamma>0\), \(\widetilde{\mathbf{W}}=\mathbf{\Phi}-\mathbf{W}^{*}\). \[\dot{V}=s\dot{s}+\frac{1}{\gamma}\widetilde{\mathbf{W}}^{T}\dot{\mathbf{\Phi}}=s(c\dot {e}+f(w)+u-\dot{w}_{d})+\frac{1}{\gamma}\widetilde{\mathbf{W}}^{T}\dot{\mathbf{\Phi}} \tag{22}\]  
The design control rate is  
\[u=-c\dot{e}-\hat{f}(w)+\dot{w}_{d}-\eta sgn(s) \tag{23}\]  
\[\dot{V} =s\Big{(}f(w)-\hat{f}(w)-\eta sgn(s)\Big{)}+\frac{1}{\gamma} \widetilde{\mathbf{W}}^{T}\dot{\mathbf{\Phi}} \tag{24}\] \[=s\Big{(}-\widetilde{\mathbf{W}}^{T}\mathbf{h}(\mathbf{w})+\epsilon-\eta sgn( s)\Big{)}+\frac{1}{\gamma}\widetilde{\mathbf{W}}^{T}\dot{\mathbf{\Phi}}\] \[=\epsilon s-\eta|s|+\widetilde{\mathbf{W}}^{T}\Big{(}\frac{1}{\gamma} \dot{\mathbf{\Phi}}-s\mathbf{h}(\mathbf{w})\Big{)}\]  
Considering \(\eta>|\epsilon|_{max}\), the adaptive rate is:  
\[\dot{\mathbf{\Phi}}=\gamma s\mathbf{h}(\mathbf{w}) \tag{25}\]  
Then, we obtained:  
\[\dot{V}=\epsilon s-\eta|s|\leq 0 \tag{26}\]  
The above analysis shows that \(V\geq 0\), \(\dot{V}\leq 0\), \(s\) and \(\mathbf{W}\) are bounded. Since \(s=0\) when \(\dot{V}=0\), it is clear from La Salle's invariance principle that this closed loop system is asymptotically stable. It follows that when \(t\to\infty\), then \(s\to 0\), \(e\to 0\) and \(\dot{e}\to 0\). However, \(V\geq 0\), \(\dot{V}\leq 0\), when \(t\to\infty\), \(V\) and \(\mathbf{\Phi}\) are bounded; however, it is not possible to prove that \(\mathbf{\Phi}\) is convergent. By setting the robust term \(\eta sgn(s)\) as the control law, the approximation error of the neural network could be overcome and the system can be stabilised.