# RBF Neural Network Based Backstepping Control for an Electrohydraulic Elastic Manipulator
## 3 Control Design
### Adaptive Backstepping Control Based RBFNN
#### 3.2.2 Proposed Control  
The virtual control (26) is represented as follows:  
\[x_{3d}=g_{1}^{-1}(-\lambda_{1}\dot{e}_{1}+\dot{x}_{2d}-f_{1}-k_{1}s_{1}-\delta_{ 1}-\eta_{1}\ \tanh(\frac{s_{1}}{\phi_{1}})) \tag{38}\]  
Additionally, the control input (27) is also rewritten as follows:  
\[u=g_{0}{}^{-1}(\dot{x}_{3d}-f_{0}-k_{2}s_{2}-g_{1}s_{1}-\lambda_{2}e_{3}- \delta_{2}-\eta_{2}\ \tanh(\frac{s_{2}}{\phi_{2}})) \tag{39}\]  
The uncertainties are estimated by \(\hat{d}_{i}(i=1,2)\) in (36), and their weights are adjusted by  
\[\hat{W}_{i}=\begin{cases}\lambda_{1}s_{i}\mu_{i}&if\left(\|\hat{W}_{i}\|<b_{w _{i}}\right)\\ &or\left(\|\hat{W}_{i}\|=b_{w_{i}}\,and\,s_{i}\hat{W}_{i}{}^{T}\mu_{i}\geq 0 \right)\\ \lambda_{1}s_{i}\mu_{i}+\frac{\lambda_{1}s_{1}\mu_{i}\hat{W}_{i}{}^{T}\hat{W }_{i}}{\|\hat{W}_{i}\|^{2}}&\text{otherwise}\end{cases} \tag{40}\]  
where \(\|.\|\) denotes the Euclidean norm; \(\lambda_{i}(i=1,2)\) are positive learning rates; and \(b_{W}\) is given positive parameter bounds.  
In order to prove the stability of the proposed control, the Lyapunov function candidate is redefined as  
\[V_{2}=\frac{1}{2}\sum_{i=1}^{2}\left(s_{i}^{2}+\frac{1}{2\lambda_{i}}\widetilde {W}_{i}{}^{T}\widetilde{W}_{i}\right) \tag{41}\]  
**Assumption 2**.: _The reconstructed error, \(\varepsilon_{1}\), in the mechanical dynamics and the uncertainties, \(\varepsilon_{2}\), in the hydraulic dynamics are bounded by \(\|\varepsilon_{i}\|  
Based on the analysis as presented above, the derivative Lyapunov function can be represented as follows:  
\[\dot{V}=\sum_{i=1}^{2}\left(s_{i}\dot{s}_{i}-\frac{1}{\lambda_{i}} \dot{\dot{W}}_{i}^{T}\widetilde{W}\right)\leq-\zeta^{T}\Xi\zeta-\sum_{i=1}^{2} \left(\varepsilon_{i}-\eta_{i}\tanh\left(\frac{s_{i}}{\phi_{i}}\right)s_{i} \right)\leq-\mu_{\min}(\Xi)\zeta^{T}\zeta \tag{46}\]  
where \(\mu_{\min}(\Xi)\) is a minimum eigenvalue of the matrix \(\Xi\). Because \(\dot{V}\) is a negative semi-definite function, \(V\left(\zeta(t),\widetilde{W}\right)\leq V\left(\zeta(0),\widetilde{W}\right)\). This means that \(s_{1},e_{3},z_{3}\) and \(\widetilde{W}\) are bounded. Let's define \(A_{v}=-\mu_{\min}(\Xi)\zeta^{T}\zeta\leq-\dot{V}\) and integrate function \(A_{V}\) in respect to time  
\[\int_{0}^{t}A_{v}(t)dt\leq V\left(\zeta(0),\widetilde{W}\right)-V\left(\zeta( t),\widetilde{W}\right) \tag{47}\]  
Since \(V\left(\zeta(0),\widetilde{W}\right)\) is a bounded function, and \(V\left(\zeta(t),\widetilde{W}\right)\) is a non-increasing and bounded function, the following result can be concluded:  
\[\lim_{t\rightarrow\infty}\int_{0}^{t}A_{v}(\tau)d\tau<\infty \tag{48}\]  
In addition, as \(\dot{A}_{V}(t)\) is bounded by [PERSON]'s lemma [25], it can be presented that \(\underset{t\rightarrow\infty}{\lim}A_{V}(t)=0\). This means that the vectors \(s_{1}\), \(e_{3}\), and \(z_{3}\) will converge to zero as \(t\rightarrow\infty\). With this result, the proposed control and adaptive laws guarantee the stability and robustness of the controlled system under the uncertainties.