# Adaptive Robust Fuzzy Impedance Control of an Electro-Hydraulic Actuator
## 4 Adaptive Robust Velocity Control
### Stability of ARC  
For simplicity, choosing nonlinear controller gains \(k_{2s1}\) and \(k_{3s1}\) large enough [45] so that the inequality conditions in (18) and (26) are satisfied for \(C_{\theta_{j}}=diag\Big{\{}c_{\theta_{j}}\Big{\}}\), \(l=1,2,3,j=3\) and \(C_{\phi_{k}}=diag\big{\{}c_{\phi_{kl}}\big{\}},k=2,3\) with \(c_{\phi_{kl}}^{2}\geq\frac{3w_{1}w_{3}}{4\varepsilon_{3_{kl}}^{2}},\forall k,l\), then the control input as (26) with the adaption law (14) guarantees that  
\[V_{4}\leq\exp(-\lambda_{V}t)V_{4}(0)+\frac{\varepsilon_{V}}{\lambda_{V}}[1- \exp(-\lambda_{V}t)] \tag{31}\]  
where \(\lambda_{V}=2\text{min}\{k_{2},k_{3}\}\) and \(\varepsilon_{V}=A_{1}w_{2}\varepsilon_{2}+w_{3}\varepsilon_{3}\).  
**Proof of (31).**  
\[\begin{split}&\bigg{|}w_{3}z_{3}\frac{\partial\alpha_{2}}{ \partial\theta}\dot{\theta}\bigg{|}=\Big{|}\sum_{k=2}^{3}w_{3}z_{3}\frac{ \partial\alpha_{2}}{\partial\theta}\frac{1}{\sqrt{3w_{3}}}C_{\theta_{3}} \sqrt{3w_{3}}C_{\theta_{3}}^{-1}\Gamma\phi_{k}w_{k}z_{k}\Big{|}\\ &\leq\sum_{k=2}^{3}\left(\frac{1}{\sqrt{3}}\sqrt{w_{3}}|z_{3}| \bigg{\|}C_{\theta_{3}}\Big{(}\frac{\partial\alpha_{2}}{\partial\theta}\Big{)} ^{T}\bigg{\|}\right)\Big{(}\sqrt{3w_{3}w_{k}}\Big{\|}C_{\theta_{3}}^{-1} \Gamma\phi_{k}\Big{\|}\sqrt{w_{k}}|z_{k}\Big{|}\Big{)}\\ &\leq\sum_{k=2}^{3}\left[\frac{1}{3}w_{3}z_{3}^{2}\bigg{\|}C_{ \theta_{3}}\Big{(}\frac{\partial\alpha_{2}}{\partial\theta}\Big{)}^{T}\bigg{\|} ^{2}+\frac{3}{4}w_{3}w_{k}\Big{\|}C_{\theta_{3}}^{-1}\Gamma\phi_{k}\Big{\|}^ {2}w_{k}z_{k}^{2}\right]\\ &\leq w_{3}\bigg{\|}C_{\theta_{3}}\Big{(}\frac{\partial\alpha_{2} }{\partial\theta}\Big{)}^{T}\bigg{\|}^{2}z_{3}^{2}+\sum_{k=2}^{3}w_{k}\Big{(} \frac{3}{4}w_{k}w_{3}\Big{\|}C_{\theta_{3}}^{-1}\Gamma\phi_{k}\Big{\|}^{2} \Big{)}z_{k}^{2}\end{split} \tag{32}\]  
If \(C_{\phi_{j}}\) and \(C_{\theta_{j}}\) satisfy the aforementioned condition, then (32) becomes  
\[\bigg{|}w_{3}z_{3}\frac{\partial\alpha_{2}}{\partial\theta}\dot{\theta}\bigg{|} \leq w_{3}\bigg{\|}C_{\theta_{3}}\Big{(}\frac{\partial\alpha_{2}}{\partial \theta}\Big{)}^{T}\bigg{\|}^{2}z_{3}^{2}+\sum_{k=2}^{3}w_{k}\Big{(}\big{\|}C_{ \theta_{3}}\Gamma\phi_{k}\big{\|}^{2}\Big{)}z_{k}^{2} \tag{33}\]  
Thus, considering the formula for \(k_{2s1}\) and \(k_{3s1}\), (30) turns into  
\[\begin{split}\dot{V}_{3}\leq&-w_{2}k_{2}z_{2}^{2}-w_{3 }k_{3}z_{3}^{2}+w_{2}A_{1}z_{2}\Big{(}\theta_{1}\alpha_{2s2}-\widetilde{\theta} \phi_{2}+\tilde{d}\Big{)}\\ &+w_{3}z_{3}\Big{[}\theta_{3}k_{q_{3}}z_{3}u_{s2}-\widetilde{ \theta}^{T}\phi_{3}-\frac{\partial\alpha_{2}}{\partial x_{2}}\tilde{d}\Big{]} \end{split} \tag{34}\]  
Noting the condition i of Equations (20) and (28) yields  
\[\dot{V}_{3}\leq-w_{2}k_{2}z_{2}^{2}-w_{3}k_{3}z_{3}^{2}+A_{1}w_{2}\varepsilon_{ 2}+w_{3}\varepsilon_{3}\leq-2\lambda_{V}V_{3}+\varepsilon_{V} \tag{35}\]which leads to (31).  
If, after a finite time, \(t_{0}\), \(\widetilde{d}=0\), i.e., only exiting parameter uncertainties, apart from results in (31), asymptotic output tracking (i.e., zero final tracking error) can be achieved [45].  
This can be proved by defining a new Lyapunov function  
\[V=V_{3}+\frac{1}{2}\widetilde{\theta}^{T}\Gamma^{-1}\widetilde{\theta} \tag{36}\]  
Assume the unknown parameter vector is constant; thus, \(\dot{\widetilde{\theta}}=\dot{\hat{\theta}}\), so the time derivative of \(V\) is  
\[\dot{V}\leq-\sum_{j=2}^{3}w_{j}k_{j}z_{j}^{2}-\widetilde{\theta}^{T}\tau+ \widetilde{\theta}^{T}\Gamma^{-1}\dot{\hat{\theta}}\leq-\sum_{j=2}^{3}w_{j}k_ {j}z_{j}^{2} \tag{37}\]  
Therefore, the stability of the ARC designed above is proved. \(\square\)