# Nonlinear Adaptive Robust Control of the Electro-Hydraulic Servo System
## 3 Controller Design
\(\tilde{\alpha}_{g}\triangleq\alpha_{g}-\hat{\alpha}_{g}\), \(\tilde{\alpha}_{h1}\triangleq\alpha_{h1}-\hat{\alpha}_{h1}\) and \(\tilde{\alpha}_{h2}\triangleq\alpha_{h2}-\hat{\alpha}_{h2}\) are the estimate errors. Then, the control input \(u\) is designed as  
\[\left\{ \begin{aligned} u=&\frac{1}{\hat{\alpha}_{h1}^{ \mathrm{T}}\overline{h_{1}}(x)}(u_{1}+u_{2})\\ u_{1}=&-\hat{\alpha}_{h2}\overline{h_{2}}(x)- \frac{1}{2}\hat{\alpha}_{g}^{\mathrm{T}}\overline{g}(x)z_{3}+\\ &\hat{\alpha}_{g}^{\mathrm{T}}\overline{g}(x)[(k_{1}+k_{1}^{2}k_ {2})z_{1}-(k_{1}k_{2}+1)z_{2}+\frac{\pi}{3}]\\ u_{2}=&-z_{2}-k_{3}z_{3}\end{aligned} \right. \tag{28}\]  
where \(u_{1}\) is the adaptive control input to track the desired trajectory and \(u_{2}\) is the robust term which deals with parametric estimate errors and external disturbance. \(k_{3}\) is a positive gain to be determined later. Then, Equation (25) can be written as  
\[\hat{V}_{3}=z_{3}[-z_{2}-k_{3}z_{3}+\hat{\alpha}_{g}^{\mathrm{T}}\phi_{g}+\hat {\alpha}_{h1}^{\mathrm{T}}\overline{h_{1}}(x)u+\hat{\alpha}_{h2}^{\mathrm{T} }\overline{h_{2}}(x)+(k_{1}+k_{2})\alpha_{g}^{\mathrm{T}}\overline{g}(x)\tilde {d}-\alpha_{g}^{\mathrm{T}}\overline{g}(x)\tilde{\overline{x}}_{3}] \tag{29}\]  
where \(\phi_{g}\triangleq\frac{1}{2}\overline{g}(x)z_{3}-\overline{g}(x)[(k_{1}+k_{1 }^{2}k_{2})z_{1}-(k_{1}k_{2}+1)z_{2}+\hat{\overline{x}}_{3}]\) and \(k_{3}=\eta_{3}+h(x,u)\). \(h(x,u)\) can be defined as follows:  
\[h(x,u)\triangleq \frac{1}{4\epsilon_{g}}\|\alpha_{gM}\|^{2}\|\phi_{g}\|^{2}+\frac{ 1}{4\epsilon_{h_{1}}}\|\alpha_{h_{1}M}\|^{2}\|\overline{h_{1}}u\|^{2}+\frac{1 }{4\epsilon_{h_{2}}}\|\alpha_{h_{2}M}\|^{2}\|\overline{h_{2}}\|^{2}\] \[+\frac{1}{4\epsilon_{d}}(k_{1}+k_{2})^{2}H_{1}^{2}\|\alpha_{g\max }\|^{2}\|\overline{g}(x)\|^{2}+\frac{1}{4\epsilon_{e}}H_{2}^{2}\|\alpha_{g\max }\|^{2}\|\overline{g}(x)\|^{2}\]  
where \(\alpha_{gM},\alpha_{h_{1}M}\) and \(\alpha_{h_{2}M}\) are the absolute error limits, respectively. \(\alpha_{gM}=\alpha_{g\max}-\alpha_{g\min}\), \(\alpha_{h_{1}M}=\alpha_{h_{1}\max}-\alpha_{h_{1}\min}\) and \(\alpha_{h_{2}M}=\alpha_{h_{2}\max}-\alpha_{h_{2}\min}\). \(\epsilon_{g}\), \(\epsilon_{h_{1}}\), \(\epsilon_{h_{2}}\), \(\epsilon_{d}\) and \(\epsilon_{e}\) are the design positive constants such that \(\epsilon=\epsilon_{g}+\epsilon_{h_{1}}+\epsilon_{h_{2}}+\epsilon_{d}+\epsilon_ {e}\). In order to consider the effect of parametric uncertainties in the hydraulic system, an augmented Lyapunov function \(V_{a}\) is designed as  
\[V_{a}=V_{2}+V_{3}+\frac{1}{2}\left(\tilde{\alpha}_{g}^{\mathrm{T}}T_{g}^{-1} \tilde{\alpha}_{g}+\tilde{\alpha}_{h1}^{\mathrm{T}}T_{h1}^{-1}\tilde{\alpha}_{ h1}+\tilde{\alpha}_{h2}^{\mathrm{T}}T_{h2}^{-1}\tilde{\alpha}_{h2}\right) \tag{30}\]  
where \(T_{g}\), \(T_{h1}\) and \(T_{h2}\) are the positive definite constant diagonal matrices. Then, the time derivative of Equation (30) is calculated as  
\[\begin{aligned} \dot{V}_{a}&=\dot{V}_{2}+\dot{V}_{3}- \left(\tilde{\alpha}_{g}^{\mathrm{T}}T_{g}^{-1}\dot{\tilde{\alpha}}_{g}+\tilde{ \alpha}_{h1}^{\mathrm{T}}T_{h1}^{-1}\dot{\tilde{\alpha}}_{h1}+\tilde{\alpha}_{ h2}^{\mathrm{T}}T_{h2}^{-1}\dot{\tilde{\alpha}}_{h2}\right)\\ &=-k_{1}z_{1}^{2}-\eta_{2}z_{2}^{2}+z_{2}\bar{d}-\frac{H_{1}^{2}} {4\delta}z_{2}^{2}-k_{3}z_{3}^{2}-(k_{1}+k_{2})\alpha_{g}^{\mathrm{T}} \overline{g}(x)\bar{d}+\alpha_{g}^{\mathrm{T}}\overline{g}(x)\tilde{\overline {x}}_{3}\\ &\quad+\tilde{\alpha}_{g}^{\mathrm{T}}\left(  
and \(V_{a}\leqslant exp(-\lambda_{\texttt{\tiny{P}}}t)V_{a}(0)+\frac{k_{\texttt{\tiny{P}}} }{\lambda_{\texttt{\tiny{P}}}}[1-\exp(-\lambda_{\texttt{\tiny{P}}}t)]\), where \(\lambda_{\texttt{\tiny{P}}}=min\left\{2k1,2\eta_{2},2\eta_{3}/(\|\alpha_{g}\| \|\overline{g}(x)\|)\right\}\) and \(k_{\texttt{\tiny{P}}}=\delta+\varepsilon\).