# Nonlinear Adaptive Robust Control of the Electro-Hydraulic Servo System
## 3 Controller Design
\begin{aligned} \alpha&\triangleq[\alpha_{1} \quad\alpha_{2}\quad\alpha_{3}\quad\alpha_{4}\quad\alpha_{5}\quad\alpha_{6} \quad\alpha_{7}\quad\alpha_{8}]^{\mathrm{T}}\\ \alpha_{1}&\triangleq V_{01}V_{02}\\ \alpha_{2}&\triangleq V_{02}A_{1}-V_{01}A_{2}\\ \alpha_{3}&\triangleq\beta V_{02}g\\ \alpha_{4}&\triangleq\beta k_{A}V_{01}g\\ \alpha_{5}&\triangleq\beta gA_{2}\\ \alpha_{6}&\triangleq C_{t}\beta(V_{02}+k_{A}V_{01})\\ \alpha_{7}&\triangleq\beta(V_{02}A_{1}+k_{A}V_{01}A_{2})\\ \alpha_{8}&\triangleq\beta A_{2}(A_{1}-A_{2})\end{aligned} \right.\]  
For simplicity, define  
\[\left\{ \begin{aligned} \alpha_{g}&\triangleq[\alpha_{1} \quad\alpha_{2}\quad 1]^{\mathrm{T}}\\ \alpha_{h1}&\triangleq[\alpha_{3}\quad\alpha_{4} \quad\alpha_{5}]^{\mathrm{T}}\\ \alpha_{h2}&\triangleq[\alpha_{6}\quad\alpha_{7} \quad\alpha_{8}]^{\mathrm{T}}\end{aligned} \right.\]and  
\[ \begin{cases}\overline{g}(x)\triangleq[1\quad x_{1}\quad-A_{1}A_{2}x_{1}^{2}]^{ \mathrm{T}}\\ \overline{h}_{1}(x)\triangleq[R_{1}\quad R_{2}\quad(R_{2}-R_{1})x_{1}]^{ \mathrm{T}}\\ \overline{h}_{2}(x)\triangleq[x_{4}-x_{3}\quad-x_{2}\quad x_{1}x_{2}]^{ \mathrm{T}}\end{cases} \]  
Obviously, the following statements hold:  
\[ \begin{cases}\begin{aligned} g(x)&=\alpha_{g}^{\mathrm{T}} \overline{g}(x)\\ h_{1}(x)&=\alpha_{h1}^{\mathrm{T}}\overline{h}_{1}(x)\\ h_{2}(x)&=\alpha_{h2}^{\mathrm{T}}\overline{h}_{2}(x)\end{aligned}\end{cases}  \tag{22}\]  
To cope with the effect of parametric estimate error, we assume that an unknown parametric vector always lies in a known bounded region, namely,  
\[\alpha\in\Omega_{\alpha}\triangleq\{\alpha:\alpha_{\min}<\alpha<\alpha_{ \max}\}\]  
where \(\alpha_{\min}=[\alpha_{1\min},\alpha_{2\min},\cdots,\alpha_{8\min}]^{\mathrm{T}}\) and \(\alpha_{\max}=[\alpha_{1\max},\alpha_{2\max}\cdots,\alpha_{8\max}]^{\mathrm{T}}\). Then, Equation (22) can be rewritten as  
\[ \begin{cases}\begin{aligned} g(x)&=\alpha_{1}+\alpha_{2}x_{1 }-A_{1}A_{2}x_{1}^{2}>0\\ h_{1}(x)&=\alpha_{3}R_{1}(x)+\alpha_{4}R_{2}(x)+\alpha_{5}[R_{2}(x)- R_{1}(x)]x_{1}\\ h_{2}(x)&=-\alpha_{6}(x_{3}-x_{4})-\alpha_{7}x_{2}+\alpha_{8}x_{1}x_{2 }\end{aligned}\end{cases}  \tag{23}\]  
Since \(g(x)>0\), the Lyapunov function is defined as  
\[V_{3}=\frac{1}{2}g(x)z_{3}^{2} \tag{24}\]  
The time derivative of Equation(24) is calculated as follows:  
\[ \begin{split}\dot{V}_{3}&=\frac{1}{2}\dot{g}(x)z_{3} ^{2}+g(x)z_{3}\dot{z}_{3}\\ &=\frac{1}{2}\dot{g}(x)z_{3}^{2}+z_{3}(h_{1}(x)u+h_{2}(x)-g(x) \dot{\overline{x}}_{3d})\end{split}  \tag{25}\]  
Here, the derivative of \(\dot{\overline{x}}_{3}\) can be calculated as  
\[\dot{\overline{x}}_{3d}=\dot{\overline{x}}_{3}-k_{1}\dot{z}_{1}-\dot{z}_{1}-k _{2}\dot{z}_{2} \tag{26}\]  
Since \(\dot{\overline{x}}_{3}\) has some uncertain parameters, a sliding mode observer can be used to estimate the derivative of state \(\overline{x}_{3}\), i.e.,  
\[ \begin{cases}\begin{aligned} \dot{w}_{1}&=-\rho( \overline{x}_{3},w_{1})\text{sign}(\overline{x}_{3}-w_{1})\\ \dot{\overline{x}}_{3}&=-\frac{\rho(\overline{x}_{3},w_{1})}{s/\tau +1}\text{sign}(\overline{x}_{3}-w_{1})\end{aligned}\end{cases}  \tag{27}\]  
where \(w_{1}\) is the estimate of \(\overline{x}_{3}\), \(\rho(\overline{x}_{3},w_{1})=\beta_{est}e^{0.3|\overline{x}_{3}-w_{1}|}\) is a variable observer gain, and \(\beta_{est}\) is a design parameter. \(\tau\) is a first-order filter time constant, and \(s\) is the Laplace operator. \(\dot{\overline{x}}_{3}\) is the estimate of \(\dot{\overline{x}}_{3}\). The observer can ensure the convergence of \(w_{1}\) to \(\overline{x}_{3}\) in a finite time \(t_{f}\), so there exists a bounded positive \(H_{2}\) such that estimate error \(\dot{\overline{x}}_{3}\triangleq\dot{\overline{x}}_{3}-\dot{\overline{x}}_{3}\) and \(H_{2}=\sup\dot{\overline{x}}_{3}|\), where \(H_{2}\) is a positive constant. Let \(\hat{\alpha}_{g}\), \(\hat{\alpha}_{h1}\), \(\hat{\alpha}_{h2}\) be the estimate of \(\alpha_{g}\), \(\alpha_{h1}\), \(\alpha_{h2}\), respectively.