## 4 Controller Design
### Backstepping Controller Design  
The the metering-in valve controls rod displacement \(x_{i1}\) to track the reference displacement signal \(x_{d}\). The metering-out valve to track the reference signal \(p_{d}\) controls the metering-out chamber pressure \(p_{2}\). For the metering-in side servo system, BSTDNDO is designed in five steps. A similar design procedure is utilized for the metering-out servo system to build a pressure regulator, which is omitted. Throughout this paper, \(\lx@sectionsign\) is used to denote the estimate of \(\bullet\), \(\overline{\bullet}\) is used to denote the estimation error of \(\bullet\). According to (23),  
Step 1: The displacement tracking error \(e_{1}\) is defined, as follows. \[e_{1}=x_{i1}-x_{d} \tag{30}\]  
where \(x_{d}\) is the given reference signal for the tracking system. Define \(e_{2}=x_{i2}-\alpha_{1}\), where \(\alpha_{1}\) is the first virtual control variable, it is presented, as follows. \[\alpha_{1}=-k_{1}e_{1}+\dot{x}_{d} \tag{31}\]  
where \(k_{1}\) is the controller gain, \(k_{1}>0\). The derivative of Equation (30) is  
\[\dot{e}_{1}=\dot{x}_{i1}-\dot{x}_{d}=x_{i2}-\dot{x}_{d}=e_{2}+\alpha_{1}-\dot{ x}_{d} \tag{32}\]  
If \(\alpha_{1}\) is treated as the virtual control input, then the subsystem (32) can converge to zero or as small as possible. Selecting the positive semi-definite Lyapunov function \(V_{1}=\frac{1}{2}e_{1}^{2}\) obtains its time derivative:  
\[\dot{V}_{1}=e_{1}\dot{e}_{1}=e_{1}(e_{2}+\alpha_{1}-\dot{x}_{d})=-k_{1}e^{2} _{1}+e_{1}e_{2} \tag{33}\]  
It is easy to know that, if \(e_{2}\) converges to zero and \(k_{1}>0\), then \(e_{1}\) can converge to zero in finite time by choosing the proper controller gain \(k_{1}\). Step 2: In this step, we need to design the controller to make sure that \(e_{2}\) can converge to zero. Define the error \(e_{3}=x_{i3}-\alpha_{2}\), where \(\alpha_{2}\) is the second virtual control variable and it is defined, as follows. \[\alpha_{2}=\frac{1}{c_{1}}(-k_{2}e_{2}-c_{2}(e_{2}+\alpha_{1})-\hat{\theta}_{ 2}-\delta_{2}-e_{1}+\hat{\alpha}_{1}) \tag{34}\]  
In Equation (34), \(k_{2}\) is the controller gain, \(k_{2}>0\). \(\hat{\alpha}_{1}\) is obtained by TD. \(\hat{\theta}_{2}\) is obtained by NDO. The derivative of \(e_{2}\) is:  
\[\dot{e}_{2}=\dot{x}_{i2}-\dot{\alpha}_{1}=c_{1}x_{3}+c_{2}x_{i2}+\theta_{2}+ \delta_{2}-\dot{\alpha}_{1} \tag{35}\]  
The following equation is obtained when the Lyapunov function signal is chosen to be \(V_{2}\ =\ V_{1}\ +\ \frac{1}{2}e_{2}^{2}\). \[\dot{V}_{2}=-k_{1}e_{1}^{2}-k_{2}e_{2}^{2}+c_{1}e_{2}e_{3}+e_{2}\overline{ \theta}_{2}-e_{2}\overline{\dot{\alpha}_{1}} \tag{36}\]  
In Equation (36), if \(e_{3}\) converge to zero, then \(e_{2}\) can converge to zero in finite time by choosing the proper controller gain \(k_{2}\). Step 3: In this step, we need to design the controller to make sure that \(e_{3}\) can converge to zero. Define \(e_{4}=x_{4}-\alpha_{3}\), where \(\alpha_{3}\) is the third virtual control variable, and it is defined, as follows. \[\alpha_{3}=\frac{1}{c_{3}}(-k_{3}e_{3}-c_{4}(e_{2}+\alpha_{1})-\hat{\theta}_{ 3}+\hat{\alpha}_{2}-c_{1}e_{2}) \tag{37}\]  
In Equation (37), \(\hat{\alpha}_{2}\) is obtained by TD, \(\hat{\theta}_{3}\) is obtained by NDO. \(k_{3}\) is the controller gain, \(k_{3}>0\). The derivative of \(e_{3}\) is obtained, as follows. \[\dot{e}_{3}=\dot{x}_{3}-\dot{\alpha}_{2}=c_{3}x_{4}+c_{4}(e_{2}+\alpha_{1})+ \theta_{3}-\dot{\alpha}_{2} \tag{38}\]The following equation is obtained when the Lyapunov function signal is chosen to be \(V_{3}\ =\ V_{2}\ +\ \frac{1}{2}e_{3}^{2}\). \[\dot{V}_{3}=-k_{1}e_{1}^{2}-k_{2}e_{2}^{2}-k_{3}e_{3}^{2}+e_{2}\overline{\theta} _{2}+e_{3}\overline{\theta}_{3}-e_{2}\hat{\alpha}_{1}-e_{3}\hat{\alpha}_{2}+c _{3}e_{3}e_{4} \tag{39}\]  
It is easy to know that if \(e_{4}\) converges to zero and \(k_{3}>0\), then \(e_{3}\) can converge to zero in finite time by choosing the proper controller gain \(k_{3}\). Step 4: In this step, we need to design the controller to make sure that \(e_{4}\) can converge to zero.