## 2 Pitch System Principles and Modeling
Controller Design**  
To ensure that each order state follows the target values, the controller design is carried out using the backstepping method, which defines \(\mathrm{x_{1d}},\mathrm{x_{2d}},\mathrm{ and }\mathrm{x_{3d}}\) as the target values of the third-order state, and \(\mathrm{e_{1}},\mathrm{e_{2}},\mathrm{and }\mathrm{e_{3}}\) as the errors of the third-order state. To address both internal and external disturbances, a parameter adaptation module is incorporated into the controller to estimate system parameters in real time and ensure system robustness. The controller structure is shown in Figure 3.  
Step 1. Define the error as follows:  
\[\mathrm{e_{1}}=\mathrm{x_{1}}-\mathrm{x_{1d}} \tag{17}\]  
\[\mathrm{e_{2}}=\dot{\mathrm{e_{1}}}+\mathrm{k_{1}}\mathrm{e_{1}}=\mathrm{x_{2 }}-\mathrm{x_{2d}} \tag{18}\]  
where \(\mathrm{k_{1}}\) is the gain, and the following is derived:  
\[\mathrm{x_{2d}}=\dot{\mathrm{x_{1d}}}-\mathrm{k_{1}}\mathrm{e_{1}} \tag{19}\]  
Figure 3: Control block diagram of pump-controlled pitch system. By defining \(x_{3d}=\dot{x}_{2d}+\overset{\wedge}{\theta}_{1}x_{2}+\overset{\wedge}{\theta}_{2} -k_{2}e_{2}\), where \(k_{2}\) is the gain factor, Equation (32) can be written as follows:  
\[\dot{V}_{2}(e_{1},e_{2})=-k_{1}^{3}e_{1}^{2}+e_{2}e_{3}+k_{1}^{2}e_{1}e_{2}-k_{ 2}e_{2}^{2}-e_{2}\Big{(}-\widetilde{\theta}_{1}x_{2}-\widetilde{\theta}_{2} \Big{)} \tag{33}\]  
where \(-k_{1}^{3}e_{1}^{2}+k_{1}^{2}e_{1}e_{2}-k_{2}e_{2}^{2}\) a is negative definite. If \(e_{3}\) is 0 and the parameter error is 0, then \(\dot{V}_{2}(e_{1},e_{2})\) is negative definite, and, therefore, the next design goal is for \(e_{3}\) to converge to 0. Step 3. To make \(e_{3}\) converge to 0, define the Lyapunov function:  
\[\mathrm{f_{c}}=\left(\frac{\mathrm{A_{1}^{2}}}{\mathrm{V_{1}}}+\frac{\mathrm{A _{2}^{2}}}{\mathrm{V_{2}}}\right)\mathrm{\beta_{e}}\mathrm{\dot{x_{p}}} \mathrm{R_{e}}\mathrm{sin}\ \alpha \tag{14}\]0  
Then,  
\[\mathrm{f_{c}}=\left(\frac{\mathrm{A_{1}^{2}}}{\mathrm{V_{1}}}+\frac{\mathrm{A _{2}^{2}}}{\mathrm{V_{2}}}\right)\mathrm{\beta_{e}}\mathrm{\dot{x_{p}}} \mathrm{R_{e}}\mathrm{sin}\ \alpha \tag{14}\]1  
Step 3:  
Bring in \(e_{3}\), \(x_{3}\), and  
\[\mathrm{f_{c}}=\left(\frac{\mathrm{A_{1}^{2}}}{\mathrm{V_{1}}}+\frac{\mathrm{A _{2}^{2}}}{\mathrm{V_{2}}}\right)\mathrm{\beta_{e}}\mathrm{\dot{x_{p}}} \mathrm{R_{e}}\mathrm{sin}\ \alpha \tag{14}\]2  
The equation is organized to obtain  
\[\mathrm{f_{c}}=\left(\frac{\mathrm{A_{1}^{2}}}{\mathrm{V_{1}}}+\frac{\mathrm{A _{2}^{2}}}{\mathrm{V_{2}}}\right)\mathrm{\beta_{e}}\mathrm{\dot{x_{p}}} \mathrm{R_{e}}\mathrm{sin}\ \alpha \tag{14}\]3  
By defining \(g_{3}u-f_{\mathrm{c}}-\overset{\wedge}{\theta}_{3}f_{\mathrm{u}}+f_{\mathrm{ L}}-\dot{x}_{3d}=-k_{3}e_{3}\), where \(k_{3}\) is the gain factor, the control rates are determined as follows:  
\[\mathrm{f_{c}}=\left(\frac{\mathrm{A_{1}^{2}}}{\mathrm{V_{1}}}+\frac{\mathrm{A _{2}^{2}}}{\mathrm{V_{2}}}\right)\mathrm{\beta_{e}}\mathrm{\dot{x_{p}}} \mathrm{R_{e}}\mathrm{sin}\ \alpha \tag{14}\]4  
Equation (37) becomes  
\[\mathrm{f_{c}}=\left(\frac{\mathrm{A_{1}^{2}}}{\mathrm{V_{1}}}+\frac{\mathrm{A _{2}^{2}}}{\mathrm{V_{2}}}\right)\mathrm{\beta_{e}}\mathrm{\dot{x_{p}}} \mathrm{R_{e}}\mathrm{sin}\ \alpha \tag{14}\]5  
Equation (39) is negatively determined when the parameter error is 0. Therefore, the next goal is to make the parameter error 0. Step 4.