## 2 Pitch System Principles and Modeling
The dynamic equation for the pressure in the two chambers of the hydraulic cylinder is as follows:  
\[\dot{\mathrm{P}}_{1}=\frac{\beta_{\mathrm{e1}}}{\mathrm{V_{01}}+\mathrm{A_{1 }x_{p}}}\left(-\mathrm{A_{1}}\dot{\mathrm{x}_{p}}-\mathrm{C_{t}P_{L}}+\mathrm{ Q_{1}}\right) \tag{5}\]  
\[\dot{\mathrm{P}}_{2}=\frac{\beta_{\mathrm{e2}}}{\mathrm{V_{02}}-\mathrm{A_{2} x_{p}}}\left(\mathrm{A_{2}}\dot{\mathrm{x}_{p}}+\mathrm{C_{t}P_{L}}-\mathrm{ Q_{2}}\right) \tag{6}\]  
where \(\mathrm{P_{1}}\) and \(\mathrm{P_{2}}\) are the actuator two-chamber pressure; \(\beta_{\mathrm{e1}}\) and \(\beta_{\mathrm{e2}}\) denote the hydraulic oil modulus of elasticity of the actuator's two chambers; \(\mathrm{V_{01}}\) and \(\mathrm{V_{02}}\) denote the initial moment volume in the hydraulic cylinder's two chambers; \(\mathrm{A_{1}}\) and \(\mathrm{A_{2}}\) represent the effective piston area in the hydraulic cylinder's two chambers; \(\mathrm{C_{t}}\) stands for the hydraulic cylinder leakage coefficient; \(\mathrm{Q_{1}}\) and \(\mathrm{Q_{2}}\) are the left and right cavities of the load flow rate. This paper takes \(\beta_{\mathrm{e1}}=\beta_{\mathrm{e2}}=\beta_{\mathrm{e^{\prime}}}\mathrm{ Q_{1}}=\mathrm{Q_{2}}\), and \(\mathrm{P_{L}}=\mathrm{P_{1}}-\mathrm{P_{2}}\). The thrust of the piston is  
\[\mathrm{F}=\mathrm{A_{1}P_{1}}-\mathrm{A_{2}P_{2}} \tag{7}\]  
and the following equation is derived:  
\[\dot{\mathrm{F}}=\mathrm{A_{1}\dot{P}_{1}}-\mathrm{A_{2}\dot{P}_{2}} \tag{8}\]  
This, in turn, gives the torque of the hydraulic cylinder acting on the paddle hub as  
\[\mathrm{T}=\mathrm{F}\;\mathrm{R_{e}}\;\mathrm{sin}\;\alpha \tag{9}\]  
The angle between the thrust force and the force arm is calculated using the following formula:  
\[\alpha=\arccos\left[\frac{\left(\mathrm{L+x_{p}}\right)^{2}+\mathrm{R_{e}^{2} }-\mathrm{H_{L}}^{2}}{2\mathrm{R_{e}}\left(\mathrm{L+x_{p}}\right)}\right] \tag{10}\]Therefore, the derivative of the torque is as follows:  
\[\dot{\mathrm{T}}=\Big{(}\mathrm{FR_{e}sin}\ \alpha\big{)}^{\prime}=\Big{(}\dot{ \mathrm{F}}\mathrm{sin}\ \alpha+\mathrm{F}\mathrm{cos}\ \alpha\Big{)}\mathrm{R_{e}} \tag{11}\]  
Define the state variables of the system as \(\mathrm{x}=[\mathrm{x_{1}},\mathrm{x_{2}},\mathrm{x_{3}}]^{\mathrm{T}}=\Big{[} \Omega,\dot{\Omega},\mathrm{T}\Big{]}^{\mathrm{T}}\); define the time-varying parameters of the system as \(\mathrm{\theta}=[\mathrm{\theta_{1}},\mathrm{\theta_{2}},\mathrm{\theta_{3}}] ^{\mathrm{T}}=[\mathrm{B},\mathrm{d},\mathrm{C_{t}}]^{\mathrm{T}}\), where \(\mathrm{B}\) is the viscous damping coefficient, and \(\mathrm{d}\) is the uncertain disturbance inside and outside the system, including the disturbance due to wind speed. The system nonlinear equation can be described as follows:  
\[\left\{ \begin{array}{c}\dot{\mathrm{x}}_{1}=\mathrm{x_{2}}\\ \mathrm{j}\dot{\mathrm{x}}_{2}=\mathrm{x_{3}}-\mathrm{\theta_{1}}\mathrm{x_{2 }}-\mathrm{\theta_{2}}\\ \dot{\mathrm{x}}_{3}=\mathrm{g_{3}}\mathrm{u}-\mathrm{f_{c}}-\mathrm{\theta_{ 3}}\mathrm{f_{u}}+\mathrm{f_{L}}\end{array} \right. \tag{12}\]  
where \(\mathrm{J}\) is the rotational inertia of the paddle; \(\mathrm{u}\) is the input pump speed; and \(\mathrm{g_{3}},\mathrm{f_{c}},\mathrm{f_{u}}\), and \(\mathrm{f_{L}}\) are known functions and are defined as follows:  
\[\mathrm{g_{3}}=\left(\frac{\mathrm{A_{1}}}{\mathrm{V_{1}}}+\frac{\mathrm{A_{2 }}}{\mathrm{V_{2}}}\right)\mathrm{\beta_{e}}\mathrm{R_{e}}\mathrm{D_{p}} \mathrm{sin}\ \alpha \tag{13}\]  
\[\mathrm{f_{c}}=\left(\frac{\mathrm{A_{1}^{2}}}{\mathrm{V_{1}}}+\frac{\mathrm{A _{2}^{2}}}{\mathrm{V_{2}}}\right)\mathrm{\beta_{e}}\mathrm{\dot{x_{p}}} \mathrm{R_{e}}\mathrm{sin}\ \alpha \tag{14}\]  
\[\mathrm{f_{u}}=\left(\frac{\mathrm{A_{1}}}{\mathrm{V_{1}}}+\frac{\mathrm{A_{2 }}}{\mathrm{V_{2}}}\right)\mathrm{\beta_{e}}\mathrm{R_{e}}\mathrm{sin}\ \alpha\mathrm{P_{L}} \tag{15}\]  
\[\mathrm{f_{L}}=(\mathrm{A_{1}P_{1}}-\mathrm{A_{2}P_{2}})\mathrm{R_{e}}\mathrm{ cos}\ \alpha \tag{16}\]  
**3.