## 2 Mathematical Modeling  
The analyzed system is depicted in Figure 1. The dynamics of the piston of the cylinder can be described by:  
\[m\ddot{x}_{p}=P_{1}A_{1}-P_{2}A_{2}-F_{l} \tag{1}\]  
where \(m\) is the mass of the load, \(x_{p}\) is the displacement of the cylinder, \(P_{1}\) and \(P_{2}\) are the pressures in the forward and return chambers of the cylinder, \(A_{1}\) is the piston area in the non-rod chamber, \(A_{2}\) is the piston area in the rod chamber, and the overall load force is denoted by \(F_{l}\). The dynamics of the cylinder's pressure can be expressed as [33]:  
\[\frac{V_{01}+A_{1}x_{p}}{\beta_{e}}\dot{P}_{1}=Q_{in}-A_{1}\dot{x}_{p}-C_{t}(P _{1}-P_{2}) \tag{2}\]  
where \(V_{01}\) and \(V_{02}\) are the total control volumes of the cylinder and the hoses connecting the cylinder to the valve, respectively, \(\beta_{e}\) is the bulk modulus of oil, \(C_{t}\) is the internal leakage coefficient, and the flow rate \(Q_{in}\) is given in [34] as the following equation:  
\[Q_{in}=Q_{in,p}+Q_{in,p}=k_{p}\alpha+k_{q}x_{v}[s_{g}(x_{v})\sqrt{P_{s}-P_{1}} +s_{g}(-x_{v})\sqrt{P_{1}-P_{t}}] \tag{3}\]  
where \(Q_{in,p}\) is the flow rate of the pump, and \(Q_{in,p}\) is the flow rate of the valve. We define the function  
\[s_{g}(x)= \begin{cases}1,\text{ if }x\ \geq\ 0\\ 0,\text{ if }x\ <\ 0\end{cases}  \tag{4}\]  
where the pump flow gain \(k_{p}\) is a constant when the pump rotation speed is fixed, \(\alpha\) is the swashplate angle, \(k_{q}\) is the flow gain of the servo valve, \(x_{v}\) is the valve spool displacement, \(P_{s}\) is the discharge pressure of the pump, and \(P_{t}\) is the tank pressure. In this paper, the swashplate angle dynamics is modeled using second-order dynamic systems, and the pump swashplate angle's magnitude and rate limitations are considered. The angle dynamics is given as  
\[\ddot{\alpha}=2\xi_{\alpha}\omega_{\alpha}\bigg{[}S_{R}\bigg{(}\frac{\omega_ {\alpha}^{2}}{2\xi_{\alpha}\omega_{\alpha}}(S_{M}(u_{\alpha})-\alpha)\bigg{)}- \dot{\alpha}\bigg{]} \tag{5}\]where \(u_{\alpha}\) is the swashplate angle, \(\omega_{\alpha}\) and \(\xi_{\alpha}\) represent the natural frequency and damping ratio of the system, and \(S_{M}(\cdot)\) and \(S_{R}(\cdot)\) are constraints on the magnitude and the rate. They are defined by:  
\[S_{M}(x)=\left\{ \begin{array}{ll}M&\text{if}\quad x\geq M\\ x&\text{if}\quad|x|<M\\ -M&\text{if}\quad x\leq-M\end{array} \right. \tag{6}\]  
The transfer function, defined as the relationship between the input \(u_{\alpha}\) and the swashplate angle \(\alpha\), is expressed as  
\[\frac{\alpha(s)}{u_{\alpha}(s)}=\frac{\omega_{\alpha}^{2}}{s^{2}+2\xi_{ \alpha}\omega_{\alpha}s+\omega_{\alpha}^{2}} \tag{7}\]  
The proportional valve dynamics can be modeled as:  
\[\ddot{x}_{v}=2\xi_{v}\omega_{v}\bigg{[}S_{R}\bigg{(}\frac{\omega_{v}^{2}}{2 \xi_{v}\omega_{v}}(S_{M}(k_{x}u_{v})-x_{v})\bigg{)}-x_{v}\bigg{]} \tag{8}\]  
where \(u_{v}\) is the input command, and \(\omega_{v}\) and \(\xi_{v}\) represent the natural frequency and damping ratio of the valve. We define \(\mathbf{x}=\left[x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7}\right]^{T}=\left[x _{p},\dot{x}_{p},P_{1},\dot{\alpha},\dot{x},x_{v},\dot{x}_{v}\right]^{T}\). The system is expressed as  
\[ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=\frac{1}{m}[A_{1}x_{3}-A_{2}P_{2}-F_{l}]\\ \dot{x}_{3}=h(x_{1})[-A_{1}x_{2}-C_{t}(x_{3}-P_{2})+Q_{in}(x_{3},x_{4},x_{6})] \\ \dot{x}_{4}=x_{5}\\ \dot{x}_{5}=2\xi_{\alpha}\omega_{\alpha}\Big{[}S_{R}\Big{(}\frac{\omega_{ \alpha}^{2}}{2\xi_{\alpha}\omega_{\alpha}}(S_{M}(u_{\alpha})-x_{4})\Big{)}-x_{ 5}\Big{]}\\ \dot{x}_{6}=x_{7}\\ \dot{x}_{7}=2\xi_{v}\omega_{v}\Big{[}S_{R}\Big{(}\frac{\omega_{\alpha}^{2}}{2 \xi_{v}\omega_{v}}(S_{M}(u_{v})-x_{6})\Big{)}-x_{7}\Big{]}\end{array}  \tag{9}\]  
with  
\[ \begin{array}{l}h(x_{1})=\frac{\xi_{\alpha}}{A_{1}x_{1}+V_{01}}\\ Q_{in}(x_{3},x_{4},x_{6})=k_{p}x_{4}+k_{q}x_{6}g(x_{3},x_{6})\\ g(x_{3},x_{6})=s_{g}(x_{6})\sqrt{P_{s}-x_{3}}+s_{g}(-x_{6})\sqrt{x_{3}-P_{t}} \end{array}