## 2 Mathematical Models of Systems  
With the continuous development and progress of industrial technology, high-performance electro-hydraulic servo systems are urgently needed as support. This system is selected as the subject of study, without losing generality, as illustrated in Figure 1. It is a typical servo valve-controlled hydraulic motor for rotary actuators. The liquid pressure is its power source, which makes the output shaft rotate. Through advanced control algorithms, the variation in the system input can be automatically, quickly and accurately reproduced by the position of the motor. To better reflect the intrinsic physical parameters of the nonlinear system, the system servo performance must be improved. Without surrendering generality, the servo system is simplified, and the following assumptions are made:  
**Assumption 1**. : _The loss in the pipeline can be ignored; the pressure of each working oil chamber is equal everywhere; every leakage is laminar flow._The nonlinear model of electro-hydraulic servo systems is established. The servo valve flow equation is  
\[Q_{L}=C_{d}\omega x_{v}\sqrt{\frac{P_{\mathrm{s}}-P_{\mathrm{L}}\mathrm{sign}(x_{ v})}{\rho}} \tag{1}\]  
where \(C_{d}\) is the pinhole throttling coefficient, \(\omega\) is the area gradient, \(P_{\mathrm{s}}\) is the pressure of the oil source, \(P_{\mathrm{L}}\) is the load pressure and \(P_{L}=P_{1}-P_{2}\), where \(P_{1}\) and \(P_{2}\) are the two chamber pressures of the oscillating motor, and \(\rho\) is oil density. The expression of the sign function \(\mathrm{sign}(x_{v})\) is  
\[\mathrm{sign}(x_{v})= \begin{cases}1&x_{v}\geq 0\\ -1&x_{v}<0\end{cases}  \tag{2}\]  
The equation of the continuity of the flow can be expressed as  
\[Q_{L}-C_{t}P_{L}+Q(t)=D_{m}\dot{\theta}+\frac{V}{4\beta_{e}}\dot{P}_{L} \tag{3}\]  
where \(C_{t}\) is the global oil leakage coefficient, \(Q(t)\) is the global uncertain leakage, \(D_{m}\) is the displacement of the oscillating motor, \(\theta\) is the rotation angle, \(V\) is the total volume, and \(\beta_{e}\) is the effective bulk modulus. The equilibrium equation can be determined using [PERSON]'s second law and is as follows:  
\[J\ddot{\theta}=P_{L}D_{m}-B\dot{\theta}+F(t) \tag{4}\]  
where \(J\) is the rotational inertia, \(F(t)\) is the external disturbance, \(B\) is the viscous damping coefficient. Servo valve is a crucial component of a hydraulic control system with high precision and a fast response. It converts a low-power electrical signal input into a high-power hydraulic energy output to achieve the displacement, speed, acceleration and force control of the actuator. Bandwidth is an index for measuring the dynamic performance of the servo valve, which is related to the response speed of servo systems. In this paper, the servo valve with a high response is used [34]. The servo valve has a high frequency bandwidth, so its dynamics are rewritten as a proportional link as follows [35]:  
\[x_{v}=k_{i}u\]  
where \(k_{i}\) is the dynamic proportionality factor and \(u\) is the control input. Figure 1: Electro-hydraulic servo systems model. Accordingly, the servo valve flow equation can be obtained as  
\[Q_{L}=k_{t}u\sqrt{P_{s}-P_{L}\text{sign}(u)} \tag{5}\]  
where \(k_{t}=C_{d}\omega k_{i}\sqrt{1/\rho}\) is the total flow gain. Selecting the state variables as \(x=[x_{1},x_{2},x_{3}]^{T}=\left[\theta,\dot{\theta},D_{m}P_{L}/J\right]^{T}\), then the state space expression can be obtained as follows:  
\[\left\{ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=x_{3}+f_{1}(x_{2})+d\\ \dot{x}_{3}=g(u,x_{3})u+f_{2}(x_{2},x_{3})+q\end{array} \right. \tag{6}\]  
where \(f_{1}(x_{2})=-B(t)x_{2}/J\), \(d=F(t)/J\), \(g(u,x_{3})=4D_{m}\beta_{e}k_{t}/(JV_{t})\sqrt{P_{s}-\frac{\text{sign}(u)x_{3}} {D_{m}}}\), \(f_{2}(x_{2},x_{3})=-4D_{m}^{2}\beta_{e}k_{t}x_{2}/(JV_{t})-4\beta_{e}C_{t}x_{3 }/V_{t}\), \(q=4D_{m}\beta_{e}Q(t)/(JV_{t})\). **Assumption 2**.