# Automation of a Hybrid Control for Electrohydraulic Servo-Actuators with Residual Dynamics
## 2 Electrohydraulic Servo-Actuator Model  
The considered system is presented in Figure 1. It is a symmetric double-acting electrohydraulic servo-drive that uses a double-rod cylinder with a stroke of 330 mm, which is controlled by a five two-way servo-valve. The characteristics of the hydraulic actuator are given in Table 1 (see Appendix A). This system disposes of some interface block between the actuator and the servo-valve. The presence of this block, which is specifically designed for the test bench to implement two servo-valves, makes it possible for the system to have three operating modes. Indeed, the actuator chambers can be fed either by a \(5/2\) single servo-valve (mode 1) or by two servo-valves in parallel to increase the flow (mode 2). The servo-valves can also be used in a three-way mode to supply flow independently to each actuator chamber (mode 3). In this paper, a single mode is considered to control the actuator by one \(5/2\) servo-valve (mode 1). Although the interface block has an advantage over the system, its presence in this last one conduces to some pressure drop between the servo-valve and the cylinder chambers. This pressure drop depends on the geometrical characteristics of different pipes that constitute the block, the velocity of the fluid and the input signal. For this reason, the output flow \((Q_{11},Q_{12})\) of the servo-valve is different from the input flow \(((Q_{1},Q_{2}))\) of the actuator (see Figure 1). In this work, this intermediate block is simply approximated to a resistive component, which is described by some constant that is determined experimentally. According to [PERSON]'s second law, the dynamics of the inertia load can be described by:  
\[Ma=S\Delta p-Mg-bv-ky+h \tag{1}\]  
where \(y\) [m], \(v\) [m\(\cdot\)s\({}^{-1}\)] and \(a\) [m\(\cdot\)s\({}^{-2}\)] are, respectively, the displacement, the velocity and the acceleration of the load, \(M\) is the mass of the moving part, \(\Delta p=p_{1}-p_{2}\) is the pressure drop across the load piston, \(S\) is the effective area of the actuator chambers, \(k\) [N/m] is the stiffness load and \(b\) [N/m\(\cdot\)s\({}^{-1}\)] represents the coefficient of the viscous friction force. Dry friction forces \(h\) depend explicitly on the velocity and can be represented by the function \(h(v(t))\). The empirical model of the friction effort, which is described by a smooth function such as \(thanh(. )\), is given in Ref. [33]. In fact, according to the Tustin friction model, \(h(v(t))\) can be defined by such nonlinear function, [36]: \(h(v(t))=[F_{sdy}+(F_{sdy}-F_{Co})e^{-C[v(t)]}thanh\left(\frac{v(t)}{v_{0}}\right)\). Where \(F_{sdy}\) and \(F_{Co}\) are the dynamic dry friction and the Coulomb friction, respectively. \(C\) and \(v_{i0}\) are the coefficient of the Stribeck effect and the velocity scaling constant, respectively. In our case, the dry friction is neglected. In a variable volume chamber, the obtained equation flows rely on some assumptions, viz. : (A1) both the temperature and pressure of the oil are homogeneous in each chamber, (A2) the oil density variation is small compared to its average density, (A3) the temperature variation is small compared to the average temperature, which is equal to the oil supply temperature. According to these assumptions, and neglecting leakage flows, the governing nonlinear equations that describe the fluid flow distribution in the servo-valve can be written as follows [1]:  
\[\left\{ \begin{array}{l}Q_{1}=\frac{V_{1}(y)}{B}\frac{dp_{1}}{dt}+\frac{dV_{1 }}{dt},\\ Q_{2}=\frac{V_{2}(y)}{B}\frac{dp_{2}}{dt}+\frac{dV_{2}}{dt}\end{array} \right. \tag{2}\]  
\(\beta\) is the effective Bulk modulus. \(V_{1}\) and \(V_{2}\) are the total volumes of the cylinder defined, respectively, by:  
\[\left\{ \begin{array}{l}V_{1}(y)=V_{0}+Sy,\\ V_{2}(y)=V_{0}-Sy,\end{array} \right. \tag{3}\]  
where:  
\[V_{0}=V_{D}+S\frac{l}{2} \tag{4}\]  
Figure 1: The schematic diagram of the hydraulic system.