# Automation of a Hybrid Control for Electrohydraulic Servo-Actuators with Residual Dynamics
## 2 Electrohydraulic Servo-Actuator Model
is the piping volume of each chamber for the zero position and \(V_{D}\) is a dead volume present on each extremity of the cylinder. Considering that the interface block is symmetric, then its piping volume is also taken account of by \(V_{0}\). In our case, the servo-valve has a large frequency of about 1 kHz. Consequently, it can be neglected in front of the slow dynamics of the actuator whose open-loop average frequency in the central position is about 250 Hz. This is why the spool valve displacement \(x_{t}\) is assumed to be directly related to the control voltage \(u\), with a proportional relationship defined by the gain \(K_{sv}\), viz: \(x_{t}=K_{sv}\)\(u\). This kind of assumption can be used in control for some operating modes [1]. Then, the flow laws can be given in affine form:  
\[\left\{ \begin{array}{c}Q_{1}=r\psi_{1}(sign(u),p_{1},p_{P},p_{T})\\ Q_{2}=-r\psi_{2}(sign(u),p_{2},p_{P},p_{T})\end{array} \right. \tag{5}\]  
where:  
\[\left\{ \begin{array}{c}\psi_{1}(.)=\alpha\Big{[}h(u)\sqrt{|p_{P}-p_{1}|} sign(p_{P}-p_{1})+h(-u)\sqrt{|p_{1}-p_{T}|}sign(p_{1}-p_{T})\\ \psi_{2}(.)=\alpha\Big{[}h(u)\sqrt{|p_{T}-p_{2}|}sign(p_{T}-p_{2})+h(-u)\sqrt{ |p_{P}-p_{2}|}sign(p_{P}-p_{2})\Big{]}\end{array} \right. \tag{6}\]  
With: \(h(u)=\frac{1+sign(u)}{2}\) and the function \(sign(u)\) defined by:  
\[sign(u)=\left\{ \begin{array}{c}1\ \ \forall u\geq 0,\\ -1\ \ \forall u<0\end{array} \right. \tag{7}\]  
The servo-valve is supposed to be a symmetric one, which justifies the presence of the same variable gain in the two flow laws. The coefficient of the flow gain \(r\) is defined by the expression \(r=\omega\psi\), is the spool valve area gradient, and \(\phi\) is the pressure drop caused by the interface block. Let us define the variable \(\alpha=C_{d\omega}K_{sv}\sqrt{2/\rho}\), which is assumed to be a constant parameter, where \(\rho\) is the fluid density and \(C_{d\omega}\) is the flow coefficient constant of each restriction. Note that the defined parameters \(\alpha,r\) are supposed to be constant. We note the physical domain of the system by:  
\[D_{\varphi}=\left\{(y,v,\Delta P)\in 1R^{3}/|y|\leq\frac{l}{2},p_{1},p_{2}\in \Omega_{p}\equiv\left]p_{T},p_{P}\right[\right\} \tag{8}\]  
Considering \(D_{\varphi}\), the terms \(sign(p_{P}-p_{j})\) and \(sign(p_{j}-p_{T}),\quad j=\{1,2\}\) introduced in the expression of \(\psi_{1}\) and \(\psi_{2}\) (Equation (6)) can be deleted as well as the absolute values. Let us define the state variables \(X=\left[y,v,\Delta P\right]^{\prime}\), and the system can then be expressed in the form below:  
\[\dot{X}=f(X)+g(X)u, \tag{9}\]  
With \(X,f(X),g(X)\in\mathbb{R}^{3}\), \(u\in\mathbb{R}\), where \(f(X),g(X)\) are locally Lipschitz vector fields, defined by:  
\[f(X)=\left( \begin{array}{c}v\\ \frac{1}{M}[S\Delta P-M\mathfrak{g}-b\upsilon-ky]\\ -\beta S\Big{[}\frac{1}{V_{1}(y)}+\frac{1}{V_{2}(y)}\Big{]}\end{array} \right)\]  
\[g(X)=\left( \begin{array}{c}0\\ 0\\ \beta\Big{[}\frac{r\psi_{1}(.)}{V_{1}(y)}+\frac{r\psi_{2}(.)}{V_{2}(y)}\Big{]} \end{array} \right)\]  
And \(b\) is the coefficient of the viscous friction force, and \(k\) is the stiffness load.