# Automation of a Hybrid Control for Electrohydraulic Servo-Actuators with Residual Dynamics
## 5 Proposed Solution  
To address the problem of our work, a reference trajectory generator of the effort is proposed, and it will be updated during the system operation. This update requires adding a dead zone \(\Delta t_{ZM}\) in the effort trajectory to enable the necessary time for the residual dynamics \(T_{res}\) to converge properly to the values imposed by \((S_{1})\). To calculate \(T_{res}\), it is necessary to have some information about the system state variables. Figure 5 describes the proposed solution. As shown in this figure, the values \(y(t_{ZM}),v(t_{ZM})\), which are, respectively, the position and the velocity at the beginning of the force static phase, must be measured. Furthermore, the error values of the position, velocity and force must be imposed in order to achieve a given accuracy. Therefore, the width of this dead zone is then used to update the switching signal. Therefore, this last one can be defined by:  
* the imposed switching times when (\(S_{1}\)) commutes to (\(S_{2}\))
* the generated switching times when (\(S_{2}\)) commutes to (\(S_{1}\))  
To compute \(T_{res}\) and \(\Delta t_{ZM}\), our proposal is developed below. The solution of (33) is defined by the following expression:  
\[X_{res}(t)=e^{A(t-t_{ZM})}.X_{res}(t_{ZM})+\int\limits_{t_{ZM}}^{t}e^{A(t-t)}B.E_{res}(t)d\tau \tag{35}\]  
where \(t_{ZM}\) is the time at which \(F_{d}=F_{0}\), with \(F_{0}=ky_{d}\big{(}t_{sp}\big{)}+Mg\) the effort given by the static phase during the position controller. The main idea of the solution is to find an approximation of \(T_{res}\) under certain considerations that will be presented. Therefore, consider that \(|e_{F}|=|F-F_{d}|<\epsilon_{F_{res}}\), where \(\epsilon_{F_{res}}\) is an imposed value. The velocity of convergence of the residual dynamics depends on the real part of the roots of the characteristic polynomial. Then, the convergence time of the residual dynamics for each kind of the characteristic equation roots will be estimated. For this, the following inequality must be solved:  
\[ \begin{array}{l}Find\ T_{res}\ with\ T_{res}\ >t_{ZM}\\ \text{where}|X(t)|=|X_{d}(t)|,\quad\forall t\geq T_{res}\end{array} \]  
\(X_{d}(t)\) represents the vector of residual dynamic values when (\(S_{1}\)) is active. Then, we can replace this last inequality by:  
\[Find\ T_{res}\ /\ T_{res}\ >t_{0}\ /\ |\epsilon_{X}(t)|\leq\epsilon_{Xd}\]  
With: \(\epsilon_{X}=\left[ \begin{array}{c}\epsilon_{X(1)}\\ \epsilon_{X(2)}\end{array} \right]=\left[ \begin{array}{c}X_{(1)}=y\\ X_{(2)}=v\end{array} \right]-\left[ \begin{array}{c}X_{d1}\\ X_{d2}\end{array} \right]\), where \(X_{d1}\) and \(X_{d2}\) are the imposed values, respectively, for the position and the velocity, which are defined by the static phase relative to the servo positioning. \(\epsilon_{X(1)}\) and \(\epsilon_{X(2)}\) are the pre-defined error values, and it is necessary to set them as low as possible. Knowing that the matrix \(A\) has a companion form, the following variable change from the state \(X_{res}\) to \(\tilde{\zeta}\) can be defined as:  
\[ \begin{array}{l}X_{res}=P\xi\Rightarrow\dot{\xi}(t)=\big{(}P^{-1}AP\big{)} \xi+P^{-1}BE_{res}\\ \Rightarrow\dot{\xi}(t)=Q\xi+B^{\prime}E_{res}\end{array}  \tag{36}\]  
where \(P\) is the transformation matrix and \(Q\) is a matrix of the diagonal or Jordan form. Then, the exponential of the matrix \(A\) is given as follows:  
\[A=PQp^{-1}\Rightarrow e^{At}=Pe^{Qt}p^{-1} \tag{37}\]  
Figure 5: Principle of the proposed solution: case (\(S_{2}\))\(\rightarrow\)(\(S_{1}\)). For each case, the convergence time of the residual dynamics \(T_{res}\) is determined and is described above. Let us define \(X_{(01)}=X_{(1)}(t_{ZM})=y(t_{ZM}),\quad X_{(02)}=X_{(2)}(t_{ZM})=v(t_{ZM})\), where \(X_{(0)}=\left[X_{(01)},X_{(02)}\right]^{T}\) and \(\lambda_{1},\lambda_{2}\) are the roots of the characteristic polynomial of (33). Therefore, depending on the sign of the discriminant of this polynomial, three cases can occur.