# Automation of a Hybrid Control for Electrohydraulic Servo-Actuators with Residual Dynamics
## 5 Proposed Solution
**Case 1:** For \(\Delta<0\),(\(\lambda_{1,2}\) conjugated complex roots)  
For the first case, the form of the matrix \(P\) can be written as:  
\[P=\left[ \begin{array}{cc}1&1\\ \lambda_{1}&\lambda_{2}\end{array} \right]\Rightarrow P^{-1}=\frac{1}{(\lambda _{2}-\lambda_{1})}\left[ \begin{array}{cc}\lambda_{2}&-1\\ -\lambda_{1}&1\end{array} \right].\]  
Assuming that \(\Gamma=\frac{1}{(\lambda_{2}-\lambda_{1})}\), then we have:  
\[|X|=\left[ \begin{array}{c}X_{(1)}\\ X_{(2)}\end{array} \right]\leq\left|e^{A\Delta t}\right|\times\left[ \begin{array} []{c}X_{(01)}\\ X_{(02)}\end{array} \right] \tag{38}\]  
Therefore,  
\[|X|\leq\left|e^{A\Delta t}X_{0}\right|+\int\limits_{t_{ZM}}^{t_{res}}\left|e^{ A(t_{res}-\theta)}B\right||\epsilon_{F_{res}}|d\theta \tag{39}\]  
From these two inequalities (38) and (39), we can extract two equations thus:  
\[\left|X_{(1)}\right| \leq\left[\left|\frac{X_{(01)}}{\Gamma}\big{(}\lambda_{1}e^{ \lambda_{1}\Delta t_{1}}-\lambda_{1}e^{\lambda_{2}\Delta t_{1}}\big{)}\right|+ \left|\frac{X_{(02)}}{\Gamma}\big{(}\lambda_{2}e^{\lambda_{2}\Delta t_{1}}- \lambda_{1}e^{\lambda_{2}\Delta t_{1}}\big{)}\right|\right]\] \[+\int\limits_{t_{ZM}}^{t_{res}}\left|\frac{\Delta t}{\Gamma M} \Big{(}e^{\lambda_{2}(t_{res}-\Delta t_{1})}+e^{\lambda_{1}(t_{res}-\Delta t _{1})}\Big{)}\right||\epsilon_{F_{res}}|\leq\epsilon_{X_{(1)}}\]  
From the first inequality of the above system, the convergence time approximation of \(y\) is given by:  
\[t_{res1}\geq t_{ZM}-\frac{1}{|d|}\log\left(\left|\frac{\epsilon_{X_{(1)}}-2 \frac{\epsilon_{F_{res}}}{M|d|\Gamma|}}{\phi_{1}}\right|\right) \tag{40}\]  
where:  
\[\phi_{1}=2\frac{\epsilon_{F_{res}}}{M|d|\left|\Gamma\right|}-2\left|X_{(01)} \right|\frac{\lambda}{\left|\Gamma\right|}-2\frac{\left|X_{(02)}\right|}{ \left|\Gamma\right|}\]  
And \(d=\Re e(\lambda_{1})=\Re e(\lambda_{2})\), \(\left|\lambda\right|=\left|\lambda_{1,2}\right|=\sqrt{\left(\Re e(\lambda_{1, 2})\right)^{2}+\left(\Im m(\lambda_{1,2})\right)^{2}}\). From the second inequality, the convergence time of \(v\) is computed by:  
\[t_{res2}\geq t_{ZM}-\frac{1}{|d|}\log\left(\left|\frac{\epsilon_{X_{(2)}}-2 \frac{\epsilon_{F_{res}}}{M|d|\Gamma|}}{\phi_{2}}\right|\right) \tag{41}\]  
where  
\[\phi_{2}=2\frac{\left|\lambda\right|\epsilon_{F_{res}}}{M|d|\left|\Gamma\right| }-2\frac{\lambda^{2}\left|X_{(01)}\right|}{\left|\Gamma\right|}-2\frac{\left| \lambda X_{(02)}\right|}{\left|\Gamma\right|}.\]  
**Case 2:** For \(\Delta<0\),(\(\lambda_{1,2}\) Real roots)For the real roots, the convergence times \(t_{res1}\),\(t_{res2}\) of \(y\) and \(v\), respectively, can be deduced. We start with the following expression for \(y\):  
\[t_{res1}\geq t_{ZM}-\frac{1}{|\lambda_{2}|}\log\biggl{(}\biggl{|}\frac{\phi_{3} }{\phi_{4}}\biggr{|}\biggr{)} \tag{42}\]  
where:  
\[\phi_{3}=\frac{1}{|\Gamma|}\biggl{(}|\lambda_{1}|-1+\frac{1}{M|\lambda_{2}|} \biggr{)};\quad\phi_{4}=\varepsilon_{X_{(1)}}+\frac{1}{M|\Gamma|}\biggl{(} \frac{1}{|\lambda_{1}|}+\frac{1}{|\lambda_{2}|}\biggr{)}\]  
For \(v\),  
\[t_{res2}\geq t_{ZM}-\frac{1}{|\lambda_{2}|}\log\biggl{(}\biggl{|}\frac{\phi_{5} }{\phi_{6}}\biggr{|}\biggr{)} \tag{43}\]  
where:  
\[\left|X_{(1)}\right| \leq\left[\left|\frac{X_{(01)}}{\Gamma}\big{(}\lambda_{1}e^{ \lambda_{1}\Delta t_{1}}-\lambda_{1}e^{\lambda_{2}\Delta t_{1}}\big{)}\right|+ \left|\frac{X_{(02)}}{\Gamma}\big{(}\lambda_{2}e^{\lambda_{2}\Delta t_{1}}- \lambda_{1}e^{\lambda_{2}\Delta t_{1}}\big{)}\right|\right]\]0  
**Case 3:** For \(\Delta=0\),(\(\lambda_{1}=\lambda_{2}=\lambda\) is the double root)  
In this particular case, we have \(\frac{k}{M}=\frac{k^{2}}{4M^{2}}\), and the transformation matrix \(P\) is written as follows:  
\[P=\left[ \begin{array}{cc}1&0\\ \lambda&1\end{array} \right]\Rightarrow P^{-1}=\left[ \begin{array}{cc}1&0\\ -\lambda&1\end{array} \right];\quad Q=\left[ \begin{array}{cc}\lambda&1\\ 0&\lambda\end{array}