# Design of Constant Speed Controller for Hydraulic Retarder Based on Robust Control
## 3 Controller Design
\begin{cases}1,&e(t)>\Delta\\ 1/\Delta,&|e(t)|\leq\Delta\\ -1,&e(t)<\Delta\end{cases} \] (27) where \(\Delta\) is the boundary layer. Then the sliding mode controller input can be further expressed as follows: \[u=ke(t)+\rho(v)\frac{|e(t)|}{e(t)}=ke(t)+\rho(v)sat(e(t))\] (28) The equation for the hydraulic retarder filling rate is expressed as follows: \[q=\frac{ke(t)+\rho(v)sat(e(t))}{-\frac{\lambda\rho p_{t}v^{2}D^{5}b_{3}^{3}{ i_{3}}^{3}}{(0.377)^{2}v^{3}m}}\] (29)
2. The high-frequency robust controller is based on the sliding mode controller for the optimization of the upper and lower limits of the positive and negative value switching of the controller, by adding a real number \(\varepsilon\) greater than \(0\) to the denominator of the auxiliary term of the controller, it makes the input switching of the high-frequency robust controller smoother and reduces the degree of switching intensity. **Lemma 2**. : _For a controlled system:_  
\[\dot{v}=f(v)+u \tag{30}\]  
_The controller input \(u\) can control the system to a given target vehicle speed \(v_{d}\), where \(u=\dot{v}_{d}+ke+\frac{\rho(v)^{2}e(t)}{\rho(v)|e(t)|+\varepsilon^{\prime}}\), \(k>0\), \(\varepsilon>0\), considering the model accuracy error and the sensor measurement error in practice, \(\rho(v)\) is usually taken as \(\rho(v)>|f(v)|\), \(e(t)=v_{d}-v\), \(v_{d}\) is the target vehicle speed [28]. In this research, the target vehicle speed is unchanged, which gives \(\dot{v}_{d}=0\)._  
The stablity of the high-frequency robust control system is demonstrated below by constructing the Lyapunov function as follows:  
\[V_{(e(t))}=\frac{1}{2}e(t)^{2} \tag{31}\]Differentiating Equation (30) gives  
\[\dot{V}_{(e(t))}=e(t)\dot{e}(t)=e(t)(-f(v)-ke(t)-\frac{\rho(v)^{2}e(t )}{\rho(v)|e(t)|+\varepsilon}) \tag{32}\] \[=-ke(t)^{2}-f(v)e(t)-e(t)\frac{\rho(v)^{2}e(t)}{\rho(v)|e(t)|+\varepsilon}\] \[\leq-ke(t)^{2}+\rho(v)|e(t)|-e(t)\frac{\rho(v)^{2}e(t)}{\rho(v)|e( t)|+\varepsilon}\] \[=-ke(t)^{2}+\frac{\rho(v)^{2}|e(t)|^{2}+\rho(v)|e(t)|-\rho(v)^{2} |e(t)|^{2}}{\rho(v)|e(t)|+\varepsilon}\] \[=-ke(t)^{2}+\varepsilon(\frac{\rho(v)|e(t)|}{\rho(v)|e(t)|+ \varepsilon})\leq-ke(t)^{2}+\varepsilon\]  
Introduce a slack variable \(s(t)>0\) to transform inequality \(\dot{V}_{(e(t))}\leq-2 kV_{(e(t))}+\varepsilon\) into an equation as follows:  
\[V_{(e(t))}=V_{(e(0))}e^{-2 kt}-e^{-2 kt}\int_{0}^{t}e^{2 kt}s(\tau)d\tau+ \varepsilon e^{-2 kt}\int_{0}^{t}e^{2 kt}d\tau \tag{33}\]  
This is a first-order linear non-homogeneous equation, and its general solution is as follows:  
\[q=\frac{ke(t)+\rho(v)sat(e(t))}{-\frac{\lambda\rho p_{t}v^{2}D^{5}b_{3}^{3}{ i_{3}}^{3}}{(0.377)^{2}v^{3}m}}\]0  
where \(s(t)>0\), \(e^{-2 kt}>0\), \(V(e(t))=\frac{1}{2}e(t)^{2}\). Integrating over \(\varepsilon e^{-2 kt}\int_{0}^{t}e^{2 kt}d\tau\), the preceding equation is simplified as follows:  
\[q=\frac{ke(t)+\rho(v)sat(e(t))}{-\frac{\lambda\rho p_{t}v^{2}D^{5}b_{3}^{3}{ i_{3}}^{3}}{(0.377)^{2}v^{3}m}}\]1  
Taking the square root on both sides of the inequality:  
\[q=\frac{ke(t)+\rho(v)sat(e(t))}{-\frac{\lambda\rho p_{t}v^{2}D^{5}b_{3}^{3}{ i_{3}}^{3}}{(0.377)^{2}v^{3}m}}\]2  
When \(t\rightarrow\infty\), \(|e(t)|\leq\sqrt{\frac{\varepsilon}{k}}\). This is a globally consistent and bounded conclusion, and the suitable \(\varepsilon>0\) can be selected to stabilize the system asymptotically. The equation for the hydraulic retarder filling rate with a high-frequency controller is expressed as follows:  
\[q=\frac{ke(t)+\rho(v)sat(e(t))}{-\frac{\lambda\rho p_{t}v^{2}D^{5}b_{3}^{3}{ i_{3}}^{3}}{(0.377)^{2}v^{3}m}}\]3