# A Novel Adaptive Gain of Optimal Sliding Mode Controller for Linear Time-Varying Systems
## 3 Illustrative Examples
### Example 4--Proposed Controller for Vibration Control with Random Bump Road Excitation (1)  
In order to demonstrate effective applicability of the proposed optimal control law, a vehicle seat suspension system featuring MR (magnetorheological) damper previously studied by the authors [16; 17; 18] is adopted. The mechanical model of the seat suspension is shown in Figure 6, and from this model, the governing equations of the system are derived as follows:  
\[m_{ls}\ddot{x}_{s}=-k_{s}(x_{s}-x_{0})-c_{ls}\Big{(}\dot{x}_{s}-\dot{x}_{0} \Big{)}+k_{I1}(x_{1}-x_{s})+c_{I1}\Big{(}\dot{x}_{1}-\dot{x}_{s}\Big{)}+F_{dMR} \tag{30}\]  
\[m_{l1}\ddot{x}_{1}=-k_{I1}(x_{1}-x_{s})+c_{I1}\Big{(}\dot{x}_{1}-\dot{x}_{s} \Big{)} \tag{31}\]  
Figure 3: Power Spectral Density (PSD) of the proposed control 1 with Equation (12): (**a**) gain function \(Q_{\Delta L}(t)\), (**b**) state variable \(x_{e}(t)\).  
Figure 4: _Cont._  
Figure 4: Simulation results of the proposed control 2 with Equation (7): (**a**) gain function \(Q_{\Delta L}(t)\), (**b**) main control \(u^{*}(t)\), (**c**) state variable \(x_{e}(t)\), (**d**) state variable \(\dot{x}_{e}(t)\), (**e**) large view of state variable \(x_{e}(t)\), (**f**) large view of state variable \(\dot{x}_{e}(t)\).  
Figure 5: Power spectral density (PSD) of the proposed control 1 with Equation (12) and the proposed control 2 with Equation (7): (**a**) gain function \(Q_{\Delta L}(t)\), (**b**) state variable \(x_{e}(t)\).  
In the above, \(x_{\rm s}\) and \(x_{1}\) represent the displacement of the sprung mass and driver mass, respectively. The variables \(k_{i}\) and \(c_{i}\) denotes the corresponding coefficients of the spring and damper. Moreover, \(x_{0}\) is the excitation displacement and \(F_{\rm MR}\) represents the magnetic field-dependent damping force of MR damper. The state-space model of the system is derived from the above equations as follows:  
\[\begin{split}\dot{x}_{l11}=\dot{x}_{\rm s}&=x_{l22} \\ \dot{x}_{l22}=f_{l11}(x_{l11},x_{l22},x_{l33},x_{l44})& +g_{l11}(x_{l11},x_{l22},x_{l33},x_{l44})u\\ \dot{x}_{l33}=\dot{x}_{1}&=x_{l44}\\ \dot{x}_{l44}=f_{l22}(x_{l11},x_{l22},x_{l33},x_{l44})\end{split} \tag{32}\]  
where:  
\[\begin{split} f_{l11}(x_{l11},x_{l22},x_{l33},x_{l44})& =-\frac{k_{l_{\rm c}}}{m_{k}}(x_{l11}-x_{0})-\frac{c_{l_{\rm c}}}{m_{k}} \Big{(}x_{l22}-\dot{x}_{0}\Big{)}+\frac{k_{l1}}{m_{k}}(x_{l33}-x_{l11})+\frac {c_{l1}}{m_{k}}(x_{l44}-x_{l22})\\ g_{l11}(x_{l11},x_{l22},x_{l33},x_{l44})&=\frac{k_{ l1}}{m_{k}}\\ u&=F_{dMR}\\ f_{l22}(x_{l11},x_{l22},x_{l33},x_{l44})&=-\frac{k_ {l1}}{m_{k}}(x_{l33}-x_{l11})-\frac{c_{l1}}{m_{l1}}(x_{l44}-x_{l22})\end{split}\]  
It is noted that \(x_{l11},x_{l22},x_{l33},x_{l44}\) are variables related to \(x_{s}\) and \(x_{1}\) of the system. The calculated result of \(u=F_{dMR}\) must be converted to the input signal to generate the required magnetic field for MR damper. The parameters of the seat suspension system used in this simulation are listed in [16; 17; 18]. The following parameters of \(t\in[0,300]\), \(M_{L\xi}(t)=20,N_{L\xi}(t)=20\), and \(\gamma=100\) are chosen for the simulation.  
In this simulation, the random road profile shown in Figure 7 is used as an excitation signal from the ground. It is seen from Figure 8a that the gain value \(Q_{\Delta L}(t)\) is always changed and its final value is the positive number. The main control \(u^{*}(t)\) shown in Figure 8b is also changed following the variation of the vibration. The results of the state system and states of the driver are shown in Figures 9-11. In Figure 9a, it is shown that