## 3 Simulations for Control Performances  
### Dynamic Model and Parameters  
In this work, a vehicle seat suspension with MR damper studied in [23] is adopted to demonstrate the effectiveness of the proposed controller for the robust control performance against disturbances. Figure 2 shows a 2-DOF (degrees of freedom) model of the vehicle seat suspension equipped with MR damper where its damping force is represented by F\({}_{\text{MR}}\) (N). The other variables are defined as follows: m\({}_{1}\) is the mass of driver (or passenger), m\({}_{1}\) = 77 (kg); k\({}_{1}\) is the stiffness coefficient of torso k\({}_{1}\) = 49.340 (N/m), c\({}_{1}\) is the damping coefficient of torso, c\({}_{1}\) = 2475 (N.s/m); m\({}_{s}\) is the mass of the seat, m\({}_{s}\) = 27 (kg); k\({}_{s}\) is the stiffness coefficient of the frame, k\({}_{s}\) = 17.830 (N/m); c\({}_{s}\) is the damping coefficient of the frame, c\({}_{s}\)= 1500 (N.s/m); x\({}_{1}\) is the displacement of the driver (m), x\({}_{s}\) is the displacement of the seat (m), x\({}_{0}\) is the displacement related excitation (m). More details on this model are given in [17; 23]. Figure 1: Control strategy of the proposed controller: (**a**) block diagram, (**b**) control process. The dynamic motion equations of the seat suspension system are obtained as follows. \[m_{s}\ddot{x}_{s}=-k_{s}(x_{s}-x_{0})-c_{s}\big{(}\dot{x}_{s}-\dot{x}_{0}\big{)}+ k_{1}(x_{1}-x_{s})+c_{1}\big{(}\dot{x}_{1}-\dot{x}_{s}\big{)}+F_{MR} \tag{34}\]  
\[m_{1}\ddot{x}_{1}=-k_{1}(x_{1}-x_{s})+c_{1}\big{(}\dot{x}_{1}-\dot{x}_{s}\big{)} \tag{35}\]  
The state-space form of Equations (34) and (35) is rewritten as follows. \[ \begin{array}{c}\dot{x}_{11}=\dot{x}_{s}=x_{22}\\ \dot{x}_{22}=f_{11}(x_{11},x_{22},x_{33},x_{44})+g_{11}(x_{11},x_{22},x_{33},x_ {44})u_{\
abla}\\ \dot{x}_{33}=\dot{x}_{1}=x_{44}\\ \dot{x}_{44}=f_{22}(x_{11},x_{22},x_{33},x_{44})\end{array}  \tag{36}\]  
In the above,  
\[ \begin{array}{c}f_{11}(x_{11},x_{22},x_{33},x_{44})=-\frac{k_{s}}{m_{s}}(x_{ 11}-x_{0})-\frac{c_{s}}{m_{s}}\big{(}x_{22}-\dot{x}_{0}\big{)}+\frac{k_{1}}{m_ {s}}(x_{33}-x_{11})+\frac{c_{1}}{m_{s}}(x_{44}-x_{22}),\\ g_{11}(x_{11},x_{22},x_{33},x_{44})=\frac{1}{m_{s}},\,u_{\
abla}=F_{MR},\,f_{ 22}(x_{11},x_{22},x_{33},x_{44})=-\frac{k_{1}}{m_{1}}(x_{33}-x_{11})-\frac{c_{ 1}}{m_{1}}(x_{44}-x_{22})\end{array} \]  
It is noted that \(x_{11},x_{22},x_{33},x_{44}\) are variables related to \(x_{s}\) and \(x_{1}\) of the system which are pointed out in both Figure 2 and Equation (36). The value of \(u_{\
abla}=F_{MR}\) is converted to the input signal to generate the magnetic field for MR damper. This can be solved by using the following equation [23]. \[F_{MR}=(c_{a}+c_{b}V)(x_{44}-x_{22})+k_{0}(x_{33}-x_{11})+(\alpha_{a}+\alpha_{ b}V) \tag{37}\]  
Thus, the voltage to be applied to MR damper is found from Equation (37) as follows:  
\[V=\frac{F_{MR}-[c_{a}(x_{44}-x_{22})+k_{0}(x_{33}-x_{11})+\alpha_{a}\phi]}{c_ {b}(x_{44}-x_{22})+\alpha_{b}\phi} \tag{38}\]  
In the above, \(k_{0}\) is the linear spring stiffness, \(k_{0}=0\) (N/m), \(c_{a}\) is the viscous damping coefficient, \(c_{a}=990\) (N\(\cdot\)s/m); \(c_{b}\) is the viscous damping coefficient influenced by V, \(c_{b}=3.095\) (Ns/m V); \(\alpha_{a}\) is the stiffness with non-voltage, \(\alpha_{a}=545\) (N/m); \(\alpha_{b}\) is the stiffness influenced by voltage V, \(\alpha_{b}=620\) (N/m V); \(\phi\) is the positive parameter of hysteresis  
Figure 2: Mechanical model of the vehicle seat suspension with magnetorheological (MR) damper. loop, \(\phi=4\). It is noted that the voltage V is directly related to the damping force of MR damper [23]. In this work, computer simulation is carried out with three controllers: the proposed controller, Compared Controller 1 [23], and Compared Controller 2 [24]. The compared controllers are chosen based on its similarity of control structure [23] and switching control logic [24].