## 3 Results
### Vehicle Suspension Simulation  
Several simulations were conducted in _Matlab_ 2024a to demonstrate the functionality of the proposed MR damper-based semi-active suspension in a vehicle. The quarter-car vehicle model (10) is simulated following the parameters in Table 1. The MR damper (1) is simulated according to the experimental parameters introduced in Table 2. Figure 8 illustrates the architectural framework of the simulation. The measured output \(y=[z_{s}-z_{u},\dot{z}_{s}-\dot{z}_{u}]^{\top}\) is transmitted to the controller, which adjusts the current supplied to the MR damper in order to modify the vertical force generated by it. The current applied to the damper is saturated, in order to be in the range of 0-1.8 A. By setting \(\varepsilon_{1}=\varepsilon_{2}=200\), a feasible solution for the controller was found by solving the proposed Theorem 1, with \(\gamma_{min}=6.82\). The _Matlab_ robust control toolbox was used for solving the LMIs. In this particular instance, the specific value of the controller gain is  
\[K=\left[-2303.97\quad 955.70\right] \tag{25}\]  
Figure 8: The architectural framework of the simulation. Figure 7: Experimental and estimated MR damper curves for an amplitude of the piston displacement of 20 mm and a current of 1.8 A at a frequency of excitation of 6 Hz: (**a**) Deflection vs. Force, (**b**) Deflection Rate vs. Force, (**c**) Estimation Error, (**d**) Correlation. In order to evaluate the system performance, two road disturbance profiles were subjected to analysis:  
1. A road bump with a height of \(H=6\) cm and a length of \(L=5\) m, at a vehicle speed of \(V=30\) km/h. The following formula represents the road bump profile over time [40]: \[z_{r}(t)= \begin{cases}\frac{H}{2}\left(1-\cos\bigl{(}\frac{2\pi V}{L}t\bigr{)} \right),&0\leq t\leq\frac{t}{V}\\ 0,&t>\frac{t}{V}\end{cases} \] (26) The bump test is necessary for evaluating the vehicle's suspension performance in mitigating the impact of high-energy disturbances. 2. A **Class A** random road profile, according to **ISO 8608**[44], is a vehicle speed of \(V=80\) km/h. The objective of evaluating the vehicle suspension under this random road profile is to guarantee that the performance is not compromised under typical highway conditions, i.e., that the semi-active suspension does not affect the vehicle when it is not required. The aforementioned road profiles are illustrated in Figure 9.  
In this work, three vehicle suspensions are compared:  
1. Passive suspension. A traditional suspension system with springs and dampers that cannot adjust to varying road conditions. The objective of this analysis is to assess the performance of a conventional vehicle. 2. Semi-active suspension with MR damper. The suspension system employs a magnetorheological damper that is capable of modifying its behavior in real time in order to adapt to the road profile. The MR damper current (13) is generated according to the required suspension control input (12). The objective of this analysis is to evaluate the potential for enhancement of vehicle behavior through the implementation of the proposed system. 3. Active suspension. Actuators are integrated into the suspension system with the objective of counteracting the impact of the varying road profile on the vehicle. The actuator force is generated according to the required suspension control input (12). The objective of this analysis is to evaluate the performance of the vehicle with the optimal suspension system installed, which would be costly and challenging to implement in practice. The bump road simulation results are presented in Figure 10. The proposed controller can achieve a lower value of body acceleration over time for the vehicle with semi-active suspension system than for the vehicle with passive one, which guarantees better ride comfort. The RMS values for the vertical acceleration are 1.101, 0.974, and 0.697 m/s\({}^{2}\) for the passive, semi-active, and active vehicle suspensions, respectively.