The effectiveness of the proposed DE-FSMC is validated through a comparative work between the proposed controller and an existing adaptive fuzzy sliding mode controller (called the AFSM). For the comparison, the DE-FSMC without the control action (called the passive) and an adaptive fuzzy sliding mode controller proposed in [
12] are considered. In this work, in order to show quantitative results, the maximum magnitude of chassis acceleration
is defined and used.
where,
P is the number of samples and
is the vertical acceleration of the chassis. Two road profiles used for surveys are the bump and sinusoidal type with disturbance surfaces. The MRD is excited to make vibrations with an amplitude of 0.0861 (m) and an angular frequency of 1.9134 (rad/s). It is noted that the speed of the car crossing over the bump profile is decided by 2.34 km/h. The vertical acceleration on the bump road profile is shown in
Figure 5. It is identified that the maximum acceleration magnitude
defined in Equation (80) related to the DE-FSMC is 0.0190 m/s
2, which is smaller than that of the fuzzy sliding mode controller with disturbance observer (FSMCD) and AFSM, whose values are 0.0256 and 0.0374 m/s
2, respectively. It is noted that the fuzzy sliding mode controller with disturbance observer (FSMCD) used in this work as a comparative controller is the modified one developed in [
1], in which the disturbance observer is also used like the proposed one. The AFSM used in this work is the modified one proposed in [
30] to adapt to the semi-active suspension system. The numerical results of acceleration identified from the figure are given in
Table 3.
Figure 6 and
Table 3 reflect the dynamic response delay
between the chassis vertical acceleration signal and the excitation signal coming from the bump-road surface analyzed via cross-correlation function (CCF). The maximum value of
is identified as 1.869 (s) for the proposed DE-FSMC, while the minimum of
is calculated by 0.264 (s) for the passive suspension. This indicates that when controlled by the proposed DE-FSMC, the vibration of the chassis is less sensitive to the road surface status compared with the passive AFSM and FSMCD. This is an advantage for controlling vibration by avoiding the resonant status. This aspect is more clearly seen via the ratio of spectral coherence
provided in
Figure 7 and
Table 4. The spectral coherence can identify the frequency-domain correlation between two databases. The values of
tending towards zero indicate that the corresponding frequency components are uncorrelated, and conversely for the tending towards 1 [
38,
39]. For the bump-road profile,
between the chassis vertical acceleration signal and the excitation signal coming from the road surface is shown in
Figure 7. The maximum values (
) and the corresponding frequency (
) related to each method are given in
Table 4. The obtained results indicate that the minimum value of
is 0.1269 at the frequency of 4.28 Hz in the proposed method. Besides,
deriving from the surveyed controllers is located in a narrow range of low frequencies.
Figure 8 presents the control force of each controller. It is clearly seen that the proposed controller can provide the better control performance with the similar magnitudes of control force as those of the FSMCD and AFSM.
In this equation,
is the vertical displacement of the road surface;
and
respectively are the amplitude and cycle of the sine-wave;
denotes the velocity of vehicle along the road; and
denotes the random value belonging to
. Related to this road profile,
indicates the road disturbance surface. By choosing
and
km/h, the road profile is achieved, as shown in
Figure 9. In order to produce this road profile, an amplitude of 0.07 (m) with an excitation frequency of 0.8333 Hz is used, which is equivalent to the angular frequency of 5.236 rad/s.
Figure 10,
Figure 11 and
Figure 12 and
Table 5 and
Table 6 present the measured results under the sinusoidal road profile.
Figure 10 and
Table 5 clearly show that the acceleration of the chassis mass is greatly reduced by activating the proposed controller. The maximum acceleration amplitudes
of the DE-FSMC, AFSM, and the passive system is identified by 0.3642, 0.5742, 0.7928, and 3.6587 m/s
2, respectively.
Table 5 shows the dynamic response delay
between the chassis vertical acceleration signal and the excitation signal coming from the sine-road surface analyzed via CCF. The maximum value of
is determined by 2.545 (s) from the proposed DE-FSMC, while the minimum of
is determined by 0.793 (s) from the passive suspension.
Figure 11 and
Table 6 provide the ratio of spectral coherence
, showing its maximum value
, and the corresponding frequency
related to each method. The minimum value of
is 0.4672 at the frequency of 24.03 Hz in the proposed control method. The required control force corresponding to each controller is shown in
Figure 12. It is clearly seen that the vibration control performance of the proposed controller is excellent with the lower input power.