# Time-Delay Vibration Reduction Control of 3-DOF Vehicle Model with Vehicle Seat
## 1 Introduction  
The suspension is an important device to ensure the comfort and safety of the ride for the vehicle running on the road [1, 2, 3, 4, 5]. This main function is to mitigate the impact and vibration caused by uneven pavement. The road excitation to the vehicle exhibits low-frequency and high-amplitude shock vibration in all directions. Many studies have shown that long-term exposure to low frequency and high-intensity vibration will seriously affect the physical and mental health of passengers. Therefore, research on vibration reduction of vehicle body and seat is of great significance to improve the driving comfort of vehicles and the physical and mental health of occupants [6, 7, 8]. The active suspension system is modified from the passive suspension in terms of the actuator (motor, hydraulic device, cylinder, etc.). Utilizing a device generating an active force, it can actively suppress the impact of uneven road surfaces on the car body. Therefore, it has obvious advantages in vibration reduction and a wide range of application prospects [9, 10]. There are unavoidable time-delay factors in the active suspension system, including the transmission delay of measurement signal from the signal sensor to a controllable computer, the calculation delay caused by calculation control method, the transmission delay of the control signal from the computer to the actuator, the actuator delay, etc. [11, 12]. On the one hand, the time delay may lead to the instability of the vibration control system, and the bifurcation phenomenon, which is harmful to safety, will occur and seriously affect the vehicle's ride comfort. On the other hand, a reasonable design of time delay can improve the control effect of the system. Some studies have been carriedout to solve the problem of delay control. [PERSON] and [PERSON] et al. [13; 14; 15; 16] first proposed using time-delay feedback to control the vibration of the main system. Its basic idea is to install a delay-state-feedback shock absorber on the main system. When the external excitation frequency is consistent with the natural frequency of the system, the vibration can be completely eliminated [17; 18; 19]. [PERSON] et al. [20; 21; 22] integral suspended Jeffcott-rotor system is an object of research. They use the time-delay displacement feedback controller to control the lateral motion of the system and analyze the stability of the controller gain and time-delay, which uses simulation analysis to verify the effectiveness of time-delay vibration reduction. [PERSON] et al. [23] carried out time-delay feedback control for cantilever beams. They use the perturbation method to obtain the appropriate time-delay feedback control parameters, which can reduce the main resonance amplitude and suppress the quasi-period of the beam. [PERSON] et al. [24] research on the time-delay problem of active control of torsion bars and building structures shows that introducing the hour delay into the system is helpful to improve the stability and dynamic performance of the system. [PERSON] and [PERSON] et al. [25; 26; 27; 28] considered time-delay in state feedback control. The time delay and control gain is adjusted to study the delay's dynamic vibration absorber system. The results show that the vibration of the main system can be almost completely suppressed with appropriate hysteresis control parameters. [PERSON] and [PERSON] [29; 30; 31; 32] et al. designed a flexible manipulator with time-delay feedback control by using numerical simulation and simulation experiments. They found that it had better motion than a flexible manipulator with no time delay. [PERSON] et al. [33; 34; 35; 36], to improve the comfort of car seats, proposed a new control strategy to improve ride comfort and applied it to the vibration control of automobile seat suspension with MR damper. The experimental results show the effectiveness of the proposed method.