## 1 Introduction  
With the development of vehicle technology, people have higher requirements for the ride comfort and handling stability of the vehicle. The suspension has an essential effect on the ride comfort and handling stability of the vehicle [1]. After the conventional passive suspension design is completed, the parameters are determined and cannot be well adapted to different road conditions. In the suspension process design, it is found that there is a contradiction between the ride comfort and handling stability of the vehicle [2]. Conventional suspension has difficulty in solving this problem, and people have had to compromise between ride comfort and handling stability [3]. To solve this problem, much research has been conducted in the past decades and many active suspension control methods have been proposed, such as adaptive control [4; 5; 6], sliding mode control [7; 8], and H\({}_{\infty}\) control [9; 10]. The active control of the suspension inevitably suffers from time delay. People always ignore the time delay for the convenience of control design [11]. However, even a small time delay can cause the actuator to energy input to the system when the system does not need it, which may reduce the control efficiency or even lead to instability of the controlled system. People have different ways of dealing with time delay. The conventional view is that time delay is an unfavorable factor that can lead to system instability [12; 13]. The inherent time delay of the system is often offset by time-delay compensation [14; 15]. However, [PERSON] et al. ingeniously found that using the time delay as an active control parameter of the system can effectively suppress the vibration of the system [16; 17]. The magnitude of time delay in a control system directly affects the stability of the system, and using time delay as an active control parameter requires us to analyze the time-delay interval corresponding to the entire control system under the premise of stability. Therefore, to take advantage of the positive effect of time delay on system vibration reduction, people must explore effective time-delay stability analysis methods. For stability analysis, time-domain analysis and frequency-domain analysis methods are generally available. The time-domain analysis method is based primarily on Lyapunov functions or functional analysis, and there is a certain conservatism in these methods. Improvements have been made to its methodology [18; 19], but there is still some conservativeness. [PERSON] et al. explored the time-delay-independent stability region and the stability switching region of systems [20]. In the time-delay-dependent stability region, the system switches between stability and instability as the time delay increases from zero to infinity, and the correctness is verified numerically on a two-degree-of-freedom model. However, with the increase in system degrees of freedom, the difficulty of stability analysis increases sharply. [PERSON] proposed a frequency domain scanning method [21], which can greatly simplify the stability analysis of multi-freedom systems, but it can only assess the stability of a certain control parameter and cannot draw the stability region of the whole system. People's research on the stability analysis of systems with time delay has developed a lot [22; 23; 24; 25], but it is still being studied in depth. The use of time delay as an active control parameter to suppress system vibrations has become an important approach. [PERSON] et al. presented an optimal time delay feedback damper that provides a minimum peak frequency response over a given frequency range [26; 27]. [PERSON] et al. [28] explored a time delay positive proportional feedback control and verified the effectiveness of a single-degree-of-freedom model. [PERSON] et al. investigated the vibration reduction mechanism of a time-delay nonlinear dynamic vibration absorber and achieved better results under simple harmonic excitation [29]. [PERSON] et al.