## 1 Introduction
proposed an adaptive fuzzy PID controller based on the modified Riccati-like Equation to suppress the vibration for vehicle seat suspension [15]. [PERSON] et al. adopted a bio-inspired reference model based fuzzy adaptive tracking control which is applied for nonlinear suspension systems to isolate vibration [16]. To suppress the vibration of the Takagi-Sugeno fuzzy suspension models with input saturation, [PERSON] et al. proposed a parallel distributed compensation-based composite nonlinear feedback control [17]. However, the establishment of the fuzzy rule base is still time and effort consuming under fuzzy control. Furthermore, these studies focus on reducing vibration of body acceleration, which improve ride comfort performance, rather than on improving the comprehensive performance between ride comfort and driving safety. Therefore, numerous studies were shown that the extension theory had the potential to improve the comprehensive performance of the suspension system and eliminate the switching condition. [PERSON] et al. applied the non-autonomous Chua's circuit to preprocess the grid signal and analyzed preprocessed voltage signal based on the fractional Lorenz chaotic system and extension theory which was used to establish the classical and sectional domains according to the dynamic errors of fault conditions [18]. [PERSON] et al. prioritized the alternative technologies for the treatment of urban sewage sludge and graded their sustainability performances based on extension theory [19]. [PERSON] et al. proposed the extension theory as a novel evaluation model for the agri-product supply chain competitiveness to amend the evaluation parameters under the current developments in society and economy [20]. [PERSON] et al. used an intelligent extension theory for human power generation system to reduce input saturation and high frequency switching in the sliding mode strategy, and increase efficiency and response speed [21]. [PERSON] proposed a smart fault-tolerant control system based on the theory of Lorenz chaotic system and extension theory for locating faults and executing tolerant control in a three-level T-type inverter. The system executed fault diagnosis based on extension theory [22]. Under mixed road conditions with a large curvature range, an extendable preview switching control method for the lateral control of autonomous vehicles was proposed by [PERSON] et al., which solved the problem of poor adaptability and low control accuracy of the lateral control system in autonomous driving vehicles [23]. However, there are few studies utilizing extension theory for the suspension system. [PERSON] and [PERSON] proposed extension control to improve vehicle suspension/steering integrated system performance [24]. To handle the optimal domain boundary problem, [PERSON] introduced the game theory to determine the domain of the extension control [25]. Notably, these studies have tended to focus on alleviating the conflicting problem between the suspension system and steering system, rather than on relieving the internal conflict for the suspension system. Meanwhile, the research about road excitation and the 2-DOF suspension model were also developed. [PERSON] et al. created an identification method of road conditions based on vertical acceleration response and proposed a new 2-DOF semi-active suspension system based on the road statistical properties [26]. [PERSON] et al. utilized an infinite Kirchhoff plate with the Kelvin foundation to establish the pavement system [27]. [PERSON] et al. presented the unscented Kalman filter to estimate the road profile by the vehicle states [28]. [PERSON] firstly analyzed the relationship between sprung mass and unsprung mass based on the 2-DOF suspension model [29]. Analyzing the 2-DOF suspension model, [PERSON] found a kind of invariant point on the suspension system firstly [30]. [PERSON] et al. applied the method of codimension-two bifurcation analysis to analyze the 2-DOF semi-active suspension model [31].