###### Abstract
[9] designed a control method using the genetic algorithm to optimize the fuzzy proportional integral derivative (PID) control algorithm and studied the vehicle ride comfort of the vehicle suspension system by taking the vertical acceleration of the vehicle body, suspension deflection, and wheel dynamic load as evaluation criteria. The nonlinear energy sink (NES) is an innovative vibration attenuation device and can absorb and dissipate the energy of the main structure [10]. Due to its superior vibration reduction performance, the NES has been widely applied in various fields, such as civil engineering [11; 12; 13], aerospace engineering [14; 15; 16], and vehicle engineering [17; 18; 19]. However, achieving optimal vibration suppression performance of the NES typically requires a larger mass, which contradicts the lightweight vehicle design principle. To address this challenge, a dual-end structural element, the inerter, has been developed to achieve a mass amplification effect, where the output force-to-relative acceleration ratio is termed as inertance [20]. Owing to the mass amplification effect of the inerter, researchers have replaced the mass block in the NES with an inerter, creating a new vibration suppression mechanism known as the nonlinear energy sink inerter (NESI) [21]. [PERSON] et al. [22] constructed a new type of NESI that successfully achieves ideal vibration damping performance with a smaller mass. [PERSON] and [PERSON] [23] developed an innovative NESI configuration in which the inerter is positioned between the mass block and the stationary base of an NES, thereby significantly enhancing its effectiveness. [PERSON] et al. [24] used an NESI to suppress the torsional vibration in the rotor system, which reduces the required rotary inertia for torsional vibration suppression. [PERSON] et al. [25] conducted a detailed analysis of the response mechanism and damping effect of the NESI and studied how NESI parameters influence the system's overall performance, particularly in terms of vibration suppression and stability under varying conditions. [PERSON] et al. [26] applied an NESI in a vehicle suspension system to enhance its vibration damping performance under various pavement excitation levels. In a previous study by the authors [27], the NESI was applied to the vehicle suspension and seat suspension in a half-vehicle-seat (HVS) system. The harmonic balance method and pseudo-arc length method were used to analyze the dynamic characteristic of the passive HVS system coupled with the NESI under pavement harmonic and random excitations, and the influence of the NESI's structural parameters on the dynamic performance was analyzed. The results indicate that the NESI can reduce the resonance peak value of seat dynamic travel, front and rear suspension dynamic travel, and front and rear dynamic tire loading; however, seat vertical acceleration, vehicle body vertical acceleration, and pitch acceleration increase slightly. Although the application of the NESI in a passive HVS system improves vehicle vibration attenuation performance [27], its structural parameters are fixed and cannot adaptively adjust to changing road impacts. Compared with passive suspension, active suspension can adjust itself in real time based on information such as vehicle speed and road conditions, ensuring optimal suspension performance across various driving conditions. [PERSON] et al. [28] developed Hoo state feedback control to minimize human head acceleration through integrated control of the seat and vehicle suspension and designed a static output feedback controller for the unmeasured driver model during driving. To address the influence of human body dynamics and uncertainties in seat skeleton dynamics, [PERSON] et al. [29] designed a sliding mode controller based on a disturbance observer and analyzed the dynamic performance of a 4-DOF human-seat model; the results indicate that the acceleration transmitted from the seat to the human body is significantly reduced. [PERSON] et al.