Various inerter systems utilizing velocity-dependent damping for vibration control have been developed. However, a velocity-dependent damping element may exhibit relatively poor performance compared to a displacement-dependent damping element (DDE) of equivalent damping ratio, when the structural deformation is small in the early stage of the seismic response. To address this issue, the advantage of DDE in generating a larger control force in the early stage of excitation is promoted here and enhanced by a supplemental inerter-spring-system, thus realizing a proposed novel displacement-dependent damping inerter system (DDIS). First, the influence of various DDIS-parameters is carried out by resorting to the stochastic linearization method to handle non-linear terms. Then, seismic responses of the DDIS-controlled system are evaluated in the time domain taking the non-linearity into account, thus validating the accuracy of the stochastic dynamic analysis. Several design cases are considered, all of which demonstrated damping enhancement and timely control achieved by the DDIS. The results show that the energy dissipation as well as reduction of structural displacement and acceleration accomplished by the proposed system are significant. DDIS suppresses structural responses in a timely manner, as soon as the peak excitation occurs. In addition, it is demonstrated that interactions among the inerter, spring, and DDE, which constitute the damping-enhancement mechanism, lead to a higher energy-dissipation efficiency compared to the DDE alone. enhancement mechanism. Arakaki et al. [27,28] utilized the rotation mechanism to amplify the effective damping force of a viscous damper, which is a type of velocity-dependent damping element (VDE). However, these devices did not explicitly use the mass enhancement effect until Ikago et al. [1] proposed the tuned viscous mass damper, which belongs to a type of velocity-dependent damping inerter system (VDIS). The performance of the tuned viscous mass damper control system was subsequently investigated via shaking table tests conducted on single-story systems equipped with scaled-down versions of the damper [29]. Garrido et al. [30] proposed a rotational inertia double-tuned mass damper by replacing the viscous damping of the tuned mass damper with a tuned viscous mass damper, which achieved significantly greater control than the tuned mass damper with similar additional mass ratio. Through the incorporation of an electromagnetic damper, which is a type of VDE, Nakamura et al. [31] developed an electromagnetic inerter mass damper with variable damping force. Asai et al. [32,33] achieved enhanced energy-harvesting performance using a tuned inerter. Zhang et al. [10,34] proposed an isolation inerter system that used an inerter and a VDE to mitigate the vibration of a storage tank. The effect of the mechanical layout of the system was also investigated and considered in the development of a demand-based optimal design method for the system. Ikago et al. [1] presented the closed-form ...