###### Abstract
[13] designed a state feedback controller by using a state-estimation function of a Kalman filter to solve the dependence problem of track-rail stiffness, while control performance was reduced when the system was far away from the equilibrium point owing to the linear approximation model adopted by the control object. Most existing maglev-control methods are based on the assumption that the control signal is transmitted without delay. Nevertheless, the levitation system must have time delay due to the existence of signal transmission, inductance delay, and controller calculation. Time delay leads to more complex dynamic characteristics, including homogeneous bifurcation, Hopf bifurcation, and limit cycle oscillation [14; 15]. The maglev train CMS04 presented by the National University of Defense Technology experienced severe rail-vehicle resonance caused by excessive time delay during testing [16; 17]. Tongji University also carried out a real vehicle test that showed that suspension failure occurred when the air-gap feedback time delay of the static suspension exceeded the critical value [18]. However, they have not proposed any control algorithm that could suppress the influence of the time delay. These phenomena may seriously affect the stability and reliability of the suspension system and may hinder the further popularization and application of the maglev transportation system. Moreover, time delay widely exists in various engineering systems. Studies have shown that a time delay can lead to performance degradation or even failure. Therefore, stability analysis and control research with time delay in maglev vehicles could significantly guide other nonlinear systems. In this paper, an adaptive robust levitation control method was developed to deal with time delay and uncertainty, but bounded disturbances for a maglev train. The nonlinear dynamic models of magnetic-suspension systems were first derived. The proposed controller utilizes a discontinuous structure to improve robustness against time delay and uncertainty on the basis of the Riccati method and sliding-mode technology. In order to eliminate the chattering phenomenon in the control input,an adaptive update law is proposed to regulate adaptive gain online. Otherwise, the stability of the overall system was assured with rigorous Lyapunov-based analysis. Both the simulation and experiment results were included to demonstrate the effectiveness of the proposed control strategy. The main contribution of this work is summarized as follows:  
1. The proposed controller can ensure better stable levitation of maglev system tackling the problems of disturbance and time delay simultaneously
2. The designed method does not require the magnitude of external disturbances, and it can attenuate chattering in control inputs effectively. 3. As verified by the experimental results, the proposed method shows increased control performance in a time-delay environment. The rest of the paper is organized as follows: in Section 2, the control problem of interest is formulated. The controller design and the closed-loop stability analysis are described in Sections 3 and 4, respectively. Sections 5 and 6 detail the simulation and experimental results and analysis. Section 7 summarizes the entire paper.