## 1 Introduction  
Linear induction motor (LIM) traction system has been widely applied in rail transit in recent years, due to its simple structure, low energy consumption, big start thrust and low cost [1; 2; 3]. The core component of the system is LIM and it is on focus for its extensive engineering application value and unique advantages like light weight, little noise, little pollution, good heat dissipation, strong climbing ability and so on [4; 5]. However, in terms of LIM's structure, it has a large magnetic circuit break and an initial gap, which makes its model more complicated than conventional induction motor. Related studies have shown that dynamic changes of system parameters, such as fluctuations in the air gap of linear motors, will lead to the first two levels of mutual inductance and LIM derivative resistance deviation [4; 5], and the dynamic end effect makes the air gap magnetic field greatly affected by the inlet vortex in the high speed region of the motor [6]. These phenomena are serious at high speed, which increases the complexity of tracking and compensation control. In recent years, a model reference adaptive indirect vector control method based on rotor time constant has been proposed and applied to LIM system with end effects to compensate for the bad influence brought by end effects in indirect vector control, achieving robust control of LIM speed [5; 7; 8]. To get better control performance, many nonlinear control methods are applied to LIMs, such as backstepping control [8; 9], adaptive control [10; 11; 12], fuzzy control, neural network control [13; 14; 15] and sliding mode variable structure control [9; 16; 17]. Among them, backstepping is a useful tool for dealing with uncertain nonlinear systems. Backstepping is widely paid attention to with its distinct controller design principle and easy combination with modern control techniques to eliminate the influence of time-varying parameter and external disturbances on the system performance. Scholars have put forward many methods to solve the computational expansion problem such as dynamic surface control [18; 19] and constrained command filter [8; 20; 21]. In [8], the adaptive backstepping control method is studied with fuzzy logic technique for LIM. However, it lacks measures to ensure the robustness and dynamic performance when facing end effects in LIM. In [18], a disturbance-observer based fast terminal sliding mode backstepping control is proposed for uncertain nonlinear systems, where the first-order filter is introduced to solve the computational expansion problem in the high-order backstepping control system. However, the design process and structure of the disturbance observer is complex, which increases the cost of computation and difficulty of controller design for LIM with end effects. Besides, the influence of the filtering error of the first-order filter on the system is not considered. Hence, more work shall be implemented against these problems on the control of LIM. Fuzzy logic system (FLS) has been widely concerned by researchers for its universal approximation property and ingenious treatment of nonlinear terms, which makes the controller design process simple and distinct [11; 22; 23]. Besides, the sliding mode control (SMC) method has low accuracy and strong robustness to the model, which has a good control effect on parameter perturbation and external disturbance [24; 25]. Among several SMC methods, the nonsingular terminal SMC has distinct advantages in finite convergence time [26; 27]. It guarantees the robustness, dynamic and static performance of the system. Comparing common terminal SMC, it can eliminate the singularity of terminal SMC and achieve better performance [28; 29]. When the system state gets far from the origin, the convergence time will be still limited under nonsingular terminal SMC and the dynamic characteristics of the system would be better [30; 31].