## 1 Introduction
The main idea of the ESO is to treat the unknown internal and external disturbances as the lumped disturbances of the system, thus expanding the disturbances into a new state variable [17]. Then, the new state variable can be estimated according to the output of the system. The ESO discards the low-pass filter and sliding mode function and improves the phase delay and chattering of the motor. Therefore, the ESO has extensive application prospects in motor control. In Ref. [18], a control strategy combining the ESO-based position estimation method and high-frequency current injection method was proposed. Compared with the traditional hybrid scheme, the two schemes were integrated into the same control structure, and the position observer was embedded in the current controller. The robustness of the control system is good in the full speed range. In Ref. [19], a linear active disturbance rejection controller (LADRC) design was proposed to reduce overshoot and improve dynamic response when a disturbance occurs. The ESO can effectively reduce the negative effects of phase delay and motor chattering. However, when the interference factors are caused by the system, the ESO cannot filter them out in time. The SMC can solve the poor adaptability of the PI controller to external disturbances. However, when the state trajectory reaches the sliding mode surface, it is difficult to slide strictly along the sliding mode to the equilibrium point, resulting in chattering. In Ref. [20], using the saturation function instead of the switching function can effectively attenuate chattering. However, the traditional reaching law is difficult to suppress chattering and keep the system stable at the origin. The backstepping control considers both the control law and the adaptive law, which can make the whole closed-loop system meet the desired dynamic and static performance. Therefore, many scholars began to study the double closed-loop control based on the backstepping method to obtain a good load disturbance suppression effect. Different from the conventional feedback control methods, the backstepping control is a cascade design method. The controller aims to decompose the complex system into subsystems that do not exceed the system order. The controller designs the state variables and the Lyapunov functions for each subsystem so that the local and global stability characteristics run through each step [21]. In general, the backstepping controller makes the control process systematic and structured through reverse design. However, keeping the convergence of backstepping control and eliminating the steady-state error still needs to be concerned when the system parameters contain uncertainties. To solve the above problems, this paper proposes a control scheme that combines an integral backstepping controller with an enhanced linear extended state observer (ELESO). Firstly, the ELESO is composed of two linear extended state observers (LESOs) cascaded, which estimate the rotor lumped interference and the internal interference of the system to eliminate the influence of the higher harmonics on position estimation. Secondly, the new Lyapunov functions are designed by adding integral terms to the backstepping controller. The integral backstepping controller can guarantee convergence and eliminate steady-state error. Moreover, the parameters of PMSM can be estimated by backstepping controller to further eliminate the internal disturbance. Finally, the effectiveness of the control algorithm is proved by simulation and experiment. The remainder of this paper is organized as follows. In Section 2, the ELESO and the integral backstepping controller are designed. In Section 3, the improved control algorithm is verified by simulation and experiment, and the results are analyzed. Finally, in Section 4, the research contents are summarized.