# Trajectory Tracking Control Method Based on Adaptive Higher Order Sliding Mode
## 1 Introduction  
In recent years, continued improvements in adversarial interception capabilities and defense levels have resulted in increases in the dimensions and difficulties of the tasks inherent in missile attacks [1], placing higher requirements on missile guidance and control systems. Traditional methods that involve control only of the terminal position of the missile can no longer meet the demands of modern operations. Currently, control of the trajectory inclination at the conclusion of the missile's mid-guidance is also necessary to ensure the control accuracy of the subsequent terminal guidance law. Traditionally, the considerations influencing mid-guidance control of the terminal angle can mainly be summarized into the following two types: feedback control, based on terminal error, and feedforward and feedback control, by tracking the nominal trajectory. The approaches based on terminal error feedback can be divided into three types: optimal control [2, 3], variable structure sliding mode control [4, 5], and variable coefficient proportional guidance control [6]. Unfortunately, the abovementioned methods do not consider feedforward control and instead only consider feedback control, which may lead to control divergence in a strong interference environment. Traditionally, the most common method for tracking the nominal trajectory in engineering is to design a proportional-integral-derivative (PID) controller based on the position error feedback for each channel using three-channel decoupling. However, this method is problematic because its ability to achieve control coupling and disturbance rejection is insufficient and the control gain cannot adaptively respond to environmental impacts. [PERSON] [7] proposed the method of tracking the nominal trajectory through a linear quadratic regulator. However, this method relies heavily on kinematic and dynamic models, resulting in low terminal control accuracy. To improve the relatively poor anti-jamming performance of the above methods, the concept of sliding mode control is introduced in this paper. [PERSON] et al. [8] proposed an adaptive integral sliding mode control strategy, ensuring control objectives and achieving fast finite-time convergence and chattering attenuation of the system; [PERSON] et al. [9] designed a stable, robust sliding mode controller in a finite time to achieve fast stabilization; [PERSON] et al. [10], using a backstepping concept, developed a novel robust second-order sliding mode control law to track line-of-sight rate profile. Although traditional sliding mode control methods have strong disturbance rejection characteristics, they still experience the problem of significant chattering. To address the problems described above, [PERSON] et al. [11] used the smooth derivative of the intermediate control command obtained by the tracking differentiator (TD); [PERSON] et al. [12] proposed a fast self-adaptive super-twisting algorithm. By introducing linear terms and a new parameter self-adaptive law, convergence speed was improved; [PERSON] et al. [13] constructed a terminal integral adaptive sliding mode control algorithm using the estimation error as the adaptive factor and demonstrated the stability of the algorithm using the Lyapunov theory. Some scholars have introduced sliding mode control into trajectory tracking. [PERSON] et al. [14], using sliding mode theory, developed a robust tracking controller for the 3D trajectory follow problem; [PERSON] et al. [15], using the fast convergence sliding mode, developed a trajectory tracking algorithm; [PERSON] et al. [16] proposed a nonsingular fast terminal sliding mode and finite-time disturbance observer methods to solve trajectory tracking problem. Although the above method has good track tracking effect, it can only be applied to the Single Input Single Output system (SISO).