# Trajectory Tracking Control Method Based on Adaptive Higher Order Sliding Mode
## 2 Design of Adaptive High-Order Sliding Mode Trajectory Tracking Controller
### Active Disturbance Rejection Control and Linear Quadratic Regulator Controller Design  
To compare and verify the superiority of the above trajectory tracking controller design method based on the adaptive high-order sliding mode, ADRC is introduced for the trajectory tracking control problem. The advantage of introducing ADRC is that it directly observes the external disturbance as a state through the full-dimensional state observer (Linear Extended State Observer, LESO) and then achieves disturbance rejection through subsequent control. Traditionally, ADRC is often used in the design of single-state controllers. [PERSON] et al. [26,27] proposed the basic idea of establishing an ADRC; [PERSON] [28]has discussed the parameter setting in ADRC in detail based on the theory of frequency domain, but the problem of the above ADRC controller is that it can only achieve a SISO control goal; [PERSON] [29], [PERSON] [30], and others used the ADRC for a multi-input single-output (MISO) controller design, but the problem with this controller is that it is only used for the zero given point (expected value converges to 0) model. To address the above problems, in Figure 3, this paper draws on the concepts used for MISO system design and achieves the control of two states (height \(y\) and trajectory inclination angle \(\theta\)) by designing an ADRC controller. Because the ADRC based on the proportional-derivative (PD) control is proposed based on the SISO control system, it is necessary to select the main control variable \(\theta\) and the secondary control variable \(y\). Considering that we can guarantee that the error of height \(y\) is not too large, provided that trajectory inclination angle \(\theta\) is controlled for convergence, \(\theta\) is consequently selected as the main control variable and \(y\) as the secondary control variable. Given that the filtering has no effect on this paper, the active disturbance rejection tracking control does not consider the tracking differentiator and is divided into three parts: controller, full-dimensional state observer, and kinematics and dynamics. In this paper, it is assumed that secondary control variable \(y\) can be accurately measured by the sensor. As a result, the block diagram of trajectory tracking control based on the active disturbance rejection tracking controller is established as follows:  
The core of the ADRC controller is derived from the PD controller, which adopts an error-based feedback control concept. Based on Equation (7), the ADRC controller in this paper is expressed as  
\[u_{0}=k_{p1}x_{1}+k_{d1}\dot{x}_{1}+k_{p2}x_{2}+k_{d2}\dot{x}_{2} \tag{25}\]  
where \(k_{p1}\) is the proportionality coefficient based on the height error, \(k_{d1}\) is the proportionality coefficient based on the trajectory inclination error, \(k_{d1}\) is the differential coefficient based on the height error, and \(k_{d2}\) is the differential coefficient based on the trajectory inclination error. It should be noted that \(u_{0}\) is not the actual deviation control. Given external disturbance factors along with the fact that the system's intermediate process cannot be measured, a full-dimensional state observer is introduced to carry out the reconstruction of the system state. Referring to [31], the LESO is established as  
\[\left\{ \begin{aligned} \dot{z}_{1}&=\beta_{1}(x_{2}-z_{ 1})+z_{2}\\ \dot{z}_{2}&=\beta_{2}(x_{2}-z_{1})+z_{3}+b_{0}u\\ \dot{z}_{3}&=\beta_{3}(x_{2}-z_{1})\end{aligned} \right. \tag{26}\]  
where \(z_{1},z_{2},z_{3}\) are the state variables of the full-dimensional state observer, \(b_{0}\) is the constant coefficient, \(\beta_{1}\), \(\beta_{2}\), \(\beta_{3}\) are the observer gains, and the observer enables the observation state to converge to the actual state using the pole assignment concept. Figure 3: Block Diagram of Active Disturbance Rejection Tracking Controller. The deviation control amount equation is expressed as  
\[u_{y}=u_{0}-\frac{z_{3}}{b_{0}} \tag{27}\]  
The design of LQR controller is close to the traditional method.