Trajectory Tracking Control Method Based on Adaptive Higher Order Sliding Mode

Abstract

To resolve the problem of high-precision trajectory tracking control under interference conditions in a missile’s mid-guidance phase, according to the constructed nominal trajectory, an improved adaptive high-order sliding mode trajectory tracking controller (AHSTC) is proposed. In this method, the open-loop nominal trajectories are established according to the nonlinear programming and Gaussian pseudospectra method. A high-precision trajectory tracking controller is developed by designing a nonlinear sliding mode surface and an adaptive high-order sliding mode approaching law combined with the trajectory tracking nonlinear error model. To verify the effectiveness and superiority of the proposed method, analysis and simulation are carried out through the example of a missile mid-guidance phase tracking control. Compared to the linear quadratic regulator (LQR) and active disturbance rejection controller (ADRC) method, the simulation results show that the proposed AHSTC method shows faster convergence and improved tracking effect. Therefore, the proposed AHSTC method has a good results and engineering application value.

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. Dukeman [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. Yogi 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; Munoz-Vazquez et al. [9] designed a stable, robust sliding mode controller in a finite time to achieve fast stabilization; Harl 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, Ming et al. [11] used the smooth derivative of the intermediate control command obtained by the tracking differentiator (TD); Liu 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; Li 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. An et al. [14], using sliding mode theory, developed a robust tracking controller for the 3D trajectory follow problem; Zhang et al. [15], using the fast convergence sliding mode, developed a trajectory tracking algorithm; Wang 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).

To solve the abovementioned problems, this paper first establishes an optimal trajectory that comprehensively considers the minimum time and the square of the control variable overload change rate as the performance index. Second, traditional sliding mode is characterized by anti-interference and strong robustness, so it is suitable for trajectory tracking control. Combined with the adaptive algorithm, a trajectory tracking control method based on high order sliding mode is proposed to solve the chattering problem of control quantity.

The rest of this paper is established as follows. In Section 2, the proposed AHSTC method is explained in detail, and the compared method is explained in brief. In Section 3, the model verification and method comparison are introduced in details. Finally, some conclusions are discussed in Section 4.

2. Design of Adaptive High-Order Sliding Mode Trajectory Tracking Controller

2.1. Establishing Trajectory Tracking Control Model

The kinematic and dynamic models of the missile based on the geodetic coordinate system in the two-dimensional plane are constructed as follows [17,18]:

$$\left\{\begin{array}{l}\dot{x}=v\cos \theta \hfill \\ \dot{y}=v\sin \theta \hfill \\ \dot{v}=g\left(n_x-\sin \theta \right)\hfill \\ \dot{\theta }=\frac{g}{v}\left(n_y-\cos \theta \right)\hfill \end{array}\right.$$

(1)

where v is the flight speed of the missile, θ is the trajectory inclination angle, x is the longitudinal position coordinate, and y is the height position coordinate, $\dot{v}$, $\dot{\theta }$, $\dot{x}$ and $\dot{y}$ are the derivatives of v, θ, x, and y, respectively, while $n_x$ and $n_y$ are the overload values in the x and y directions, respectively. The acceleration of gravity is g and is equal to 9.81 m/s². $n_x$ is expressed as

$$n_x=\frac{X}{mg}$$

(2)

where $m=1030 \mathrm{kg}$ is the mass of the missile, and resistance X is expressed as

$$X=\frac{1}{2}\rho v^2{C}_{x}S$$

(3)

where $C_x=0.1$, $S=0.5 {\mathrm{m}}^2$, and the density ρ is expressed as

$$\rho =1.225e^{-\frac{y}{7110}}$$

(4)

According to the modeling methods reported in [18], the equation of motion of a particle and the model of the error tracking controller are obtained as follows:

$$\left(\begin{array}{l}\dot{y}\hfill \\ \dot{\theta }\hfill \end{array}\right)=\left(\begin{array}{l}v\sin \theta \hfill \\ -\frac{g\cos \theta }{v}\hfill \end{array}\right)+\left(\begin{array}{c}0\\ \frac{g}{v}\end{array}\right)n_y$$

(5)

$$x=\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)=\left(\begin{array}{c}y-y_n\\ \theta -{\theta }_n\end{array}\right)$$

(6)

where state with subscript n represents the nominal state value.

$$\begin{array}{ll}\dot{x}& =\left(\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\end{array}\right)=\left(\begin{array}{c}\dot{y}-{\dot{y}}_n\\ \dot{\theta }-{\dot{\theta }}_n\end{array}\right)\\ & =\left(\begin{array}{c}{v}_n\sin \theta -v_n\sin {\theta }_n\\ \frac{g}{v_n}{n}_y-\frac{g}{v_n}\cos \theta -\frac{g}{v_n}{n}_{y_n}+\frac{g}{v_n}\cos {\theta }_n\end{array}\right)\\ & =\left(\begin{array}{c}{v}_n\sin \theta -v_n\sin {\theta }_n\\ -\frac{g}{v_n}\cos \theta +\frac{g}{v_n}\cos {\theta }_n\end{array}\right)+\left(\begin{array}{c}0\\ \frac{g}{v_n}\end{array}\right)\mathsf{\Delta }{n}_y\end{array}$$

(7)

where the uncontrolled variable v uses the state value on the nominal trajectory at the corresponding moment and is marked with subscript n. For the overload deviation, $\mathsf{\Delta }{n}_y=n_y-n_{y_n}$ is taken as the control variable. Let $u_y=\mathsf{\Delta }{n}_y$, then the above model can be transformed into

$$\left\{\begin{array}{l}{\dot{x}}_1=V_n\sin \theta -V_n\sin {\theta }_n\hfill \\ {\dot{x}}_2=-\frac{g}{V_n}\cos \theta +\frac{g}{V_n}\cos {\theta }_n+\frac{g}{V_n}{u}_y\hfill \end{array}\right.$$

(8)

At this point, the mid-guidance for trajectory tracking control model based on the nominal trajectory has been established.

2.2. Establishing the Optimal Nominal Trajectory

Given the initial state and terminal state constraints, the establishment of the optimal nominal trajectory is a typical two-point boundary value problem. To reduce the probability of interception, the flight time of the missile must be as short as possible [19]. Moreover, considering the stability issues affecting the control quality, the optimized performance indicators designed for this issue are

$$J=K{t}_f+{\displaystyle \int {\dot{n}}_y^{2}dt}$$

(9)

Here, K is a constant and t_f is the terminal time, that is, the total flight time. The selection of performance indicators not only ensures the shortest terminal time but also ensures that the nominal trajectory control is relatively smooth. It also ensures the feasibility and rationality of using steering gear for control during engineering implementation.

In this paper, a nominal trajectory satisfying the initial and terminal constraints is established by a nonlinear programming function based on the discrete state equation of the Gaussian pseudospectral method. To improve the somewhat inadequate numerical calculations that arise when using the numerical iteration method, this paper introduces a concept of normalization [20,21] where the nominal trajectory in Equation (1) is established by normalizing the state variables x, y, and v into the state equation.

Nominal initial conditions: initial velocity: $v=1360 \mathrm{m}/\mathrm{s}$, initial trajectory inclination: $\theta =0^0$, initial range: $x=0 \mathrm{km}$, initial altitude: $y=10 \mathrm{km}$.

Desired terminal state: desired terminal trajectory inclination: ${\theta }_n=-{20}^0$, desired terminal height: $y_n=8 \mathrm{km}$.

The nominal trajectory established in this paper is shown in Figure 1 and Figure 2. The entire nominal trajectory is relatively smooth, and there is no significant control chattering. It should be noted that the established nominal trajectory is the optimal open-loop nominal trajectory. As a result, a tracking control law must be designed to track the nominal trajectory to ensure that the terminal state meets the requirements.

2.3. Adaptive Higher-Order Sliding Mode Algorithms

The nonlinear system is defined as

$$\begin{array}{l}\dot{z}=f\left(z\right)+g\left(z\right)\upsilon \\ y=h\left(z,t\right)\end{array}$$

(10)

where $z\in Z\subset R^n$ is the state variable, $\upsilon \in U\subset R$ is the input variable (it should be noted that the input variable is not the actual control variable), functions $f\left(z\right)$ and $g\left(z\right)$ are unknown but are bounded nonlinear functions, and $y$ is the output observation of the system. To counteract problems related to strong nonlinearity, external disturbance, and the presence of deviation in the initial tension state, this paper introduces a robust, relatively strong sliding mode control. The main idea behind the sliding mode control is the design of the input variable $\upsilon$ to cause the sliding mode variable s to converge to 0. If $s=0$, the system trajectory will be attracted to the plane and the desired state will tend to be 0 [22].

To simplify the design of the controller, the following two reasonable assumptions are made for the system nonlinear model (10).

Assumption 1: The relative degree of the system (10) is $\rho$, where $\rho >1$ and is defined as

$$s^{\left(\rho \right)}=a\left(z,t\right)+b\left(z,t\right)\upsilon$$

(11)

Assumption 2: $a\left(z,t\right)$ and $b\left(z,t\right)$ are unknown but bounded functions. Moreover, $\left|a\right|\le a_M$, $0\le b_m\le b\le b_M$ in regards to $z\in Z$ and $t>0$; $a_M$,$b_m$ and $b_M$ are positive constants.

Based on the above assumptions, this paper seeks to enable sliding mode variable $s$ to approach 0 during a finite time by designing input $\upsilon$ under the conditions of model uncertainty and external disturbances. Zakaria et al. [23] proposed a class of higher-order sliding mode control algorithms based on the Lyapunov function stability proof. The controller is expressed as

$$\begin{array}{l}\upsilon =-k{\mathrm{sig}}^0\left(s\right)\\ s_i={\mathrm{sig}}^{\frac{r_1}{r_i}}\left(s^{\left(i-1\right)}\right)+k_{i-1}^{\frac{r_1}{r_i}}{s}_{i-1}\hspace{1em},i=2,\cdots ,\rho \end{array}$$

(12)

where ${\mathrm{sig}}^k\left(s\right)={\left|s\right|}^k\mathrm{sign}\left(\mathrm{s}\right)$, relative degree $\rho \ge 2$, $r=\left(r_1,\cdots ,r_{\rho }\right)=\left(\rho ,\rho -1,\cdots ,1\right)$, and $s_1=s$, $\left(k_1,\cdots ,k_{\rho }\right)$ is the controller gain. To establish the $\rho$-order sliding mode control, assuming that gain k₁ has been defined, controller gain k₂ should satisfy the following conditions:

• When $\rho =2$

  $$b_m{k}_2-a_M\ge {\gamma }_1{k}_1^2$$

  (13)
• When $\rho >2$

  $$\begin{array}{l}{k}_i={\gamma }_{i-1}{k}_1^{\frac{\rho }{\rho -\left(i-1\right)}},\forall i=2,\cdots ,\rho -1\\ b_m{k}_{\rho }-a_M\ge {\gamma }_1{k}_1^{\rho }\end{array}$$

  (14)

  where parameter γ_i−1 can be obtained from

  Table 1. Value of Parameter γ_i−1.

  $\mathit{\rho }$	Parameters
  2	${\gamma }_1=1.26$
  3	${\gamma }_2=9.62,{\gamma }_1=1.5$
  4	${\gamma }_3=739.5,{\gamma }_2=8.1,{\gamma }_1=2$

  .

To simplify the controller model, using $\rho =2$ in this paper, we obtain

$$\left\{\begin{array}{c}{s}_1=s, s_2={\mathrm{sig}}^2\left(\dot{s}\right)+k_1^{2}s,{\gamma }_1=1.26\\ \upsilon =-k_2\mathrm{sign}\left(s_2\right)\end{array}\right.$$

(15)

To address the problem of different control gains required under different control precisions, this paper introduces an adaptive algorithm [24]. Based on Equation (12), parameter index $\overline{\alpha }$ is introduced as follows:

$$\upsilon =-k_{\rho }{\mathrm{sig}}^{\overline{\alpha }}\left(s_{\rho }\right)$$

(16)

here, the parameters $k_{\rho }$ and $s_{\rho }$ are adjusted as expressed in Equations (12)–(14), and $\overline{\alpha }\in \left[0,1\right]$ satisfies

$$\overline{\alpha }=\max \left(-\overline{\beta }{\displaystyle \sum _{i=1}^{\rho }\frac{\left|s^{i-1}\right|}{\left|s^{i-1}\right|+{\epsilon }_{s_i}}+1,0}\right)$$

(17)

where ${\epsilon }_{s_i}$ is a constant and $\overline{\beta }>1$.

The stability proof of the adaptive method is given in [24,25], and the core concepts are as follows:

• If the value of the time derivative of $\left|s\right|$ and $s$ is sufficiently small, a higher-order sliding mode surface is established; that is, $s=\dot{s}=\cdots =s^{\left(\rho -1\right)}=0$, and the exponential term $\overline{\alpha }$ will approach 1. The controller is transformed into the linear controller $\upsilon =-k_{\rho }{s}_{\rho }$, thereby reducing the control chattering.
• If the value of the time derivative of $\left|s\right|$ and $s$ is sufficiently small, a higher-order sliding mode surface is established; that is, $s=\dot{s}=\cdots =s^{\left(\rho -1\right)}=0$, and the exponential term $\overline{\alpha }$ will approach 1. The controller is transformed into the linear controller $\upsilon =-k_{\rho }{s}_{\rho }$, thereby reducing the control chattering.
• Parameters ${\epsilon }_{s_i}$ and $\overline{\beta }$ are used to adjust the accuracy of the controller.

2.4. Design of the Tracking Controller Based on the Adaptive Super-Twisting Algorithm

For Equation (8), the sliding surface used in this paper is given by

$$s=\beta {\mathrm{sig}}^{\alpha }\left(x_1\right)+{\lambda }_1{x}_1+x_2$$

(18)

where $\alpha \in \left(1,2\right)$, ${\mathrm{sig}}^{\alpha }\left(x_1\right)={\left|x_1\right|}^{\alpha }\mathrm{sign}\left(x_1\right)$.

By taking the derivative of Equation (18) with respect to time t, we obtain

$$\begin{array}{ll}\dot{s}& =\alpha \beta {\left|x_1\right|}^{\alpha -1}{\dot{x}}_1+{\lambda }_1{\dot{x}}_1+{\dot{x}}_2\\ & =\underset{a}{\underbrace{\left(\alpha \beta {\left|x_1\right|}^{\alpha -1}+{\lambda }_1\right)\left(V\sin \theta -V_n\sin {\theta }_n\right)-\frac{g}{V}\cos \theta +\frac{g}{V_n}\cos {\theta }_n}}+\underset{b}{\underbrace{\frac{g}{V_n}}}{u}_y\end{array}$$

(19)

It should be noted that from Equation (19), we can see that the relative order of the system is equal to 1, and the above algorithm requires the relative order of the system to be greater than or equal to 2. To solve this problem, we take the derivative of Equation (19):

$$\ddot{s}=\overline{a}+\overline{b}{\dot{u}}_y$$

(20)

where $\overline{a}$ and $\overline{b}$ can be calculated using Equation (19), and ${\dot{u}}_y$ is defined as the new system input; namely, a system with relative order 2 is obtained. Based on the above algorithm, the controller is designed as follows:

$${\dot{u}}_y=\frac{1}{\overline{b}}(-\overline{a}+\upsilon )$$

(21)

$$\begin{array}{ll}{u}_y& ={\displaystyle \int {\dot{u}}_y}d\tau \\ & ={\displaystyle \int \frac{1}{\overline{b}}}(-\overline{a}+\upsilon )d\tau \end{array}$$

(22)

where $\upsilon$ is the same as that shown in Equation (12). The controller designed to verify Equation (21) can satisfy the requirement of the sliding mode variable s approaching 0, so that Equation (21) is substituted into Equation (19). Equation (19) can be transformed into

$$\ddot{s}=\overline{a}+\overline{b}{\dot{u}}_y=\upsilon$$

(23)

If the controller shown in Equation (12) is used, and Equation (12) is substituted into Equation (23), we obtain

$$\ddot{s}=-k\mathrm{s}\mathrm{i}{\mathrm{g}}^0\left(s\right)$$

(24)

Equation (24) shows that when $s>0$, $\ddot{s}<0$. At this point, regardless of its value, ultimately $\dot{s}<0$; when $s<0$, $\ddot{s}>0$, regardless of the value of $\dot{s}$, ultimately $\dot{s}>0$. Therefore, the above derivation shows that $\dot{s}$ can approach 0 under any initial conditions, thus ensuring that the desired state approaches 0. Similarly, this conclusion can be obtained by using adaptive Equation (16).

2.5. 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. Han et al. [26,27] proposed the basic idea of establishing an ADRC; Gao [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; Wang [29], Wu [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 θ) 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 θ 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 θ is controlled for convergence, θ 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

$$u_0=k_{p1}{x}_1+k_{d1}{\dot{x}}_1+k_{p2}{x}_2+k_{d2}{\dot{x}}_2$$

(25)

where $k_{p1}$ is the proportionality coefficient based on the height error, $k_{p2}$ 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 u₀ 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

$$\left\{\begin{array}{c}{\dot{z}}_1={\beta }_1\left(x_2-z_1\right)+z_2\\ {\dot{z}}_2={\beta }_2\left(x_2-z_1\right)+z_3+b_{0}u\\ {\dot{z}}_3={\beta }_3\left(x_2-z_1\right)\end{array}\right.$$

(26)

where z₁, z₂, z₃ are the state variables of the full-dimensional state observer, b₀ is the constant coefficient, β₁, β₂, β₃ are the observer gains, and the observer enables the observation state to converge to the actual state using the pole assignment concept.

The deviation control amount equation is expressed as

$$u_y=u_0-\frac{z_3}{b_0}$$

(27)

The design of LQR controller is close to the traditional method. Firstly, the feedback coefficient matrix is obtained by selecting several discrete points and linearizing. Secondly, the deviation control quantity is obtained by combining the feedback matrix and the deviation state quantity obtained by the all-state sensor. Finally, the actual control quantity is obtained by combining the deviation and nominal control quantity.

3. Case Analysis

3.1. Model Verification and Method Comparsion

According to [18], the drag characteristics of the entire flight stage have been considered in the overall design of the missile, so that the x directional position and velocity v can be considered uncontrolled. Therefore, the x-directional position and velocity v in this paper assume nominal values, and only the y-directional position and trajectory inclination θ can be controlled. To verify the effectiveness of this method, this paper conducts simulation analysis using different initial deviations.

Sliding mode controller parameters: $\alpha =1.01$, $\beta =0.01$, $\gamma =11.9$, $\lambda =0.01$, $\overline{\beta }=1.5$, $k_1=1.2$, ${\epsilon }_i=0.5$.

Simulation Analysis 1: only the initial position deviation is present: $\Delta y=-100 \mathrm{m}$.

For the case where only an initial position deviation is present, the simulation results obtained using the three methods designed in this paper are shown in Figure 4, Figure 5 and Figure 6.

The Figure 4, Figure 5 and Figure 6 reveal that in the case of an initial position deviation, compared to LQR and ADRC, the AHSTC controller designed in this paper can achieve good tracking control under the initial deviation, and the control accuracy is high. In addition,

Table 2. Tracking error based on Simulation Analysis 1.

Terminal State	AHSTC	LQR	ADRC
y coordinate/m	4.9789	−14.2031	−0.82315
Trajectory inclination θ/°	0.0443	0.0262	0.0995

reveals that the AHSTC method has relatively small terminal state error, compared to the LQR method. It is noted that the terminal state y and θ are measured by all-state sensors.

In the AHSTC method, although the control quantity has some slight chatter, the terminal height error is significantly smaller than the LQR method.

Simulation analysis 2: an initial trajectory inclination deviation: $\mathsf{\Delta }\theta =3^0$.

For the case where an initial trajectory inclination deviation is present, the simulation results in this paper are shown in Figure 7, Figure 8 and Figure 9.

The Figure 7, Figure 8 and Figure 9 reveal that when an initial trajectory inclination deviation is present, the AHSTC controller designed in this paper can achieve good tracking control under the initial deviation, and the control accuracy is high. The position tracking can achieve fast convergence, the angle tracking can achieve convergence in approximately 7 s, and the model convergence speed is fast.

Table 3. Tracking error based on Simulation Analysis 2.

Terminal State	AHSTC	LQR	ADRC
y coordinate/m	−2.8481	5.1634	−0.75889
Trajectory inclination θ/°	−0.0057	−0.0046	0.1101

reveals that the trajectory inclination error between the AHSTC and LQR methods is small, but the AHSTC method has a small terminal height error.

Simulation analysis 3: an initial position deviation is present: $\mathsf{\Delta }y=-100\mathrm{m}$, and the initial trajectory inclination deviation: $\Delta \theta =3^0$.

For the cases with an initial position deviation and initial trajectory inclination deviation, the simulation results using three methods designed in this paper are shown in Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14.

In the AHSTC method, the control tracking curve is presented in Figure 11, which shows that the control curve quickly converges to the nominal control curve, the overall tracking control variable is relatively smooth, and the tracking effect is good. The tracking curves of the two state quantities are shown in Figure 12 and Figure 13, indicating that the tracking trajectory is not only smooth overall but also achieves good tracking of the nominal trajectory. Figure 14 shows the sliding mode surface change curve and the sliding mode surface change rate curve, respectively. The sliding mode surface designed in this paper can achieve rapid convergence under the condition of initial deviation. After the system converges, the stability accuracy is high, and the sliding mode chattering is small.

Table 4. Tracking error based on Simulation Analysis 3.

Terminal State	AHSTC	LQR	ADRC
y coordinate/m	4.3235	−9.1456	3.6223
Trajectory inclination θ/°	0.0403	0.01858	1.4914

draws the same conclusions as Table 2 and Table 3.

3.2. Analysis of Simulation Results

In the ADRC method, the plots of the simulation results reveal that the control convergence does not occur in the ADRC method, resulting in the nonconvergence of the state quantity, so that the simulation effect is relatively poor.

In the LQR method, trajectory tracking curve and control curve are smooth. However, the problem is that the terminal position error is large, which is unacceptable. The method should balance the terminal state error and control instruction smoothness.

In the AHSTC method, although the control quantity has some slight chatter, the terminal state error and the control quantity converge. The simulation results also show that the terminal state errors of the three methods are not very large, but the overall simulation performance based on the AHSTC method is good, and has good applicability, feasibility, and robustness.

3.3. Ways to Improve Accuracy of ADRC and LQR

The traditional ADRC method is used for a SISO problem, rather than a MISO problem. Because the proposed method is an improved ADRC method, the way to improve accuracy is to determine major and minor controlled state, select the appropriate feedback control coefficient and improve the observer bandwidth.

In the LQR method, the way to improve accuracy is to carefully increase the number of discrete points and set weighting matrix, which influences the results significantly.

4. Conclusions

By establishing an optimal nominal trajectory that takes into account the minimum attack time and the rate of change in the control quantity, combined with the approach of high-order sliding mode control and self-adaptation, this paper proposes a trajectory tracking control method based on the adaptive high-order sliding mode, thus achieving good tracking control for nominal trajectories. The main conclusions of this paper are as follows:

• This paper establishes the nominal trajectory that takes into account the flight time and the rate of change in the control quantity by using the Gaussian pseudospectral method and nonlinear programming;
• Based on the sliding mode control disturbance rejection, the high-order sliding mode control introduced in this paper reduces the chattering problem of the control variables to a greater extent than in the ADRC method;
• By introducing the adaptive control method, this paper ensures that the controller design maintains a good control effect in the presence of initial state interference.
• The control variables and state variables for the trajectory tracking control method, based on the adaptive high-order sliding mode, can effectively converge to the nominal trajectory, and the terminal error is small.