Nonsingular Fast Terminal Sliding Mode Neural Network Decentralized Control of a Quadrotor Unmanned Aerial Vehicle

Mao, Yuqing; Chen, Jing

doi:https://doi.org/10.1155/2023/3288944

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Abstract Introduction Discussion Conclusion Abbreviations Data Availability Additional Points Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 3288944 | https://doi.org/10.1155/2023/3288944

Nonsingular Fast Terminal Sliding Mode Neural Network Decentralized Control of a Quadrotor Unmanned Aerial Vehicle

Yuqing Mao^1,2and Jing Chen²

Academic Editor: Qingdu Li

Received15 Feb 2023

Revised17 Aug 2023

Accepted25 Aug 2023

Published09 Sept 2023

Abstract

A nonsingular terminal sliding mode decentralized controller that can ensure the tracking errors of the trajectories and attitude rapid convergence in finite time is proposed for an insufficient driven and strongly coupled nonlinear four-rotor unmanned aerial vehicle (UAV). The total lift of the UAV system is decomposed into three virtual drive separation forces corresponding to the three positions. The insufficient drive UAV system is transformed into a virtual full-drive model for research. The three position states and the three attitude states of UAV are placed correspondingly to the six subsystems by variable substitution. The model uncertainty and unknown disturbance term for each subsystem serves as total coupling terms among the subsystems. The upper bounds of the total coupling terms are considered as unknown ordinary higher order polynomials varying with the six states of the system under the action of time change. With the help of Cauchy inequality, the estimates of the upper bounds are obtained from the approximation performance of the RBF neural network. Finally, the decentralized controller is designed for each attitude subsystem and the virtual decentralized controller for each position subsystem. It is also mapped to the tracking total lift controller by using the virtual decentralized position controller. The controller design process uses the nonsingular terminal sliding mode control technology to ensure that the quadrotor attitude and position variables can quickly converge to the desired value in a short time. Simulation experiments verify that the proposed control method is effective and feasible.

1. Introduction

1.1. Background

Being light sensitive and highly mobile, the quadrotor UAV can automatically perform tasks, such as image processing, video acquisition, environmental condition detection, site monitoring, and target tracking, which can be widely used in aerial photography, production supervision, public safety supervision, cargo distribution, combat reconnaissance, and other fields. The quadrotor UAV performs various missions by tracking the desired track. The accuracy of the tracking controller directly determines the quality of its task completion. However, the quadrotor is a complex nonlinear system with strong coupling, insufficient drive, multiple inputs, and multiple outputs. There is also a strong coupling effect between its posture and position. The task subjects to parameter uncertainty of the system and external environment interference, and all these factors greatly increase the difficulty of controller design.

1.2. Literature Review of Related Works

Recently, many scholars have proposed different effective control methods for tracking flight control problem of UAVs.

The authors of [1] presented a heuristic approach for optimal path planning that the optimization strategy is based on the indirect solution of the open-loop optimal control problem. In [2], a disturbance-observer-based adaptive fuzzy tracking control scheme is proposed for a medium-scale unmanned helicopter of six degrees of freedom in the presence of system uncertainties, flight boundary constraints, and external disturbances. In [3], a disturbance observer is employed to estimate time-varying external disturbances and uncertainties for ducted fan aerial vehicles. The unknown disturbance is estimated as one of the unmeasured states based on the expansion state observer in [4–7].

The above literature uses different methods to solve the control problem of UAV affected by external disturbances. However, the UAV is a complex nonlinear system. For the complex nonlinear problem, literature [8] uses backstepping technology to design effective controllers.

In order to avoid the integral explosion problem in the backstepping control and to improve system robustness, the authors of [9] used the dynamic surface control method to design the controller. To solve the system uncertainty of the quadrotor, literature [4, 10–12] introduced the sliding mode control technology to design an adaptive sliding mode trajectory-tracking controller to compensate for model uncertainty and ensure the target-tracking performance. In literature [13], an event-triggered sliding mode solution is presented to reduce the update frequency of the control input and reduce the communication burden.

The above literature solves the problem of integral explosion of backstepping control by sliding mode control technology. However, the control process of the UAV usually needs to be achieved in a limited time, and attitude and position variables can achieve stability in a short time. Therefore, terminal sliding mode control methods are used in [14–19] literature.

In literature [14], a terminal sliding mode attitude-position quaternion based control of a quadrotor unmanned aerial vehicle is presented. In [15], robust nonlinear control strategies for attitude and position control are innovatively proposed based on a new continuous nonsingular terminal sliding mode control (CNTSMC) scheme. Literature [16] proposed a global fast terminal sliding mode controller for trajectory tracking. Literature [17] combines recursive control and robust control to design the finite-time adaptive integration reverse fast terminal sliding mode control method (FT-AIBFTSMC), which improves the external wind interference and unknown tracking performance by using virtual control laws and adaptive switch gain. In literature [18], a nonsingular fast terminal sliding mode control (NFTSMC) strategy based on a finite-time observer and an improved reaching rate is proposed to solve the control problem of aerial robot systems subject to actuator faults and internal and external disturbances. In [19], a new sliding mode surface-like variable-based position tracking control scheme and a novel nonsingular terminal sliding mode-based attitude synchronization control scheme are developed to drive the UAV tracking the reference trajectory with obstacle avoiding.

1.3. Contributions

Inspired by the above research, this paper is to solve control difficulty and complexity brought about by strong coupling. Quadrotor system uncertainties are affected by external time-varying perturbations during flight missions.

Because the radial-based function (RBF) neural network has global approximation ability, it can approximate any nonlinear function with arbitrary accuracy and has a simple structure and fast approximation speed, and we have designed a nonsingular fast sliding mode decentralized control method based on the radial fundamental neural network in this paper. The main contributions of this work are summarized as follows:(1)Six interconnected subsystems were constructed from the three pose variables and three position variables. The uncertainty terms of each subsystem and external perturbations are discussed as lumped coupling terms between the subsystems. By considering that total coupling terms have upper bounds under time-varying action, and assuming that the upper bounds are unknown, ordinary higher order polynomials about the six state-tracking errors of the quadrotor, the estimates are obtained by using the deformation of the Cauchy inequality and the approximation power of the RBF neural network.(2)To overcome the strong coupling effect between pose subsystems, pose decentralized controllers are designed so that any attitude controller is related to none of the other two poses and the three position variables but only the corresponding pose variable. When the quadrotor attitude is deflected on a certain space surface, we can adjust only the attitude angle on the corresponding space surface to achieve the desired control effect, which greatly reduces control difficulty caused by the mutual coupling of the subsystems. In the design of the attitude decentralized controller, position virtual decentralized controllers are also designed. The total lift controller is finally obtained according to the mapping relationship between the position virtual controller and the total lift. Then, the angular speed controllers of the four rotors are obtained.(3)In order to ensure that quadrotor attitude and position variables can quickly converge to the expected value in a short time, nonsingular terminal sliding mode control technology is used in the controller design process. The Lyapunov theory proves the stability of the system, and the simulation experiment verifies that the proposed control method is effective and feasible.

The rest of this paper is organized as follows: Section 2 details the modeling and problem description of the UAV system. Section 3 shows the controller design. Section 4 shows the stability analysis. Simulation results are presented in Section 5 to demonstrate the effectiveness of the proposed control scheme. Finally, the article is summarized in Section 6.

Remark 1. Existing works in the literature have proposed and implemented sliding mode controllers on real quadrotor hardware. Considering that the actual UAV is sometimes implemented for new kinds of mission in which unmodeled dynamics or disturbances are present, so as future directions, these control strategies will be extended for systems with unmodeled dynamics and uncertainties.

2. Problem Statement

The structure of the quadrotor UAV consists mainly of “+” and “X” types. In this paper, an “X” quadrotor UAV is studied. The specific structure and motion state are shown in Figure 1. The four motors of the quadrotor provide power, and the motor speed change controls the rotor speed change; thus, the three positions of front and back, left and right, up and down, as well as the three poses of pitch, roll, and yaw are adjusted. The coordinate origin of the ground coordinate system is the take-off point of the UAV, and the coordinate origin of the body coordinate system is the geometric center of the four rotors of the UAV. According to the Newtonian acceleration formula and the Newton–Euler theorem of motion, the kinetic model of a quadrotor UAV can be described by the following equations which are obtained under the “small angle approximation” assumption [14]:

Here, . The mapping between and is shown in formula (3):that is,

Here, are the lift coefficient and force to torque scale factor of the rotor, respectively.

To achieve attitude decentralized control, the three positions and three attitude angles of the UAV are considered as states corresponding to the six subsystems, represented by six-dimensional vectors, which are

The first derivative of the subsystem vector is denoted as , which is

is the external time-varying disturbance of the subsystem.

is the expected tracking trajectory of each subsystem. The quadrotor UAV system described in equation (1) can be converted into the following model:

Here, are the control inputs to each subsystem of the system, are the total coupling terms of each subsystem composed of external perturbations, model uncertainty, and strong coupling terms between the subsystems, and they are given by formulas (7) and (8), respectively:

In this paper, the attitude decentralized controllers named and the position decentralized virtual controllers named are designed for the strongly coupled four-rotor UAV described in (1)–(8), and then, the total lift control law is obtained through the mapping relationship of (8).

3. Controller Design

The trajectory-tracking error of each subsystem of the quadrotor UAV is described by

Here, is the model (6) that can be changed into the following formula:

Then, the nonsingular sliding mode surface of each subsystem is shown in the following formula:

Here, and are all in positive and odd numbers, and .

In order to easily represent the relationship of the variables in the controller design process, the following assumptions are made on the system-related variables.

Hypothesis 2. The total coupling terms of each subsystem are bounded, and the upper bounds are unknown, ordinary higher order polynomials for the six state-tracking errors corresponding to the UAV position and attitude, i.e.,

Hypothesis 3. The expected trajectory of each subsystem and its first and second-order derivatives are bounded; that is, the existence of can make the following inequalities hold true:

Theorem 4. According to the dynamic model, the tracking error for the six subsystems of the three position variables and the three pose variables of a quadrotor robot is shown in formula (10), the fast nonsingular terminal sliding mode controllers are shown in formula (14), and the parameter adaptive laws shown in formulas (15) and (16) are designed, respectively, to ensure that the tracking error of each subsystem converges in a finite time:

Here, are the parametric adaptive rates, are the design constants, and I is a design value.

Remark 5. As can be seen from the controllers given by equation (14), the controllers of the three quadrotor attitude variables are only related to each attitude variable itself, which greatly improves the control performance.
Remark: As existing works shown in literature [12], the saturation function can be used to eliminate shaking vibration when to realize sliding mode controllers in practice. Since in the sliding mode control, the value of the switching control function is in the boundary layer for most of the time, and the feedback control rather than the switching control is used in the control law, so the vibration is well suppressed.

4. Stability Analysis

are all positive and odd, and so .The Lyapunov function of the i-th subsystem can be defined as

The derivative of equation (19) is given by

Here, and are all positive and odd, and , so we can get the derivative of shown by

The Lyapunov function defining the quadrotor UAV system is given by

The derivative of equation (22) is given by

Let and if then

Letwhere are continuous functions. Therefore, the approximation can be performed with radial-based neural networks, i.e.,

Here, is the input signal of a neural network, is the optimal weight of the neural network, is the estimate of , and .

are Gaussian-type basis functions, where and are, respectively, center and basis width parameters of the j-th neuron basis function in the hidden layer, is the approximation error of the neural network, and , and are the estimates of and .The neural network model for approximating the continuous function is shown in Figure 2.

Thus,

For , , and then, makes

When the following results can be obtained by (27):

When , the following results can be obtained by (27):

When the first line of (15) and (16) is established, the following results can be obtained by (28) and (29):

When the second line of (15) and (16) is established, the following results can be obtained by (28) and (29):where Therefore,and the following results can be obtained by (30)–(32):

Here, .

We know V is bounded from inequality (33) and the results in literature [20, 21], and convergence to 0 is achieved exponentially.

We suppose that when the subsystem reaches the sliding mode surface, i.e., .When , the dynamic equation of tracking error of the first subsystem obtained by equation (11) is given by

We deform (34) to (35) as follows:

Derivatives on both sides of (35) can obtain

The following results can be obtained by (36):

are all positive and odd numbers, and and then, . is in the process of convergence to 0; i.e., decreases monotonically as grows. It is assumed that the subsystem converges to 0 after reaching the sliding mode surface for time, i.e., . Thus,

In summary, the tracking error of the subsystem can then converge rapidly to zero in finite-time .

5. Simulation Results and Discussion

To verify the effectiveness of the nonsingular fast terminal sliding mode neural network decentralized control (FTSMNNDC) method designed for the quadrotor UAV described in (1), this section will then test the control algorithm by simulation based on the numerical analysis method. The number of neurons in the hidden layer of the neural network is 6, and the j-th neuron basis function center and the base width parameter are taken as

The physical parameters of the quadrotor UAV are shown in Table 1 [14]. The settings of the position, posture, attitude, and reference heading angle are shown in Table 2 [9]; considering that the altitude of UAV flight is limited, the trajectory of is optimized on the basis of literature [9] as follows:where the constant is the highest flight altitude that the drone can reach. External unknown perturbations are shown in Table 3 [9], and the relevant parameters of the designed controller and parameter adaptive laws are shown in Table 4.

In order to verify that the designed decentralized control method has better performance, the nonsingular fast terminal sliding mode neural network decentralized control (FTSMNNDC) method and finite-time adaptive integral backstepping fast terminal sliding mode control (FT-AIBFTSMC) method proposed by literature [17] are compared for simulation analysis.

Figures 3 and 4 show the three-channel position tracking error response curves and the three-channel attitude angle tracking error response curves of the quadrotor UAV under two different control methods, respectively. The comparative analysis of two different control methods is conducted from two aspects:(1)Tracking speed: Although both methods can track the desired track for a limited time, it is clear that the proposed method is much faster than the literature [17] method. In this paper, we have proved that is in the process of convergence to 0, i.e., . From formula (38), we can obtain that as long as Then, So the three position variables and the three attitude variables can effectively track the desired track within 1 s. Indeed, since is close to 0, is much less than 1; therefore, is much less than 1, and the tracking error converges to 0 almost instantly. The simulation results also confirm this conclusion very well. While simulation conclusions in [16] show that the fastest tracking speed of literature [17] is close to 5 s, the slowest tracking speed is nearly 30 s.(2)Tracking effect: Both methods have a good tracking effect, and both can ensure that the tracking error is close to zero, but the proposed method of this paper is more stable.

(a)

(b)

(c)

(a)

(b)

(c)

Figure 5 shows the quadrotor position and attitude virtual decentralized controller and the resulting total lift, rolling torque, pitch torque, and yaw torque. In simulation, the controllers are chosen by formula (14), whose parameters are given in Table 4.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 6 shows the control inputs for the actual angular velocity of the four propellers obtained by mapping equation (2). It can be seen that the actual control inputs of both the virtual decentralized controller and the actual control inputs of the propellers are bounded, which meet the requirements of the actual control task.

(a)

(b)

(c)

(d)

6. Conclusion

This paper designs the attitude decentralized controller and the position virtual decentralized controller for the strongly coupled and insufficient driven quadrotor UAV which is disturbed by the external wind field. It reduces the control complexity caused by external interference and the coupling of the system itself and improves the control performance of the controller. By making a comprehensive use of the nonsingular fast terminal sliding mode technique and the neural network approximation technique, we accelerate the trajectory-tracking speed of the controller and enable the proposed controller to quickly track the desired trajectories. The simulation results not only show the effectiveness of the proposed control scheme but also show the faster and more accurate tracking performance by comparing with the existing research results.

In the next step, we will further study the design of effective controllers if the quadrotor actuator is limited or the UAV attitude and track state cannot be observed.

Abbreviations

:	The ground frame
:	The body frame
:	Displacement of the UAV during a horizontal forward, horizontal right, and vertical upward movement, m
:	First derivative of , m/s
:	Second derivative of , m/s²
:	The roll, pitch, and yaw angles
:	First derivative of
:	Second derivative of
:	The rotation speeds of the four rotors, rad/s
:	The lift coefficient of the rotor
:	The four lifts provided by the four rotors, N
:	The corresponding position in the body coordinate system of the UAV when flying
:	Vertical upward control law of the total lift of the body, N
:	Rolling moment, pitch moment, and yaw moment inputs, N
:	The position decentralized virtual controllers, N
:	The attitude decentralized controllers, N.m
:	The total coupling terms of each subsystem composed of external perturbations, model uncertainty, and strong coupling terms between the subsystems
:	Total mass of UAV, kg
:	The moment of inertia of the three axes,
:	Propeller inertia,
:	The drag coefficients,
:	The external unknown time-varying perturbations, N , N.m
:	The distance from the center of the drone to the center of the rotor, m
:	States corresponding to the six subsystems, represented by six-dimensional vectors
:	The first derivative of the subsystem vector
:	The external time-varying disturbance of the subsystem
:	The expected tracking trajectory of each subsystem
:	The trajectory-tracking error of each subsystem of the quadrotor UAV
:	The nonsingular sliding mode surface of each subsystem
:	Sliding mode surface parameters
:	Coefficients of the quadratic function
:	Upper bound for tracking traces and their first and second derivatives
:	The symbolic function
:	The saturation function
:	The optimal weight of the neural network
:	The estimate of
:	Estimation error of the neural network weights
:	The basis function vector of the neural network
:	Basis functions
:	Center and basis width parameters of the j-th neuron basis function in the hidden layer
:	The approximation errors of the neural network
:	Upper bound for the neural network approximation error
:	The estimates of
:	Estimation error for the upper bound of the neural network approximation error
:	The parametric adaptive rates of the neural network weights and approximation error
:	The design constants.

Data Availability

No data were used to support this study.

Additional Points

The parameters of UAV refer to the parameters commonly used in the existing research literature. The initial value of neural network approximation and the design parameters of the system are selected within the conditions of the design of the controller.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was supported by the 2022 Annual Domestic Visiting Scholars Program of Zhejiang Province (Zhejiang Education Office Letter [2022] No. 215), the National Natural Science Foundation of China (Nos. 61973137 and 62073082), and the Natural Science Foundation of Jiangsu Province (No. BK20201339).

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Copyright © 2023 Yuqing Mao and Jing Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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