VGESO-Based Finite-Time Fault-Tolerant Tracking Control for Quadrotor Unmanned Aerial Vehicle

Shen, Haidong; Du, Jiwei; Yan, Kun; Liu, Yanbin; Chen, Jinbao

doi:https://doi.org/10.1155/2024/2541698

International Journal of Aerospace Engineering

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Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 2541698 | https://doi.org/10.1155/2024/2541698

VGESO-Based Finite-Time Fault-Tolerant Tracking Control for Quadrotor Unmanned Aerial Vehicle

Haidong Shen,¹Jiwei Du,²Kun Yan,²Yanbin Liu,¹and Jinbao Chen¹

Academic Editor: Binbin Yan

Received31 Oct 2023

Revised06 Jan 2024

Accepted18 Jan 2024

Published02 Feb 2024

Abstract

Based on the variable gain extended state observer, a finite-time fault-tolerant control strategy is developed for the quadrotor unmanned aerial vehicle with actuator faults and external disturbances. Firstly, a novel variable gain extended state observer is designed to estimate the unknown external disturbances, which mitigates the initial peaking phenomenon existing in traditional extended state observer-based methods. Meanwhile, the neural networks are applied to accurately approximate unknown couplings online. Moreover, with the help of the projection operator technique, the unknown actuator faults are observed in real time. Combined with the backstepping framework, the finite-time robust fault-tolerant control scheme is constructed and the stability is strictly proved via Lyapunov’s theory. Finally, the validity of the developed control scheme is demonstrated through numerical simulations.

1. Introduction

With the reformation and development of automatic control theory, the quadrotor unmanned aerial vehicle (UAV) has attracted wide attention in different fields. Owing to the advantages of low-speed flight, low-altitude hovering, vertical takeoff, and landing, it can not only be applied to many civilian areas but also has pivotal practical utility in the military and national defense fields [1, 2]. However, the quadrotor UAV is not only a typical underactuated system but also impressionable to the actuator faults and external disturbances because of the unique rotor structures [3]. Therefore, high-efficiency robust fault-tolerant control (FTC) design for the quadrotor UAV is a challenging topic worthy of intensive study.

Disturbances exist in almost all industrial systems and adversely affect control performance. Consequently, various antidisturbance methods have been presented to assure the system tracking performance in recent years, such as robust control [4, 5], sliding mode control [6–9], disturbance observer-based control [10, 11], and active disturbance rejection control (ADRC) [12–16]. Sliding mode control is widely used due to its robustness and fast response. Among these methods, the ADRC technique has been extensively used because of its ability to estimate total unknown uncertainties and disturbances [12]. As an important part of the ADRC technique, the high-quality extended state observer (ESO) can enhance the robustness of the system. In [13], a high-order ESO-based trajectory tracking control method was proposed for the quadrotor UAV in the presence of position constraints and uncertainties. In [14], an ESO-based robust deadbeat current controller was developed for the permanent magnet synchronous machine system with mismatch parameters and unmodeled nonlinear elements. In [15], the sliding mode approach was combined with the ESO approach to stabilize the pneumatic servo system subject to unknown disturbances. In [16], a deep forest algorithm-based fault diagnosis and location algorithm was presented for the quadrotor UAV system by means of the ESO technique. Nevertheless, most of the existing research results related to the ADRC methods were based on the constant observation gain, which always requires large values to ensure the rapid convergence of observation errors. For this reason, there is a so-called initial peaking phenomenon in the early operation of the observer existing in the traditional ESO method, and it is unfavorable to the transient dynamic response of the system. Therefore, novel variable gain extended state observer (VGESO) needs further exploration to enhance the antidisturbance performance.

Furthermore, actuator fault is another important factor that threatens the safe flight of quadrotor UAV. If the actuator fault cannot be handled in a timely manner, it will not only affect the flight control performance but also cause property losses in serious cases. At present, the adaptive estimation scheme is widely adopted to deal with the unknown fault due to its direct observation capability. In [17], an adaptive FTC strategy was developed for quadrotor UAV systems under actuator faults. In [18], an output feedback-based robust adaptive fault estimation strategy was presented for the quadrotor attitude system with actuator faults. In [19], sensor faults were studied for the switched uncertain nonlinear systems based on fuzzy control technology. In [20], a distributed fault estimation method was presented for the heterogeneous multiagent systems. Radial basis function neural networks (RBFNNs) are commonly utilized to approach continuous unknown functions. In [21], a neural FTC strategy was developed to address uncertainties existing in the helicopter system. In [22], a fuzzy neural network PID control method was presented to restrain the adverse impact of actuator faults. However, most of the existing works focus on the asymptotic stability of the quadrotor UAV under actuator faults, and the finite-time stability needs further consideration.

Since the operational missions of the quadrotor UAV become more and more complicated, it is of practical significance to address the finite-time convergence problem. In [8], the problem of finite-time stability was discussed for the variable sweep morphing aircraft based on the adaptive supertwisting sliding mode control method. In [23], a finite-time terminal sliding mode control scheme was developed for UAV systems with load suspension. In [24], a backstepping-based finite-time controller was presented for the disturbed quadrotor UAV system. Considering the dynamic obstacle disturbances, a finite-time controller was developed for the quadrotor UAV system in [25]. In [26], the issue of time-triggered-based finite-time control was investigated for the quadrotor system. In [27], an adaptive sliding mode finite-time stabilization control method was designed for the UAV system with parametric uncertainties. However, reviewing the reported literature, the high-quality finite-time FTC design for quadrotor UAV subject to external disturbance and actuator fault deserves more attention.

In general, a VGESO-based finite-time FTC algorithm is developed for the quadrotor UAV to guarantee flight safety and mission success. The main contributions of this work are summarized as follows: (1)A novel VGESO is developed to deal with the unknown disturbance, which can overcome the initial peaking phenomenon existing in the traditional ESO approach(2)The adaptive fault observer is combined with the RBFNN technique to estimate the unknown actuator fault directly(3)The presented antidisturbance FTC strategy can make the quadrotor UAV accomplish the tracking mission in finite time

The rest of this article is organized as follows. In Section 2, the nonlinear dynamic equations of the quadrotor UAV are established and some necessary assumptions are given. In Section 3, the robust fault-tolerant controller design and stability analysis are introduced. In Section 4, contrastive numerical simulations are conducted to demonstrate the effectiveness of the developed technique. In Section 5, conclusions and prospects are given.

2. Problem Description

The schematic diagram of the quadrotor UAV is given in Figure 1, where defines the earth frame fixed on a point on Earth and denotes the body coordinate frame fixed on the centroid of the quadrotor UAV. Then, taking both actuator faults and external disturbances into account, the complete dynamic equations of quadrotor UAVs are derived based on Newton-Euler theory as follows [28]: where denotes the position vector defined in ; represents the attitude angle including roll angle , pitch angle , and yaw angle ; is the angular rate defined in ; is the total thrust; is the control moment; is the mass; is the acceleration of gravity; is the moment of inertia matrix; and define the constant partial loss of effectiveness (LOE) fault factor of corresponding actuator; and represents unknown external disturbances.

The primary control objective of this study is to develop an effective robust fault-tolerant controller, which simultaneously guarantees that (1)all errors of the closed-loop system are bounded(2)the desired trajectories can be tracked in finite time

Meanwhile, the following assumptions and lemmas are given.

Assumption 1 (see [29]). The external disturbances are assumed to be bounded satisfying , , , and , where , , , and are positive constants. Moreover, the actuator LOE fault factors and are assumed to be constant and belong to , where is the lower bound.

Lemma 2 (see [21]). RBFNNs are commonly used for approximate unknown continuous functions , which can be written in the form of where is the input variable vector and is the resulting approximation error, stands for the estimation of the optimum weight vector , and represents the basis function. The optimal weight vector of the RBFNN is defined as where is a valid set with being a constant and is an acceptable set of the state. Substituting the optimal weight value results in where is the optimal approximation error satisfying with being a constant.

Lemma 3 (see [30]). For any real numbers , the following inequalities hold: where and .

Lemma 4 (see [30]). For arbitrary positive constants , , and , the following inequation holds: where and are real values.

Lemma 5 (see [30, 31]). For the given nonlinear system, if there exists a smooth positive definite function satisfying , where real numbers satisfy , , and , the system states convergence in finite time, with the settling time defined as , where is a constant.

3. Finite-Time Fault-Tolerant Controller Design

In this section, the design process of the proposed finite-time robust fault-tolerant controller is introduced elaborately, with the flow chart presented in Figure 2.

3.1. VGESO Design of Position Loop

For the sake of clarity, we rewrite the position loop governing equations of quadrotor UAV as where is the velocity vector, , , , , , and .

Based on Lemma 2, the following approximation of the unknown coupling term can be obtained: where defines the positive diagonal matrix, is the optimal weight matrix satisfying , is the Gaussian basis function vector and , and is the approximate error.

Define and . Then, we have

According to (9), the VGESO is expressed as where represents the estimation of ; ; ; ; , , and are positive constants; and is the estimation of .

Considering (9) and (10), the observation errors of the VGESO can be expressed as where and .

Define . Then, we can obtain

For the purpose of ensuring is Hurwitz matrix, the parameters satisfy and . To put it differently, there is a positive definite matrix which satisfies where represents the positive definite matrix.

Remark 6. In the initial phase of ESO estimation, owing to the large initial error between the estimated signal and actual value, a large overshoot will appear in the early adjustment process. This is the so-called initial peaking phenomenon [32, 33]. The VGESO designed above can solve the initial peaking problem by using a small gain at first and then maintaining a high gain, which upgrades the practical application and ensures observation accuracy.

3.2. Robust Fault-Tolerant Controller Design of Position Motion

Tracking errors of position motion are defined as where denotes the desired position and is the virtual controller.

Combining (7) and (17), the derivative of (16) is

The virtual controller is designed as where and .

Considering (19), equation (18) can be further described as

Then, differentiating (17) yields

The position loop finite-time controller is proposed as where is the proposed positive definite matrix.

Define . Since , it can be seen that . Then, (22) becomes where is the estimate of .

Combining (23), equation (21) can be expressed as where .

Select the Lyapunov candidate function as where and are the appropriate parameters.

Considering (20) and (24), the derivative of (25) is given by where .

The parameter update law and adaptive fault observer are designed as where and are designed positive constants, is the projection operator, and its role is to project into [34].

Defining , , and , we can get where and are designed positive parameters, , and .

Substituting (28), (29), and (30) into (26), we have where and is the identity matrix.

Define . With the desired yaw angle given, the corresponding attitude angles and and required propeller lift can be calculated as [35]

3.3. VGESO Design of Attitude Motion

Similarly, the attitude equation of the quadrotor UAV can be rewritten as where is the attitude transition matrix and .

Considering the unknown actuator fault , RBFNNs are adopted to approximate the coupling term , which is in the form of where is the designed parameter, is the optimal weight satisfying , is the Gaussian function which satisfies , and is the approximate error.

Define and . Then, we have

Based on (36), the VGESO is established as where and are the estimations of and ; ; ; ; is the estimation of ; and , , and are the prepared positive constants.

Considering (36) and (37), the observation errors of the VGESO are given by where and .

Let . Then, we can obtain

For the purpose of ensuring that is Hurwitz matrix, the parameters satisfy and . To put it differently, there is a positive definite matrix which satisfies where is the selected positive definite matrix.

3.4. Robust Fault-Tolerant Controller Design of Attitude Motion

The tracking errors of attitude motion are defined as where and is the virtual control law.

Combining (34) and (44), the derivative of (43) can be reformulated as

The virtual control law is given as where is the positive design matrix.

Substituting (46) into (45) gives

Then, differentiating (44) yields

The attitude loop finite-time controller is proposed as where is the designed positive definite matrix.

Define . Since , it can be seen that . Then, (49) becomes where and is the estimation of .

Substituting (50) into (48) yields where .

Choose the Lyapunov candidate function as where and are the designed parameters.

Considering (47) and (51), the derivative of (52) is given by where .

The parameter update law and adaptive fault observer are designed as where and are designed positive constants, is the projection operator, and are the elements of and .

Defining , , and , we have where and are designed positive parameters, , and .

Substituting (55), (56), and (57) into (53), one has where

3.5. Closed-Loop Stability Analysis

Theorem 7. For the given quadrotor UAV system (1) containing actuator faults and external disturbances, the VGESOs are designed as (10) and (37). By applying the proposed robust fault-tolerant tracking controllers (23) and (50), all closed-loop tracking errors are bounded and convergent in finite time.

Proof. Select the Lyapunov function as Considering (31) and (58), the derivative of (60) can be expressed as where By the utilization of Lemma 3, we can get where , , , , , ,, and .
By using Lemma 4 with , , , , and , we can get Combining (63), (64), (65), (66), and (67), equation (61) can be written as where and .
According to Lemma 5, all error signals converge in finite time. Meanwhile, the upper bound of setting time is calculated by with being a constant. This concludes the above proof.

4. Simulation Results

In this section, numerical simulations of the quadrotor UAV with actuator faults and external disturbances are carried out. The mass and inertia matrix are selected as kg and Nm. The reference trajectories are chosen as m, m, m, and rad. The parameters during the control design are chosen as , , , , , , , , , , , , , , , , , , and .

To implement related numerical simulations and evaluate the observer performance, the actuator LOE fault factors are introduced

The external disturbances are assumed as

The comparison results of the developed VGESO and traditional ESO are presented in Figure 3, where the blue lines define the observation error of VGESO and the red lines represent that of traditional ESO. From Figure 3, it can be observed that despite the traditional ESO has the ability of estimating disturbances, the so-called initial peaking phenomenon is unavoidable, which is unfavorable to the transient performance of the system. As a contrast, the developed VGESO overcomes the shortcoming by selecting a small gain at the initial phase and increasing gradually to a high value. Furthermore, to validate the feasibility of the proposed adaptive fault observer, LOE fault factors are selected as , , , and . Since the output of the adaptive fault observer is the reciprocal of and , actual estimation values should be , , , and . From Figure 4, it can be concluded that the adaptive fault observer can estimate the unknown actuator fault with both high accuracy and speed. Meanwhile, the norms of the NN weight matrix are displayed in Figure 5 and the corresponding control inputs are presented in Figure 6, respectively, which indicate that all of the simulation results are convergent and vary within reasonable limits.

The trajectory tracking results under the proposed finite-time antidisturbance FTC tactics are shown in Figure 7. From Figure 7, we can see that all states of quadrotor UAVs follow the desired trajectories, indicating the efficacy of the developed algorithm. Moreover, the comparative position tracking results under different control methods are presented in Figure 1, where trajectory A is the tracking result under the presented method, trajectory B is the tracking result under ESO-based backstepping sliding mode controller, trajectory C is the tracking result without handling the disturbance and fault, and trajectory D is the tracking result under PID controller. From Figure 8, if the negative effects derived from unknown disturbance and fault cannot be eliminated in time, the desired trajectory cannot be tracked. Meanwhile, compared with the ESO-based backstepping sliding mode controller and PID controller, the given method can ensure that the quadrotor UAV has better tracking accuracy and faster convergence speed. To sum up, the developed robust adaptive finite-time FTC scheme guarantees satisfactory performance of the quadrotor UAV suffering from actuator faults and external disturbances.

5. Conclusions

In this study, a finite-time FTC strategy based on the VGESO technique has been established to solve the trajectory tracking problem of quadrotor UAV with actuator faults and unknown disturbances. Firstly, a quadrotor UAV nonlinear model has been established. Then, the VGESO has been designed to estimate unknown disturbances. Subsequently, combined with the Lyapunov stability theory, the adaptive fault observer combined with RBFNNs has been employed to estimate the fault factors. Finally, the fault-tolerant tracking controller with finite-time convergence capability has been proposed. Simulation results indicate that the developed method has superior fault tolerance and antidisturbance properties. In the future, the proposed control algorithm will be tested through quadrotor UAV flight experiments.

Data Availability

The underlying data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 62103315 and 52272369; the Fundamental Research Funds for the Central Universities under Grant NS2022081; the “Chunhui Plan” Cooperative Scientific Project from the Ministry of Education under Grant HZKY20220523; the Industrial Project from the Science and Technology Department of Shaanxi Province under Grants 2022GY-243, 2023-ZDLNY-61, and 2022QFY01-16; and the Xi’an University Institutes Science and Technology Staff Service Enterprise Project under Grant 22GXFW0045.

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Copyright © 2024 Haidong Shen et al. 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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