Adaptive Fault-Tolerate Lateral Control for Autonomous Electric Vehicles with Unknown Parameters and Actuator Faults

Liu, Haiyan; Wang, Yadong; Wang, Kunfeng

doi:https://doi.org/10.1155/2023/5548144

Mathematical Problems in Engineering

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

Research Article | Open Access

Volume 2023 | Article ID 5548144 | https://doi.org/10.1155/2023/5548144

Adaptive Fault-Tolerate Lateral Control for Autonomous Electric Vehicles with Unknown Parameters and Actuator Faults

Haiyan Liu,¹Yadong Wang,¹and Kunfeng Wang¹

Academic Editor: Zhizheng Wu

Received22 Nov 2022

Revised22 Feb 2023

Accepted20 Mar 2023

Published13 Apr 2023

Abstract

In this paper, the fault-tolerant lateral control problem is considered for autonomous electric vehicles with unknown parameters and actuator faults. In the real world, the physical characteristics of vehicles are time-varying, causing that the parameters are unknown and even piecewise. Besides, the actuators may encounter with intermittent actuator faults, which can destroy the system’s performance. To deal with the piecewise unknown parameters and actuator faults, quadratic damping terms are designed in the controller and estimators are introduced to estimate those uncertainties. Then, by utilizing the backstepping technique, a novel adaptive fault-tolerant lateral control scheme is proposed, which guarantees that the tracking errors converge into a compact set. Simulation results are provided to validate the effectiveness and robustness of the proposed control scheme.

1. Introduction

To deal with the serious environmental problems, autonomous electric vehicles (AEVs) are gradually widely used by the public due to the advantages of zero-emission and pollution-free-new energy [1]. The safety and fast responses of AEVs are quite important for the further improvements in energy efficiency, vehicle handling, comfort, and safety fields [2]. Since the lateral stability is an important issue for AEVs [3], it has attracted considerable attentions of researchers in recent years. Some inspiring results can be seen in literature [4, 5]. For a linearized vehicle model with known parameters, a feedback-feedforward controller based on the frequency domain characteristics is presented [6]. A model predictive controller (MPC) is proposed for the discretized model [7]. Based on previous results, the lateral stability of vehicles can be realized in a relative ideal circumstance with known parameters. Besides, an active disturbance rejection controller for the discrete system model with known parameters is designed to tackle the external disturbance [8]. The unmodeled tire dynamics is considered, and a finite-time control scheme is proposed [9]. The lateral error model is transformed into a second-order integrator system [10]. By utilizing backstepping technique, an observer-based control scheme is proposed to guarantee the lateral tracking performance. However, the system parameters are exactly known in previous references. In reference [11], a composite adaptive backstepping robust tracking controller is presented, which aims to improve the lateral dynamics stability for an electric vehicle (EV) while considering the uncertain parameters and external disturbances. The kinematic controller is proposed as an integrated backstepping controller for an autonomous tracked vehicle [12]. An adaptive path tracking control approach for autonomous vehicle systems is proposed in the presence of dynamical uncertainties and actuator failures. In reference [13], an adaptive path tracking controller is designed by adopting the adaptive backstepping approach.

Since it is difficult to accurately measure most parameters of AEVs in practical applications, the control scheme, which can handle uncertain parameters, is more significant. To solve this problem, adaptive methods are introduced in the controller design by many scholars. An event-triggered adaptive control mechanism is designed [14]. Using discrete state information, a controller is designed to fulfill the lateral stabilization objective, of which the control gains are obtained by solving LMIs. An adaptive controller of the overtaking vehicle with unknown velocities is proposed [15]. Based on the low-speed vehicle kinematics model, an adaptive control scheme is designed [16], in which the uncertain parameters are estimated by using parameter adaption laws. Noting that system parameters can be changed with the vehicle operation, it is more significant to consider and tackle time-varying unknown parameters.

It is worth noting that the stabilization under actuator faults is also critical to guarantee the desired lateral performances [17–19]. In reference [20], a fault-tolerant control scheme is proposed. By solving LMI, the control gain matrix can be obtained. However, the fault coefficients must be known to construct the LMI, which is difficult to be implemented in practice. Adaptive fault-tolerate control schemes are proposed to tackle unknown fault coefficients by introducing adaptive estimators [21–23]. The fault-tolerant sliding mode controller is designed considering that the fault coefficient is constant [24]. A meta-RL-basedfault-tolerant control (FTC) method is proposed to improve the tracking performance of vehicles in the case of actuator faults [25]. In reference [26], antisaturation fault-tolerant controllers in presence of model parametric uncertainties and actuator faults are designed. However, the considered actuator faults must be continuously differentiable [18], and the boundedness of fault coefficient estimates cannot be guaranteed [22, 23, 25, 26].

In most of the previous literature, considered unknown parameters are constant and fault coefficients are required to be continuously differentiable. However, in practical applications, these parameters and fault coefficients can be time-varying and even piecewise. To tackle with this problem, an adaptive fault-tolerant lateral control scheme is proposed for AEVs in this paper. The main contributions of this paper are given as follows:(1)An adaptive lateral control scheme is proposed for AEVs. Compared with the results [14–16], the proposed control scheme can handle time-varying and piecewise parameters. Based on this control scheme, it can be guaranteed that all the closed-loop signals are locally uniformly bounded and the lateral tracking errors converge into an adjustable compact set.(2)The proposed control scheme can also deal with actuator faults. In contrast to reference [20], the actuator faults with piecewise coefficients are tackled by introducing the quadratic damping terms and coefficient estimators. Compared with the results in references [22, 23], the boundedness of the estimate values for the fault coefficients can be guaranteed.

The rest of this paper is given as follows. The adaptive fault-tolerant lateral control problem for AEVs is formulated in the part of Problem Formulation. In the part of Controller Design, the control scheme is designed based on the backstepping technique. In the part of Stability Analysis, the stability analysis of the closed-loop system is provided. Simulation studies are given in the part of Simulation Results to validate the theoretical results, followed by a conclusion of this paper in the part of Conclusions.

2. Problem Formulation

2.1. Lateral Tracking Error Model

The lateral tracking error model [10, 27] is expressed as follows:where

and are unknown parameters. is the longitudinal velocity. is the desired yaw rate. is a state vector and the elements stand for the tracking error in the Y direction, the velocity tracking error in the Y direction, yaw angle error, and the angular velocity error, respectively.

Figure 1 illustrates the lateral motion for the AEVs. We derive the model in terms of the lateral position and the heading errors with respect to the road. The vehicle is maintained at the road center by regulating both the lateral position and heading errors.

2.2. Intermittent Actuator Fault Model

In this paper, we consider the case that the vehicle actuator suffers from faults in an intermittent manner, which is described as follows:withwhere is the designed control input. and are unknown positive constants. is the starting time instant for the kth fault. and so forth. Equations (3) and (4) indicate that the vehicle actuator loses of its effectiveness from till .

2.3. Control Object

The control objective of this paper is to design an adaptive fault-tolerant lateral tracking controller for AEVs, such that the boundedness of the tracking errors can be guaranteed in the presence of intermittent actuator faults and piecewise unknown parameters.

To achieve the control objective, the following assumptions are imposed: Assumption 1: the signs of and are known Assumption 2: all unknown parameters are bounded

Remark 1. and are values related to , , , and , where is mass of the vehicle, is yaw moment of inertia, is cornering stiffness of the front tires, and is distance of the front axle from center of gravity of the vehicle. Since the range of the variable can be roughly measured, the signs of and can be obtained. It also shows that assumption 2 is reasonable.

3. Controller Design

From equation (1), we can obtain

Define , , and as follows:

We have

When the yaw angle error is small, the look-ahead lateral error can be defined as follows [28]:where is a constant related with the length of the vehicle.

Computing the derivatives of and yields that

Then,where and .

Based on equation (10), the control-design procedure can be divided into two steps by adopting the backstepping technique [29–31]. A block diagram of the proposed method is shown in Figure 2. For easier reading, the notation for the time-varying signals is omitted hereafter, if no confusion arises.

Step 1. In this step, the virtual controller is designed.
Before the control-design process, the following error variables are defined:where is a virtual controller and designed as follows:with a positive constant .
Define a candidate Lyapunov function as follows:Substituting equations (10)–(12) into the derivative of equation (13) yields the following equation:

Step 2. In this step, we introduce estimators and to tackle piecewise unknown parameters and control gains. Based on these estimate values, an adaptive fault-tolerate controller is designed.
Definewhere is the estimate of the upper bound of .
Designwhere is a positive constant and , , and .
Letwhere is the estimator of , is the estimator of , and is the estimator of . is the sign, which is defined as follows:To ensure the boundedness of and , the projection operator [32] is introduced as follows:where , , , , and are positive constants.
Based on the property of projection operator technique, it is shown that

Remark 2. The unknown parameters are constant, and the fault coefficients are continuously differentiable [6–8, 22, 23]. Note that the intermittent actuator faults induce infinite number of sudden changes for the control coefficient. Hence, the time derivative of becomes unbounded for some time instants. Besides, the parametric uncertainties can be piecewise with the operation of vehicles. Above all, the piecewise unknown parameters and control coefficients are piecewise, causing that the controller design and selection of Lyapunov function are difficult. To handle this problem, the nonlinear damping term in the quadratic form is designed in the controller (16) and adaptive laws (19), (20).

Remark 3. It should be noted that if the controller in this paper is used in a higher order system it may lead to differential explosion. The articles [33–35] provide solutions to this problem and interested authors can refer to them.

4. Stability Analysis

With the designed controllers and adaptive laws, the main result of this paper is proposed in the following theorem.

Theorem 1. Consider the closed-loop system consists of the lateral error model (1), the virtual controllers (11), the control input (15) and (16), and parameter estimators (17), (19), and (20). Then, it is guaranteed that all the closed-loop signals are locally uniformly bounded and the lateral tracking errors converge into a compact set.

Proof. We choose a Lyapunov function as follows:where . . . . is the lower bound of . and .
The time derivative of is as follows:Substituting equations (10) and (15) into the time derivative of equation (11) yields thatThen,From equation (16), we can obtainSince is always negative and according to equation (22), then we haveBecause of , we obtainBased on Young’s inequality, we haveThen,whereBy solving the differential equation, we can obtainAccording to equations (13) and (23), we can obtainFrom equation (34), it can be concluded that , , , and are bounded. From reference [11], the boundedness of , , , and can be guaranteed. Besides, based on the bounded parameters, we can know that and are bounded. Then, according to the controller and adaptive laws, it is shown that the control input and adaptive laws are bounded. Since the initial yaw angle error needs to be located in a small compact set, all the closed-loop signals are locally uniformly bound. In accordance with the definition of and , the change of controller parameters has no effect on . Besides, decreasing and will make smaller.
According to the previous analysis, it is shown that the designed control scheme can guarantee the local stability of the closed systems with bounded tracking errors.

5. Simulation Results

In this section, simulation results are provided to validate the effectiveness of the designed control scheme.

The physical parameters of AEVs are given as follows: , , , , , , , , , and .

The actuator faults for AEVs are designed as follows:

The designed control parameters are chosen as follows: and .

The unknown parameter for AEVs is designed as follows:

Case 1. The actuator is normal and the unknown parameters are constants.
Simulation results are plotted in Figures 3–8. Figure 1 shows that the tracking error can converge to 0 after a period of time using the PID controller, Bstp BLF controller [36], and the controller proposed in this paper. , , and are the same as . Figure 7 shows the estimate of . Figure 8 shows the estimate of .
The simulation results show that PID controller, Bstp BLF controller, and the controller proposed in this paper can guarantee the system performance under time-invariant parameters and fault-free case.

Case 2. The actuator suffers from intermittent faults and the unknown parameters change suddenly.
Simulation results are plotted in Figures 9–14. It can be seen from Figure 9 that when the first fault occurs in 3.0 s, the PID method can keep in a nondivergent state, but it cannot converge after 6.0 s and tends to diverge. , , and are the same as . When the system suffers faults twice, the Bstp BLF controller makes converge to a wrong value and keeps , , and converge to 0 after a period of time. Using the method proposed in this paper, Figure 9 shows that even if the system suffers faults twice, it still converges to 0. After a period of time, the error is close to 0. , , and are the same as . Figure 13 shows the estimate of . Figure 14 shows the estimate of .
The simulation results show that PID controller and Bstp BLF controller cannot guarantee the system performance under time-varying parameters and fault coefficients. However, the controller proposed in this paper can guarantee the system performance.
It is shown in the simulation results that the control objective can be achieved with the designed control scheme. The proposed method can still keep convergence when the fault occurs.
The values of the control parameters chosen for the simulation using the proposed method are and . In order to explain the criteria used to choose them for parameter choices, several comparative tests with different parameters are executed. Simulation results are plotted in Figures 15–19. The control parameters affect the convergence speed of the controller and amplitude of controller input. According to the controller design, increasing and will make amplitude of controller input larger. It can be seen from Figure 15 that increasing and makes amplitude of controller input larger before 3.0 s.

6. Conclusions

The fault-tolerant adaptive lateral tracking control problem for AEVs is investigated in this paper. To overcome the difficulty induced by the piecewise parametric uncertainties and intermittent actuator faults, a novel adaptive control scheme is proposed by introducing estimators and quadratic damping terms. Simulation results are given to verify the effectiveness of this control scheme. The boundedness of the tracking errors can be guaranteed in the presence of piecewise parametric uncertainties and intermittent actuator faults. It should be noted that this paper can provide theoretical support for the design of lateral control system and the analysis can be carried out in engineering practice.

Data Availability

The data supporting the current study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported in part by the National Key Research and Development Program of China under Grant 2020YFC2003900 and National Natural Science Foundation of China under Grants 62076020 and U22A2061.

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Copyright © 2023 Haiyan Liu 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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