Robust <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.3443pt" id="M1" height="15.7437pt" version="1.1" viewBox="-0.0657574 -11.3994 26.6531 15.7437" width="26.6531pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-73" d="M828 650H570L564 622C646 614 650 609 634 524L604 366H253L281 524C296 608 308 615 382 622L388 650H132L126 622C211 615 214 609 198 524L121 122C106 42 102 35 23 28L17 0H276L282 28C196 35 191 40 206 122L243 324H597L559 123C544 42 537 35 452 28L446 0H710L716 28C632 35 628 43 643 122L719 524C735 610 741 615 822 622L828 650Z"/></g><g transform="matrix(.012,0,0,-0.012,13.299,4.134)"><path id="g50-30" d="M1000 227C1000 351 905 451 780 451C658 451 590 380 530 279C465 385 388 451 261 451C168 451 38 376 38 210C38 98 121 -12 261 -12C381 -12 448 59 508 160C573 53 652 -12 780 -12C879 -12 1000 64 1000 227ZM485 199C434 109 370 25 271 25C188 25 133 117 133 241C133 355 189 414 249 414C357 414 426 302 485 199ZM903 204C903 70 852 25 790 25C681 25 611 136 552 240C603 330 669 414 768 414C855 414 903 321 903 204Z"/></g></svg> Control for Path Tracking of Network-Based Autonomous Vehicles

Chen, Changfang; Shu, Minglei; Wang, Yinglong; Liu, Ruixia

doi:https://doi.org/10.1155/2020/2537086

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2020 | Article ID 2537086 | https://doi.org/10.1155/2020/2537086

Robust Control for Path Tracking of Network-Based Autonomous Vehicles

Changfang Chen,¹Minglei Shu,¹Yinglong Wang,¹and Ruixia Liu¹

Academic Editor: Qiuye Sun

Received09 Aug 2019

Revised17 Jan 2020

Accepted28 Jan 2020

Published09 Mar 2020

Abstract

This paper investigates the robust path-tracking control problem of network-based autonomous vehicles (AVs) with delay and packet dropout. Generally, both network-induced delay and packet dropout bring negative effects on the system stability and performance. A robust control scheme is proposed to realize the desired path tracking and lateral stability. The closed-loop system is asymptotically stable with the prescribed disturbance attention level if there exist some matrices satisfying certain linear matrix inequality (LMI) conditions. Furthermore, the proposed controller is robust to the parameter uncertainties and external disturbances. Simulation results are presented to verify the effectiveness of the proposed control scheme.

1. Introduction

With the advantages of improved security, better road utilization, and greatly reduced mobility costs, autonomous vehicles (AVs) offer the possibility of fundamentally changing the conventional transportation system [1]. One of the rudimentary control issues for AVs is the path-tracking control. The objective is to control the vehicle to track the predefined desired path with zero steady-state tracking errors, including lateral offset and heading error [2, 3]. Generally, to reduce the traffic accidents, the path-tracking errors (especially the lateral offset) must be guaranteed in reasonable and safe regions, either in mitigative or extreme driving conditions.

Several control strategies are proposed for the traditional vehicle stability and handling control, for example, robust control [4, 5], model predictive control methods [6, 7], and Lyapunov-based control approaches [8–10]. In [11], a robust linear quadratic regulator-based controller is proposed to improve stability and handling of four-wheel independently actuated electric vehicles with the norm-bounded uncertainties. A robust gain-scheduled controller is developed for lateral stability control of electric vehicles via linear parameter-varying technique [12], and the quadratic D-stability is applied to improve the transient response of the closed-loop system. In [13], a distributed robust control method is presented for multivehicle systems to balance the performance of robust stability, tracking performance, and string stability of a platoon. In [14], a hierarchical adaptive path-tracking controller is designed for an autonomous vehicle to track a reference path in the presence of uncertainties in both tire-road condition and external disturbance. In [15], the authors address the problem of position trajectory tracking and path-following control for underactuated autonomous vehicles in the presence of possibly large modeling parametric uncertainty. In [16], a composite nonlinear feedback control is investigated for path-tracking control of four-wheel independently actuated AVs considering the tire force saturations.

All of these aforementioned control methods consider the autonomous vehicle as a centralized control system. However, with the development of in-vehicle networks, the autonomous vehicle is actually a networked control system rather than a centralized control system. The control and measurement signals from controllers and sensors are exchanged through communication network, which imposes the effects of network-induced delays and packet dropout on the control loop due to the bandwidth limitation. The unavoidable delay and packet dropout during the signal transmissions will probably degrade the control effect and deteriorate the system stability [17–19]. Thus, this paper aims to solve the path-tracking control problem for AVs in presence of the delay and packet dropout. And we can obtain the useful inspirations from some well-studied theoretical results. In [20], new results on stability and performance are obtained for time-delay systems with two successive delay components by exploiting a new Lyapunov–Krasovskii functional. In [21], the fault detection problem is addressed for time-delay delta operator systems with random two-channel packet losses and limited communication. The sufficient conditions for the asymptotical stability and performance of the delta operator systems are provided by the Lyapunov functional technique. In order to save the communication resources with limited bandwidth, the event-triggered schemes are developed for networked control systems [22–24], and the stability and performance for the closed-loop system are obtained in terms of a group of LMIs. In [25–27], the tracking performance of networked control systems under the packet dropouts and channel noise is investigated. It is shown that the optimal tracking performance is related to the nonminimum phase zeros, unstable poles of the given plant, and the packet dropout probability. In [28], a novel characteristic equation-based small-signal modeling approach is proposed for converter-dominated ac microgrids to assess the system low-frequency stability. The distributed energy management algorithms are presented for the energy system formed by many energy bodies, and the main purpose is to maximize the resource allocation while meeting the coupled matrix constraint and the system operation constraints [29–31].

The robust control synthesis problem is investigated in the paper to achieve the desired path tracking for network-based autonomous vehicles with delay and packet dropout, parameter uncertainties, and external disturbances. The main contributions of the paper are addressed as follows. Different from the traditional vehicle stability and handling control, the networked-induced delay and packet dropout are considered in the control design, and the yaw rate and the sideslip angle are regulated simultaneously to improve the vehicle stability. The existence conditions of the robust path-tracking controller are obtained in terms of LMIs, and the closed-loop system is asymptotically stable with a guaranteed performance. In addition, the uncertainty effects of the tire cornering stiffness and external disturbance are considered to enhance the robustness of the proposed control scheme.

The paper is organized as follows. In Section 2, the autonomous vehicle model and path-tracking model are described, where the parametric uncertainties and delay and packet dropout are introduced. In Section 3, the problem of robust path-tracking control design for network-based autonomous vehicles is solved and the corresponding LMI conditions for the existence of desired state-feedback controllers are presented. The simulation results are given in Section 4. Finally, the paper concludes in Section 5.

Notation. The notations used in the paper are summarized as follows. The superscripts T and denote matrix transposition and matrix inverse, respectively. The notation means that Q is symmetric and positive definite (semidefinite). If the dimensions of matrices are not explicitly stated, it is assumed to be compatible for algebraic operations. In symmetric block matrices, the notation represents a term that is induced by symmetry. And notation stands for a block-diagonal matrix.

2. System Modeling

The dynamics of the autonomous vehicle is shown in Figure 1, which has the following lateral and yaw degrees of freedom (DOF). Assuming that the front-wheel steering angle is small, the dynamic equations can be described aswhere is the longitudinal velocity of center of gravity (), β is the vehicle sideslip angle, and γ is the yaw rate. and model the external disturbance. m and are the mass of vehicle and the inertia about the z-axis, respectively. and denote the distances from to the front and rear axles.

The lateral tire forces of the front and rear wheels can be described aswhere the proportionality constants and are the front and rear cornering stiffness and and are the tire slip angles of the front and rear wheels. Supposing the vehicle sideslip angle is sufficiently small, it follows that . And and can be calculated aswhere is the wheel steering angle used for the control scheme. From (1)–(3), the vehicle dynamics can be expressed as follows:where

The path-tracking model of the autonomous vehicle is shown in Figure 2, where is the lateral offset of the vehicle with respect to the lane centerline at a preview distance and is the angle between vehicle longitudinal axis and the road centerline. The relationship between y, , and can be described aswhere is the horizontal distance from the to the sensor.

Denote as the yaw angle of the road centerline with respect to the global coordinate frame; then, the vehicle yaw angle can be written as

To track the reference path of road curvature at a longitudinal velocity , the absolute desired yaw rate should be , i.e., . In practice, can be measured using a combined GPS/GIS system.

By differentiating (6) and (7), the path-tracking model is given as follows:

Define the state vector and the control input . Combining (4) and (8), the dynamics of the overall system in the state-space form can be denoted bywhere

Due to the change of the road conditions, the tire-cornering stiffness is assumed to be time varying, which can be expressed as multiplicative perturbations:where and are the time-varying parameters, satisfying , , and and are the nominal values of and , respectively. Thus, it follows thatwhere and are the nominal values of A and B, respectively. and are the variations of A and B, and and , where

To deal with the uncertainties, the parameter matrices can be expressed aswhere , is the diagonal matrix of the uncertain parameters, and and are the and constant matrices, respectively. For the uniform road conditions of the front and rear wheels, we can suppose to reduce the possible design conservation. So the path-tracking model can be rewritten as

As the active safety systems become more advanced, much information about the state of the vehicle can be obtained by direct measurement. For example, vehicle sideslip angle, lateral offset, and heading error can be obtained by GPS, while the yaw rate can be measured by the vehicle on-board inertial measurement unit [32]. However, due to the network congestion or node failure, there unavoidably exist delay and packet dropout in the vehicle control system.

The networked path-tracking control scheme is shown in Figure 3. The whole delay is denoted by at each sampling period, where is the controller-to-actuator delay and is the sensor-to-controller delay at sampling time . Considering the behavior of the zero-order-hold in the control system, the data packets received by the networked controller are expressed aswhere h denotes the sampling period and represents the number of the packet dropout at the time . Thus, by lumping network-induced delay and packet dropout together, it can be deduced that

Let ; we can obtain

Assuming that the time-varying delay satisfies , where is the upper bound of delay. Due to the transmission of the system state to the controller with network-induced delay and packet dropout, the state received by the controller at sampling instant is , so by utilizing parallel distributed compensation scheme, the state-feedback controller can be designed as follows:where K is the control gain to be designed.

To finish the path-tracking task of autonomous vehicles, it is needed to control the lateral offset and heading error as small as possible. Besides, the yaw rate γ and sideslip angle β should be well regulated to assure the vehicle lateral stability. Choose the controlled outputs as , and combining (15) and (19), the closed-loop path-tracking vehicle model can be expressed as

The control objective is to design a robust path-tracking controller such that the closed-loop controlled system in (20) is asymptotically stable with an disturbance attenuation level , i.e.,

3. Controller Design

This section is devoted to solve the problem of robust path-tracking control of autonomous vehicles with network-induced delay and packet dropout. We study the conditions under which the closed-loop path-tracking control system is asymptotically stable with the prescribed disturbance attention performance. Before proceeding further, the following lemmas are introduced.

Lemma 1. Given matrices , , and of appropriate dimensions, , and W satisfies , then the following condition:holds if and only if there exists a positive scalar such that

Lemma 2. If there exist matrices P and , then the following conditions are equivalent:

Theorem 1. Given positive constants , , and control gain matrix K, closed-loop system (20) is asymptotically stable with an disturbance attention level if there exist matrices , , , general matrices , , and a scalar satisfyingwhere .

Proof. Define a Lyapunov–Krasovskii functional aswhere P, Q, and R are the symmetric positive definite matrices. Along the trajectory of system (20), the time derivative of , , can be computed as follows:Define and for any matrix with appropriate dimension, we have by using the Newton-Leibniz formula. Thus,It is true thatFrom (26)–(28), we can obtainwhereBy the Schur complement, guarantees thatwhereTo take the parameter uncertainty into consideration, rewrite (32) in the form of (22) as follows:where and stands for . According to Lemma 1, holds if and only if there exists a positive scalar such thatThen, by using a Schur complement operation, equation (25) can be obtained, and the proof is completed.

It is noted that the matrix inequalities in Theorem 1 are not in the form of LMIs, and in the following, an LMI condition for the existence of desired state-feedback controller will be presented. Thus, the problem of robust path-tracking control for network-based vehicle systems will be solved.

Theorem 2. Given positive constants and , there exists a robust state-feedback controller in the form of (19) such that closed-loop system (20) is asymptotically stable with an disturbance attention level if there exist matrices , , , general matrices Y, , , and a scalar satisfyingwhereMoreover, the control gain matrix is given by .

Proof. From Theorem 1, it is known that there exists a state-feedback controller in the form of (19) such that closed-loop system (20) is asymptotically stable with an disturbance attention level if there exist matrices , , , general matrices , , and scalars satisfying (25). Performing a congruence transformation to (25) by , it follows thatwhereDue to the existence of the nonlinear term , the above inequality cannot be solved by the standard numerical software. Note that , it follows that , which is equivalent to by Lemma 2. Define the following matrix variables:and perform a Schur complement operation to the term in the (1, 1) block. Equation (36) can be obtained, and this completes the proof of Theorem 2.

Remark 1. It is noted that (36) is an LMI not only over the matrix variables but also over the scalar . Thus, for the optimization variable , the minimum disturbance attention level bound can be obtained with the proposed controller in terms of the feasibility of (36). This means to solve the following convex optimization problem: minimize subject to inequality (36) over matrices , , , general matrices Y, , , and scalar .
When there are no delay and packet dropout in the signal transmission, the vehicle path-tracking model in (20) readsThe state-feedback controller is expressed aswhere is the control gain to be designed. And the closed-loop system can be written asThen, following the similar lines as in the proof of Theorem 2, the following corollary can be obtained.

Corollary 1. Given a positive constant , there exists a robust state-feedback controller in the form of (42) such that closed-loop system (43) is asymptotically stable with an disturbance attention level if there exist matrix , general matrix , and a scalar satisfyingwhere and the control gain is given by .

4. Simulation Results

In this section, two simulation manoeuvers are presented to examine the effectiveness of the proposed method. The data of the vehicle system are listed as follows: , , , , , , and . Suppose the parameter uncertainty of the cornering stiffness is of its nominal value. The maximum value of the measurement and transmission delay is assumed to be . The maximum number of packet dropout is 5, and the sampling period is ; then, the maximum delay is . The obtained minimum performance index in terms of the feasibility of (36) is 0.6256. For the sake of convenience, the proposed robust controller is named as DC, and the robust controller without delay and packet dropout is denoted as RC.

In the first simulation, the vehicle is assumed to finish a single-lane change maneuver. The vehicle velocity is set as , and the initial lateral displacement is . The road curvature is shown in Figure 4, and it can be seen that in this case the road curvature varies smoothly. The simulation results for the lateral offset in the single-lane change maneuver are presented in Figure 5. It can be observed that with the same time-delay , the lateral offset of the vehicle by DC is smaller than that of RC and better than that of RC . The heading errors for both controllers are compared in Figure 6; it is shown that the DC yields more smooth responses for vehicle orientation control, while RC causes much larger overshoot. The delay and packet dropout cause large oscillations in the path-tracking states and deteriorate the performance of RC. The front-wheel steering angle is plotted in Figure 7, and it shows a similar variation trend to the road curvature and is controlled in reasonable regions with several fluctuations. The global trajectory results are shown in Figure 8; it can be seen that the vehicle tracks the desired path more accurately with the proposed robust controller than RC in presence of delay and packet dropout, parameter uncertainties, and external disturbances.

In the second simulation, a double-lane change maneuver is chosen to simulate the severe driving conditions. The initial velocity is , and the road curvature is plotted in Figure 9, where some step variations are included in the road curvature. The simulation results for the lateral offset and heading error in the double-lane change maneuver are shown in Figures 10 and 11. It can be seen that even with some step variations introduced in the road curvature, there are less number of oscillations of the path-tracking errors by using DC than RC , considering delay and packet dropout. Figure 12 shows that the path-tracking control effect of steering input of DC is good as that of RC . The global trajectory results are presented in Figure 13, and it can be observed that the path-tracking task is finished better by the controller DC than the controller RC. Thus, the vehicle can be well controlled even on the tough driving conditions, and this justifies the effectiveness and robustness of the proposed controller.

5. Conclusions

In the paper, a robust controller is proposed to control network-based autonomous vehicles to track the desired path in presence of time-delay and packet dropout. The yaw rate and sideslip angle are also regulated to improve the vehicle lateral stability. The closed-loop path-tracking control system is asymptotically stable with the predefined disturbance attention performance if certain LMIs conditions are satisfied. The proposed controller is robust to the parameter uncertainties and external disturbances. Furthermore, the superiority of the proposed control scheme can be validated by comparison with no delay and packet dropout case. Simulation results under different manoeuvres are presented to illustrate the effectiveness of the control scheme. The analysis on how to further reduce the conservatism of the obtained result is considered to be topics of our future work.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

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

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 61603224 and the Natural Science Foundation of Shandong Province under Grant ZR2017MF029.

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Copyright © 2020 Changfang Chen 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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