# RobustH∞ Control for Path Tracking of Network-Based Autonomous Vehicles

Authors: Changfang Chen; Minglei Shu; Yinglong Wang; Ruixia Liu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper investigates the robustH∞ 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 H∞ control scheme is proposed to realize the desired path tracking and lateral stability. The closed-loop system is asymptotically stable with the prescribed H∞ 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.


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## Body

## 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, robustH∞ control [4, 5], model predictive control methods [6, 7], and Lyapunov-based control approaches [8–10]. In [11], a robust linear quadratic regulator-based H∞ controller is proposed to improve stability and handling of four-wheel independently actuated electric vehicles with the norm-bounded uncertainties. A robust gain-scheduled H∞ 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 H∞ 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 H∞ performance are obtained for time-delay systems with two successive delay components by exploiting a new Lyapunov–Krasovskii functional. In [21], the H∞ 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 H∞ 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 H∞ 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 robustH∞ 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 H∞ path-tracking controller are obtained in terms of LMIs, and the closed-loop system is asymptotically stable with a guaranteed H∞ 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 Section2, 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 H∞ 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 −1 denote matrix transposition and matrix inverse, respectively. The notation Q>0≥0 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 diag⋯ stands for a block-diagonal matrix.


## 2. System Modeling

The dynamics of the autonomous vehicle is shown in Figure1, 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 as(1)β˙=1mvxFyf+Fyr−γ+w1,γ˙=1IzlfFyf−lrFyr+w2,where vx is the longitudinal velocity of center of gravity (CG), β is the vehicle sideslip angle, and γ is the yaw rate. w1 and w2 model the external disturbance. m and Iz are the mass of vehicle and the inertia about the z-axis, respectively. lf and lr denote the distances from CG to the front and rear axles.Figure 1
Schematic diagram of the vehicle model.The lateral tire forces of the front and rear wheels can be described as(2)Fyf=2Cfαf,Fyr=−2Crαr,where the proportionality constants Cf and Cr are the front and rear cornering stiffness and αf and αr are the tire slip angles of the front and rear wheels. Supposing the vehicle sideslip angle is sufficiently small, it follows that vy=vxβ. And αf and αr can be calculated as(3)αf=δf−vy+lfγvx,αr=vy−lrγvx,where δf is the wheel steering angle used for the control scheme. From (1)–(3), the vehicle dynamics can be expressed as follows:(4)β˙=a11β+a12γ+b1δf+w1,γ˙=a21β+a22γ+b2δf+w2,where(5)a11=−2Cf+2Crmvx,a12=−1−2Cflf−2Crlrmvx2,a21=−2Cflf−2CrlrIz,a22=−2Cflf2+2Crlr2Izvx,b1=2Cfm,b2=2CflfIz.The path-tracking model of the autonomous vehicle is shown in Figure2, where ye is the lateral offset of the vehicle with respect to the lane centerline at a preview distance ls and ϕe is the angle between vehicle longitudinal axis and the road centerline. The relationship between y, ye, and ϕe can be described as(6)ϕe=ye−yls,where ls is the horizontal distance from the CG to the sensor.Figure 2
Schematic diagram of path-tracking model.Denoteϕd as the yaw angle of the road centerline with respect to the global coordinate frame; then, the vehicle yaw angle can be written as(7)ϕ=ϕe+ϕ.To track the reference path of road curvatureρref at a longitudinal velocity vx, the absolute desired yaw rate should be vxρref, i.e., ϕ˙d=vxρref. In practice, ρref can be measured using a combined GPS/GIS system.By differentiating (6) and (7), the path-tracking model is given as follows:(8)ϕ˙e=γ−vxρref,y˙e=vxβ+ϕe+lsγ−vxρref.Define the state vectorx=β,γ,ϕe,yeT and the control input ut=δf. Combining (4) and (8), the dynamics of the overall system in the state-space form can be denoted by(9)x˙t=Axt+But+Bwwt,where(10)A=−2Cf+2Crmvx−1−2Cflf−2Crlrmvx200−2Cflf−2CrlrIz−2Cflf2+2Crlr2Izvx000100vxlsvx0,B=2Cfm2CflfIz00T,Bw=I,w=w1w2−vxρref−lsvxρrefT.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:(11)Cf=1+κfCf0,Cr=1+κrCr0,where κf and κr are the time-varying parameters, satisfying κi≤1, i=f,r, and Cf0 and Cr0 are the nominal values of Cf and Cr, respectively. Thus, it follows that(12)A=A0+ΔA,B=B0+ΔB,where A0 and B0 are the nominal values of A and B, respectively. ΔA and ΔB are the variations of A and B, and ΔA=κfAcf+κrAcr and ΔB=κfBcf, where(13)A0=a¯11a¯1200a¯21a¯22000100vxlsvx0,B0=b¯1b¯200T,Acf=−2Cf0mvx−2lfCf0mvx200−2lfCf0Iz−2lf2Cf0Izvx0000000000,Acr=−2Cr0mvx2lrCr0mvx2002lrCr0Iz−2lr2Cr0Izvx00000000,Bcf=2Cf0m2lfCf0Iz00T,a¯11=−2Cf0+2Cr0mvx,a¯12=−1−2Cf0lf−2Cr0lrmvx2a¯21=−2Cf0lf−2Cr0lrIz,a¯22=−2Cf0lf2+2Cr0lr2Izvxb¯1=2Cf0m,b¯2=2Cf0lfIz.To deal with the uncertainties, the parameter matrices can be expressed as(14)ΔAΔB=HΛE1E2,where H=AcfAcrBcf, Λ is the 9×9 diagonal matrix of the uncertain parameters, and E1 and E2 are the 9×4 and 9×1 constant matrices, respectively. For the uniform road conditions of the front and rear wheels, we can suppose κf=κr to reduce the possible design conservation. So the path-tracking model can be rewritten as(15)x˙t=A0+ΔAxt+B0+ΔBut+Bwwt.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 Figure3. The whole delay is denoted by τi=τca+τsc at each sampling period, where τca is the controller-to-actuator delay and τsc is the sensor-to-controller delay at sampling time ti. Considering the behavior of the zero-order-hold in the control system, the data packets received by the networked controller are expressed as(16)x^ti=xti,ifni=0,x^ti=xti−h,ifni=1,x^ti=xti−mh,ifni=m,where h denotes the sampling period and ni represents the number of the packet dropout at the time ti. Thus, by lumping network-induced delay and packet dropout together, it can be deduced that(17)x^ti=xti−mh−τi.Figure 3
Schematic diagram of the networked path-tracking control system.Letτt=t−ti−mh−τi; we can obtain(18)x^ti=xt−τt.Assuming that the time-varying delay satisfies0<τt≤τmax, where τmax 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 ti is x^ti, so by utilizing parallel distributed compensation scheme, the state-feedback controller can be designed as follows:(19)ut=Kx^ti,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 offsetye and heading error ϕe 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 z=Cx=βγϕeyeT, and combining (15) and (19), the closed-loop path-tracking vehicle model can be expressed as(20)x˙t=A0+ΔAxt+B0+ΔBKxt−τt+Bwwt,zt=Cxt.The control objective is to design a robustH∞ path-tracking controller such that the closed-loop controlled system in (20) is asymptotically stable with an H∞ disturbance attenuation level γ0, i.e.,(21)∫0tzTtztdt≤γ02∫0twTtwtdt.


## 3. Controller Design

This section is devoted to solve the problem of robustH∞ 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 H∞ disturbance attention performance. Before proceeding further, the following lemmas are introduced.Lemma 1.
Given matricesΣ1, Σ2, and Σ3 of appropriate dimensions, Σ1=Σ1T, and W satisfies WWT<I, then the following condition:(22)Σ1+Σ2WΣ3+Σ3TWTΣ2T<0,holds if and only if there exists a positive scalar ε>0 such that(23)Σ1+ϵΣ2Σ2T+ε−1Σ3TΣ3<0.Lemma 2.
If there exist matricesP and W>0, then the following conditions are equivalent:(24)1P−WTP−1P−W≥0,2−WP−1W≤P−2W.Theorem 1.
Given positive constantsγ0, τmax, and control gain matrix K, closed-loop system (20) is asymptotically stable with an H∞ disturbance attention level γ0 if there exist matrices P>0, Q>0, R>0, general matrices Ni, i=1,…,4, and a scalar ε>0 satisfying(25)Π¯1PB0K+N2T−N1N3TPBw+N4TA0TN1εPHE1T∗−N2−N2T−N3T−N4TB0KTN2N2E2KT∗∗−Q00N300∗∗∗−γ02IBwTN400∗∗∗∗−τmax−1R−10εH0∗∗∗∗∗−τmax−1R00∗∗∗∗∗∗−εI0∗∗∗∗∗∗0−εI<0,where Π¯1=PA0+A0TP+N1+N1T+Q+CTC.Proof.
Define a Lyapunov–Krasovskii functional as(26)Vt=V1t+V2t+V3t,V1t=xTtPxt,V2t=∫t−τmaxtxTsQxsds,V3t=∫−τmax0∫t+θtx˙TsRx˙sdsdθ,where P, Q, and R are the symmetric positive definite matrices. Along the trajectory of system (20), the time derivative of Vit, i=1,2,3, can be computed as follows:(27)V˙1t=xTtPA0+ΔA+A0+ΔATPxt+2xTtPB0+ΔBKxt−τ+2xTtPBwwt,V˙2t=xTtQxt−xTt−τmaxQxt−τmax,V˙3t=τmaxx˙TtRx˙t−∫t−τmaxtx˙TsRx˙sds.
Defineξ=xTtxTt−τtxTt−τmaxwTtT and for any matrix N=N1TN2TN3TN4TT with appropriate dimension, we have ξTNxt−xt−τmax−∫t−τmaxtx˙sds=0 by using the Newton-Leibniz formula. Thus,(28)−∫t−τmaxtx˙TsRx˙sds=−∫t−τttx˙TsRx˙sds−2ξTN∫t−τttx˙sds−τmaxξTNR−1NTξ+2ξTNxt−xt−τt+τmaxξTNR−1NTξ=−∫t−τttξTN+x˙TRR−1NTξ+Rx˙ds+2ξTNxt−xt−τt+τmaxξTNR−1NTξ.
It is true that(29)−∫t−τttξTN+x˙TRR−1NTξ+Rx˙ds≤0.
From (26)–(28), we can obtain(30)V˙t+zTtzt−γ02wTtwt=xTtPA0+ΔA+A0+ΔATPxt+2xTtPB0+ΔBKxt−τ+2xTtPBwwt+xTtQxt−xTt−τmaxQxt−τmax−∫t−τttξTN+x˙TRR−1NTξ+Rx˙ds+2ξTNxt−xt−τt+τmaxξTNR−1NTξ+τmaxx˙TtRx˙t+xTtCTCxt−γ02wTtwt≤xTtPA0+ΔA+A0+ΔATP+Q+CTCxt+2xTtPB0+ΔBKxt−τ+2xTtPBwwt−xTt−τmaxQxt−τmax+τmaxx˙TtRx˙t+2ξTNI0−I0ξ+τmaxξTNR−1NTξ−γ02wTtwt=ξTΦ1ξ,where(31)Φ1=Ω1+Ξ1+Ξ1T+τmaxNR−1NT+τmaxΞ2TRΞ2,Ω1=Π1PB0+ΔBK0PBw∗000∗∗−Q0∗∗∗−γ02I,Π1=PA0+ΔA+A0+ΔATP+Q+CTC,Ξ1=NI0−I0,Ξ2=A0+ΔAB0+ΔBK0Bw.
By the Schur complement,Φ1<0 guarantees that(32)Φ2=Ω2Ξ2TN∗−τmax−1R−10∗∗−τmax−1R<0,where(33)Ω2=Π1+N1+N1TPB0+ΔBK−N1+N2TN3TPBw+N4T∗−N2−N2T−N3T−N4T∗∗−Q0∗∗∗−γ02I.
To take the parameter uncertainty into consideration, rewrite (32) in the form of (22) as follows:(34)Φ2=Π¯1PB0K+N2T−N1N3TPBw+N4TA0TN1∗−N2−N2T−N3T−N4TB0KTN2∗∗−Q00N3∗∗∗−γ02IBwTN4∗∗∗∗−τmax−1R−10∗∗∗∗∗−τmax−1R+sysPH000H0ΛE1E2K0000<0,where Π¯1=PA0+A0TP+N1+N1T+Q+CTC and sysZ stands for Z+ZT. According to Lemma 1, Φ2<0 holds if and only if there exists a positive scalar ε>0 such that(35)Π¯1PB0K+N2T−N1N3TPBw+N4TA0TN1∗−N2−N2T−N3T−N4TB0KTN2∗∗−Q00N3∗∗∗−γ02IBwTN4∗∗∗∗−τmax−1R−10∗∗∗∗∗−τmax−1R+εPH000H0PHT000HT0+ε−1E1TE2KT0000E1E2K0000<0.
Then, by using a Schur complement operation, equation (25) can be obtained, and the proof is completed.It is noted that the matrix inequalities in Theorem1 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 H∞ path-tracking control for network-based vehicle systems will be solved.Theorem 2.
Given positive constantsγ0 and τmax, there exists a robust H∞ state-feedback controller in the form of (19) such that closed-loop system (20) is asymptotically stable with an H∞ disturbance attention level γ0 if there exist matrices X>0, Q¯>0, R¯>0, general matrices Y, N¯i, i=1,…,4, and a scalar ε>0 satisfying(36)Ξ11Ξ12Ξ13Ξ14Ξ15Ξ16Ξ17Ξ18Ξ19∗Ξ22Ξ23Ξ24Ξ25Ξ260Ξ280∗∗Ξ3300Ξ36000∗∗∗Ξ44Ξ45Ξ46000∗∗∗∗Ξ550Ξ5700∗∗∗∗∗Ξ66000∗∗∗∗∗∗Ξ7700∗∗∗∗∗∗∗Ξ880∗∗∗∗∗∗∗∗Ξ99<0,where(37)Ξ11=A0X+XA0T+Q¯+N¯1+N¯1T,Ξ12=B0Y+N¯2T−N¯1,Ξ13=N¯3T,Ξ14=Bw+N¯4T,Ξ15=XA0T,Ξ16=N¯1,Ξ17=εH,Ξ18=XE1T,Ξ19=XCT,Ξ22=−N¯2−N¯2T,Ξ23=−N¯3T,Ξ24=−N¯4T,Ξ25=B0YT,Ξ26=N¯2,Ξ28=E2YT,Ξ33=−Q¯,Ξ36=N¯3,Ξ44=−γ02I,Ξ45=BwTΞ46=N¯4,Ξ55=−τmax−1R¯,Ξ57=εH,Ξ66=τmax−1R¯−2X,Ξ77=−εI,Ξ88=−εI,Ξ99=−I.
Moreover, the control gain matrix is given byK=YX−1.Proof.
From Theorem1, 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 H∞ disturbance attention level γ0 if there exist matrices P>0, Q>0, R>0, general matrices Ni, i=1,…,4, and scalars ε>0 satisfying (25). Performing a congruence transformation to (25) by diagP−1,P−1,P−1,I,I,P−1,I,I, it follows that(38)Θ11Θ12Θ13Θ14Θ15Θ16Θ17Θ18∗Θ22Θ23Θ24Θ25Θ260Θ28∗∗Θ3300Θ3600∗∗∗Θ44Θ45Θ4600∗∗∗∗Θ550Θ570∗∗∗∗∗Θ6600∗∗∗∗∗∗Θ770∗∗∗∗∗∗∗Θ88<0,where(39)Θ11=A0P−1+P−1A0T+P−1N1P−1+P−1N1TP−1+P−1QP−1+P−1CTCP−1,Θ12=B0KP−1+P−1N2TP−1−P−1N1P−1,Θ13=P−1N3TP−1,Θ14=Bw+P−1N4T,Θ15=P−1A0T,Θ16=P−1N1P−1,Θ17=εH,Θ18=P−1E1T,Θ22=−P−1N2P−1−P−1N2TP−1,Θ23=−P−1N3TP−1,Θ24=−P−1N4T,Θ25=P−1KTB0T,Θ26=P−1N2P−1,Θ28=P−1KTE2T,Θ33=−P−1QP−1,Θ36=P−1N3P−1,Θ44=−γ02I,Θ45=BwT,Θ46=N4P−1,Θ55=−τmax−1R−1,Θ57=εH,Θ66=−τmax−1P−1RP−1,Θ77=−εI,Θ88=−εI.
Due to the existence of the nonlinear term−τmax−1P−1RP−1, the above inequality cannot be solved by the standard numerical software. Note that R>0, it follows that R−1−P−1RR−1−P−1≥0, which is equivalent to −τmax−1P−1RP−1≤τmax−1R−1−2P−1 by Lemma 2. Define the following matrix variables:(40)X=P−1,Y=KP−1,Q¯=P−1QP−1,R¯=R−1,N¯1=P−1N1P−1,N¯2=P−1N2P−1,N¯3=P−1N3P−1,N¯4=N4P−1,and perform a Schur complement operation to the term XCTCX 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 γ0. Thus, for the optimization variable γ0, the minimum H∞ 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 γ0 subject to inequality (36) over matrices X>0, Q¯>0, R¯>0, general matrices Y, N¯i, i=1,…,4, and scalar ε>0.
When there are no delay and packet dropout in the signal transmission, the vehicle path-tracking model in (20) reads(41)x˙t=A0+ΔAxt+B0+ΔBut+Bwwt.
The state-feedback controller is expressed as(42)ut=Kzdxt,where Kzd is the control gain to be designed. And the closed-loop system can be written as(43)x˙t=A0+ΔA+B0+ΔBKzdxt+Bwwt,zt=Cxt.
Then, following the similar lines as in the proof of Theorem2, the following corollary can be obtained.Corollary 1.
Given a positive constantγ0, there exists a robust H∞ state-feedback controller in the form of (42) such that closed-loop system (43) is asymptotically stable with an H∞ disturbance attention level γ0 if there exist matrix P¯>0, general matrix X¯, and a scalar ε0>0 satisfying(44)Ξ¯1Bwε0HP¯E1T+X¯TE2TP¯CT∗−γ02I000∗∗−ε0I00∗∗∗−ε0I0∗∗∗∗−I<0,where Ξ¯1=A0P¯+B0X¯+P¯A0T+B0X¯T and the control gain is given by Kzd=X¯P¯−1.


## 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:m=1500kg, Iz=2500kgm2, lf=1.3m, lr=1.4m, ls=0.8m, Cf=40,000Nm/rad, and Cr=40,000Nm/rad. Suppose the parameter uncertainty of the cornering stiffness is 20% of its nominal value. The maximum value of the measurement and transmission delay is assumed to be 30ms. The maximum number of packet dropout is 5, and the sampling period is 2ms; then, the maximum delay is 40ms. The obtained minimum H∞ performance index in terms of the feasibility of (36) is 0.6256. For the sake of convenience, the proposed robust H∞ 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 as20m/s, and the initial lateral displacement is 0m. 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 τ=0.04s, the lateral offset of the vehicle by DC is smaller than that of RC τ=0.04s and better than that of RC τ=0.00s. 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 τ=0.04s 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.Figure 4
Road curvature of the single-lane change maneuver.Figure 5
Lateral offset results of the single-lane change maneuver.Figure 6
Heading error results of the single-lane change maneuver.Figure 7
Control inputs of the single-lane change maneuver.Figure 8
Global trajectory results of the single-lane change maneuver.In the second simulation, a double-lane change maneuver is chosen to simulate the severe driving conditions. The initial velocity is20m/s, 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 τ=0.04s, considering delay and packet dropout. Figure 12 shows that the path-tracking control effect of steering input of DC τ=0.04s is good as that of RC τ=0.00s. 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.Figure 9
Road curvature of the double-lane change maneuver.Figure 10
Lateral offset results of the double-change maneuver.Figure 11
Heading error results of the double-change maneuver.Figure 12
Control inputs of the double-lane change maneuver.Figure 13
Global trajectory results of the double-lane change maneuver.


## 5. Conclusions

In the paper, a robustH∞ 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 H∞ 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.



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