Robust H‘ Control for Path Tracking of Network-Based Autonomous Vehicles

)is paper investigates the robust H∞ 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. )e 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.


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. e 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 H ∞ control [4,5], model predictive control methods [6,7], and Lyapunov-based control approaches [8][9][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 parametervarying technique [12], and the quadratic D-stability is applied to improve the transient response of the closedloop 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. e 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. e unavoidable delay and packet dropout during the signal transmissions will probably degrade the control effect and deteriorate the system stability [17][18][19].
us, 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 wellstudied 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. e 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][23][24], and the stability and H ∞ performance for the closed-loop system are obtained in terms of a group of LMIs. In [25][26][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. e 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][30][31].
e robust H ∞ 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. e 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. e 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. e 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 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. e simulation results are given in Section 4. Finally, the paper concludes in Section 5.
Notation. e notations used in the paper are summarized as follows. e superscripts T and − 1 denote matrix transposition and matrix inverse, respectively. e 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.

System Modeling
e 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 as where v x is the longitudinal velocity of center of gravity (CG), β is the vehicle sideslip angle, and c is the yaw rate. w 1 and w 2 model the external disturbance. m and I z are the mass of vehicle and the inertia about the z-axis, respectively. l f and l r denote the distances from CG to the front and rear axles. e lateral tire forces of the front and rear wheels can be described as where the proportionality constants C f and C r are the front and rear cornering stiffness and α f and α r are the tire slip angles of the front and rear wheels. Supposing the vehicle sideslip angle is sufficiently small, it follows that v y � v x β. And α f and α r can be calculated as where δ f is the wheel steering angle used for the control scheme. From (1)-(3), the vehicle dynamics can be expressed as follows: where e path-tracking model of the autonomous vehicle is shown in Figure 2, where y e is the lateral offset of the vehicle with respect to the lane centerline at a preview distance l s and ϕ e is the angle between vehicle longitudinal axis and the road centerline. e relationship between y, y e , and ϕ e can be described as where l s is the horizontal distance from the CG to the sensor. Denote ϕ d 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 ρ ref at a longitudinal velocity v x , the absolute desired yaw rate should In practice, ρ ref 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 x � [β, c, ϕ e , y e ] T and the control input u(t) � δ f . Combining (4) and (8), the dynamics of the overall system in the state-space form can be denoted by where Due to the change of the road conditions, the tirecornering stiffness is assumed to be time varying, which can be expressed as multiplicative perturbations: where κ f and κ r are the time-varying parameters, satisfying |κ i | ≤ 1, i � f, r, and C f0 and C r0 are the nominal values of C f and C r , respectively. us, it follows that

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where A 0 and B 0 are the nominal values of A and B, respectively. ΔA and ΔB are the variations of A and B, and To deal with the uncertainties, the parameter matrices can be expressed as where H � A cf A cr B cf , Λ is the 9 × 9 diagonal matrix of the uncertain parameters, and E 1 and E 2 are the 9 × 4 and 9 × 1 constant matrices, respectively. For the uniform road conditions of the front and rear wheels, we can suppose κ f � κ r to reduce the possible design conservation. So the pathtracking 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. e networked path-tracking control scheme is shown in Figure 3. e whole delay is denoted by τ i � τ ca + τ sc at each sampling period, where τ ca is the controller-to-actuator delay and τ sc is the sensor-to-controller delay at sampling time t i . Considering the behavior of the zero-order-hold in the control system, the data packets received by the networked controller are expressed as where h denotes the sampling period and n(i) represents the number of the packet dropout at the time t i . us, by lumping network-induced delay and packet dropout together, it can be deduced that Assuming that the time-varying delay satisfies 0 < τ(t) ≤ τ max , where τ max 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 t i is x(t i ), 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 y e and heading error ϕ e as small as possible. Besides, the yaw rate c and sideslip angle β should be well regulated to assure the vehicle lateral stability. Choose the controlled outputs as z � Cx � β c ϕ e y e T , and combining (15) and (19), the closed-loop path-tracking vehicle model can be expressed as e control objective is to design a robust H ∞ pathtracking controller such that the closed-loop controlled system in (20) is asymptotically stable with an H ∞ disturbance attenuation level c 0 , i.e.,

Controller Design
is section is devoted to solve the problem of robust H ∞ path-tracking control of autonomous vehicles with networkinduced 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 matrices Σ 1 , Σ 2 , and Σ 3 of appropriate dimensions, Σ 1 � Σ T 1 , and W satisfies WW T < I, then the following condition: holds if and only if there exists a positive scalar ε > 0 such that Lemma 2. If there exist matrices P and W > 0, then the following conditions are equivalent:  Figure 3: Schematic diagram of the networked path-tracking control system. where Proof. Define a Lyapunov-Krasovskii functional as where P, Q, and R are the symmetric positive definite matrices. Along the trajectory of system (20), the time derivative of V i (t), i � 1, 2, 3, can be computed as follows: x(s)ds) � 0 by using the Newton-Leibniz formula. us, It is true that From (26)-(28), we can obtain where By the Schur complement, Φ 1 < 0 guarantees that

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where To take the parameter uncertainty into consideration, rewrite (32) in the form of (22) as follows: where Π 1 � PA 0 + A T 0 P + N 1 + N T 1 + Q + C T C and sys Z { } stands for Z + Z T . According to Lemma 1, Φ 2 < 0 holds if and only if there exists a positive scalar ε > 0 such that en, by using a Schur complement operation, equation (25) can be obtained, and the proof is completed.
□ It is noted that the matrix inequalities in eorem 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. us, the problem of robust H ∞ path-tracking control for network-based vehicle systems will be solved.

Theorem 2.
Given positive constants c 0 and τ max , there exists a robust H ∞ state-feedback controller in the form of (19) such that closed-loop system (20) is asymptotically stable with an general matrices Y, N i , i � 1, . . . , 4, and a scalar ε > 0 satisfying

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where Moreover, the control gain matrix is given by K � YX − 1 .
□ Remark 1. It is noted that (36) is an LMI not only over the matrix variables but also over the scalar c 0 . us, for the optimization variable c 0 , the minimum H ∞ disturbance attention level bound can be obtained with the proposed controller in terms of the feasibility of (36). is means to solve the following convex optimization problem: minimize c 0 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 e state-feedback controller is expressed as where K zd is the control gain to be designed. And the closedloop system can be written as en, following the similar lines as in the proof of eorem 2, the following corollary can be obtained.
where � 1 � A 0 P + B 0 X + PA T 0 + (B 0 X) T and the control gain is given by K zd � X P − 1 .

Simulation Results
In this section, two simulation manoeuvers are presented to examine the effectiveness of the proposed method. e data of the vehicle system are listed as follows: m � 1500 kg, I z � 2500 kgm 2 , l f � 1.3 m, l r � 1.4 m, l s � 0.8 m, C f � 40,000 Nm/rad, and C r � 40,000 Nm/rad. Suppose the parameter uncertainty of the cornering stiffness is 20% of its nominal value. e maximum value of the measurement and transmission delay is assumed to be 30 ms. e maximum number of packet dropout is 5, and the sampling period is 2 ms; then, the maximum delay is 40 ms. e 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. e vehicle velocity is set as 20 (m/s), and the initial lateral displacement is 0 m. e road curvature is shown in Figure 4, and it can be seen that in this case the road curvature varies smoothly. e 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 � 0.04 s, the lateral offset of the vehicle by DC is smaller than that of RC (τ � 0.04 s) and better than that of RC (τ � 0.00 s). e 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 � 0.04 s) causes much larger overshoot. e delay and packet dropout cause large oscillations in the path-tracking states and deteriorate the performance of RC. e 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. e 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. e initial velocity is 20 (m/s), and the road curvature is plotted in Figure 9, where some step variations are included in the road curvature. e 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 pathtracking errors by using DC than RC (τ � 0.04 s), considering delay and packet dropout. Figure 12 shows that the path-tracking control effect of steering input of DC (τ � 0.04 s) is good as that of RC (τ � 0.00 s). e 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. us, the vehicle can be well controlled even on the tough driving conditions, and this justifies the effectiveness and robustness of the proposed controller.

Conclusions
In the paper, a robust H ∞ controller is proposed to control network-based autonomous vehicles to track the desired path in presence of time-delay and packet dropout. e yaw rate and sideslip angle are also regulated to improve the vehicle lateral stability. e closed-loop path-tracking control system is asymptotically stable with the predefined H ∞ disturbance attention performance if certain LMIs conditions are satisfied. e 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. e analysis on how to further reduce the conservatism of the obtained result is considered to be topics of our future work.

Data Availability
e data used to support the findings of this study are included within the article.