Comparative Study of Trajectory Tracking Control for Automated Vehicles via Model Predictive Control and Robust H-infinity State Feedback Control

A comparative study of model predictive control (MPC) schemes and robust H∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\infty }$$\end{document} state feedback control (RSC) method for trajectory tracking is proposed in this paper. The main objective of this paper is to compare MPC and RSC controllers’ performance in tracking predefined trajectory under different scenarios. MPC controller is designed based on the simple longitudinal-yaw-lateral motions of a single-track vehicle with a linear tire, which is an approximation of the more realistic model of a vehicle with double-track motion with a non-linear tire mode. RSC is designed on the basis of the same method as adopted for the MPC controller to achieve a fair comparison. Then, three test cases are built in CarSim-Simulink joint platform. Specifically, the verification test is used to test the tracking accuracy of MPC and RSC controller under well road conditions. Besides, the double lane change test with low road adhesion is designed to find the maximum velocity that both controllers can carry out while guaranteeing stability. Furthermore, an extreme curve test is built where the road adhesion changes suddenly, in order to test the performance of both controllers under extreme conditions. Finally, the advantages and disadvantages of MPC and RSC under different scenarios are also discussed.


Introduction
The increasing demands on mobility, efficiency and safety have extremely promoted the development of intelligent transportation system (ITS) in recent years [1][2][3][4]. As one of the most promising technologies of automotive industry, automated vehicles (AVs) with the improved security, has great potential applications in many fields and has attracted from both industry and academic communities [5][6][7], many attentions. Generally, the typical architecture of an AV system mainly incorporates two parts [8,9], namely, perception system and decision-making. Each of them is, in general, divided into many subsystems. Trajectory tracking control is a fundamental issue for AVs, which is devoted to track a predefined path and velocity profiles accurately and the errors (i.e. the lateral offset, heading and velocity errors) need to converge to zero during the manipulating process [10].
In recent years, much research work on trajectory tracking have been studied [11][12][13][14] and the main challenges of achieving accurate trajectory tracking lie in the following aspects: (1) non-holonomic property and multi-constrains of AGVs; (2) trade-off between vehicle model accuracy and computation efficiency; (3) uncertainties and time-varying parameters of vehicle dynamic model; (4) external disturbances of complex driving environment. Besides, the interaction between road and tires is also an important source of coupling. According Yang et al. Chin. J. Mech. Eng. (2021) 34:74 to the vehicle dynamics, it's known that the maximal longitudinal and lateral tire force are determined based on the friction ellipse under certain road conditions [15]. Furthermore, for most existing literatures of trajectory tracking, the bicycle vehicle model which describes longitudinal, lateral and yaw motion is usually adopted for the controller design. However, owing to the yaw motion caused by wheels steering, dynamic and kinematic couplings will put huge impact on the controllers' performance [16]. From literatures, model predictive control (MPC) and robust H ∞ control are two effective techniques to address the problems mentioned above [17][18][19][20][21][22][23].
Recently, MPC has become one of the most popular optimal control methods due to its well performance in processing multi-constrained linear or non-linear system [24,25], and it is easy to be used at different levels of the process control structures [18,26]. Owing to its advantages, MPC has been widely implemented in autonomous industry including trajectory tracking issues. The trajectory controller is designed by taking the tracking task as a multi-constrained model predictive control (MMPC) issue [27,28]. On the other hand, robust H ∞ control also has certain advantages in handling the presence of model uncertainties and external disturbances, which results from variations of environment parameters as well as the vehicle states. Ref. [29] managed to devise a robust H ∞ path following control strategy for AVs with considering signal transmission delay and data dropouts. In addition, the time-varying parameters, steering system backlashtype hysteresis, and uncertainties of nonlinear tire model were also considered in Refs. [30][31][32].
Though MPC and robust H ∞ control are two effective strategies in tackling trajectory tracking issues, they show different features. Therefore, the objective of this paper is to show the difference between MPC and robust H ∞ state feedback control (RSC) under different driving conditions. To the best of the authors' knowledge, there is no comparative study of MPC and RSC techniques for trajectory tracking. Both MPC and RSC controllers are designed based on the same longitudinal-yawlateral motions of a single-track vehicle with a linear tire model to achieve a fair comparison. The first key part of this paper is to design appropriate test cases to compare the difference between MPC and RSC controller. We re-emphasize that the main objective of this paper is a comparative study of path-following control employing different manoeuvres. Thus, the second part is to find the maximum velocity that both controllers can execute and to give corresponding reasons, which can give guidance for researchers to design controllers.
The rest of the paper are organized as follows. In Section 2, the trajectory tracking error model and 3-DOF vehicle dynamics model are deduced. In Section 3, the controllers based on MPC and RSC are designed to track expected trajectory. Then, the performance of MPC and RSC controllers is demonstrated in Section 4. Finally, the conclusions are illustrated in Section 5.

Trajectory Tracking Model
The trajectory tracking model is shown in Figure 1 and the definition of parameters used in the design of controllers is listed in Table 1.
At any given time, it's assumed that the reference trajectory is given by a tuple [x r , y r , ϕ r , v r ] T and subscript r represents reference profiles. From Figure 1, the lateral offset e y can be calculated by: The heading error e ϕ and velocity error e v are defined as follows Given the road curvature C R , the desired yaw rate is obtained: Besides, the derivatives of CG's position can be obtained from the kinematic relationship: Thus, combined Eq. (1) with Eqs. (2)-(4), we can get The schematic diagram of single-track vehicle model coupled 3-DOF vehicle dynamic model is also shown in Figure 1. According to Newton's laws, the vehicle dynamics in the yaw plane can be described by the following differential equations: where F x means the generalized longitudinal force of AVs including wind drag and tire rolling resistance. F xf and F xr are the longitudinal force in front wheels and rear wheels, respectively, δ f is front wheel steering angle.
The front and rear lateral force F yf and F yr are the function of tire slip angles that can be calculated as follows: where α f and α r denote the wheel slip angles, which can be calculated as follows: Therefore, combined Eq. (6) with Eqs. (7), (8), then where To facilitate the controller design, rewrite the Eq. (9) to form the following linear state-space equation, which can be represented as: where the state variables is x(t) = v y , r, e y , e ϕ , e v T and the output variables is

Control Law Allocation
In the previous section, the trajectory tracking model has been established. Then, in this section, the trajectory tracking controller will be designed based on MPC and robust theory, respectively. Specifically, both controllers will be built with model described in Eq. (10) to compare the differences between them.

Formulation of Augmented System
As analyzed above, the trajectory tracking task can be taken as a predictive control problem with multi-constraints. Note that the model described in Eq. (10) is a continuous-time system. To facilitate MPC controller design, the continuous-time system should be transformed into a discrete state-space mode with the fixed sampling period. Here, the zero-order hold (ZOH) method is applied and the Eq. (10) can be converted as: where A d ∈ R 5×5 , B d ∈ R 5×2 and C d ∈ R 4×5 are the system coefficient matrices and k means the time step. Furthermore, to achieve a better control performance, the state vector x(k) and the increment input �u(k) are usually coupled in an augmented vector, which can be represented as And the control input can be calculated by (11) Thus, the system Eq. (11) can be rewritten as the following equations: where The sequence of future incremental inputs and disturbances at time step k are denoted as �U (k) and D(k) , respectively.
where N p is the prediction horizon and N c means the control horizon.
Therefore, the predicted output can be calculated by the following formulations: Here, define the predicted outputs of the predictive state-space model at time step k as: Denote the performance outputs over the prediction horizon N p as a compact matrix form:

Formulation of Trajectory Tracking Problem Using MPC
The aim of MPC-based trajectory tracking controller is to make the predicted outputs as close as possible to the reference trajectory within the predictive horizon, and the reference trajectory Y r (k) is assumed to remain unchanged during current optimization window. The reference signals are described as: Thus, a typical tracking accuracy and control smoothness-oriented cost function over the predictive horizons is defined: where diagonal matrices Q and R are the positive definite weight matrices (i.e., Q > 0 and R > 0 ) which can be regulated to achieve desired closed-loop performance. The first item in cost function reflects the tracking error between the predictive outputs and the reference trajectory. The second one refers to the penalty on the control inputs that is to make the control process smoother.
On the other hand, vehicles have many inherent physical limitations. These limitations on the capacity of control actuators result in the hard constrains of the tracking system. Besides, there are also some restrictions imposed on output variables due to environment conditions (i.e., road boundary, speed limit, etc.). Here, the constrains for this trajectory tracking problem are specified as: Then, combined Eq. (19) with Eq. (20), the trajectory tracking problem is converted to solve the following optimal problem with multi-constrains: It's found that the optimization problem Eq. (21) is based on linear system and the cost function is convex quadratic, which means it's can be remade as quadratic program (QP) issues if the constrains are convex too. The problem described in Eq. (21) needs to be translated into standard form of quadratic problems: Then, at time step k, the above quadratic problem can be solved (i.e., quadprog function in Matlab) and will get a solution U * k . Once the solution is obtained, the first element of solution vector in vector, namely, u * k will be used and the optimal control inputs at this time step is computed as: Similarly, at next time step k + 1 , new input measurements and updated system states produce new quadratic problem that needs to solve. In conclusion, by solving a quadratic problem at each time step, the MPC controller will get an optimal input for next time step.

Preliminary Knowledge of RSC
We re-emphasize that the main objective of this paper is a comparative study of different control strategies. Both MPC-based and RSC-based tracking controllers use standard design methods, and specifically, the RSC-based tracking controller design mainly adopts the theory applied in Refs. [10,21]. The design of RSC is also based on the linearized vehicle model Eq. (10) and the objective function is defined as follows: where the main task is to obtain a control law that can make the closed-loop system satisfy the asymptotically stability and fulfill the H ∞ performance index γ that attenuate the effect of the external disturbance D(t).
Before the controller design, some essential lemmas and theorems also need to be introduced. Consider a polytopic linear parameter varying (LPV) system, which is described as follows: It's equal to the following equations: where the A 0i , B 0i and C 0i are the values of A 0 (q) , B 0 (q) and C 0 (q) at the vertex q i of the parameters polytope. The proof of this theorem can be found in Ref. [21]. The RSC-based controller design is on account of linear matrix inequality (LMI) method. So far, the preliminary knowledge has been prepared sufficiently for the RSCbased controller design.

Formulation of Trajectory Tracking Problem
Similar to MPC-based controller, the constrains imposed on inputs should be taken into consideration [33]. Define Lyapunov equation as follows: And it's assumed that the following conditions are satisfied where K is the gain of the controller and η is a given constant. The maximum inputs are denoted as u max , then, where θ max is the largest eigenvalue of the matrix (P − 1 2 K T KP − 1 2 ). Based on Schur complement lemma [34] and Eq. (31), the following conditions can be obtained, For the system depicted by Eq. (10), it can be found that time-varying parameters v x , 1 v x , r and C R are coupled in the system matrices. These parameters need to be processed before carrying out the design of RSC control law. Here, the polytope model method in Ref. [10] is applied to cover the rang of time-varying parameters. Specifically, in this system, the longitudinal velocity is usually bounded. It can be assumed that v x varies in the range of [v xmin , v xmax ] , 1 v x is bound in the range of 1 v xmax , 1 v xmin and v xmin , v xmax denote the minimum and maximum values of longitudinal velocity. Similarly, the same assumptions are imposed on r and C R , namely, r and C R vary in the range of [r max , r min ] and [C Rmin , C Rmax ] , respectively. Therefore, the time-varying parameters v x , 1 v x , r and C R can be expressed by a linear combination of the vertices as follows: where σ represents the vertices coordinates of the polytope and ρ means the weighting factors, which can be denoted as: Combined Eq. (10) and Eq. (33), the system plant of trajectory tracking system can be rewritten in the polytypic form as: 2 n=1 ρ 1i ρ 2j ρ 3m ρ 4n . Define β = [β 1 , β 2 , · · · , β 16 ] T , y = r, e y , e ϕ , e v T and the LPV model of trajectory tracking system can be expressed as: with Then, the state-feedback controller is proposed as: where K i is the control gain.
Combined the Eq. (36) with Eq. (37), the closed-loop system can be obtained: According to theorem 1 and Eq. (38), if the following condition is fulfilled, the controller will achieve the aim of ensuring system stability and H ∞ performance index of the closed-loop system.
where the P i is a symmetric positive definite matrix and S is a matrix with proper dimensions. γ represents the performance index of system in attenuating disturbance.
So far, the key to solve the trajectory tracking issue has converted to calculate the controller gain K (β) by solving the inequality problems above. In this paper, we use the yalmip toolbox to solve this issue because it has simple syntax and is easy to use.

Comparative Study of MPC-Based and RSC-Based Controllers
In this section, three driving scenarios will be designed to compare the difference between MPC-based and RSC-based controllers. To begin with, the simulation description and section are detailed. Then, the function verification test is displayed to show that the parameters of both controllers had been tuned as optimal as possible. Finally, double lane change and extreme curve test are illustrated to compare the performance of both controllers in the harsh driving environment.

Simulation Description and Setup
Here, three simulation tests were designed to compare the different performance of MPC and RSC control schemes, which are carried out on the CarSim-Simulink platform. Specifically, these tests are (1) function verification, (2) double lane change manoeuvre with low adhesion, (3) curve test (curvature: 0.02) with adhesion changing abruptly. Besides, to quantify the tracking accuracy, the root mean square error (RMSE) was used, which is expressed as where M is the number of time periods, y i is the measured outputs and y r donates reference values. The parameters of AVs and controllers used in simulation are listed in Table 2.

Case 1: Function Verification
In this section, the presented controllers were verified to demonstrate that the parameters of both controllers had been tuned as optimal as possible. Here, a straight line with expected velocity 25 m/s was applied as the reference trajectory. The initial lateral, angular and velocity errors of AVs were set as − 1 m, 0, − 5 m/s, respectively. Besides, the road adhesion is set as 0.85. As shown in Figures 2 and 3, it can be seen that MPCbased controller showed a faster response than RSCbased one and both controllers tracked the expected trajectory exactly. To be specific, the path tracking error was MPC-based controller is 0.2372 m, which was smaller than that of RSC-based one 0.2582 m. Besides, the velocity errors of MPC and RSC controller were 1.4520 m/s and 1.4470 m/s, respectively. It can be concluded that in this test, MPC showed better performance that RSC in terms of tracking accuracy. The reason why it occurred was that in well driving conditions, MPC was able to predict system's future states precisely based on current states, and it takes the tracking issue as an optimal issue in current time. Owing to that the optimal issue is convex as shown in Eq. (21), of course, it would be better. On the other hand, constraints of actuators should also be considered. Here, from Figures 4 and 5, it is shown that the steer angle and longitudinal acceleration were limited in an expected range.
Overall, in this section, both controllers were verified and the parameters of them had been tuned through trial and error method, trying to ensure that both controllers can achieve well performance. In next sections, some extreme conditions were carried out based on these parameters.

Case 2: Double Lane Change Test
In last section, both MPC-based and RSC-based controllers were tested in well driving conditions. The results show that both controllers tracked the expected trajectory exactly.
Obviously, it's better to choose MPC controller for its accuracy. However, in this section, we were concerned with the performance of both controllers in the harsh driving environment, because some unexpected cases may occur that will put huge effects on vehicle's safety and stability.
Therefore, in this section, MPC and RSC controllers were tested in the road with low adhesion. Specifically, both controllers carried out the double lane change Table 2 Symbol and definition of parameters

Definition Symbol Value
Vehicle mass m (kg) 1750 Inertia moment of the vehicle about yaw axis I z (kg·m 2 ) 2500 Distance of the front/rear axle from CG l f /l r (m) 1.24/1.46 Nominal cornering stiffness of front/rear wheels C f /C r (kN/rad) 60/60 Minimal/maximal longitudinal velocity v min /v max (m/s) 5/30 Minimal/maximal yaw rate r min /r max (rad/s) − 0.5/0.5 Minimal/maximal curvature of the path C Rmin /C Rmax ( m −1 ) − 0.02/0.02 manoeuvre as shown in Figure 6. Here, the road adhesion was set as 0.2, corresponding to snow and ice road. It's known that vehicles are prone to lose lateral stability on low adhesion roads, especially when the velocity is high. Thus, the goal of this section is to find the maximum velocity that both controllers can carry out with guaranteeing that the vehicle is under control. The initial lateral, angular and velocity errors of AVs were all set as 0.       Figure 10, in this case, it can be known that the maximum acceleration equaled to 0.2g approximately, which means the vehicle was under the critical states. Compared to RSC controller, the phases of MPC that acceleration approached to 0.2g was short. Thus, it can be seen in Figure 9 that the vehicle controlled by RSC obviously slipped, while the vehicle controlled by MPC didn't experience that. The reason why MPC shows well performance is that it can predict the future states based on current states. Note that is this case, the states MPC utilized in its prediction model was correct, in other words, the current states were still in the accepted range of the model. For RSC, due to that the gain of controller was calculated offline, so the tracking issue couldn't be optimized specifically for the current states compared with MPC, but the vehicle was still under control. However, when the velocity increased to 11 m/s, from the Figure 11, it can be seen that the vehicle controlled by MPC was out of control that means the controller couldn't track the expected trajectory anymore. On the contrary, RSC showed a more robust performance though the vehicle still slipped and the vehicle tracked the expected trajectory in the end. Figure 12 demonstrates the simulation results of acceleration. It can be found that compared with RSC, the vehicle controlled by MPC was in critical states for a significantly long time. If the vehicle was in a critical states for a long time, the state information used in its prediction model may be inaccurate as the system was prone to be unstable at this time, which will cause the prediction to fail. For RSC, the REMS of path tracking error is 0.04690 m in this case. Thus, from the analysis above, it can be concluded that the maximum velocity MPC can carry out was 10 m/s. Next, the velocity will be increased for the test of RSC. The step applied in this simulation was 1 m/s.
To simplify, the simulation results of 12 m/s and 13 m/s were not displayed here. Figures 13 and 14 display the simulation results of RSC when the velocity was 14 m/s and 15 m/s, respectively. Obviously, the vehicle was out of control when the velocity was 15 m/s. When the expected velocity was 14 m/s, the vehicle tracked trajectory after experiencing slipping. Besides, it can be found that the period that acceleration exceeded to 0.2g is shorter in Figure 15 compared with Figure 16. Therefore, the vehicle was out of control when the velocity was increased to 15 m/s.

Case 3: Extreme Curve Test
The goal of this section is to study the performance of MPC and RSC when the road adhesion and road curvature change abruptly. Thus, as shown in Figure 17, a curve road was built in CarSim, the maximum curvature was set as 0.02 m −1 , the road adhesion was set as 0.85 if global X was in the range of [0, 20 m], otherwise, it was set as 0.2. Similarly, the initial lateral, angular and velocity errors of AVs were all set as 0.
When the velocity was set as 9 m/s, the simulation results are shown in Figures 18 and 19. Both MPC controller and RSC controller tracked the expected trajectory well, where the REMS of path tracking error is 0.025 m and 0.0412 m, respectively. From Figure 19, it can be found that the maximum acceleration didn't exceed 0.2g, so it can be considered that the current conditions were in the stable regions of both controllers.
However, the vehicle controlled by MPC was out of control when the velocity was up to 10 m/s on this kind of roads. The results are shown in Figures 20 and 21, and it's obvious that the vehicle controlled by MPC deviated seriously from the road and the controller lost its stability finally. From Figure 21, it can be known that the vehicle controlled by MPC and RSC slipped from 3 s to 7 s, but vehicle controlled by MPC was out of control in the end. On the other hand, though the vehicle controlled by RSC also experiencing slipping for a long time, it tracked the expected trajectory in the end, where the REMS of path tracking error is 0.8266 m. Therefore, it can be concluded that if the system is in an unstable state for a long time, RSC will show better performance.
With the purpose to test the maximum velocity that RSC can carry out in this case, we set the velocity as 11 m/s, 12 m/s and 13 m/s, respectively. Figures 22, 23 and 24 demonstrate the simulation results of three test. In the tests where velocity was set as 11 m/s and 12 m/s, though the RSC controller wasn't out of control, the tracking performance got worse and worse as velocity increased. When the velocity increased to 13 m/s, the RSC controller couldn't track the expected trajectory anymore and it lost stability quickly. Figures 25, 26 and  27 show the acceleration profiles of three tests.
Overall, in this case, the maximum velocity MPC can carry out is 9 m/s, while that of RSC is 12 m/s.