Sliding Mode Integrated Control for Vehicle Systems Based on AFS and DYC

(is paper has investigated an integrated control of active front steering (AFS) and direct yaw-moment control (DYC) for vehicle systems. First of all, the desired yaw rate and sideslip angle are estimated by using a two-degree-of-freedom (2-DOF) model of the vehicle system. On this basis, the actual sideslip angle is estimated by means of an observer. (en, the sliding mode control (SMC) is developed for AFS and DYC, respectively, to guarantee that the actual yaw rate and the sideslip angle track their reference signals. Additionally, the disturbance observer (DOB) technique is introduced to further improve the control performance. Finally, the simulation results validate the superiority of the AFS and DYC integrated control by using CarSim software during the following conditions: double lane change and side wind disturbance.


Introduction
With the development of electronic technology, the use of control technology to improve the active safety of automobiles has become a hot topic in the automotive field. Active safety can minimize or avoid traffic accidents through vehicle design, while passive safety means that vehicles are designed to minimize the damage to passengers after an accident occurs. In the 90s of the last century, the concept of vehicle stability was proposed [1][2][3]. en, the active safety control received considerable attention in the vehicle stability control, such as [4][5][6][7]. Among them, AFS catches researchers' eyes since it can directly adjust the steering angle of the drive. Nevertheless, when the moment of the vehicle in control limit is under the sections of high-speed turning, heavy braking, or acceleration, steering will have no effect or limited effect, the car loses the ability to turn, and the initiative will have little effect on the steering. On the contrary, the effect of DYC is very obvious at this time, and it is easy to realize. It does not require a great change in the original structure of the vehicle. is is because the goal of DYC is to adjust the vehicle yaw motion. us, the integrated control of AFS and DYC can not only further improve the lateral stability of the vehicle but also reduce the influence of braking on longitudinal dynamics and improve the driving comfort.
In [7], the LQR method was used in the integrated control of AFS and DYC and compared with DYC control. In [8], an integrated control of AFS and DYC with forward and feedback controllers was presented, and the parameters of the feedback controller were obtained by the optimum control theory. en, the control strategy of AFS based on sliding mode theory and the control strategy of EPS (combined control on both direct yaw moment and variable slip ratio) were presented in [9]. An integrated AFS and DYC control system was developed in [10] based on the fuzzy logic control, which was used for the yaw rate controller to keep the yaw rate in its ideal value. In [11], the model predictive control was used to adopt the hierarchical integrated control structure.
In addition, on the one hand, the sliding mode control (SMC) is popular for rejecting the uncertainties [12][13][14][15][16][17][18]. On the other hand, the accurate mathematical model is indeed necessary for SMC [19,20]. Hence, it is obvious that the SMC method is a very useful tool for active safety control, such as [21][22][23].
is paper focuses on the investigation of the integrated SMC control for the AFS and DYC system, which is designed to make sure the actual yaw rate and sideslip angle track the desired signals. First of all, the proposed control can drive the error of the yaw rate and the sideslip angle to zero within a finite time. en, a disturbance observer (DOB) [24] is constructed for the proposed control to reduce the control gain so as to reduce the chattering. e effectiveness of the proposed control is illustrated by the simulation on MATLAB and CarSim.
As compared with the existing results, the contributions of this paper are twofold. On the one hand, an integrated control of AFS and DYC for vehicle systems has been studied. On the other hand, the Lyapunov stability analysis and simulation results have been given to demonstrate the effectiveness of the proposed strategy. e rest of the paper is organized as follows. Section 2 introduces the dynamic model of the vehicle and problem statement. e process of control design is given in Section 3.
e simulation results are shown in Section 4. Section 5 concludes this paper.

2-DOF Model.
e 2-DOF model is called as the "bicycle model" [25,26], which is shown in Figure 1. en, the model is described as where C f and C r , respectively, are the front and rear tire cornering stiffness, F is the tire force, x means the longitudinal position, y means the lateral position, β is the sideslip angle, a and b mean the distances from the center of gravity to the front and rear axles, V is the velocity, f means "front," r means "rear," r is the yaw rate, I z is the moment of inertia, δ f is the front-wheel steering angle, and m is the mass.

Problem Formulation.
It is important to make sure what the ideal reference signal is. Usually, the yaw rate and the sideslip angle are the two important parameters to measure vehicle stability. According to the 2-DOF model, the desired yaw rate and sideslip angle can be calculated as [27] with where μ is the tire-road friction coefficient, K is a positive constant, δ f d is the angle input from the steering wheel to the front wheel, β max � arctan(0.02μg), and g is the gravitational constant. e diagram of the integrated control of active front steering and direct yaw moment (AFS + DYC) is depicted in Figure 2, and the vehicle inputs include the vehicle speed V x and the steering wheel angle δ f d commanded by the driver. According to the 2-DOF vehicle model, the ideal sideslip angle β d and yaw rate r d can be calculated, respectively. e CarSim vehicle model is regarded as a real vehicle model, which can output the actual yaw rate r. As a matter of fact, it is difficult to measure the actual sideslip angle directly by some special sensors, and the sensors needed are expensive. On the contrary, the accuracy and stability need to be further improved, thus designing the state observer to estimate the exact value is a good solution.

Remark 1.
e DYC algorithm cannot be imposed on the 2-DOF model.
is is because the implementation of DYC algorithm is based on the yaw moment M z generated by the torque difference between left and right wheels, while the 2-DOF model regards the left and right wheels as one wheel such that there is no M z in the 2-DOF model.

Sideslip Angle Observer.
e lateral acceleration a y is expressed as Introduce the variables e vehicle model described by equations (1) and (2) can be rewritten as en, the observer is constructed as where c 1 and c 2 are positive constants. en, the following lemma is obtained, whose proof is similar to that in [23]. us, it is omitted here.

Lemma 1.
e output x 2 of (8) will track x 2 within a finite time.
Remark 2. It should be pointed out that the values of the front and rear tire cornering stiffness C f and C r are assumed to be constants in the vehicle dynamic model. In fact, their values are dependent on the road condition and normal force on tires, and thus, there are some perturbations for the two parameters. In addition, the model error also exists in the vehicle dynamics. ese factors affect the accuracy of the estimation. To fix this problem, the lateral acceleration error a y − a y is introduced in the observer (8) to compensate the model error and parameter perturbations.

Baseline Controller
e AFS is to control the steering angle of the front wheel in the linear range of the tire. erefore, the design of the controller is based on 2-DOF. e purpose of the controller is to eliminate the error between the actual vehicle and the ideal vehicle by controlling the front steering angle and to make sure the yaw rate follows the ideal model well.
e active front steering 2-DOF vehicle model is described as e difference between the actual yaw rate and the ideal value is Taking the derivative of (10) yields We choose the sliding surface as s 1 � e. Taking the time derivative of s 1 along system (9) gives where Theorem 1. If the AFS controller is constructed as where k 1 > c D 1 and k 2 > 0, r will finite-time converge to r d .
Proof. Putting (14) into (12) yields Let V(s 1 ) � (1/2)s 2 1 . e time derivative of V(s 1 ) along system (15) is . From [28][29][30], the sliding variable s 1 will finite-time converge to zero. Hence, the corrective steer angle Δδ f generated by the controller is determined as in which (19) can be rewritten as follows: Substituting (20) into (18) produces en, we select a sliding surface, which is composed of the deviation of the yaw rate and the deviation of the sideslip angle, i.e., with ξ > 0.
Combining the time derivative of (22) with (21) gives with

Theorem 2. If the DYC controller is constructed as
where K 1 > c D 2 and K 2 > 0, s 2 will finite-time converge to zero.
Proof. Putting (25) into (23) yields Choose the Lyapunov function as V(s 2 ) � (1/2)s 2 2 . Differentiating V(s 2 ) along system (26) gives , and s 2 finite-time converges to zero. □ 3.3. Composite Controller Design. In the following, by using DOB technique and baseline controllers, we will give the composite controllers to enhance the control performance.
Proof. Putting (41) into (23) yields From NDOB (37), it is clear that the error e 2 (t) is bounded. is means that a constant c e 2 > 0 can be found such that |e 2 (t)| � |D 2 (t) − D 2 | ≤ c e 2 . e remainder of the proof is similar to that in eorem 2.

Torque Distribution Controller.
On the basis of the ideal yaw moment obtained by the DYC controller, the main function of the whole control strategy is to assign the expected yaw moment calculated by the DYC controller to four wheels. e torque distributor designed in this paper is based on the vertical load distribution, taking into account the motor output and road surface conditions of the distribution constraints. First of all, the relationship between the longitudinal force of the tire and the torque of the motor is which leads to Finally, the torques obtained by formula (44) are constrained as follows: Mathematical Problems in Engineering where F zij is the driving or braking torque, T ij is the vertical load, T max is the maximum motor output torque, μ is the adhesion coefficient of the road, and R represents the effective radius of the tire.

Simulation Results
e validity of the AFS and DYC control is shown by using the cosimulation of MATLAB/Simulink and CarSim. e responses are compared with those without any control, with yaw-moment control only, and with the AFS and DYC integrated control. e parameters are given in Table 1.
In the simulation, the initial speed of the vehicle is 80 km/h, and μ is 0.3 in the double lane-changing maneuver, and ξ � 0.5.      5 that, without any control, the yaw rate and the sideslip angle are larger values deviating from their ideal value, even cannot guarantee the stability of the vehicle. On the contrary, under the other three controllers, the vehicle trajectory can track the expected trajectory, the actual yaw rate can follow the ideal value, and the actual sideslip angle can be controlled in a stable region. Compared with other controllers, the integrated control of AFS and DYC based on the DOB technique can improve the vehicle response, which can also be verified by Figure 3.

Response to Side Wind Disturbance.
e simulation is given under a side wind force, as shown in Figure 6. It can be seen from Figures 7 and 8 that the case without any control fails to keep vehicle's stability, while the case with the yawmoment control only cannot control well compared with the integrated control. e properties can be clearly seen in Figure 9, which reflects the double lane-change maneuvering under the proposed controllers. At last, it can also be observed from Figure 9 that the integrated control of AFS and DYC based on the DOB technique has better control performance compared with the pure AFS and DYC integrated control.

Conclusion
In our study, it has been shown that the AFS and DYC integrated control has better control performance than the DYC control. At the same time, we also confirm that the integrated control of AFS and DYC based on the DOB technique can suppress the large disturbance and have better robustness in comparison with the pure AFS and DYC integrated control [31].

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest.    Mathematical Problems in Engineering 7