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

Authors: Chao Lu; Jing Yuan; Genlong Zha
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This 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. Then, 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.


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## Body

## 1. 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 [1–3]. Then, the active safety control received considerable attention in the vehicle stability control, such as [4–7]. 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. This is because the goal of DYC is to adjust the vehicle yaw motion. Thus, 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. Then, 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 [12–18]. 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 [21–23].This 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. Then, a disturbance observer (DOB) [24] is constructed for the proposed control to reduce the control gain so as to reduce the chattering. The 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.The rest of the paper is organized as follows. Section2 introduces the dynamic model of the vehicle and problem statement. The process of control design is given in Section 3. The simulation results are shown in Section 4. Section 5 concludes this paper.


## 2. Dynamical Model and Problem Statement

### 2.1. 2-DOF Model

The 2-DOF model is called as the “bicycle model” [25, 26], which is shown in Figure 1.Figure 1
2-DOF vehicle model.Then, the model is described as(1)mVxβ˙+r=−2Cf+Crβ+−2aCf−bCrVxr+2Cfδf,(2)Izr˙=−2aCf−bCrβ+−2a2Cf+b2CrVxr+2aCfδf,where Cf and Cr, 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, Iz is the moment of inertia, δf is the front-wheel steering angle, and m is the mass.


### 2.2. 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](3)rd=rt,rt<0.85μgVx,0.85μgVxsignrt,rt≥0.85μgVx,(4)βd=βt,βt<βmax,βmaxsignβt,βt≥βmax,with(5)rt=Vxa+b1+KVx2δfd,βt=b−amVx2/2Cfa+ba+b1+KVx2δfd,where μ is the tire-road friction coefficient, K is a positive constant, δfd is the angle input from the steering wheel to the front wheel, βmax=arctan0.02μg, and g is the gravitational constant.The diagram of the integrated control of active front steering and direct yaw moment (AFS + DYC) is depicted in Figure2, and the vehicle inputs include the vehicle speed Vx and the steering wheel angle δfd commanded by the driver. According to the 2-DOF vehicle model, the ideal sideslip angle βd and yaw rate rd can be calculated, respectively. The 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.Figure 2
Structure of the AFS + DYC control system.Remark 1.
The DYC algorithm cannot be imposed on the 2-DOF model. This is because the implementation of DYC algorithm is based on the yaw momentMz 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 Mz in the 2-DOF model.


## 2.1. 2-DOF Model

The 2-DOF model is called as the “bicycle model” [25, 26], which is shown in Figure 1.Figure 1
2-DOF vehicle model.Then, the model is described as(1)mVxβ˙+r=−2Cf+Crβ+−2aCf−bCrVxr+2Cfδf,(2)Izr˙=−2aCf−bCrβ+−2a2Cf+b2CrVxr+2aCfδf,where Cf and Cr, 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, Iz is the moment of inertia, δf is the front-wheel steering angle, and m is the mass.


## 2.2. 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](3)rd=rt,rt<0.85μgVx,0.85μgVxsignrt,rt≥0.85μgVx,(4)βd=βt,βt<βmax,βmaxsignβt,βt≥βmax,with(5)rt=Vxa+b1+KVx2δfd,βt=b−amVx2/2Cfa+ba+b1+KVx2δfd,where μ is the tire-road friction coefficient, K is a positive constant, δfd is the angle input from the steering wheel to the front wheel, βmax=arctan0.02μg, and g is the gravitational constant.The diagram of the integrated control of active front steering and direct yaw moment (AFS + DYC) is depicted in Figure2, and the vehicle inputs include the vehicle speed Vx and the steering wheel angle δfd commanded by the driver. According to the 2-DOF vehicle model, the ideal sideslip angle βd and yaw rate rd can be calculated, respectively. The 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.Figure 2
Structure of the AFS + DYC control system.Remark 1.
The DYC algorithm cannot be imposed on the 2-DOF model. This is because the implementation of DYC algorithm is based on the yaw momentMz 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 Mz in the 2-DOF model.


## 3. Control Design

### 3.1. Sideslip Angle Observer

The lateral accelerationay is expressed as(6)ay=Vxβ˙+r=−2Cf+Crmβ+−2aCf−bCrmVxr+2Cfmδf.Introduce the variablesx1=r, x2=β, X=r,βT, Y=y1,y2T=r,ayT, and u=δf. The vehicle model described by equations (1) and (2) can be rewritten as(7)X˙=AX+Bu,Y=CX+Du,A=−2a2Cf+b2CrIzVx−2aCf−bCrIz−2aCf−bCrmVx2−1−2Cf+CrmVx,B=2aCfIz2CfmVx,C=10VxA21+1VxA22,D=0VxB2.Then, the observer is constructed as(8)x^˙1=A11y1+A12x^2+B1u+c1y1−x^11/2signy1−x^1,x^˙2=A21y1+A22x^2+B2u+c2signy1−x^1+1Vxay−a^y,a^y=VxA21+1y1+VxA22x^2+VxB2δf,where c1 and c2 are positive constants.Then, the following lemma is obtained, whose proof is similar to that in [23]. Thus, it is omitted here.Lemma 1.
The outputx^2 of (8) will track x2 within a finite time.Remark 2.
It should be pointed out that the values of the front and rear tire cornering stiffnessCf and Cr 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. These factors affect the accuracy of the estimation. To fix this problem, the lateral acceleration error ay−a^y is introduced in the observer (8) to compensate the model error and parameter perturbations.


### 3.2. Baseline Controller

#### 3.2.1. AFS Controller

The AFS is to control the steering angle of the front wheel in the linear range of the tire. Therefore, the design of the controller is based on 2-DOF. The 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.The active front steering 2-DOF vehicle model is described as(9)r˙β˙=A11A12A21A22rβ+B1B2δf.The difference between the actual yaw rate and the ideal value is(10)e=r−rd.Taking the derivative of (10) yields(11)e˙=r˙−rd..We choose the sliding surface ass1=e. Taking the time derivative of s1 along system (9) gives(12)s˙1=A11r+A12β^+B1δf+D1t,where D1=−r˙d+A12β−β^. One can find a constant γD1 such that(13)|D1t|≤γD1.Theorem 1.
If the AFS controller is constructed as(14)δf=1B1−A11r−A12β^−k1signs1−k2s1,where k1>γD1andk2>0, r will finite-time converge to rd.Proof.
Putting (14) into (12) yields(15)s1.=−k1signs1−k2s1+D1t.
LetVs1=1/2s12. The time derivative of Vs1 along system (15) is(16)V˙=s1s1.=−k1signs1s1−k2s12+D1ts1≤−k1s1−k2s12+D1ts1≤−k1−γD1s1−k2s12≤−k1−γD1s1.
Note thatk1>γD1. It can be proved that V˙≤−CV1/2, C=2k1−γD1. From [28–30], the sliding variable s1 will finite-time converge to zero. Hence, the corrective steer angle Δδf generated by the controller is determined as(17)Δδf=δf−δfd.


#### 3.2.2. DYC Controller

DYC control is to utilize the present antilock brake system to obtain the demanded longitudinal force and steady yaw moment. When the yaw moment is taken into account, the equation of state (9) becomes(18)r˙=A11r+A12β+B1δf+1IzM,(19)β˙=A21r+A22β+B2δf,in which (19) can be rewritten as follows:(20)δf=1B2β˙−A21r−A22β.Substituting (20) into (18) produces(21)r˙=A11−B1B2A21r+A12−B1B2A22β+B1B2β˙+1IzM.Then, we select a sliding surface, which is composed of the deviation of the yaw rate and the deviation of the sideslip angle, i.e.,(22)s2=r−rd+ξβ−βd,with ξ>0.Combining the time derivative of (22) with (21) gives(23)s˙2=A11−B1B2A21r+A12−B1B2A22β^+1IzM+D2t,with D2t=A12−B1/B2A22β−β^+B1/B2β˙+ξβ˙−βd.−r˙d. Note that β is often very small, and from the definition of βd and rd, βd. and rd. are bounded. Hence, a constant γD2 can be found such that(24)D2t≤γD2.Theorem 2.
If the DYC controller is constructed as(25)M=Iz−A11−B1B2A21γ−A12−B1B2A22β^−K1signs2−K2s2,where K1>γD2andK2>0, s2 will finite-time converge to zero.Proof.
Putting (25) into (23) yields(26)s2.=−K1signs2−K2s2+D2t.
Choose the Lyapunov function asVs2=1/2s22. Differentiating Vs2 along system (26) gives(27)V˙=s2,s2.=−K1signs2s2−K2s22+D2ts2≤−K1s2−K2s22+D2ts2≤−K1−γD2s2−K2s22≤−K1−γD2s2.
Note thatK1>γD2. It can be proved that V˙≤−CγV1/2, Cγ=2K1−γD2, and s2 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.


#### 3.3.1. AFS Controller Based on a DOB

We can rewrite system (12) as(28)s1.=Fs1+G1s1δf+G2s1D1t,where Fs1=A11r+A12β^, G1s1=B1, G2s1=1, and D1t is regarded as the unknown disturbance and satisfies D˙1t≤γD˙1 with γD1.>0 being a constant, which at least holds locally.Design the nonlinear DOB (NDOB) as(29)P˙=−L1G2P−L1G2L1s1+Fs1+G1δf,D1=P+L1s1^,where P and L1 are the state and constant.Lete1t=D1t−D^1t. Differentiating e1t along systems (28) and (29) gives(30)e˙1=D˙1−D^˙1=D˙1−−L1G2P−L12G2s1+L1G2D1=D˙1−L1G2e1.LetVe1=1/2e12. Then, one has(31)V˙e1=e1e1.=e1D˙1−L1G2e1≤e1γD˙1−L1G2e12.Define a region asQ1=e1:e1≤γD˙1/L1G2. For any e1t∈R/Q1, one has e1>γD˙1/L1G2. This, together with (31), yields V˙e1≤−|e1|L1G2|e1|−γD˙1<0. Note that G2=1, which means that e1t will reach and stay in the domain(32)Q1=e1:e1≤γD˙1L1.Theorem 3.
Provided that the composite AFS controller is constructed as(33)δf=1B1−A11r−A12β^−D^1−k1signs1−k2s1,where k1>0, k2>0, and L1>γD˙1, s1 will finite-time converge to zero.Proof.
Putting (33) into (12) yields(34)s1.=−K1signs1−K2s1+D1t−D^1=−K1⋅signs1−K2s1+e1t.
From NDOB (29), it is clear that the error e1t is bounded. This means that a constant γe1 can be found such that e1t=D1t−D^1≤γe1.
Hence, the corrective steer angleΔδf generated by the controller is determined as follows:(35)Δδf=δf−δfd.


#### 3.3.2. DYC Controller Based on a DOB

System (23) can be rewritten as(36)s˙2=fs2+g1s2M+g2s2D2t,where fs2=A11−B1/B2A21r+A12−B1/B2A22β^, g1s2=1/Iz, g2s2=1, and D2t is considered as the unknown perturbation, and D˙2t≤γD˙2. Then, a NDOB is constructed as(37)p˙=−L2g2p−L2g2L2s2+fs2+g1M,D2^=p+L2s2.L2 is a positive constant.Lete2t=D2t−D^2t. Taking the derivative of e2t along systems (36) and (37) gives(38)e˙2=D˙2−D^˙2=D˙2−−L2g2p−L22g2s2+L2g2D2=D˙2−L2g2e2.We choose a Lyapunov function asVe2=1/2e22, whose time derivative along (38) is(39)V˙e2=e2e2.=e2D˙2−L2g2e2≤e2γD˙2−L2g2e12.Define a region asQ2=e2:e2≤γD˙2/L2g2. For any e2t∈R/Q2, we have e2>γD˙2/L2g2. This, together with (39), yields V˙e2≤−e2L2g2e2−γD˙2<0. It is noted that g2=1. This means that e2t will stay in the domain(40)Q2=e2:e2≤γD˙2L2.Theorem 4.
Provided that the composite DYC controller is constructed as(41)M=Iz−A11−B1B2A21r−A12−B1B2A22β^−K1signs2−K2s2−D2^,where k1>0, k2>0, and L1>γD˙1, s2 will finite-time converge to zero.Proof.
Putting (41) into (23) yields(42)s2.=−k1signs2−k2s2+D2t−D^2=−k1⋅signs2−k2s2+e2t.
From NDOB (37), it is clear that the error e2t is bounded. This means that a constant γe2>0 can be found such that e2t=D2t−D^2≤γe2. The remainder of the proof is similar to that in Theorem 2.


### 3.4. 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. The 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(43)Fxij=TijR,which leads to(44)Tfl=FzflFzMz−df/2cosδfd+asinδfdR,Tfr=FzfrFzMzdf/2cosδfd+asinδfdR,Trl=−FzrlFzMz2drR,Tfl=FzrrFzMz2drR.Finally, the torques obtained by formula (44) are constrained as follows:(45)Tij≤minμRFzij,Tmax,where Fzij is the driving or braking torque, Tij is the vertical load, Tmax is the maximum motor output torque, μ is the adhesion coefficient of the road, and R represents the effective radius of the tire.


## 3.1. Sideslip Angle Observer

The lateral accelerationay is expressed as(6)ay=Vxβ˙+r=−2Cf+Crmβ+−2aCf−bCrmVxr+2Cfmδf.Introduce the variablesx1=r, x2=β, X=r,βT, Y=y1,y2T=r,ayT, and u=δf. The vehicle model described by equations (1) and (2) can be rewritten as(7)X˙=AX+Bu,Y=CX+Du,A=−2a2Cf+b2CrIzVx−2aCf−bCrIz−2aCf−bCrmVx2−1−2Cf+CrmVx,B=2aCfIz2CfmVx,C=10VxA21+1VxA22,D=0VxB2.Then, the observer is constructed as(8)x^˙1=A11y1+A12x^2+B1u+c1y1−x^11/2signy1−x^1,x^˙2=A21y1+A22x^2+B2u+c2signy1−x^1+1Vxay−a^y,a^y=VxA21+1y1+VxA22x^2+VxB2δf,where c1 and c2 are positive constants.Then, the following lemma is obtained, whose proof is similar to that in [23]. Thus, it is omitted here.Lemma 1.
The outputx^2 of (8) will track x2 within a finite time.Remark 2.
It should be pointed out that the values of the front and rear tire cornering stiffnessCf and Cr 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. These factors affect the accuracy of the estimation. To fix this problem, the lateral acceleration error ay−a^y is introduced in the observer (8) to compensate the model error and parameter perturbations.


## 3.2. Baseline Controller

### 3.2.1. AFS Controller

The AFS is to control the steering angle of the front wheel in the linear range of the tire. Therefore, the design of the controller is based on 2-DOF. The 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.The active front steering 2-DOF vehicle model is described as(9)r˙β˙=A11A12A21A22rβ+B1B2δf.The difference between the actual yaw rate and the ideal value is(10)e=r−rd.Taking the derivative of (10) yields(11)e˙=r˙−rd..We choose the sliding surface ass1=e. Taking the time derivative of s1 along system (9) gives(12)s˙1=A11r+A12β^+B1δf+D1t,where D1=−r˙d+A12β−β^. One can find a constant γD1 such that(13)|D1t|≤γD1.Theorem 1.
If the AFS controller is constructed as(14)δf=1B1−A11r−A12β^−k1signs1−k2s1,where k1>γD1andk2>0, r will finite-time converge to rd.Proof.
Putting (14) into (12) yields(15)s1.=−k1signs1−k2s1+D1t.
LetVs1=1/2s12. The time derivative of Vs1 along system (15) is(16)V˙=s1s1.=−k1signs1s1−k2s12+D1ts1≤−k1s1−k2s12+D1ts1≤−k1−γD1s1−k2s12≤−k1−γD1s1.
Note thatk1>γD1. It can be proved that V˙≤−CV1/2, C=2k1−γD1. From [28–30], the sliding variable s1 will finite-time converge to zero. Hence, the corrective steer angle Δδf generated by the controller is determined as(17)Δδf=δf−δfd.


### 3.2.2. DYC Controller

DYC control is to utilize the present antilock brake system to obtain the demanded longitudinal force and steady yaw moment. When the yaw moment is taken into account, the equation of state (9) becomes(18)r˙=A11r+A12β+B1δf+1IzM,(19)β˙=A21r+A22β+B2δf,in which (19) can be rewritten as follows:(20)δf=1B2β˙−A21r−A22β.Substituting (20) into (18) produces(21)r˙=A11−B1B2A21r+A12−B1B2A22β+B1B2β˙+1IzM.Then, we select a sliding surface, which is composed of the deviation of the yaw rate and the deviation of the sideslip angle, i.e.,(22)s2=r−rd+ξβ−βd,with ξ>0.Combining the time derivative of (22) with (21) gives(23)s˙2=A11−B1B2A21r+A12−B1B2A22β^+1IzM+D2t,with D2t=A12−B1/B2A22β−β^+B1/B2β˙+ξβ˙−βd.−r˙d. Note that β is often very small, and from the definition of βd and rd, βd. and rd. are bounded. Hence, a constant γD2 can be found such that(24)D2t≤γD2.Theorem 2.
If the DYC controller is constructed as(25)M=Iz−A11−B1B2A21γ−A12−B1B2A22β^−K1signs2−K2s2,where K1>γD2andK2>0, s2 will finite-time converge to zero.Proof.
Putting (25) into (23) yields(26)s2.=−K1signs2−K2s2+D2t.
Choose the Lyapunov function asVs2=1/2s22. Differentiating Vs2 along system (26) gives(27)V˙=s2,s2.=−K1signs2s2−K2s22+D2ts2≤−K1s2−K2s22+D2ts2≤−K1−γD2s2−K2s22≤−K1−γD2s2.
Note thatK1>γD2. It can be proved that V˙≤−CγV1/2, Cγ=2K1−γD2, and s2 finite-time converges to zero.


## 3.2.1. AFS Controller

The AFS is to control the steering angle of the front wheel in the linear range of the tire. Therefore, the design of the controller is based on 2-DOF. The 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.The active front steering 2-DOF vehicle model is described as(9)r˙β˙=A11A12A21A22rβ+B1B2δf.The difference between the actual yaw rate and the ideal value is(10)e=r−rd.Taking the derivative of (10) yields(11)e˙=r˙−rd..We choose the sliding surface ass1=e. Taking the time derivative of s1 along system (9) gives(12)s˙1=A11r+A12β^+B1δf+D1t,where D1=−r˙d+A12β−β^. One can find a constant γD1 such that(13)|D1t|≤γD1.Theorem 1.
If the AFS controller is constructed as(14)δf=1B1−A11r−A12β^−k1signs1−k2s1,where k1>γD1andk2>0, r will finite-time converge to rd.Proof.
Putting (14) into (12) yields(15)s1.=−k1signs1−k2s1+D1t.
LetVs1=1/2s12. The time derivative of Vs1 along system (15) is(16)V˙=s1s1.=−k1signs1s1−k2s12+D1ts1≤−k1s1−k2s12+D1ts1≤−k1−γD1s1−k2s12≤−k1−γD1s1.
Note thatk1>γD1. It can be proved that V˙≤−CV1/2, C=2k1−γD1. From [28–30], the sliding variable s1 will finite-time converge to zero. Hence, the corrective steer angle Δδf generated by the controller is determined as(17)Δδf=δf−δfd.


## 3.2.2. DYC Controller

DYC control is to utilize the present antilock brake system to obtain the demanded longitudinal force and steady yaw moment. When the yaw moment is taken into account, the equation of state (9) becomes(18)r˙=A11r+A12β+B1δf+1IzM,(19)β˙=A21r+A22β+B2δf,in which (19) can be rewritten as follows:(20)δf=1B2β˙−A21r−A22β.Substituting (20) into (18) produces(21)r˙=A11−B1B2A21r+A12−B1B2A22β+B1B2β˙+1IzM.Then, we select a sliding surface, which is composed of the deviation of the yaw rate and the deviation of the sideslip angle, i.e.,(22)s2=r−rd+ξβ−βd,with ξ>0.Combining the time derivative of (22) with (21) gives(23)s˙2=A11−B1B2A21r+A12−B1B2A22β^+1IzM+D2t,with D2t=A12−B1/B2A22β−β^+B1/B2β˙+ξβ˙−βd.−r˙d. Note that β is often very small, and from the definition of βd and rd, βd. and rd. are bounded. Hence, a constant γD2 can be found such that(24)D2t≤γD2.Theorem 2.
If the DYC controller is constructed as(25)M=Iz−A11−B1B2A21γ−A12−B1B2A22β^−K1signs2−K2s2,where K1>γD2andK2>0, s2 will finite-time converge to zero.Proof.
Putting (25) into (23) yields(26)s2.=−K1signs2−K2s2+D2t.
Choose the Lyapunov function asVs2=1/2s22. Differentiating Vs2 along system (26) gives(27)V˙=s2,s2.=−K1signs2s2−K2s22+D2ts2≤−K1s2−K2s22+D2ts2≤−K1−γD2s2−K2s22≤−K1−γD2s2.
Note thatK1>γD2. It can be proved that V˙≤−CγV1/2, Cγ=2K1−γD2, and s2 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.


### 3.3.1. AFS Controller Based on a DOB

We can rewrite system (12) as(28)s1.=Fs1+G1s1δf+G2s1D1t,where Fs1=A11r+A12β^, G1s1=B1, G2s1=1, and D1t is regarded as the unknown disturbance and satisfies D˙1t≤γD˙1 with γD1.>0 being a constant, which at least holds locally.Design the nonlinear DOB (NDOB) as(29)P˙=−L1G2P−L1G2L1s1+Fs1+G1δf,D1=P+L1s1^,where P and L1 are the state and constant.Lete1t=D1t−D^1t. Differentiating e1t along systems (28) and (29) gives(30)e˙1=D˙1−D^˙1=D˙1−−L1G2P−L12G2s1+L1G2D1=D˙1−L1G2e1.LetVe1=1/2e12. Then, one has(31)V˙e1=e1e1.=e1D˙1−L1G2e1≤e1γD˙1−L1G2e12.Define a region asQ1=e1:e1≤γD˙1/L1G2. For any e1t∈R/Q1, one has e1>γD˙1/L1G2. This, together with (31), yields V˙e1≤−|e1|L1G2|e1|−γD˙1<0. Note that G2=1, which means that e1t will reach and stay in the domain(32)Q1=e1:e1≤γD˙1L1.Theorem 3.
Provided that the composite AFS controller is constructed as(33)δf=1B1−A11r−A12β^−D^1−k1signs1−k2s1,where k1>0, k2>0, and L1>γD˙1, s1 will finite-time converge to zero.Proof.
Putting (33) into (12) yields(34)s1.=−K1signs1−K2s1+D1t−D^1=−K1⋅signs1−K2s1+e1t.
From NDOB (29), it is clear that the error e1t is bounded. This means that a constant γe1 can be found such that e1t=D1t−D^1≤γe1.
Hence, the corrective steer angleΔδf generated by the controller is determined as follows:(35)Δδf=δf−δfd.


### 3.3.2. DYC Controller Based on a DOB

System (23) can be rewritten as(36)s˙2=fs2+g1s2M+g2s2D2t,where fs2=A11−B1/B2A21r+A12−B1/B2A22β^, g1s2=1/Iz, g2s2=1, and D2t is considered as the unknown perturbation, and D˙2t≤γD˙2. Then, a NDOB is constructed as(37)p˙=−L2g2p−L2g2L2s2+fs2+g1M,D2^=p+L2s2.L2 is a positive constant.Lete2t=D2t−D^2t. Taking the derivative of e2t along systems (36) and (37) gives(38)e˙2=D˙2−D^˙2=D˙2−−L2g2p−L22g2s2+L2g2D2=D˙2−L2g2e2.We choose a Lyapunov function asVe2=1/2e22, whose time derivative along (38) is(39)V˙e2=e2e2.=e2D˙2−L2g2e2≤e2γD˙2−L2g2e12.Define a region asQ2=e2:e2≤γD˙2/L2g2. For any e2t∈R/Q2, we have e2>γD˙2/L2g2. This, together with (39), yields V˙e2≤−e2L2g2e2−γD˙2<0. It is noted that g2=1. This means that e2t will stay in the domain(40)Q2=e2:e2≤γD˙2L2.Theorem 4.
Provided that the composite DYC controller is constructed as(41)M=Iz−A11−B1B2A21r−A12−B1B2A22β^−K1signs2−K2s2−D2^,where k1>0, k2>0, and L1>γD˙1, s2 will finite-time converge to zero.Proof.
Putting (41) into (23) yields(42)s2.=−k1signs2−k2s2+D2t−D^2=−k1⋅signs2−k2s2+e2t.
From NDOB (37), it is clear that the error e2t is bounded. This means that a constant γe2>0 can be found such that e2t=D2t−D^2≤γe2. The remainder of the proof is similar to that in Theorem 2.


## 3.3.1. AFS Controller Based on a DOB

We can rewrite system (12) as(28)s1.=Fs1+G1s1δf+G2s1D1t,where Fs1=A11r+A12β^, G1s1=B1, G2s1=1, and D1t is regarded as the unknown disturbance and satisfies D˙1t≤γD˙1 with γD1.>0 being a constant, which at least holds locally.Design the nonlinear DOB (NDOB) as(29)P˙=−L1G2P−L1G2L1s1+Fs1+G1δf,D1=P+L1s1^,where P and L1 are the state and constant.Lete1t=D1t−D^1t. Differentiating e1t along systems (28) and (29) gives(30)e˙1=D˙1−D^˙1=D˙1−−L1G2P−L12G2s1+L1G2D1=D˙1−L1G2e1.LetVe1=1/2e12. Then, one has(31)V˙e1=e1e1.=e1D˙1−L1G2e1≤e1γD˙1−L1G2e12.Define a region asQ1=e1:e1≤γD˙1/L1G2. For any e1t∈R/Q1, one has e1>γD˙1/L1G2. This, together with (31), yields V˙e1≤−|e1|L1G2|e1|−γD˙1<0. Note that G2=1, which means that e1t will reach and stay in the domain(32)Q1=e1:e1≤γD˙1L1.Theorem 3.
Provided that the composite AFS controller is constructed as(33)δf=1B1−A11r−A12β^−D^1−k1signs1−k2s1,where k1>0, k2>0, and L1>γD˙1, s1 will finite-time converge to zero.Proof.
Putting (33) into (12) yields(34)s1.=−K1signs1−K2s1+D1t−D^1=−K1⋅signs1−K2s1+e1t.
From NDOB (29), it is clear that the error e1t is bounded. This means that a constant γe1 can be found such that e1t=D1t−D^1≤γe1.
Hence, the corrective steer angleΔδf generated by the controller is determined as follows:(35)Δδf=δf−δfd.


## 3.3.2. DYC Controller Based on a DOB

System (23) can be rewritten as(36)s˙2=fs2+g1s2M+g2s2D2t,where fs2=A11−B1/B2A21r+A12−B1/B2A22β^, g1s2=1/Iz, g2s2=1, and D2t is considered as the unknown perturbation, and D˙2t≤γD˙2. Then, a NDOB is constructed as(37)p˙=−L2g2p−L2g2L2s2+fs2+g1M,D2^=p+L2s2.L2 is a positive constant.Lete2t=D2t−D^2t. Taking the derivative of e2t along systems (36) and (37) gives(38)e˙2=D˙2−D^˙2=D˙2−−L2g2p−L22g2s2+L2g2D2=D˙2−L2g2e2.We choose a Lyapunov function asVe2=1/2e22, whose time derivative along (38) is(39)V˙e2=e2e2.=e2D˙2−L2g2e2≤e2γD˙2−L2g2e12.Define a region asQ2=e2:e2≤γD˙2/L2g2. For any e2t∈R/Q2, we have e2>γD˙2/L2g2. This, together with (39), yields V˙e2≤−e2L2g2e2−γD˙2<0. It is noted that g2=1. This means that e2t will stay in the domain(40)Q2=e2:e2≤γD˙2L2.Theorem 4.
Provided that the composite DYC controller is constructed as(41)M=Iz−A11−B1B2A21r−A12−B1B2A22β^−K1signs2−K2s2−D2^,where k1>0, k2>0, and L1>γD˙1, s2 will finite-time converge to zero.Proof.
Putting (41) into (23) yields(42)s2.=−k1signs2−k2s2+D2t−D^2=−k1⋅signs2−k2s2+e2t.
From NDOB (37), it is clear that the error e2t is bounded. This means that a constant γe2>0 can be found such that e2t=D2t−D^2≤γe2. The remainder of the proof is similar to that in Theorem 2.


## 3.4. 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. The 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(43)Fxij=TijR,which leads to(44)Tfl=FzflFzMz−df/2cosδfd+asinδfdR,Tfr=FzfrFzMzdf/2cosδfd+asinδfdR,Trl=−FzrlFzMz2drR,Tfl=FzrrFzMz2drR.Finally, the torques obtained by formula (44) are constrained as follows:(45)Tij≤minμRFzij,Tmax,where Fzij is the driving or braking torque, Tij is the vertical load, Tmax is the maximum motor output torque, μ is the adhesion coefficient of the road, and R represents the effective radius of the tire.


## 4. Simulation Results

The validity of the AFS and DYC control is shown by using the cosimulation of MATLAB/Simulink and CarSim. The responses are compared with those without any control, with yaw-moment control only, and with the AFS and DYC integrated control. The parameters are given in Table1.Table 1
Parameters of the vehicle model.
SymbolValuem1429 (Kg)b1.569 (m)dr1.565 (m)Iz1765 (Kg⋅m2)Cr87002 (N/rad)a1.05 (m)df1.565 (m)R0.35 (m)Cf79240 (N/rad)n20In the simulation, the initial speed of the vehicle is80km/h, and μ is 0.3 in the double lane-changing maneuver, and ξ=0.5.


### 4.1. Double Lane-Change Maneuver

The simulation results are depicted in Figures3–5. It can be seen from Figures 4 and 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.Figure 3
The vehicle trajectory.Figure 4
Time history of the yaw rate.Figure 5
Time history of the sideslip angle.


### 4.2. Response to Side Wind Disturbance

The simulation is given under a side wind force, as shown in Figure6. 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 yaw-moment control only cannot control well compared with the integrated control. The 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.Figure 6
The side wind disturbance input.Figure 7
The response curve of the yaw rate.Figure 8
Time history of the sideslip angle.Figure 9
The vehicle trajectory.


## 4.1. Double Lane-Change Maneuver

The simulation results are depicted in Figures3–5. It can be seen from Figures 4 and 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.Figure 3
The vehicle trajectory.Figure 4
Time history of the yaw rate.Figure 5
Time history of the sideslip angle.


## 4.2. Response to Side Wind Disturbance

The simulation is given under a side wind force, as shown in Figure6. 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 yaw-moment control only cannot control well compared with the integrated control. The 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.Figure 6
The side wind disturbance input.Figure 7
The response curve of the yaw rate.Figure 8
Time history of the sideslip angle.Figure 9
The vehicle trajectory.


## 5. 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].



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*Source: 8826630-2020-12-10.xml*