# Research on Model Predictive Control Method for Vehicle Lateral Stability Based on Hardware-in-the-Loop Test

Authors: Jian Wang; Shifu Liu; Jian Wu; Jun Yang; Aijuan Li
Journal: Mathematical Problems in Engineering
 (2020)

---

## Abstract

With the rapid development of the vehicle chassis control and autonomous driving technology, it is more and more urgent to realize the active steering technology of autonomous driving stability control. Under emergency conditions, the adhesion constraints, the model uncertainty, and the strong nonlinearity of vehicle bring great challenges to active steering control. In this paper, a model predictive control method for an active steering system based on a nonlinear vehicle model is proposed to solve the problems of adhesion constraint, model uncertainty, and external disturbance in the active steering system. Based on the real-time measurement of vehicle state, a new optimization method is proposed in this paper, which has good performance in dealing with the uncertainty and nonlinearity of the model. The control method transforms the constraint problem into quadratic programming and nonlinear programming. In order to ensure the control accuracy when the vehicle enters the nonlinear area, the control model is built with the combination of the nonlinear tire model and the 2DOF model. The control model is built based on Simulink, and the effectiveness of the controller is the verified joint simulation of Simulink and CarSim. The hardware-in-the-loop (HIL) test bench based on LabVIEW RT is built and tested in order to verify the feasibility and real effect of the controller. Simulation and HIL test results demonstrate that, compared with PID controller, the model predictive controller can accomplish the driving task well and improve the vehicle handling stability.


---

## Body

## 1. Introduction

The vehicle stability can be easily affected by the large lateral winds or other unpredictable conditions when running on a slippery road surface. In this situation, the ordinary driver may operate the vehicle improperly with panic so that vehicle goes into a state of instability. With the rapid development of the vehicle chassis control and autonomous driving technology, it is more and more urgent to realize the active steering technology of autonomous driving stability control [1]. Active front steering (AFS) is one of the main technologies in vehicle active safety and intelligent control systems [2], and it helps the driver to enhance the vehicle stability and maintain the vehicle mobility by generating additional front-wheel angle [3, 4]. AFS indirectly controls the vehicle’s yaw rate through adjusting the steering wheel angle which can greatly improve the stability of the vehicle. In addition, the correction of the front-wheel angle is a continuous process that will not reduce the ride comfort and can ensure the lateral stability [5].For AFS control strategies, a large number of relevant studies have been done by many scholars at home and abroad. Classical control theories, modern control theories, and advanced control theories have been applied to the studies on AFS systems one after another. The classical control theory is applied to AFS system for the first time. Considering the constraints of ground adhesive force, Li and Yu [6] designed the chassis steering and braking integrated controller using fuzzy PID. Chu et al. [3] used PID to implement AFS control and then used genetic algorithm to design the integrated controller for active steering and ESP. Based on the fuzzy PID, Jin et al. [7] devised the AFS controller to enhance the vehicle’s stability under varieties of disturbances. These AFS control systems based on classical control theories can greatly improve the vehicle’s operating stability and then the modern control theories are applied to further develop the AFS control. Peng and Chen [8, 9] achieved vehicle stability control through four-wheel steering and direct yaw moment control to ensure safe driving. Li et al. [10] designed active disturbance rejection controller to obtain better control effect, improving the route maintenance capacity. Huo et al. [11] proposed a linear steering system based on genetic algorithm to optimize BP neural network.The above researches can significantly enhance the vehicle’s handling stability. In order to further improve the anti-interference ability of the controller, robust control theories are applied to the active steering research by more scholars. Wu et al. [12] designed a robust controller to coordinate the contradiction between the performance and stability of the controller. It improved the controller’s performance and robustness by compensating vehicle parameter uncertainty. Tan et al. [13] researched the vehicle stability control based on H∞ control and designed the AFS controller for the driver's lane maintenance. Wang et al. [14] devised the active front steering robust controller applied to the vehicle, and the controller availably resisted external interference, ensuring the system’s robustness. Zhao et al. [1] studied the AFS system based on hybrid H2/H∞ control, improving and optimizing the anti-interference capacity of the system. Considering the uncertainty of the model and the influence of external interference, the AFS robust controller is designed to effectively improve the robustness of the controller [15, 16].The above research can improve the stability of the vehicle and ensure the robustness of the system. In order to further solve the control of the restraint system and improve the robustness, the model predictive control of active steering system has been studied by many scholars. Model predictive control (MPC) is an effective control strategy to solve the optimal control of the constraints [17–20]. The MPC is an optimal control method developed from the Bellman optimal control theories [21]. As it performs well in handling the problems of nonlinearity, uncertainty, and constraints of the vehicle dynamic model, many studies have used MPC control method to solve the AFS problem. Jalali et al. [22] studied the hierarchical control of AFS and differential braking and designed the MPC controller. For the vehicle stability control problems with ground adhesive constraints and motor drive force constraints, Ren et al. [23] designed the MPC controller for hierarchical control of AFS and driving torque. Zhao et al. [24] studied the integration of electric power steering and AFS and developed an AFS system with coupled force and displacement control, achieving the harmonization of the safety and flexibility of vehicles.Because there is a gap between the steering model used in the algorithm design and the actual steering system, the true effect of the algorithm cannot be verified. Therefore, considering the constraints of ground surface adhesion and the influence of model uncertainties, an MPC controller with the desired yaw rate as the tracking target is designed in this paper. Finally, the true control effect is verified on the steering hardware-in-the-loop test bench.Model predictive control is widely used in the industrial production field. This control method adopts receding horizon optimization to predict the system state of a period of time in the future and obtain the control input that minimizes the objective function. As an effective control method, MPC control solves the constraints problem by converting the constraints into the quadratic programming and nonlinear programming. The contributions of this article are as follows:(1)
In this paper, the ground adhesion is regarded as the hard constraints to the control system, and the stability control problem can be described as a limited tracking problem. And the time-varying linear model predictive AFS controller is designed to obtain the optimization solution method for the constrained active steering system.(2)
In this paper, the hardware in the ring testbed is built, and the MPC controller is applied to the testbed, and the real control effect is obtained. The experimental results show that the designed active steering controller has a good control effect.


## 2. System Modeling and Problem Statement

In order to design a controller for AFS, a nonlinear model based on vehicle dynamics is developed in this section, which consists of vehicle model and tire model. Symbols of the vehicle model are shown in Table1.Table 1
Model parameters.
SymbolsDescriptionβSideslip angle in vehicle bodyαSideslip angle of tireωYaw rate in vehicle bodyδfFront-wheel steering anglemVehicle massIzYaw moment of vehicle inertialf,lrDistance from the center of gravity to front or rear axleCf,CrCornering stiffness at front or rear each tire


### 2.1. System Modeling for Control Design

#### 2.1.1. Vehicle Model

In order to facilitate the study of the characteristics of vehicle handling stability, this paper simplifies the vehicle model into a two-degree-of-freedom vehicle model, which only considers the motion state of the vehicle along the horizontal direction of they-axis and around the z-axis. In this paper, the influence of the longitudinal force of the tire is ignored, and only the influence of the lateral force on the stability of the vehicle is considered, so the longitudinal speed is 0. On the basis of vehicle dynamics theories in [25], a simplified model of vehicle with two degrees of freedom is presented in Figure 1.Figure 1
Two-degree-of-freedom vehicle model.The model’s motion differential equation is(1)mv˙y+vxωr=Fy,f+Fy,r,Izω˙r=lfFy,f−lrFy,r,where Fy,f and Fr,f are the lateral force of the front and rear tires, respectively. Considering that the front-wheel angle δf of the vehicle is relatively small, there is an approximate relation cosδf=1.


#### 2.1.2. Tire Model

In this paper, the influence of tire’s longitudinal forces was ignored, and the side forces of tires were calculated by Pacejka tire model in the pure sideslip condition [26]. The calculation formula of tire lateral force is expressed as follows:(2)Fy=DsinCarctanBα−EBα−arctanBα,C=a0,D=a1Fz2+a2Fz,B=a3sin2arctanFZ/a4C∗D,E=a5Fz+a6,where the variable α is the tire sideslip angle; a0=1.5; a1=0; a2=1050; a3=1200; a4=7; a5=0; a6=0.2.The tire sideslip angles and vertical load of the front and rear wheels are written as(3)αf=β+ωrlfvx−δf,αr=β−ωrlrvx,Fz,f=mglrlf+lr,Fz,r=mglflf+lr.However, the nonlinear vehicle model is too complex to be written as a state-space form with which the controller can be designed. We simplify the model and deal with tire nonlinearity by the method of local linearization in this paper.At each sampling moment, the lateral forces of front and rear tires were locally linearized, and the linearized tire force equation was obtained as follows:(4)Fy,i=αiCi∗,where i=f,r represents the front and rear wheels; Ci∗ is the cornering stiffness of the tire at the front-wheel slip angle αi∗. The data Ci∗ is updated once at each sampling time, and the calculation formula is written as(5)Ci∗=∂Fy,i∂αiαi=αi∗,where Fy,i∗ is the lateral force of the current tire side angle αi∗.By substituting (3)–(5) into (1), the system of the vehicle dynamics can be expressed as follows:(6)X˙=AX+BuU+Bdd,Yc=CcX.The model is of second order with state vector asX=vyωrT, with the steering angle U=δf as control input, and with the yaw rate as output (y=ω). d=dt is white Gaussian noise input matrix.G0 is road irregularity coefficient. The longitudinal velocity of the vehicle is constant because the longitudinal force of the tire is ignored. One has(7)A=Cf∗+Cr∗mvxlfCf∗−lrCr∗mvx−vxlfCf∗−lrCr∗Izvxlf2Cf∗+lr2CrIzvx;Bu=−Cf∗m−lfCf∗Iz;Bd=02πG0vx;Cc=1001.


### 2.2. Control Problem Statement

#### 2.2.1. Nominal Yaw Rate

The nominal state value of vehicle is also called the expected state value of the vehicle, which is determined by the vehicle predictive model, the driver's input, and the changing driving environment. In order to keep the vehicle stable, the active front steering controller can let the actual vehicle track the ideal vehicle state and make the actual state of the vehicle reach the expected value. In the active front steering controller, the target value of the yaw rate is set to the yaw rate response value of the 2DOF vehicle model [27]. It is shown that(8)ωr=vx/L1+Kvx2.The stability factor isK=m/L2lf/Cr∗−lr/Cf∗. After considering the influence of the ground adhesion, formula (8) is modified as(9)ωref1=minvx/L1+Kvx2δf,μgvx⋅sgnδf.When the upper limit is too high, the vehicle may enter the nonlinear zone before reaching the boundary. Further correction for the nominal yaw rate can be written as(10)ωref2=minvx/L1+Kvx2δf,0.85μgvx⋅sgnδf.


#### 2.2.2. Nominal Vehicle Sideslip Angle

In the research of vehicle active front steering control, there are two kinds to determine the nominal sideslip angle of vehicle. The first point considers that setting the target vehicle sideslip angle to 0 conforms to the driver’s expectation; the second point is that the expected sideslip angle is calculated by the linear 2DOF model; that is to say, the response characteristics of the 2DOF model accord with the driver's expectation. If the target vehicle sideslip angle is set to zero, it will cause the controller unnecessary intervention control. The second method of determining the vehicle sideslip angle accords with the actual driving features; therefore, this article uses the second method.By using the linear 2DOF model, the vehicle sideslip angle is calculated as follows:(11)βref0=lf+mlrvx2/Cr∗LL1+Kvx2δf.Similar to the yaw rate, the effect of surface adhesion should be considered to correct the nominal vehicle sideslip angle as follows:(12)βref=minlr+mlfvx2/Cr∗LL1+Kvx2δf,arctan0.02μg⋅sgnδf.


## 2.1. System Modeling for Control Design

### 2.1.1. Vehicle Model

In order to facilitate the study of the characteristics of vehicle handling stability, this paper simplifies the vehicle model into a two-degree-of-freedom vehicle model, which only considers the motion state of the vehicle along the horizontal direction of they-axis and around the z-axis. In this paper, the influence of the longitudinal force of the tire is ignored, and only the influence of the lateral force on the stability of the vehicle is considered, so the longitudinal speed is 0. On the basis of vehicle dynamics theories in [25], a simplified model of vehicle with two degrees of freedom is presented in Figure 1.Figure 1
Two-degree-of-freedom vehicle model.The model’s motion differential equation is(1)mv˙y+vxωr=Fy,f+Fy,r,Izω˙r=lfFy,f−lrFy,r,where Fy,f and Fr,f are the lateral force of the front and rear tires, respectively. Considering that the front-wheel angle δf of the vehicle is relatively small, there is an approximate relation cosδf=1.


### 2.1.2. Tire Model

In this paper, the influence of tire’s longitudinal forces was ignored, and the side forces of tires were calculated by Pacejka tire model in the pure sideslip condition [26]. The calculation formula of tire lateral force is expressed as follows:(2)Fy=DsinCarctanBα−EBα−arctanBα,C=a0,D=a1Fz2+a2Fz,B=a3sin2arctanFZ/a4C∗D,E=a5Fz+a6,where the variable α is the tire sideslip angle; a0=1.5; a1=0; a2=1050; a3=1200; a4=7; a5=0; a6=0.2.The tire sideslip angles and vertical load of the front and rear wheels are written as(3)αf=β+ωrlfvx−δf,αr=β−ωrlrvx,Fz,f=mglrlf+lr,Fz,r=mglflf+lr.However, the nonlinear vehicle model is too complex to be written as a state-space form with which the controller can be designed. We simplify the model and deal with tire nonlinearity by the method of local linearization in this paper.At each sampling moment, the lateral forces of front and rear tires were locally linearized, and the linearized tire force equation was obtained as follows:(4)Fy,i=αiCi∗,where i=f,r represents the front and rear wheels; Ci∗ is the cornering stiffness of the tire at the front-wheel slip angle αi∗. The data Ci∗ is updated once at each sampling time, and the calculation formula is written as(5)Ci∗=∂Fy,i∂αiαi=αi∗,where Fy,i∗ is the lateral force of the current tire side angle αi∗.By substituting (3)–(5) into (1), the system of the vehicle dynamics can be expressed as follows:(6)X˙=AX+BuU+Bdd,Yc=CcX.The model is of second order with state vector asX=vyωrT, with the steering angle U=δf as control input, and with the yaw rate as output (y=ω). d=dt is white Gaussian noise input matrix.G0 is road irregularity coefficient. The longitudinal velocity of the vehicle is constant because the longitudinal force of the tire is ignored. One has(7)A=Cf∗+Cr∗mvxlfCf∗−lrCr∗mvx−vxlfCf∗−lrCr∗Izvxlf2Cf∗+lr2CrIzvx;Bu=−Cf∗m−lfCf∗Iz;Bd=02πG0vx;Cc=1001.


## 2.1.1. Vehicle Model

In order to facilitate the study of the characteristics of vehicle handling stability, this paper simplifies the vehicle model into a two-degree-of-freedom vehicle model, which only considers the motion state of the vehicle along the horizontal direction of they-axis and around the z-axis. In this paper, the influence of the longitudinal force of the tire is ignored, and only the influence of the lateral force on the stability of the vehicle is considered, so the longitudinal speed is 0. On the basis of vehicle dynamics theories in [25], a simplified model of vehicle with two degrees of freedom is presented in Figure 1.Figure 1
Two-degree-of-freedom vehicle model.The model’s motion differential equation is(1)mv˙y+vxωr=Fy,f+Fy,r,Izω˙r=lfFy,f−lrFy,r,where Fy,f and Fr,f are the lateral force of the front and rear tires, respectively. Considering that the front-wheel angle δf of the vehicle is relatively small, there is an approximate relation cosδf=1.


## 2.1.2. Tire Model

In this paper, the influence of tire’s longitudinal forces was ignored, and the side forces of tires were calculated by Pacejka tire model in the pure sideslip condition [26]. The calculation formula of tire lateral force is expressed as follows:(2)Fy=DsinCarctanBα−EBα−arctanBα,C=a0,D=a1Fz2+a2Fz,B=a3sin2arctanFZ/a4C∗D,E=a5Fz+a6,where the variable α is the tire sideslip angle; a0=1.5; a1=0; a2=1050; a3=1200; a4=7; a5=0; a6=0.2.The tire sideslip angles and vertical load of the front and rear wheels are written as(3)αf=β+ωrlfvx−δf,αr=β−ωrlrvx,Fz,f=mglrlf+lr,Fz,r=mglflf+lr.However, the nonlinear vehicle model is too complex to be written as a state-space form with which the controller can be designed. We simplify the model and deal with tire nonlinearity by the method of local linearization in this paper.At each sampling moment, the lateral forces of front and rear tires were locally linearized, and the linearized tire force equation was obtained as follows:(4)Fy,i=αiCi∗,where i=f,r represents the front and rear wheels; Ci∗ is the cornering stiffness of the tire at the front-wheel slip angle αi∗. The data Ci∗ is updated once at each sampling time, and the calculation formula is written as(5)Ci∗=∂Fy,i∂αiαi=αi∗,where Fy,i∗ is the lateral force of the current tire side angle αi∗.By substituting (3)–(5) into (1), the system of the vehicle dynamics can be expressed as follows:(6)X˙=AX+BuU+Bdd,Yc=CcX.The model is of second order with state vector asX=vyωrT, with the steering angle U=δf as control input, and with the yaw rate as output (y=ω). d=dt is white Gaussian noise input matrix.G0 is road irregularity coefficient. The longitudinal velocity of the vehicle is constant because the longitudinal force of the tire is ignored. One has(7)A=Cf∗+Cr∗mvxlfCf∗−lrCr∗mvx−vxlfCf∗−lrCr∗Izvxlf2Cf∗+lr2CrIzvx;Bu=−Cf∗m−lfCf∗Iz;Bd=02πG0vx;Cc=1001.


## 2.2. Control Problem Statement

### 2.2.1. Nominal Yaw Rate

The nominal state value of vehicle is also called the expected state value of the vehicle, which is determined by the vehicle predictive model, the driver's input, and the changing driving environment. In order to keep the vehicle stable, the active front steering controller can let the actual vehicle track the ideal vehicle state and make the actual state of the vehicle reach the expected value. In the active front steering controller, the target value of the yaw rate is set to the yaw rate response value of the 2DOF vehicle model [27]. It is shown that(8)ωr=vx/L1+Kvx2.The stability factor isK=m/L2lf/Cr∗−lr/Cf∗. After considering the influence of the ground adhesion, formula (8) is modified as(9)ωref1=minvx/L1+Kvx2δf,μgvx⋅sgnδf.When the upper limit is too high, the vehicle may enter the nonlinear zone before reaching the boundary. Further correction for the nominal yaw rate can be written as(10)ωref2=minvx/L1+Kvx2δf,0.85μgvx⋅sgnδf.


### 2.2.2. Nominal Vehicle Sideslip Angle

In the research of vehicle active front steering control, there are two kinds to determine the nominal sideslip angle of vehicle. The first point considers that setting the target vehicle sideslip angle to 0 conforms to the driver’s expectation; the second point is that the expected sideslip angle is calculated by the linear 2DOF model; that is to say, the response characteristics of the 2DOF model accord with the driver's expectation. If the target vehicle sideslip angle is set to zero, it will cause the controller unnecessary intervention control. The second method of determining the vehicle sideslip angle accords with the actual driving features; therefore, this article uses the second method.By using the linear 2DOF model, the vehicle sideslip angle is calculated as follows:(11)βref0=lf+mlrvx2/Cr∗LL1+Kvx2δf.Similar to the yaw rate, the effect of surface adhesion should be considered to correct the nominal vehicle sideslip angle as follows:(12)βref=minlr+mlfvx2/Cr∗LL1+Kvx2δf,arctan0.02μg⋅sgnδf.


## 2.2.1. Nominal Yaw Rate

The nominal state value of vehicle is also called the expected state value of the vehicle, which is determined by the vehicle predictive model, the driver's input, and the changing driving environment. In order to keep the vehicle stable, the active front steering controller can let the actual vehicle track the ideal vehicle state and make the actual state of the vehicle reach the expected value. In the active front steering controller, the target value of the yaw rate is set to the yaw rate response value of the 2DOF vehicle model [27]. It is shown that(8)ωr=vx/L1+Kvx2.The stability factor isK=m/L2lf/Cr∗−lr/Cf∗. After considering the influence of the ground adhesion, formula (8) is modified as(9)ωref1=minvx/L1+Kvx2δf,μgvx⋅sgnδf.When the upper limit is too high, the vehicle may enter the nonlinear zone before reaching the boundary. Further correction for the nominal yaw rate can be written as(10)ωref2=minvx/L1+Kvx2δf,0.85μgvx⋅sgnδf.


## 2.2.2. Nominal Vehicle Sideslip Angle

In the research of vehicle active front steering control, there are two kinds to determine the nominal sideslip angle of vehicle. The first point considers that setting the target vehicle sideslip angle to 0 conforms to the driver’s expectation; the second point is that the expected sideslip angle is calculated by the linear 2DOF model; that is to say, the response characteristics of the 2DOF model accord with the driver's expectation. If the target vehicle sideslip angle is set to zero, it will cause the controller unnecessary intervention control. The second method of determining the vehicle sideslip angle accords with the actual driving features; therefore, this article uses the second method.By using the linear 2DOF model, the vehicle sideslip angle is calculated as follows:(11)βref0=lf+mlrvx2/Cr∗LL1+Kvx2δf.Similar to the yaw rate, the effect of surface adhesion should be considered to correct the nominal vehicle sideslip angle as follows:(12)βref=minlr+mlfvx2/Cr∗LL1+Kvx2δf,arctan0.02μg⋅sgnδf.


## 3. Model Predictive Controller Design

In this section, we present a model predictive controller to solve the above active front steering tracking problem. First, the vehicle control model is discretized. Then, the predictive equation is deduced and the predictive output is obtained. Finally, the optimal control sequence is solved. The brief active front steering control scheme is shown in Figure2.Figure 2
AFS stability control structure.


### 3.1. Discretization of Vehicle Model

In the practical application, the controller runs at a certain time interval, so the continuous-time system model can be transformed into a discrete-time system model in order to establish the controller. The discrete-time system is obtained as follows:(13)xk+1=Axk+Buuk+Bddk,yck=Ccxk,where(14)A=eAcTs,Bu=∫0TseAcτdτ⋅Bcu,Bd=∫0TseAcτdτ⋅Bcd,Cc=C,where Ts is the sample time for the system.


### 3.2. Predictive Output Equation and Optimization

In order to introduce integrals to reduce or eliminate static errors, rewrite (13) in an incremental form. The predictive functions are described as(15)Δxk+1=AΔxk+BuΔuk+BdΔdk,yck=CcΔxk+yk−1.Let the predictive horizon and control horizon benp and nc, respectively, and nc<np. To derive the system predictive equation, we need to make the following assumptions:(1)
Controlled variables do not change beyond the control horizon:(16)Δuk+i=0,i=nc,nc+1,…,np−1.(2)
Predictable interferences do not change after time stepk:(17)Δdk+i=0,i=1,2,…,np−1.The predictive output of the system is as follows:(18)yck+1k=CcAΔxk+CcBuΔuk+CcBdΔdk+yk,yck+2k=CcA2+CcAΔxk+CcABu+CcBuΔuk+CcBuΔuk+1+CcABd+CcBdΔdk+yk,yck+nck=∑i=1ncCcAiΔxk+∑i=1ncCcAi−1BuΔuk+∑i=1nc−1CcAi−1BuΔuk+1+⋯+CcBuΔuk+m−1+∑i=1ncCcAi−1BdΔdk+yk,yck+npk=∑i=1npCcAiΔxk+∑i=1npCcAi−1BuΔuk+∑i=1np−1CcAi−1BuΔuk+1+⋯+∑i=1np−nc+1CcAi−1BuΔuk+m−1+∑i=1npCcAi−1BdΔdk+yk.Define the predictive output and control input of the system at time stepk as(19)Ypk=yck+1kyck+2k⋮yck+npknp×1,Uk=ukuk+1⋮uk+np−1np×1.According to (13), the predicted outputs of the system in the next np steps can be formulated as(20)Ypk+1k=SxΔxk+Iyk+SdΔdk+SuΔUk,where(21)Sx=CcA∑i=12CcAi⋮∑i=1npCcAinp×1;I=Im×mIm×m⋮Im×mnp×1;Sd=CcBd∑i=12CcAi−1Bd⋮∑i=1npCcAi−1Bdnp×1;Su=CcBu∑i=12CcAi−1Bu⋮0CcBu⋮⋯⋯⋱00⋮∑i=1ncCcAi−1Bu∑i=1nc−1CcAi−1Bu⋯CcBu⋮⋮⋱⋮∑i=1npCcAi−1Bu∑i=1np−1CcAi−1Bu⋯∑i=1np−nc+1CcAi−1Bunp×nc.Longitudinal acceleration and lateral acceleration are limited by the ground adhesion and have the following relationships:(22)ax2+ay2≤μg,where ax and ay are longitudinal acceleration and lateral acceleration, respectively. Considering the speed of the vehicle being a constant, (22) can be further simplified as(23)ay≤μg.In this paper, the solver will dynamically adjust the constraint condition according to the solution of each control cycle. As shown in(24)ay,min−ε≤ay≤ay,max+ε,where ay,min and ay,max are the acceleration constraint.Introducing the relaxation factor into the objective function, the formula is depicted as follows:(25)J=ΓyYpk+1k−Rk+12+ΓuΔUk2+ρε2,where ρ is the relaxation factor.The reference output isR as follows:(26)Rk+1=rk+1rk+2⋯rk+np1×npTSubj. toumin≤uk+i≤umax,i=0,1,…,m−1Δumin≤Δuk+i≤Δumax,i=0,1,…,m−1ymin−ε≤ysc≤ymax+εε>0,where Γy and Γu are weighted matrices of output and control inputs, respectively, as follows:(27)Γy=diagΓy,1,Γy,2,…,Γy,np,Γu=diagΓu,1,Γu,2,…,Γu,nc.Optimal control sequence for constrained optimization problems and the incremental form of the sequence can be written as(28)ΔU∗k=Δum∗k,Δum∗k+1,…,Δum∗k+np−1,εT.The first sample ofum∗ can be used to compute the optimal steering angle control law:(29)umk=umk−1+um∗k.


## 3.1. Discretization of Vehicle Model

In the practical application, the controller runs at a certain time interval, so the continuous-time system model can be transformed into a discrete-time system model in order to establish the controller. The discrete-time system is obtained as follows:(13)xk+1=Axk+Buuk+Bddk,yck=Ccxk,where(14)A=eAcTs,Bu=∫0TseAcτdτ⋅Bcu,Bd=∫0TseAcτdτ⋅Bcd,Cc=C,where Ts is the sample time for the system.


## 3.2. Predictive Output Equation and Optimization

In order to introduce integrals to reduce or eliminate static errors, rewrite (13) in an incremental form. The predictive functions are described as(15)Δxk+1=AΔxk+BuΔuk+BdΔdk,yck=CcΔxk+yk−1.Let the predictive horizon and control horizon benp and nc, respectively, and nc<np. To derive the system predictive equation, we need to make the following assumptions:(1)
Controlled variables do not change beyond the control horizon:(16)Δuk+i=0,i=nc,nc+1,…,np−1.(2)
Predictable interferences do not change after time stepk:(17)Δdk+i=0,i=1,2,…,np−1.The predictive output of the system is as follows:(18)yck+1k=CcAΔxk+CcBuΔuk+CcBdΔdk+yk,yck+2k=CcA2+CcAΔxk+CcABu+CcBuΔuk+CcBuΔuk+1+CcABd+CcBdΔdk+yk,yck+nck=∑i=1ncCcAiΔxk+∑i=1ncCcAi−1BuΔuk+∑i=1nc−1CcAi−1BuΔuk+1+⋯+CcBuΔuk+m−1+∑i=1ncCcAi−1BdΔdk+yk,yck+npk=∑i=1npCcAiΔxk+∑i=1npCcAi−1BuΔuk+∑i=1np−1CcAi−1BuΔuk+1+⋯+∑i=1np−nc+1CcAi−1BuΔuk+m−1+∑i=1npCcAi−1BdΔdk+yk.Define the predictive output and control input of the system at time stepk as(19)Ypk=yck+1kyck+2k⋮yck+npknp×1,Uk=ukuk+1⋮uk+np−1np×1.According to (13), the predicted outputs of the system in the next np steps can be formulated as(20)Ypk+1k=SxΔxk+Iyk+SdΔdk+SuΔUk,where(21)Sx=CcA∑i=12CcAi⋮∑i=1npCcAinp×1;I=Im×mIm×m⋮Im×mnp×1;Sd=CcBd∑i=12CcAi−1Bd⋮∑i=1npCcAi−1Bdnp×1;Su=CcBu∑i=12CcAi−1Bu⋮0CcBu⋮⋯⋯⋱00⋮∑i=1ncCcAi−1Bu∑i=1nc−1CcAi−1Bu⋯CcBu⋮⋮⋱⋮∑i=1npCcAi−1Bu∑i=1np−1CcAi−1Bu⋯∑i=1np−nc+1CcAi−1Bunp×nc.Longitudinal acceleration and lateral acceleration are limited by the ground adhesion and have the following relationships:(22)ax2+ay2≤μg,where ax and ay are longitudinal acceleration and lateral acceleration, respectively. Considering the speed of the vehicle being a constant, (22) can be further simplified as(23)ay≤μg.In this paper, the solver will dynamically adjust the constraint condition according to the solution of each control cycle. As shown in(24)ay,min−ε≤ay≤ay,max+ε,where ay,min and ay,max are the acceleration constraint.Introducing the relaxation factor into the objective function, the formula is depicted as follows:(25)J=ΓyYpk+1k−Rk+12+ΓuΔUk2+ρε2,where ρ is the relaxation factor.The reference output isR as follows:(26)Rk+1=rk+1rk+2⋯rk+np1×npTSubj. toumin≤uk+i≤umax,i=0,1,…,m−1Δumin≤Δuk+i≤Δumax,i=0,1,…,m−1ymin−ε≤ysc≤ymax+εε>0,where Γy and Γu are weighted matrices of output and control inputs, respectively, as follows:(27)Γy=diagΓy,1,Γy,2,…,Γy,np,Γu=diagΓu,1,Γu,2,…,Γu,nc.Optimal control sequence for constrained optimization problems and the incremental form of the sequence can be written as(28)ΔU∗k=Δum∗k,Δum∗k+1,…,Δum∗k+np−1,εT.The first sample ofum∗ can be used to compute the optimal steering angle control law:(29)umk=umk−1+um∗k.


## 4. Simulation Results and HIL Implementation

In this section, firstly, the control feasibility and effectiveness of the controller are verified by the joint simulation of CarSim and Simulink. Then, the hardware-in-the-loop (HIL) was tested on the test bench, and the real control effect of the controller was obtained. The double lance change (DLC) is chosen for the simulation and test conditions.


### 4.1. Simulation Results

In order to evaluate the performance of the proposed model predictive controller, a simulation of emergency avoidance is carried out. CarSim’s parameters are set as follows: A-class car is chosen as the simulation experimental vehicle and the vehicle speed is set to a constant speed of 120 km/h. A vehicle without control means that there is no function of yaw stability. To compare control effects better, the simulation experiments of the model predictive control, PID control, and uncontrolled control are carried out. The comparison of the simulation data is shown as follows.Figure3 shows the comparison curves of vehicle state parameters under the three control modes. It can be seen from Figures 3(a) and 3(b) that the movement track of the uncontrolled vehicle deviates largely from the target path, and the steering angle exceeded the limit. The vehicle with active front steering controller can follow the target path, but the PID controller overshoot is larger due to the accumulated error. Compared with the PID controller, the MPC controller has better response and stronger anti-interference ability. Figure 3(c) is the comparison curve of the lateral acceleration of the vehicle. It can be clearly seen that the lateral acceleration of the vehicle without the active steering controller cannot converge, which proves that the vehicle is seriously unstable and unable to run normally. The vehicles with active steering controller can drive steadily, while vehicles with MPC controller have smaller overshoot and more stable running. It proves that MPC controller can improve vehicle stability and driving safety very well. In Figure 3(d), the yaw rate of the uncontrolled vehicle has exceeded the 20°/s during 5 s∼6 s, proving the vehicle has lost its stability. The model predictive controller adjusts the vehicle to stable state during 10 s and holds it. The vehicle with active front steering controller is always in stable state, but the model predictive controller has better effect than PID controller. Because the PID control has little robustness and the external disturbances input may result in violent vibrations of the vehicle at high speed, PID controller’s control effect is slightly poorer. Figure 3(e) is the contrast curve of the sideslip angle of the vehicle's center of mass. It can be seen clearly that the vehicle state parameters controlled by MPC controller are smoother, which proves that the vehicle runs smoother and can control the vehicle's rapid stability.Figure 3
Comparison of vehicle status parameters. (a) Steering wheel angle. (b) Lateral displacement. (c) Lateral acceleration. (d) Yaw rate. (e) Sideslip angle.
(a)(b)(c)(d)(e)


### 4.2. HIL Implementation

In order to verify the real control effect of the controller, an HIL experimental setup based on LabVIEW is built. By combining the real steering system with the vehicle simulation and road model, the HIL experimental setup is built to evaluate the performance of the designed controller. The deviation between the actual steering angle measured by Bosch sensor and the expected steering angle is taken as the control target and the motor is driven by the driver signal generated by the controller. The controller model is established by Simulink and then encoded into the dSPACE; the vehicle real-time parameters required by the controller are received by CAN communication. The HIL experimental setup is shown in Figure4.Figure 4
HIL experimental setup.The testing vehicle speed is set to 120 km/h and the double lance change (DLC) is chosen for the test conditions. The comparison of the test data is shown in the following.Figures5(a) and 5(b) show the steering column angle and lateral displacement comparisons under three different kinds of controls. Because of the influence of the noise of the test bench on the sensor, the actual steering wheel angle data collected have slight vibration. The results show that the uncontrolled steering column angle increases sharply at 5 s and the vehicle loses its lateral stability. The change range of the steering angle with both control strategies is small; however, the experimental data obtained by model predictive controller is better than that of PID. HIL test results show that both the MPC controller and the PID controller can successfully complete the DLC task, and the MPC controller has better path tracking performance. The optimization process of the model predictive control strategy is not conducted offline but repeated and conducted online. In addition, it can take the uncertainties by model mismatch and interference into account and timely make up for them.Figure 5
Comparison of vehicle status parameters. (a) Actual steering wheel angle. (b) Lateral displacement. (c) Lateral acceleration. (d) Yaw rate. (e) Sideslip angle.
(a)(b)(c)(d)(e)Figures5(c) and 5(d) show the lateral acceleration and yaw rate of the vehicles. It can be seen that, from the diagram, the vehicle’s yaw rate without the controller has exceeded the limit and the vehicle has lost its stability. Compared with PID controller, the variable range of vehicle yaw rate of model predictive controller is smaller. The model predictive controller responds faster than the PID controller. Due to the accumulation of errors in PID control, the closed-loop system becomes sluggish and easily to generate vibration; compared with PID, the rolling optimization strategy adopted by the model predictive control can repeatedly correct the prediction deviation online, thus to improve the stability of the system.Figure5(e) is the comparison of vehicle sideslip angle data. It is obvious that the vehicle sideslip angle overshoot without active steering controller is serious. It breaks through the normal vehicle state value at 4S and cannot return to the normal value. The overshoot of the vehicle sideslip angle controlled by MPC is smaller at the peak value than that of the vehicle controlled by PID, so the MPC controller can improve the vehicle stability better. It can be proved that the control performance of MPC controller is better than that of PID controller.Figure6 shows the current of motor and root mean square value of motor currents under three control modes. Figure 6(a) shows the motor current data controlled by MPC. It can be clearly seen that the current is within the range of ±15 A and can drive the steering motor to rotate normally and smoothly to assist the driver to control the vehicle. Figure 6(b) is the motor current data controlled by PID. It can be seen that the current is within the range of ±20 A, and the shaking situation of the steering column when the motor drives affects the operator's operating comfort. Figure 6(c) shows the motor current without control. It can be seen that the current value has exceeded ±20 A at 6 s, and the motor cannot be normally driven, so that the vehicle is out of control and cannot run normally.Figure 6
Comparison of motor current. (a) MPC controlled motor current. (b) PID controlled motor current. (c) Uncontrolled motor current. (d) Mean square root of motor current.
(a)(b)(c)(d)Figure6(d) shows the root mean square value of current under the three control modes, which can reflect the energy consumed. It can be seen that the MPC controller consumes the least energy, which can ensure the control effect and give consideration to the optimal energy consumption at the same time. PID controller takes second place, although it can achieve the control goal to achieve the requirements of vehicle stability control, but it needs to consume too much energy. The energy consumption without controller is the most, which proves that the steering system has completely lost its maneuverability and the vehicle cannot run normally.


## 4.1. Simulation Results

In order to evaluate the performance of the proposed model predictive controller, a simulation of emergency avoidance is carried out. CarSim’s parameters are set as follows: A-class car is chosen as the simulation experimental vehicle and the vehicle speed is set to a constant speed of 120 km/h. A vehicle without control means that there is no function of yaw stability. To compare control effects better, the simulation experiments of the model predictive control, PID control, and uncontrolled control are carried out. The comparison of the simulation data is shown as follows.Figure3 shows the comparison curves of vehicle state parameters under the three control modes. It can be seen from Figures 3(a) and 3(b) that the movement track of the uncontrolled vehicle deviates largely from the target path, and the steering angle exceeded the limit. The vehicle with active front steering controller can follow the target path, but the PID controller overshoot is larger due to the accumulated error. Compared with the PID controller, the MPC controller has better response and stronger anti-interference ability. Figure 3(c) is the comparison curve of the lateral acceleration of the vehicle. It can be clearly seen that the lateral acceleration of the vehicle without the active steering controller cannot converge, which proves that the vehicle is seriously unstable and unable to run normally. The vehicles with active steering controller can drive steadily, while vehicles with MPC controller have smaller overshoot and more stable running. It proves that MPC controller can improve vehicle stability and driving safety very well. In Figure 3(d), the yaw rate of the uncontrolled vehicle has exceeded the 20°/s during 5 s∼6 s, proving the vehicle has lost its stability. The model predictive controller adjusts the vehicle to stable state during 10 s and holds it. The vehicle with active front steering controller is always in stable state, but the model predictive controller has better effect than PID controller. Because the PID control has little robustness and the external disturbances input may result in violent vibrations of the vehicle at high speed, PID controller’s control effect is slightly poorer. Figure 3(e) is the contrast curve of the sideslip angle of the vehicle's center of mass. It can be seen clearly that the vehicle state parameters controlled by MPC controller are smoother, which proves that the vehicle runs smoother and can control the vehicle's rapid stability.Figure 3
Comparison of vehicle status parameters. (a) Steering wheel angle. (b) Lateral displacement. (c) Lateral acceleration. (d) Yaw rate. (e) Sideslip angle.
(a)(b)(c)(d)(e)


## 4.2. HIL Implementation

In order to verify the real control effect of the controller, an HIL experimental setup based on LabVIEW is built. By combining the real steering system with the vehicle simulation and road model, the HIL experimental setup is built to evaluate the performance of the designed controller. The deviation between the actual steering angle measured by Bosch sensor and the expected steering angle is taken as the control target and the motor is driven by the driver signal generated by the controller. The controller model is established by Simulink and then encoded into the dSPACE; the vehicle real-time parameters required by the controller are received by CAN communication. The HIL experimental setup is shown in Figure4.Figure 4
HIL experimental setup.The testing vehicle speed is set to 120 km/h and the double lance change (DLC) is chosen for the test conditions. The comparison of the test data is shown in the following.Figures5(a) and 5(b) show the steering column angle and lateral displacement comparisons under three different kinds of controls. Because of the influence of the noise of the test bench on the sensor, the actual steering wheel angle data collected have slight vibration. The results show that the uncontrolled steering column angle increases sharply at 5 s and the vehicle loses its lateral stability. The change range of the steering angle with both control strategies is small; however, the experimental data obtained by model predictive controller is better than that of PID. HIL test results show that both the MPC controller and the PID controller can successfully complete the DLC task, and the MPC controller has better path tracking performance. The optimization process of the model predictive control strategy is not conducted offline but repeated and conducted online. In addition, it can take the uncertainties by model mismatch and interference into account and timely make up for them.Figure 5
Comparison of vehicle status parameters. (a) Actual steering wheel angle. (b) Lateral displacement. (c) Lateral acceleration. (d) Yaw rate. (e) Sideslip angle.
(a)(b)(c)(d)(e)Figures5(c) and 5(d) show the lateral acceleration and yaw rate of the vehicles. It can be seen that, from the diagram, the vehicle’s yaw rate without the controller has exceeded the limit and the vehicle has lost its stability. Compared with PID controller, the variable range of vehicle yaw rate of model predictive controller is smaller. The model predictive controller responds faster than the PID controller. Due to the accumulation of errors in PID control, the closed-loop system becomes sluggish and easily to generate vibration; compared with PID, the rolling optimization strategy adopted by the model predictive control can repeatedly correct the prediction deviation online, thus to improve the stability of the system.Figure5(e) is the comparison of vehicle sideslip angle data. It is obvious that the vehicle sideslip angle overshoot without active steering controller is serious. It breaks through the normal vehicle state value at 4S and cannot return to the normal value. The overshoot of the vehicle sideslip angle controlled by MPC is smaller at the peak value than that of the vehicle controlled by PID, so the MPC controller can improve the vehicle stability better. It can be proved that the control performance of MPC controller is better than that of PID controller.Figure6 shows the current of motor and root mean square value of motor currents under three control modes. Figure 6(a) shows the motor current data controlled by MPC. It can be clearly seen that the current is within the range of ±15 A and can drive the steering motor to rotate normally and smoothly to assist the driver to control the vehicle. Figure 6(b) is the motor current data controlled by PID. It can be seen that the current is within the range of ±20 A, and the shaking situation of the steering column when the motor drives affects the operator's operating comfort. Figure 6(c) shows the motor current without control. It can be seen that the current value has exceeded ±20 A at 6 s, and the motor cannot be normally driven, so that the vehicle is out of control and cannot run normally.Figure 6
Comparison of motor current. (a) MPC controlled motor current. (b) PID controlled motor current. (c) Uncontrolled motor current. (d) Mean square root of motor current.
(a)(b)(c)(d)Figure6(d) shows the root mean square value of current under the three control modes, which can reflect the energy consumed. It can be seen that the MPC controller consumes the least energy, which can ensure the control effect and give consideration to the optimal energy consumption at the same time. PID controller takes second place, although it can achieve the control goal to achieve the requirements of vehicle stability control, but it needs to consume too much energy. The energy consumption without controller is the most, which proves that the steering system has completely lost its maneuverability and the vehicle cannot run normally.


## 5. Conclusions

This paper presents and studies a new yaw stability control strategy for active front steering system. The main contributions are as follows:(1)
The model predictive control strategy of the active front-wheel steering system is derived by using the linear optimal theory to improve the lateral stability of the vehicle. On this basis, a model predictive controller is designed to deal with multivariable optimization problems with constraints.(2)
The HIL experimental device was established based on LabVIEW RT, and the controller was tested on the experimental platform in combination with the hardware to verify the actual control effect. The designed model predictive controller can effectively resist interference and improve the robustness of the controller. It can safely complete the driving task and improve the vehicle’s maneuvering stability.In the future, we will further study and design a vehicle AFS model predictive controller equipped with wire-controlled steering system and apply it to the actual vehicle test.



---

*Source: 4712327-2020-07-24.xml*