CN109050658B - Model Predictive Control-Based Adaptive Adjustment Method for Vehicle Active Front Wheel Steering - Google Patents

Abstract

The model predictive control-based automobile active front wheel steering self-adaptive adjustment method is characterized by comprising a reference model, a tire data processor, an MPC controller and a CarSim automobile model; the reference model is used to determine a desired yaw rate of the vehicle; the tire data processor is used for determining a slip angle, a lateral force and a lateral force gradient of the tire; the CarSim automobile model is used for outputting actual motion state information of an automobile, wherein the actual motion state information comprises automobile longitudinal speed, yaw velocity, mass center slip angle and road adhesion coefficient; and the MPC controller optimally solves the additional turning angle of the front wheel of the automobile according to the expected yaw velocity of the automobile and the actual motion state information of the automobile, outputs the additional turning angle to the CarSim automobile model, and controls the automobile to realize yaw stability control.

Description

Model Predictive Control-Based Adaptive Adjustment Method for Vehicle Active Front Wheel Steering

Technical field:

The invention relates to the field of vehicle yaw stability control, in particular to an adaptive adjustment method for vehicle active front wheel steering based on model predictive control.

Background technique:

As people pay more and more attention to the driving safety of automobiles, the active safety system of automobiles has been developed rapidly, among which Active Front Steering (AFS) technology is widely used as an effective yaw stability control system. At present, the control methods used by AFS mainly include PID control, sliding mode variable structure control and Model Predictive Control (MPC), etc. Among them, MPC can better handle multi-objective tasks and system constraints. Sexual control has been widely used.

According to the different prediction models and optimization methods used, MPC can be divided into linear MPC and nonlinear MPC. Linear MPC is widely used because of its low computational burden and fast calculation speed. However, linear MPC cannot characterize the tire cornering characteristics in the nonlinear region, and the nonlinear MPC that can characterize the nonlinear dynamic characteristics of automobiles has too heavy computational burden and poor real-time performance. , difficult to apply in practice. Therefore, many scholars began to linearize the tire lateral force and proposed a linear time-varying MPC control method. Paper [Chen Jie, Li Liang, Song Jian. Research on Vehicle Stability Control Based on LTV-MPC [J]. Automotive Engineering, 2016, 38(3):308-316.] Using the linear time-varying MPC method to realize the vehicle The stability control takes into account the nonlinear characteristics of the system and the computational burden. However, the method of linearizing the tire lateral force in the paper is too simple to represent the actual change of the tire lateral force, and the control effect of the controller is not ideal under the extreme conditions; It remains unchanged in the time domain and cannot represent the actual changing trend of the car during the rolling forecasting process. The paper [Choi M, Choi S B. MPC for vehicle lateral stability via differential braking and active front steering considering practical aspects[J]. Proceedings of the Institution of Mechanical Engineers Part D Journal of Automobile Engineering, 2016, 230(4).] is based on linear The optimized tire model gives the control strategy when the tire lateral force reaches saturation, and realizes the vehicle stability control under extreme conditions. However, the prediction model designed in this paper also remains unchanged in the prediction time domain. Under the extreme conditions, the prediction model cannot accurately represent the actual motion of the car during the rolling prediction process, which leads to the poor control effect of the controller.

Invention content:

In order to solve the problem that the prediction model of the existing linear time-varying MPC method cannot reflect the nonlinear dynamic characteristics of the vehicle in the rolling prediction process, resulting in poor control effect of the AFS system under extreme working conditions. The invention provides an adaptive adjustment method for vehicle active front wheel steering based on model predictive control, which adopts a linear time-varying method to convert a nonlinear predictive control problem into a linear predictive control problem. The trend automatic adjustment prediction model can accurately characterize the nonlinear dynamic characteristics of the vehicle while reducing the computational burden of the system, ensure the stability of the AFS controller under extreme working conditions, and realize the stability control of the vehicle.

The technical scheme adopted by the present invention to solve the technical problem is as follows:

An adaptive adjustment method for vehicle active front wheel steering based on model predictive control, characterized in that the method includes a reference model, a tire data processor, an MPC controller, and a CarSim vehicle model; the reference model is used to determine a desired vehicle yaw rate; The tire data processor is used to determine the tire's slip angle, lateral force and lateral force gradient; the CarSim car model is used to output the actual motion state information of the car, including the car's longitudinal speed, yaw rate, center of mass slip angle and road adhesion coefficient; the MPC controller optimizes and solves the additional turning angle of the front wheel of the car according to the expected yaw rate of the car and the actual motion state information of the car, and outputs it to the CarSim car model to control the car to achieve yaw stability control;

The method includes the following steps:

Step 1. Establish a reference model to determine the expected vehicle yaw rate. The process includes the following sub-steps:

Step 1.1. The linear two-degree-of-freedom vehicle model is used as the reference model, and the differential equation of motion is expressed as follows:

Among them: β is the side slip angle of the car mass center; γ is the yaw angular velocity of the car; I _z is the yaw moment of inertia around the vertical axis of the car mass center; U _x is the longitudinal speed of the car; l _f and l _r are the car mass center to front , the distance of the rear axle; C _f and C _r are the cornering stiffnesses of the front and rear tires of the car, respectively; δ _f,dri is the front wheel turning angle generated by the driver's steering input;

Step 1.2. Convert the motion differential equation of the linear two-degree-of-freedom vehicle model into a transfer function, the form is as follows:

In order to achieve the ideal closed-loop effect, the desired vehicle yaw rate is obtained based on formula (2):

where: γ _ref is the desired vehicle yaw rate; _wn is the natural frequency of the system; ξ is the system damping; G _ω (s) is the transfer function gain; w _d = k ₁ w _n , ξ _d = k ₂ ξ, G _kω (s)=k ₃ G _ω (s); k ₁ , k ₂ , k ₃ are parameters to improve the system phase delay and response speed;

Step 2. Design a tire data processor, and the process includes the following sub-steps:

Step 2.1. Design the tire side slip angle calculation module. The front and rear tire side slip angles are calculated by the following formula:

Among them: α _f and α _r are the side slip angles of the front and rear tires of the car, respectively; δ _f is the front wheel rotation angle of the car;

Step 2.2. Design the tire lateral force and tire lateral force gradient calculation module. In order to obtain the nonlinear characteristics of the tire, based on the Pacejka tire model, obtain the relationship curve between the tire lateral force and the tire side slip angle under different road adhesion coefficients, Obtain the three-dimensional map of tire cornering characteristics; obtain the relationship curve of the tire lateral force to the tire side-slip angle derivative under different road adhesion coefficients, and obtain the three-dimensional map of the tire lateral force gradient; the tire data processor calculates the actual tire cornering at the current moment The angle and road adhesion coefficient are input into the three-dimensional map of tire cornering characteristics and the three-dimensional map of tire lateral force gradient, respectively, and the current tire lateral force and tire lateral force gradient are obtained through linear interpolation, and output to the MPC controller. ; The tire data processor updates the tire lateral force and tire lateral force gradient values once in each control cycle;

Among them: Pacejka tire model is as follows:

F _y,j = μDsin(Catan(Bα _j -E(Bα _j -α _j tan(Bα _j ))))

where: j=f, r, representing the front and rear wheels; F _{y, j} is the tire lateral force, α _j is the tire slip angle; B, C, D and E depend on the vertical wheel load F _z ; a ₀ =1.75; a ₁ =0; a ₂ =1000; a ₃ =1289; a ₄ =7.11; a ₅ =0.0053; a ₆ =0.1925;

Step 3. Design the MPC controller, and its process includes the following sub-steps:

Step 3.1, establish a prediction model, the process includes the following sub-steps:

Step 3.1.1. Linearize the tire model, and its expression is as follows:

是在当前侧偏角

的轮胎侧向力梯度值；

是轮胎的残余侧向力，通过如下公式计算：in:

is at the current slip angle

The tire lateral force gradient value;

is the residual lateral force of the tire, calculated by the following formula:

是基于轮胎侧偏特性三维图，通过线性插值法获得的轮胎侧向力；

是基于轮胎侧偏刚度特性三维图，通过线性插值法获得的轮胎侧向力梯度；

是当前时刻实际的轮胎侧偏角；in:

is the tire lateral force obtained by the linear interpolation method based on the three-dimensional map of the tire cornering characteristics;

is the tire lateral force gradient obtained by the linear interpolation method based on the three-dimensional map of the tire cornering stiffness characteristics;

is the actual tire slip angle at the current moment;

Based on formula (5), in the rolling prediction process, the lateral force of the designed tire is expressed as follows:

in:

和

变化的权重因子；Among them: P is the prediction time domain; the superscript "k+i|k" indicates the i-th time in the future predicted at the current time k; ρ ^k+i|k and ξ ^k+i|k are the adjustment

and

changing weighting factors;

Step 3.1.2. Establish a prediction model. The differential equation of motion of the prediction model is expressed as:

Bringing formula (7) into formula (9), the prediction model in the rolling prediction process is obtained as:

Step 3.1.3. Establish a prediction equation for predicting the future output of the system, and write formula (10) as a state space equation for designing the prediction equation, as follows:

in:

x=γ; _u =δf;

In order to realize the tracking control of the vehicle yaw rate, the prediction model of the continuous time system is converted into the incremental model of the discrete time system:

Wherein: sampling time k=int(t/T _s ), t is the simulation time, and T _s is the simulation step size;

Step 3.2, design optimization objectives and constraints, the process includes the following sub-steps:

Step 3.2.1. Use the two-norm of the expected vehicle yaw rate and the actual vehicle yaw rate error as the yaw rate tracking performance index to reflect the trajectory tracking characteristics of the vehicle, and its expression is as follows:

Where: _γref is the expected vehicle yaw rate; γ is the actual vehicle yaw rate; P is the prediction time domain; k is the current moment; Q is the weighting factor;

Step 3.2.2. Use the second norm of the change rate of the control variable as the steering smoothing index to reflect the smoothing characteristics of the steering during the tracking process of the yaw rate. The control variable u is the front wheel angle of the car, and the discrete quadratic steering smoothing index is established as:

Among them: M is the control time domain; △u is the variation of the control quantity; k is the current moment; S is the weighting factor;

Step 3.2.3. Set the physical constraints of the actuator to meet the requirements of the actuator:

Using the linear inequality to limit the upper and lower limits of the front wheel rotation angle and its variation, the physical constraints of the steering actuator are obtained, and its mathematical expression is:

Among them: δ _fmin is the lower limit of the front wheel rotation angle, δ _fmax is the upper limit of the front wheel rotation angle; _{Δδ fmin} is the lower limit of the front wheel rotation angle variation; _{Δδ fmax} is the upper limit of the front wheel rotation angle variation;

Step 3.3. Solve the predicted output of the system, and the process includes the following sub-steps:

Step 3.3.1. Use the linear weighting method to convert the tracking performance index described in step 3.2.1 and the steering smoothness index described in step 3.2.2 into a single index to construct a multi-objective optimal control problem for vehicle yaw stability. The physical constraints of the steering actuator, and the input and output conform to the predictive model:

subject to

i) Predictive model

ii) The constraint condition is formula (15)

Step 3.3.2. In the controller, the quadratic programming algorithm is used to solve the multi-objective optimal control problem (16), and the optimal open-loop control sequence _Δδf is obtained as:

Select the first element Δδ _f (0) in the optimal open-loop control sequence at the current moment for feedback, and linearly superimpose it with the previous moment to obtain the front wheel angle δ _f , which is output to the CarSim car model to realize the yaw of the car Stability Control.

The beneficial effects of the invention are: the method uses a linear time-varying method to convert the nonlinear predictive control problem into a linear predictive control problem, which can reduce the computational burden of the system; The predictive model of the adaptive adjustment system can achieve the control effect of nonlinear MPC and improve the control effect of AFS under extreme working conditions.

Description of drawings

FIG. 1 is a schematic structural diagram of the control system of the present invention.

Figure 2 is a schematic diagram of a linear two-degree-of-freedom vehicle model.

FIG. 3 is a three-dimensional view of tire cornering characteristics.

Figure 4 is a three-dimensional map of the tire lateral force gradient.

Figure 5 is a schematic diagram of the linearization of the tire model.

Figure 6 is a schematic diagram of tire model linearization during rolling prediction.

Detailed ways

The present invention will be described in detail below with reference to the accompanying drawings and embodiments.

Fig. 1 is the system structure schematic diagram of the vehicle active front wheel steering adaptive adjustment method based on model predictive control of the present invention, this system mainly comprises reference model 1, tire data processor 2, MPC controller 3, Carsim car model 4; Reference model 1 is used to determine the expected yaw rate of the car; the tire data processor 2 is used to determine the slip angle, lateral force and lateral force gradient of the tire; the CarSim car model 4 is used to output the actual motion state information of the car, including the car Longitudinal speed, yaw rate, center of mass slip angle, and road adhesion coefficient; MPC controller 3 The controller optimizes and solves the additional front wheel angle of the vehicle according to the expected vehicle yaw rate and the actual motion state information of the vehicle, and outputs it to CarSim Car model 4, control the car to achieve yaw stability control.

The method of the present invention is specifically described below with a certain car model of the CarSim automobile simulation software as a platform, and its main parameters are shown in Table 1:

Table 1 Main parameters of CarSim car

The establishment of reference model 1 includes two parts: 1.1 establish a linear two-degree-of-freedom vehicle model; 1.2 determine the expected vehicle yaw rate;

In Section 1.1, the linear two-degree-of-freedom vehicle model is shown in Figure 2, and its motion differential equation is expressed as follows:

Among them: β is the side slip angle of the car mass center; γ is the yaw angular velocity of the car; I _z is the yaw moment of inertia around the vertical axis of the car mass center; U _x is the longitudinal speed of the car; l _f and l _r are the car mass center to front , the distance between the axles; C _f and C _r are the cornering stiffnesses of the front and rear tires of the car, respectively; δ _f,dri is the front wheel turning angle generated by the driver's steering input.

In Section 1.2, the differential equation of motion of the linear two-degree-of-freedom vehicle model is converted into a transfer function in the following form:

In order to achieve the ideal closed-loop effect, the desired vehicle yaw rate is obtained based on formula (2):

where: γ _ref is the desired yaw rate; _wn is the natural frequency of the system; ξ is the system damping; G _ω (s) is the transfer function gain; w _d = k ₁ w _n ,ξ _d =k ₂ ξ,G _kω (s)=k ₃ G _ω (s); k ₁ , k ₂ , k ₃ are parameters to improve the system phase delay and response speed; the calculation process of _wn , ξ, G _ω (s), K _ω is as follows:

The design of the tire data processor 2 includes two parts: 2.1 Design the tire slip angle calculation module; 2.2 Design the tire lateral force and tire lateral force gradient calculation module;

In section 2.1, the front and rear tire slip angles are calculated by the following formula:

Among them: α _f and α _r are the side slip angles of the front and rear tires of the car, respectively; δ _f is the front wheel rotation angle of the car;

In section 2.2, in order to obtain the nonlinear characteristics of the tire, based on the Pacejka tire model, the relationship curve between the tire lateral force and the tire side slip angle under different road adhesion coefficients was obtained, and the three-dimensional map of the tire side slip characteristics was obtained, as shown in Figure 3; Obtain the relationship curve between the tire lateral force and the tire slip angle derivative under different road adhesion coefficients, and obtain a three-dimensional map of the tire lateral force gradient, as shown in Figure 4. The tire data processor 2 inputs the actual tire side slip angle and road adhesion coefficient at the current moment into the three-dimensional graph of tire cornering characteristics and the three-dimensional graph of tire lateral force gradient, respectively, and obtains the tire lateral force at the current moment through the linear interpolation method. and tire lateral force gradient, and output to MPC controller 3. The tire lateral force and tire lateral force gradient values are updated once per control cycle by the tire data processor.

Among them: Pacejka tire model is as follows:

where: j=f, r, representing the front and rear wheels; F _{y, j} is the tire lateral force, α _j is the tire slip angle; B, C, D and E depend on the vertical wheel load F _z ; a ₀ =1.75; a ₁ =0; a ₂ =1000; a ₃ =1289; a ₄ =7.11; a ₅ =0.0053; a ₆ =0.1925.

The design of MPC controller 3 consists of three parts: 3.1 Establish prediction model and prediction equation; 3.2 Design optimization objectives and constraints; 3.3 Solve system prediction output;

In section 3.1, the establishment of prediction model and prediction equation includes three parts: 3.1.1 Linearized tire model; 3.1.2 Establishment of prediction model; 3.1.3 Establishment of prediction equation;

处，如图5所示，对轮胎模型进行线性化，其表达式如下：In section 3.1.1, at the current slip angle

As shown in Figure 5, the tire model is linearized, and its expression is as follows:

是在当前侧偏角

的轮胎侧向力梯度值；

是轮胎的残余侧向力，如图5所示，通过如下公式计算：in:

is at the current slip angle

The tire lateral force gradient value;

is the residual lateral force of the tire, as shown in Figure 5, calculated by the following formula:

是基于轮胎侧偏特性三维图(图3)，通过线性插值法获得的轮胎侧向力；

是基于轮胎侧偏刚度特性三维图(图4)，通过线性插值法获得的轮胎侧向力梯度；

是当前时刻实际的轮胎侧偏角。in:

is the tire lateral force obtained by the linear interpolation method based on the three-dimensional map of the tire cornering characteristics (Fig. 3).

is the tire lateral force gradient obtained by linear interpolation based on the three-dimensional map of tire cornering stiffness characteristics (Figure 4).

is the actual tire slip angle at the current moment.

Based on formula (5), in the rolling prediction process, as shown in Figure 6, the lateral force expression of the designed tire is as follows:

in:

和

变化的权重因子。Among them: P is the prediction time domain; the superscript "k+i|k" indicates the i-th time in the future predicted at the current time k; ρ ^k+i|k and ξ ^k+i|k are the adjustment

and

Changed weighting factor.

In section 3.1.2, the prediction model adopts the linear two-degree-of-freedom vehicle model shown in Figure 2, and its motion differential equation is expressed as:

Bringing formula (7) into formula (9), the prediction model in the rolling prediction process is obtained as:

In Section 3.1.3, formula (10) is written as a state-space equation for designing the prediction equation as follows:

in:

x=γ; _u =δf;

In order to realize the tracking control of the vehicle yaw rate, the prediction model of the continuous time system is converted into the incremental model of the discrete time system:

C＝1。Wherein: sampling time k=int(t/T _s ), t is the simulation time, and T _s is the simulation step size;

C=1.

In Section 3.2, the design of optimization objectives and constraints includes three parts: 3.2.1 Design yaw rate tracking performance index; 3.2.2 Design steering smoothness index; 3.2.3 Set actuator physical constraints;

In section 3.2.1, the two-norm of the expected vehicle yaw rate and the actual vehicle yaw rate error is used as the yaw rate tracking performance index to reflect the vehicle's trajectory tracking characteristics, and its expression is as follows:

Among them: _γref is the expected vehicle yaw rate; γ is the actual vehicle yaw rate; P is the prediction time domain; k is the current moment; Q is the weighting factor.

In section 3.2.2, the second norm of the control variable rate of change is used as the steering smoothing index to reflect the steering smoothing characteristics in the process of yaw rate tracking. The control variable u is the front wheel angle of the car, and a discrete quadratic steering smoothing index is established. for:

Among them: M is the control time domain; △u is the variation of the control quantity; k is the current moment; S is the weighting factor.

In section 3.2.3, the upper and lower limits of the front wheel rotation angle and its variation are limited by linear inequalities, and the physical constraints of the steering actuator are obtained. The mathematical expression is:

Where: δ _fmin is the lower limit of the front wheel rotation angle, δ _fmax is the upper limit of the front wheel rotation angle; _{Δδ fmin} is the lower limit of the front wheel rotation angle variation; _{Δδ fmax} is the upper limit of the front wheel rotation angle variation.

In Section 3.3, the solution of the predicted output of the system includes two parts: 3.3.1 Constructing the multi-objective optimal control problem of vehicle yaw stability; 3.3.2 Solving the multi-objective optimal control problem;

In section 3.3.1, the yaw rate tracking performance index of formula (13) and the steering smoothness index of formula (14) are transformed into a single index by the linear weighting method, and the multi-objective optimal control problem of vehicle yaw stability is constructed. The problem is to satisfy the physical constraints of the steering actuator, and the input and output conform to the predictive model:

subject to

i) Predictive model

ii) The constraint condition is formula (15)

In section 3.3.2, in the controller, the quadratic programming algorithm is used to solve the multi-objective optimal control problem (16), and the optimal open-loop control sequence Δδ _f is obtained as:

Select the first element Δδ _f (0) in the optimal open-loop control sequence at the current moment for feedback, and linearly superimpose it with the previous moment to obtain the front wheel angle δ _f , which is output to the CarSim vehicle model 4 to realize the horizontal Pendulum stability control.

Claims (1)

1. the auto active front wheel steering adaptive adjustment method based on model predictive control, is characterized in that, this method comprises reference model, tire data processor, MPC controller, CarSim car model; Reference model is used to determine expected car yaw Angular velocity; the tire data processor is used to determine the slip angle, lateral force and lateral force gradient of the tire; the CarSim car model is used to output the actual motion state information of the car, including the car longitudinal velocity, yaw angular velocity, center of mass slip angle and Road adhesion coefficient; MPC controller optimizes and solves the additional turning angle of the front wheel of the car according to the expected yaw rate of the car and the actual motion state information of the car, and outputs it to the CarSim car model to control the car to achieve yaw stability control; The method includes the following steps: Step 1. Establish a reference model to determine the expected vehicle yaw rate. The process includes the following sub-steps: Step 1.1. The linear two-degree-of-freedom vehicle model is used as the reference model, and the differential equation of motion is expressed as follows:

Among them: β is the side slip angle of the car mass center; γ is the yaw angular velocity of the car; I _z is the yaw moment of inertia around the vertical axis of the car mass center; U _x is the longitudinal speed of the car; l _f and l _r are the car mass center to front , the distance of the rear axle; C _f and C _r are the cornering stiffnesses of the front and rear tires of the car, respectively; δ _f,dri is the front wheel turning angle generated by the driver's steering input; Step 1.2. Convert the motion differential equation of the linear two-degree-of-freedom vehicle model into a transfer function, the form is as follows:

In order to achieve the ideal closed-loop effect, the desired vehicle yaw rate is obtained based on formula (2):

where: γ _ref is the desired vehicle yaw rate; _wn is the natural frequency of the system; ξ is the system damping; G _ω (s) is the transfer function gain; w _d = k ₁ w _n , ξ _d = k ₂ ξ, G _kω (s)=k ₃ G _ω (s); k ₁ , k ₂ , k ₃ are parameters to improve the system phase delay and response speed; Step 2. Design a tire data processor, and the process includes the following sub-steps: Step 2.1. Design the tire side slip angle calculation module. The front and rear tire side slip angles are calculated by the following formula:

Among them: α _f and α _r are the side slip angles of the front and rear tires of the car, respectively; δ _f is the front wheel rotation angle of the car; Step 2.2. Design the tire lateral force and tire lateral force gradient calculation module. In order to obtain the nonlinear characteristics of the tire, based on the Pacejka tire model, obtain the relationship curve between the tire lateral force and the tire side slip angle under different road adhesion coefficients, Obtain the three-dimensional map of tire cornering characteristics; obtain the relationship curve of the tire lateral force to the tire side-slip angle derivative under different road adhesion coefficients, and obtain the three-dimensional map of the tire lateral force gradient; the tire data processor calculates the actual tire cornering at the current moment The angle and road adhesion coefficient are input into the three-dimensional map of tire cornering characteristics and the three-dimensional map of tire lateral force gradient, respectively, and the current tire lateral force and tire lateral force gradient are obtained through linear interpolation, and output to the MPC controller. ; The tire data processor updates the tire lateral force and tire lateral force gradient values once in each control cycle; Among them: Pacejka tire model is as follows: F _y,j = μD sin(Ca tan(Bα _j -E(Bα _j -α _j tan(Bα _j ))))

where: μ is the road adhesion coefficient; j = f, r, representing the front and rear wheels; F _{y, j} is the tire lateral force, α _j is the tire slip angle; B, C, D and E depend on the wheel vertical load F _z ; a ₀ =1.75; a ₁ =0; a ₂ =1000; a ₃ =1289; a ₄ =7.11; a ₅ =0.0053; a ₆ =0.1925; Step 3. Design the MPC controller, and its process includes the following sub-steps: Step 3.1, establish a prediction model, the process includes the following sub-steps: Step 3.1.1. Linearize the tire model, and its expression is as follows:

in:

is at the current slip angle

The tire lateral force gradient value;

is the residual lateral force of the tire, calculated by the following formula:

in:

is the tire lateral force obtained by the linear interpolation method based on the three-dimensional map of the tire cornering characteristics;

is the tire lateral force gradient obtained by the linear interpolation method based on the three-dimensional map of the tire cornering stiffness characteristics;

is the actual tire slip angle at the current moment; Based on formula (5), in the rolling prediction process, the lateral force of the designed tire is expressed as follows:

in:

Among them: P is the prediction time domain; the superscript "k+i|k" indicates the i-th time in the future predicted at the current time k; ρ ^k+i|k and ξ ^k+i|k are the adjustment

and

changing weighting factors; Step 3.1.2. Establish a prediction model. The differential equation of motion of the prediction model is expressed as:

Bringing formula (7) into formula (9), the prediction model in the rolling prediction process is obtained as:

Step 3.1.3. Establish a prediction equation for predicting the future output of the system, and write formula (10) as a state space equation for designing the prediction equation, as follows:

in: x=γ; _u =δf;

In order to realize the tracking control of the vehicle yaw rate, the prediction model of the continuous time system is converted into the incremental model of the discrete time system:

Wherein: sampling time k=int(t/T _s ), t is the simulation time, and T _s is the simulation step size;

c = 1; Step 3.2, design optimization objectives and constraints, the process includes the following sub-steps: Step 3.2.1. Use the two-norm of the expected vehicle yaw rate and the actual vehicle yaw rate error as the yaw rate tracking performance index to reflect the trajectory tracking characteristics of the vehicle, and its expression is as follows:

Where: _γref is the expected vehicle yaw rate; γ is the actual vehicle yaw rate; P is the prediction time domain; k is the current moment; Q is the weighting factor; Step 3.2.2. Use the second norm of the change rate of the control variable as the steering smoothing index to reflect the smoothing characteristics of the steering during the tracking process of the yaw rate. The control variable u is the front wheel angle of the car, and the discrete quadratic steering smoothing index is established as:

Among them: M is the control time domain; Δu is the variation of the control quantity; k is the current moment; S is the weighting factor; Step 3.2.3. Set the physical constraints of the actuator to meet the requirements of the actuator: Using the linear inequality to limit the upper and lower limits of the front wheel rotation angle and its variation, the physical constraints of the steering actuator are obtained, and its mathematical expression is:

Where: δ _fmin is the lower limit of the front wheel rotation angle, δ _fmax is the upper limit of the front wheel rotation angle; Δδ _fmin is the lower limit of the front wheel rotation angle variation; Δδ _fmax is the upper limit of the front wheel rotation angle variation; Step 3.3. Solve the predicted output of the system, and the process includes the following sub-steps: Step 3.3.1. Use the linear weighting method to convert the tracking performance index described in step 3.2.1 and the steering smoothness index described in step 3.2.2 into a single index to construct a multi-objective optimal control problem for vehicle yaw stability. The physical constraints of the steering actuator, and the input and output conform to the predictive model:

subject to i) Predictive model ii) The constraint condition is formula (15) Step 3.3.2. In the controller, the quadratic programming algorithm is used to solve the multi-objective optimal control problem (16), and the optimal open-loop control sequence Δδ _f is obtained as:

Select the first element Δδ _f (0) in the optimal open-loop control sequence at the current moment for feedback, and linearly superimpose it with the previous moment to obtain the front wheel angle δ _f , which is output to the CarSim car model to achieve the yaw stability of the car Sexual control.

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