Combined with the characteristics of the distributed-drive electric vehicle and direct yaw moment control, a double-layer structure direct yaw moment controller is designed. The upper additional yaw moment controller is constructed based on model predictive control. Aiming at minimizing the utilization rate of tire adhesion and constrained by the working characteristics of motor system and brake system, a quadratic programming active set was designed to optimize the distribution of additional yaw moments. The road surface adhesion coefficient has a great impact on the reliability of direct yaw moment control, for which joint observer of vehicle state parameters and road surface parameters is designed by using unscented Kalman filter algorithm, which correlates vehicle state observer and road surface parameter observer to form closed-loop feedback correction. The results show that compared to the “feedforward + feedback” control, the vehicle’s error of yaw rate and sideslip angle by the model predictive control is smaller, which can improve the vehicle stability effectively. In addition, according to the results of the docking road simulation test, the joint observer of vehicle state and road surface parameters can improve the adaptability of the vehicle stability controller to the road conditions with variable adhesion coefficients.

When the vehicle is sharp turning at high speed, the limited tire adhesion force cannot provide enough lateral force, which will lead to the tire lateral force exceeding the adhesion limit and cause rollover and sideslip. For vehicle instability in extreme conditions, the researchers put forward the direct yaw moment control strategy. The control strategy adjusts the driving and brake torque of each wheel on basis of the current vehicle state for producing the yaw moment to improve the vehicle stability. So the yaw rate and sideslip angle of the vehicle can be controlled within the scope of the stability to maintain the vehicle stability. Distributed-drive electric vehicle which has the flexible driving form creates the ideal conditions for vehicle stability control. But compared with the traditional fuel vehicles, the complexity in the dynamic characteristics, actuator response characteristics and the actuator of distributed-drive electric vehicles are increased. For the stability control system of distributed-drive electric vehicle, we need to conduct specialized research.

According to the structure of the control system, the vehicle stability control system can be divided into the centralized vehicle stability controller and the hierarchical vehicle stability controller. The hierarchical controller can realize the decoupling between different systems, improve the transient control performance of each systems and reduce the controller complexity [

Wang et al. [

Boada et al. [

Considering tire saturation characteristics, Chilali et al. [

Demirci et al. [

Barbarisi et al. [

All in all, compared with the robust control, fuzzy control, neural network and other control theory, model predictive control algorithm is more receptive. Through feedback loop optimization approach MPC can realize control target and the continuous control of controlled object. Therefore, this paper studies the hierarchical structure direct yaw moment controller based on MPC to get better vehicle stability. Considering the influence of road adhesion coefficient on stability control system, a joint observer of vehicle state parameters and road surface parameters is also studied.

The contributions of this paper are as follows: (1) Based on the trackless Kalman filter algorithm, a joint observer is designed to monitor the vehicle state parameters and road parameters in real time. (2) A hierarchical structure direct yaw moment controller is designed. The upper layer proposes the decision of additional yaw moment based on the model predictive control method and the lower layer distributes the torque between wheels based on the quadratic programming set method.

The organization of this paper is as follows: Section 2 introduces the structure of direct yaw moment controller. Section 3 designs the joint observer of vehicle state parameters and road parameters. Section 4 constructs the direct yaw moment controller. Section 5 carries on the control strategy simulation verification. Finally, conclusions of this research and future works are given in Section 6.

As is shown in

The upper layer controller includes vehicle state parameter estimator model based on dual unscented Kalman filter, stability control reference model, longitudinal driving force controller and additional yaw moment decision model. The upper layer controller calculated the steady-state yaw rate

Accurate acquisition of vehicle state information can improve the control effect of vehicle stability controller. The vehicle stability control strategy usually takes the yaw rate and the sideslip angle as the control targets. Yaw rate represents the vehicle’s steering dynamic characteristics and the sideslip angle reflects the vehicle’s driving trajectory. The yaw rate can be directly collected by the gyroscope. But the direct measurement of the vehicle sideslip angle is more difficult. Meanwhile, the road parameters also limit the reference yaw rate and sideslip angle. Consider the above two reasons, a joint observer of vehicle state parameters and road parameters is designed based on UKF which has good adaptability to nonlinear system.

As shown in

According to the vehicle dynamic model, the dynamic equations of longitudinal, lateral and yaw motion of the vehicle can be obtained [

In the formula

According to the magic tire model [

where,

The vertical load of each wheel is as follows [

where,

In the joint observer, vehicle state vector

The input vector

The measurement vector consists of vehicle longitudinal acceleration

According to the 3-DOF vehicle dynamics model and the magic tire model, the nonlinear vehicle state observation equation can be obtained:

In the sampling time

The vehicle sideslip angle could be calculated by the longitudinal speed and lateral speed.

The observation equation and state equation of road surface parameters observer is:

In the formula,

(1) Establish initial 2n + 1 sigma point set

where,

Weight of sigma sampling point set’s is as follows:

(2) Vehicle state parameters time update: Calculate the one step predicted sigma points according to the state transfer function

Calculate the predicted value and covariance matrix according to the set of predicted sampling points:

where,

(3) Establish the initial sigma point set

where,

Weight of sigma sampling point set’s is as follows:

(4) Road parameters time update: calculate the one step predicted Road parameters.

Update the parameter prediction error covariance matrix.

where,

(5) Vehicle state measurement update:

Calculate covariance matrix of the vehicle state observation information:

Calculate cross covariance matrix of the vehicle state observation information:

(6) Road parameters measurement update:

Calculate covariance matrix of the road parameters observation information:

Calculate cross covariance matrix of the road parameters observation:

(7) Calculate the Kalman filter gain matric

Calculate the optimal estimation of vehicle state parameters based on the vector of vehicle state parameters

Update the covariance matrix of the vehicle state parameters error:

(8) Calculate the Kalman filter gain matric

Calculate the optimal estimation of the road parameters vector

Update covariance matrix of the road parameters error:

Let k = K + 1 repeat the above steps to realize the joint observation of vehicle state and road parameters.

The 2-DoF vehicle model can well describe the steady-state characteristics of the vehicle. Therefore, the yaw rate and the sideslip angle under the stable operating condition are selected as the control targets of the controller in this paper. The linear 2-DOF model is shown in

The vehicle differential equation is as follows:

When vehicle is in the steady state,

When the vehicle is driving on the low adhesion coefficient road, such as rain, snow and sands, the adhesion force provided by the road adhesion condition is small, which cannot produce the high yaw rate required by the vehicle in the stable state. Therefore, when the vehicle linear 2-DOF model is selected as the reference model, the reference yaw rate and sideslip angle must be limited by the tire and road adhesion coefficient.

The upper boundary of the reference yaw rate is:

Therefore, the reference yaw rate is:

The upper bound of the reference sideslip angle must be specified. This paper adopts empirical formula

Therefore, the reference sideslip angle is:

Thus, the basic control target, the reference yaw rate

Adding additional yaw moment

System state equation is as follows:

where,

In order to meet the discrete control requirements of model predictive control, Euler method is used to discretization the above system space state equation [

where,

Utilize the discretized state transfer equation to predict the system state in time domain P:

where,

According to the predicted system state, the corresponding output of predicted system can be obtained.

where,

The optimization objective function is shown as follows:

where,

The constraint function are as follows:

(1) Due to the motor power limit, the control input of the vehicle will be limited:

(2) Control increments also need to be limited to prevent vehicle instability caused by excessive energy fluctuation:

(3) The system output constraint

The optimal control can be obtained by solving the above quadratic programming problem:

And apply the first term in

Taking the minimum tire adhesion coefficient utilization rate as the optimal allocation target of additional yaw moment [

Introducing the following constraint conditions:

(1) Tire adhesion limit constraint:

(2) Motor system performance constraint:

where,

(3) Braking system constraints:

After torque optimization, if the torque is negative, it is necessary to apply braking torque. The braking torque should be less than the limit braking torque

This paper adopts the active set method to solve the above quadratic programming problem [

There is a method based on determining its optimal solution and corresponding multiplier at the same time, namely Lagrange function:

It can be obtained from its matrix form:

Considering the long development cycle and high cost of controller, a test platform based on experimental distributed-drive electric vehicle using A&D 5435 hardware-in-the-loop simulation system has been set up. Test on low adhesion coefficient road and joint pavement simulation conditions are conduct respectively. At the same time, the direct yaw moment controller based on “feedforward + feedback” control is designed as a comparison, so as to verify the feasibility and accuracy of the proposed stability control strategy.

The target track of the double lane change condition is shown in

Maximum | Maximum error | Average error | Root mean square error | ||||||
---|---|---|---|---|---|---|---|---|---|

MPC | Feedforward + feedback | Non-control | MPC | Feedforward + feedback | MPC | Feedforward + feedback | MPC | Feedforward + feedback | |

Yaw rate (°/s) | 11.972 | 12.217 | 15.821 | 1.220 | 3.125 | 0.030 | 0.049 | 0.597 | 1.372 |

Sideslip angle (°) | 1.003 | 1.522 | 2.247 | 0.121 | 0.641 | 0.016 | 0.023 | 0.340 | 0.363 |

It can be seen from

At the same time, according to the error analysis results, the maximum error, the mean error and the root mean square error of the vehicle yaw rate and the sideslip under MPC-based stability control system are all smaller than those under the “feedforward + feedback” controller. As can be seen from the actual driving track of the vehicle in

Meanwhile, according to the observation data of adhesion coefficient shown in

This paper selects the snake test on joint pavement pylon as test condition. The vehicle speed is 85 km/h and the road adhesion coefficient is shown in

Simulation section | 0~350 m | 300~700 m |

Road adhesion coefficient |
0.9 | 0.52 |

Error analysis was performed on the yaw rate and the sideslip angle response value after the simulation time of 12 s (road junction). The error analysis of yaw rate, sideslip angle and target value under the model prediction controller and the “feedforward + feedback” controller is shown in

Maximum | Maximum error | Average error | Root mean square error | |||||
---|---|---|---|---|---|---|---|---|

MPC | feedforward + feedback | Non-control | MPC | feedforward + feedback | MPC | feedforward + feedback | MPC | feedforward + feedback |

17.534 | 21.053 | 89.273 | 4.384 | 12.333 | 0.026 | 0.049 | 3.676 | 7.101 |

2.217 | 3.672 | 24.985 | 1.153 | 2.119 | 0.013 | 0.028 | 0.737 | 1.294 |

According to the

At the same time, according to the error analysis results, the maximum error, the mean error and the root mean square error of the vehicle yaw rate and the sideslip under the model predictive controller are all smaller than the error values under the “feedforward + feedback” controller. It can be seen from the

Meanwhile, according to the observation data of adhesion coefficient in

A double-layer structure direct yaw moment controller consisting of the additional yaw moment decision layer based on the MPC and the additional yaw moment distribution layer based on quadratic programming active set is designed. Considering the influence of road adhesion coefficient on stability control system, a joint observer of vehicle state parameters and road surface parameters is established. According to the low adhesion coefficient road and joint pavement simulation test results, the vehicle stability can get significantly improved with the vehicle stability controller based on the joint observer. In addition, compared with the “feedforward + feedback” controller, the yaw rate and sideslip angle’s response values of the distributed driven electric vehicle are closer to the steady-state value under the model prediction controller, which has more reliable control effect in terms of vehicle stability control. Future works will focus on the coordinated control of the line control system chassis.