This paper focuses on the problem of tracking control for vehicle lateral dynamic systems and designs an adaptive robust controller (ARC) based on backstepping technology to improve vehicle handling and stability, in the presence of parameter uncertainties and external nonlinearities. The main target of controller design has two aspects: the first target is to control the sideslip angle as small as possible, and the second one is to keep the real yaw rate tracking the desired yaw rate. In order to compromise the two indexes, the desired sideslip angle is planned as a new reference signal, instead of the ideal “zero.” As a result, the designed controller not only accomplishes the control purposes mentioned above, but also effectively attenuates both the changes of vehicle mass and the variations of cornering stiffness. In addition, to overcome the problem of “explosion of complexity” caused by backstepping method in the traditional ARC design, the dynamic surface control (DSC) technique is used to estimate the derivative of the virtual control. Finally, a nonlinear vehicle model is employed as the design example to illustrate the effectiveness of the proposed control laws.
Because of the large number of traffic accidents occurring daily, vehicle safety control has been a hot research topic. In recent years, vehicle lateral dynamic control has been studied widely for contributing to the car’s handling and keeping vehicle ride safe. Especially the yawmoment control is proved to be an important approach to improve safety performance and has a great potential to meet the requirements demanded by users. To this end, a considerable amount of research has been carried out [
In vehicle dynamic systems, inevitable uncertainties often emerge, which will bring considerable difficulties in the process of controller design. For example, because of the change of the number of passengers or the payload, vehicle load is easily varied, which will accordingly change the vehicle mass as a varying parameter. On the other side, the moment of inertia is usually an unknown parameter. Besides, since the yawmoment control relies on the tire lateral force and the tire force strongly depends on tire vertical load, which is very sensitive to vehicle motion and environmental conditions, the tire cornering stiffness inevitably obtains uncertainties that need to be coped with. Roughly speaking, the abovementioned uncertainties can be classified into two categories: parametric uncertainties (e.g., car body mass for vehicle dynamic control) and general uncertainties (coming from modeling errors and external disturbances). To handle this situation, a number of control techniques have been proposed, such as robust
The main target of yawmoment control can be divided into two aspects. The first target is to control the sideslip angle to converge to zero, and the second one is to keep the real yaw rate tracking the desired trajectory. However, these two requirements are conflicting, and it is difficult to achieve both these two indexes simultaneously, especially for the systems with uncertainties. Therefore, a compromise of the requirements must be reached. In this paper, the problem of tracking control for vehicle lateral dynamic systems is investigated, and a backsteppingtype adaptive robust controller is designed to improve vehicle handling and stability, in the presence of parameter uncertainties and external nonlinearities. In order to compromise the two tracking indexes, the desired sideslip angle is planned as a new reference signal, instead of the ideal “zero.” As a result, the designed controller not only accomplishes the required control purposes, but also effectively attenuates both the changes of vehicle mass and the variations of cornering stiffness. In addition, to overcome the problem of “explosion of complexity” caused by backstepping method in the traditional ARC design, the DSC technique is used to estimate the derivative of the virtual control. Furthermore, the adaptive law is designed to estimate the real value of the moment of inertia. Finally, a nonlinear vehicle model is employed as the design example to illustrate the effectiveness of the proposed control law.
The rest of this paper is organized as follows. The problem to be solved is formulated mathematically in Section
In this paper, a “bicycle model” is used to model the dynamics of the car, as shown in Figure
Simplified vehicle ‘‘bicycle model.”
The equations of the vehicle’s handling dynamics in the yaw plane are given as
For the lateral dynamics systems in (
In this section, an adaptive robust controller is presented to track the desired trajectories of the lateral dynamic systems with uncertain parameters and external disturbances, and the detailed process of controller design is given as follows.
Choose
Starting with the equation of tracking error, we have
Substituting
Substituting (
Synthesize an adaptive robust control law for
Differentiating the error dynamics for
Then, substituting (
Structure diagram of the adaptive robust controller design.
If
In general (i.e., the system is subjected to parametric uncertainties, unmodelled uncertainties, and external disturbances), both the tracking errors
If, after a finite time, the system is subjected to parametric uncertainties only (i.e., all the disturbances vanish after a finite time), both the tracking errors
Firstly, the proof of statement (a) is given. Choose a positive definite function as
If, after a finite time, the system is only subjected to parametric uncertainties, the dynamic systems can be written as
As stated above, our main target has two aspects: the first target is to control the sideslip angle to converge to zero, and the second one is to keep the real yaw rate tracking the desired trajectory. However, these two requirements are conflicting, and it is difficult to achieve both these two indexes simultaneously, especially for the systems with uncertainties and nonlinearities. Therefore, a compromise of the requirements must be reached. To handle this situation, in this paper, the desired sideslip angle is replanned as follows:
Next, we will give the proof that
Actuator saturation appears frequently in engineering systems, which is also a source of performance degradation and the closedloop system instability. Roughly speaking, all actuators of physical devices are subject to amplitude saturation. Although, in some applications, it may be possible to ignore this fact, the reliable operation and acceptable performance of most control systems must be assessed in light of actuator saturation. From the analysis above, it is known that all the signals are bounded within the known ranges, and thus the actuator saturation constraints will be guaranteed as long as the initial values and tuning gains (
From the design process above, it can been seen that it is quite complicated to obtain
Differentiating the error dynamics for
Before the main result is given, the following definitions are given. For any
If the virtual input, control input, and adaptive law are designed as (
Define a positive function as shown in (
Next, we will show that the steadystate tracking errors
In this section, we provide an example to illustrate the effectiveness of the proposed approach. The vehicle model parameters are listed in Table
The model parameters of vehicle lateral systems.
Parameter  Value 


1100–1300 kg 

1700 kgm^{2} 

20000 N/m 

1 m 

0.03 

1600 kgm^{2} 

1500 kgm^{2} 

20000 N/m 

1.5 m 

20 m/s 
The initial state values are assumed as zeros, and
The controller parameters of adaptive robust controller.
Parameter  Value 


1000 

10 

10 

10 

10 

10 

10 
For checking the vehicle lateral dynamic performance in terms of the change of vehicle mass, three different masses are tested to illustrate the effectiveness of robust control:
closedloop systems with the vehicle mass
closedloop systems with the vehicle mass
closedloop systems with the vehicle mass
openloop systems without controller.
In order to illustrate the effectiveness of the proposed control law, in this paper, it is assumed that the steering angle
The sideslip angle responses of the openloop system, closedloop systems with designed ARC controller for different vehicle masses are compared in Figure
The sideslip angle responses of the openloop system, closedloop systems with designed ARC controller for different vehicle masses.
The local enlargement of Figure
Figures
The yaw rate responses of the openloop system, closedloop systems with designed ARC controller for different vehicle masses.
The local enlargement of Figure
The tracking errors between real yaw rate and desired yaw rate of the openloop system, closedloop systems with designed ARC controller for different vehicle masses.
The local enlargement of Figure
The simulation result in Figure
The moment of inertia about the yaw axis,
The control inputs (the yaw moment,
Control inputs,
Abrupt demand signal can be generally assumed as discrete events of relatively short duration and high intensity, and the corresponding function is given by
The sideslip angle responses of the openloop system, closedloop systems with designed ARC controller for different vehicle masses.
The yaw rate responses of the openloop system, closedloop systems with designed ARC controller for different vehicle masses.
The tracking errors between real yaw rate and desired yaw rate of the openloop system, closedloop systems with designed ARC controller for different vehicle masses.
Control inputs,
The time histories of yaw rate for both openloop systems and closedloop systems with ARC controller in the case of different vehicle masses are shown in Figure
In this paper, an adaptive robust control strategy has been proposed for vehicle lateral dynamic systems to improve vehicle handling and stability, where parameter uncertainties and external nonlinearities are considered in a unified framework. In order to manage two conflicting indexes, a compromise of the requirements is made, and the desired sideslip angle is replanned as a new reference signal, instead of the ideal “zero.” To the end, the designed controller can not only accomplish the required control purposes, but also effectively attenuate both the changes of vehicle mass and the variations of cornering stiffness. In addition, to overcome the problem of “explosion of complexity” caused by backstepping method in the traditional ARC design, the dynamic surface control (DSC) technique has been used to estimate the derivative of the virtual control. Finally, a nonlinear vehicle model is employed as the design example to illustrate the effectiveness of the proposed control law.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported in part by the selfplanned task (no. SKLRS201308B) of the State Key Laboratory of Robotics and System (HIT), the Fundamental Research Funds for the Central Universities (no. HIT.NSRIF.2015032), China.