Abstract
Aiming at the differences of vehicle chassis key subsystems influence on vehicle handling stability and effective acting regions, comprehensive considering of the nonlinear characteristic of the tires and the dynamic coupling among suspension, steering, and braking subsystems in vehicle chassis, the 14-DOF full vehicle model is built. Based on the control characteristic local optimum of each subsystem, multilevel hierarchical control theory is adopted and the vehicle stability coordinated control system including organization, coordination, and execution level is established. Using sliding mode control theory and the inverse tire model, the generalized target forces and moments from organization level are translated into the tire sideslip angle and slip ratio. And then, based on the principle of functional allocation, the control functions of each subsystem are coordinated and the function decoupling of vehicle chassis complex system is realized. The Matlab/Simulink platform is used and the full vehicle stability coordinated control system is simulated. The results show that the full vehicle coordinated control system based on multilevel hierarchical control theory can improve the vehicle stability preferably than the subsystem combined control and uncontrolled system.
1. Introduction
Vehicle chassis is a complex system which consists of steering, braking, and suspension subsystems. In order to improve the active safety, operation stability, and riding comfort of vehicle, many advanced control subsystems have appeared since the beginning of 1980s [1–5]. For the full vehicle, due to nonlinear characteristics of the tires, there is a serious dynamic coupling relationship in the vehicle longitudinal, lateral, and vertical directions. Because the active control subsystems in vehicle chassis have different objectives, this will lead to the function conflict, interference, or making the control target lack comprehensiveness.
In recent years, with the further focus on vehicle instabilities in extreme conditions and the development of the microelectronic technology and modern control theory, the vehicle chassis subsystems coordinated control have become a hot topic in the field of vehicle active safety control. Karbalaei proposed to integrate the control of active front steering (AFS) and direct yaw moment control (DYC) by the fuzzy control method; a better control effect is achieved by coordinated control wheel steering angle and yaw moment [6]. Hwang et al. studied the integrated control of AFS and ESP by supervised control strategy [7]. Ting and Lin studied integrated control of ABS and active suspension by the back stepping control [8]. Chen et al. researched on integrated control of ASS and AFS, ASS and ABS respectively by predictive control, interference suppression control and optimization control. The vehicle handling stability and riding comfort are improved significantly [9–11]. Although the integrated control of the vehicle chassis had done a lot of researches and has made full achievements, the integrated control of subsystems is achieved by the multiobjective optimization algorithm. Due to the fact that the vehicle system dynamic model is complex relatively, the integration of control strategy is very high and the controller is more difficult to design. The control system lacks flexibility and is not conducive to system expansion.
In this study, comprehensive considering of the nonlinear characteristic of tires and dynamic coupling among suspension, steering, and braking subsystems in vehicle chassis, based on the control characteristic local optimum of each subsystem, the multilevel hierarchical control theory is adopted and the vehicle stability coordinated control system including organization, coordination, and execution level is established. According to the driver input, real-time vehicle driving condition, and ideal reference model, the upper organization level controller can get generalized target control force and moment which is the vehicle stability control needed. Based on the principle of function allocation, the control functions of subsystem are distributed by the coordination level controller and the executive level controller executes the control instructions, which are sent by coordination level. By the simulation analysis, the effectiveness of coordinated control system is verified.
2. Vehicle Nonlinear Dynamic Model
2.1. Full Vehicle Dynamic Model
Taking the steering, braking, and suspension system as the vehicle chassis key subsystems, the 14-DOF full vehicle dynamic models which contain the vehicle longitudinal, lateral, and vertical dynamics, vertical jump and the rotation of the wheels are established. According to the actual situation of vehicle system, the vehicle dynamic model assumptions and simplifications are listed as follows:(I)assuming that the full vehicle consists of chassis, body, and tire, the vehicle quality is simplified for the sprung mass and four unsprung masses which are connected flexibly;(II)assuming that the tires and the road are real-time contact, ignoring the impact of the vehicle air resistance, road slope, and inclination, the tire location parameters and the suspension deformation on handling stability are not considered.
Combining the force analysis of Figures 1 and 2, according to the Newton’s second law, the vehicle nonlinear dynamics equations are shown as where and () are the tire longitudinal and lateral force, respectively, which are calculated by magic formula of nonlinear tire model. is vertical loads of the suspension. The vertical loads of the suspension are In the equations above, are vehicle longitudinal velocity, lateral velocity, yaw angle, pitch angle, roll angle, steering wheel angle, and the wheel angle speed separately; is the quality of full vehicle; is sprung mass; is unsprung mass; are rotary inertia of around the axis separately; is the rotary inertia of wheel; are suspension stiffness and damping; is the tire radial stiffness; is tire damping; is the vertical displacement of vehicle body; is the vertical displacement of suspension; is the vertical displacement of tires; , are the front and rear suspension roll stiffness; is the wheel effective rolling radius; and are distances from centroid to the front and rear axle; , are the distances of front wheel and rear; is height of center of mass; is the roll center height; is pitch center height; , are tire longitudinal and lateral force separately; is the wheel braking torque; is suspension vertical load; is the semiactive suspension control force.
2.2. Nonlinear Dynamic Model of the Tire
The magic formula of tire model is adopted in this study, which can express the longitudinal and the lateral force accurately by the same forms of the formula; the general expression of magic formula as follows [12]: where is the curve shape factor; is the peak; is the stiffness factor; is curvature factor; is horizontal shift; is vertical shift. When represents slip ratio ,is longitudinal force of the tire; when represents the tire sideslip angle , is lateral force . Figure 3 is original graph of the magic formula.
The longitudinal and lateral forces of the single condition are calculated by Formula (2). According to the theory of friction elliptic, the longitudinal force and lateral force are calculated by Formula (4) joint in the cornering and braking conditions. Figure 4 shows the tire dynamic coupling characteristics under joint working condition. Consider where ; .
3. Identification of the Vehicle Stable Driving
Yaw rate and sideslip angle are important common quantities to represent the driving stability of the vehicle. Ignoring the difference of steering angle and the tire force between the left and right front wheels, the 2-DOF vehicle lateral dynamic model is simplified as follows:
is yaw moment of external effect; ,
According to state equation (5) of the reference model, the characteristic equation is obtained by , where
According to the Hurwitz stability criterion, constant and a coefficient are greater than zero to make system stabilize; that is, , . Apparently ; therefore, is only considered. Due to the fact that its denominator is nonnegative, stability condition of system is obtained by the numerator greater than 0. Consider where is the vehicle characteristic velocity ().
When the vehicle steering radius is a certain value, the front wheel steering angle is ; when the lateral acceleration is close to zero (), the front wheel steering angle is ; when it has a lateral acceleration, then the steering angle ratio of the steering wheel is
According to Formulas (9) and (10), the state identification and stability criterion of the vehicle are achieved in different driving conditions as shown in Table 1. Where is the limit value of front wheel steering; , are longitudinal acceleration and its expectations; is the expectations yaw rate.
4. Vehicle Chassis Based on Multimodel and Multilevel Hierarchical Coordination Control
According to the driving condition identification and stability criterion, we take the vehicle driving stability in limiting conditions as the control target. Based on the multilevel hierarchical control theory, the multiple subsystems closed loop coordinated control system of vehicle chassis is established as shown in Figure 5.
4.1. Organizational Level Controller Based on Sliding Mode Variable Structure Control
In the limiting conditions, the main goal of vehicle driving stability control is the yaw rate and sideslip angle can well track the ideal reference value of the mode, meanwhile, keeping the wheel slip ratio near the corresponding slip ratio of the peak adhesion coefficient [13–16]. Therefore, according to the deviation between the actual value of driving condition and the ideal value of reference model, using the sliding mode variable structure control theory, the organization level controller is designed and the generalized target control force and torque are obtained so as to maintain vehicle driving stability. The generalized target control forces are converted into ideal slip angle and slip ratio of the tires by the tire inverse dynamics model. They are all the ultimate control targets of each subsystem in the multiple-model coordination control.
For the nonlinear dynamics model of full vehicle, Formula (1) is simplified appropriately and the state variables are selected. Consider
Then, the related equations to characterize vehicle stability in Formula (1) can be expressed as follows:
Based on the sliding variable structure control theory, the switching function is selected as follows: where is control variable error; is the integral of control variable error; the parameter . The main objective of the integral term is to limit the steady state error of the control variable.
Applying the constant speed reaching law, we get
Then, the differential of Formula (13) is
Combining Formula (13) with Formula (15), we get
Let , in order to further eliminate the high-frequency chattering of the control input, using instead of saturation function. Consider where is the boundary layer thickness. Then, the generalized target control force and torque to achieve vehicle stability control are shown in the following formula:
Because the wheel longitudinal and lateral force are functions about slip ratio and slip angle of tires, according to the tire inverse model [17, 18], the longitudinal target control force and the lateral target control force are transformed into the tire longitudinal target slip ratio and sideslip angle .
Assuming the generalized target control forces , , are shown in the following equation: where is tire force vector; is matrix of the tire force. Consider
Suppose is the middle control input. It includes the sideslip angle and slip ratio of four tires. Consider where are longitudinal slip ratio and are sideslip angle of four tires.
The control inputs of tires including the front wheel steering angle and the control torque are obtained by controlling the steering angle of the active steering system and longitudinal slip ratio of the braking system.
With Formula (19), the matrix of tire force is as follows:
Because the matrix of tire force is not a square matrix, the assigned tire forces are not obtained directly. So, using the unconstrained optimization, the generalized target control forces for tires can be distributed.
From the nonlinear dynamic model of tires, the ideal control forces or torques are got from organizational level in the hierarchical control of vehicle chassis [4]. It is a nonlinear function about the control system configuration parameters and control input elements as follows: where ; are vertical loads of the tire; is road friction coefficient; is steering angle.
Formula (23) is linearized about working identification point as follows:
In Formula (25), is the Jacobian about the point ; that is, it is a working matrix of operation system, which is obtained by the tire model of magic formula.
Let the incremental of middle control input be
At the th sampling time, the vehicle control force is generated by the four wheels, and it can be approximated as follows:
According to the control objectives and constraints of the tire force distribution, the objective function is proposed as follows: where , , and are weight matrices which represent force tracking error, control input increment, and control input amplitude, respectively.
The difference between vehicle target control force and current control forces is Then,
Plug Formulas (28) and (31) into Formula (29), with the minimum objective function; let ; then, the optimal control variable increments of tire can be achieved as follows:
Then, the target control input for tire actuator is assigned as follows:
Translating the longitudinal target control force into the tire slip ratio, it can be realized by ABS and SAS active control. At the same time, changing the lateral control target force into the tire sideslip angle, it can be realized by AFS system. Vehicle yaw generalized target torque can be realized by coordinated control of steering and braking systems. The vertical, roll, and pitch generalized objective control forces and torques , , and are translated into equivalent control force of four semiactive suspensions, and they participate in coordinated control among the other subsystems and improve the vehicle driving stability and safety.
4.2. Coordination Level Controller Based on Multimodel Hierarchical Control
According to the vehicle driving state identification and the stability judgment rules as shown in Table 1, the vehicle of each subsystem is coordinated control and the vehicle real states are identified and decided. Then, the main control objectives are confined to the corresponding working conditions. Meanwhile, they are an important basis for subsystem task assignment. Consider the following.
(I) When the vehicle is in stable condition, according to the requirements of the actual driving status, the vehicle active subsystems of AFS, ABS, and SAS are controlled with each optimal control.
(II) When the vehicle is in an unsteady operation such as the separation coefficient road, high-speed steering, and braking, because of the AFS control performance and structure limitation, the tracking control of larger yaw moment cannot achieve well. At this moment, ABS can put the different braking torque to the wheel to obtain larger yaw moment. Meanwhile, SAS takes part in the vehicle stability control to control the vehicle vertical load. According to the control strategy of coordination level controller and the generalized control target of organizational level controller, the control goals are assigned to ABS, AFS, and SAS.
(III) When the vehicle chassis subsystems are coordinated control, the additional yaw moment generated by ABS through wheels assist brake is
Then, the goal yaw moment achieved by AFS is where is a weight coefficient. When ABS controls independently, ; when AFS works independently, ; when the vehicle chassis subsystems are coordinated control, in order to prevent generating great shock in the switch process, is calculated by continuous function Sigmoid. The expression of Sigmoid function is shown as follows: where , are constant greater than zero, and they represent the control shape and position of function, respectively.
According to the lateral acceleration, the additional control force of the SAS system is given as follows: where the parameter is adjusted with the lateral acceleration to achieve coordination control with other subsystems.
4.3. Execution Level Controller Based on Multimodel Hierarchical Control
4.3.1. AFS Controller Based on Robust Control
According to the performance requirements of AFS steering system and the vehicle steering stability control target, the robust control theory is used to study its control characteristics [7, 19]. The standardized form of AFS steering system with control is shown in Figure 6.
In Figure 6, are external input variables; where is road noise; is measuring noise; is steering wheel control torque; are output variables of the system performance evaluation; is the measured variable; the control input signal is electromagnetic torque. is transfer function from the input signal , to the output signal , ; is the controller.
In order to ensure drivers road-feel and the sensitivity of EPS, let the weighting function , be as follows:
In order to make controller of AFS system meet the AFS assisted steering requirements and the vehicle handling stability control objectives, the choice principles about system evaluation output variables , and are complied with the following rules:(I)make minimal, to ensure the offset between the actual motor assist torque generated by motor and the ideal motor assist torque is minimum;(II)make minimal to reduce the interference of the road surface and ensure a good steering feel;(III)make minimal to access the vehicles real yaw rate to the ideal yaw rate and ensure a good handling stability.
4.3.2. SAS Controller Based on Fuzzy Control
Taking the body vertical acceleration and velocity, pitch angle acceleration and velocity, and roll angle acceleration and velocity as the controller inputs, where the vertical, pitch, and roll equivalent control force and moment are the controller outputs, three independent fuzzy controllers are designed. And then, by pseudoinverse calculation, the active control forces of four suspensions are achieved. By this way, the vehicle riding comfort and handling stability are improved. The structure of semiactive suspension fuzzy controller is shown in Figure 7.
Combine the vehicle system dynamics equations in Formula (1), . Using the pseudoinverse operation, the goal control force of four semiactive suspensions is given as follows:
4.3.3. ABS Controller Based on Fuzzy Control.
Based on the fuzzy control theory, taking the slip ratio and vehicle deceleration as the controller inputs, adjusting torque as the output, the ABS fuzzy controller is designed. The structure of ABS controller is shown in Figure 8.
5. Simulation and Analysis
Applying the simulation software of Matlab/Simulink, taking a vehicle as an example, the control strategy of vehicle chassis integrated control based on multimodel and multilevel hierarchical control is simulated. The parameters of vehicle are shown in Table 2.
5.1. Simulation and Analysis in Single Line Condition
Assuming the vehicle velocity is 30 m/s, driving at class road surface, the front wheel steering input is a sine signal, in which frequency is 0.5 Hz and amplitude is , as shown in Figure 9.
Figures 10, 11, 12, and 13 are showing that the response curves of the vehicle when yaw-rate, the sideslip angle, roll angle, and pitch angle, respectively. As shown in these figures, using different coordination control strategy between ABS, AFS, and SAS can improve the vehicle lateral stability. For ABS and SAS combined control, the vehicle stability control effect is poor relatively. This is because the main target of SAS vertical auxiliary role is to enhance the braking safety in this control strategy, that is, the vertical load and braking torque with the same phase changing. In addition, the simulation results also show that AFS can improve the vehicle handling stability effectively when the yaw rate is large, also the yaw rate amplitude jitter is minimal when ABS, AFS, and SAS are comprehensive coordination control. The change trend of vehicle side-slip angle is the same as the yaw rate. Figure 14 shows that the vehicle body vertical acceleration amplitude is reduced with coordinated control method. It indicates that ABS, AFS, and SAS subsystem integrated control based on multimodel and multilevel hierarchical control not only improves the vehicle handling stability, but also it improves the riding comfort.
5.2. Simulation and Analysis with Step Steering Angle
Assuming the vehicle velocity is 30 m/s, driving at B class road surface, the front wheel steering input is a step signal, in which amplitude is , and after 0.3 s, the vehicle starts braking. When the vehicle is in emergency turning and braking condition, the multisubsystem coordinated control strategy is simulated and the results are shown in Figures 15, 16, 17, 18, 19, 20, 21, and 22.
Figures 15 to 18 are showing that the response curves of the vehicle when yaw-rate, the sideslip angle, roll angle and pitch angle, respectively. As shown in these figures, the vehicle handling stability with the vehicle chassis subsystem of ABS and SAS joint control is poor relatively. The reason for this is the vehicle has worked in the unstable state in the condition of steering and braking. The lateral auxiliary control function of ABS is limited. While, the main target of SAS is to improve the vehicle braking safety and comfort. In addition, the simulation results indicate that AFS can improve the vehicle handing stability and reduce the vehicle roll and pitch angle obviously. The yaw rate amplitude jitter is minimal with multisubsystem coordinated control and enables the vehicle to reach steady state quickly.
Figures 19 to 22 represent the response curves of vehicle braking distance and the longitudinal slip ratio of front and rear wheels, respectively. The figures show that the braking distance is reduced, the braking time is shortened, and the slip ratio keeps near the target value 0.2 with the coordinated control strategy of ABS, AFS, and SAS. This is due to the wheel vertical load consistent with the braking torque with SAS control. Therefore, it improves the braking ability of the full vehicle. Figure 22 shows that the vertical acceleration of vehicle body is reduced. It indicates that the vehicle chassis integrated control based on multimodel and multilevel hierarchical control can improve the vehicle handling stability and the riding comfort effectively.
6. Conclusions
(I)Based on the multilevel hierarchical control theory and function assignment principle, the generalized target control force or torque for controlling the vehicle stability is transformed into the control target of vehicle chassis key subsystem with the tires inverse dynamics model and pseudoinverse calculation. So, the function decoupling of the vehicle nonlinear dynamics coupling and the vehicle chassis multimodel coordinated control in the extreme conditions are realized.(II)The vehicle chassis multimodel and multilevel hierarchical control strategy can suppress the interference among multisubsystems to some extent and improve the vehicle handling stability, active safety, and riding comfort effectively.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
Authors would like to thank anonymous reviewers and the editor for their valuable comments and the National Natural Science Foundation of China (no. 51278514) and the Special Fund of Chongqing Education Commission (KJ120415) and the Special Fund of Shaanxi Education Department (12JK0693) for their sponsorship.