Robust Tracking Control for Vehicle Lateral Dynamics with Uncertain Parameters and External Nonlinearities

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 backsteppingmethod 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.


Introduction
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 yaw-moment 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 [1][2][3][4][5][6][7][8][9][10][11][12], and many vehicle lateral control approaches have been proposed, based on various control techniques, such as fuzzy logic control [13][14][15],  ∞ control [16], adaptive control [17][18][19][20], and nonlinear robust control [21,22].Such strategies could considerably enhance vehicle handling and active safety during severe driving maneuvers and, at the same time, allow the driver to keep control of the vehicle when the vehicle is at the physical limit of adhesion between the tires and the road.
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 yaw-moment 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  ∞ or  2 control [23][24][25][26], optimal control [27], fuzzy control [28,29], sliding mode control [30,31], neural network control [32][33][34][35][36][37][38], adaptive control [39], and fault tolerant control [40,41].Besides, during the past decade, a mathematically rigorous nonlinear adaptive robust control theory has also been developed to lay a solid foundation for the design of a new generation of controllers which will help industry build modern machines of great performance and high intelligence [42][43][44].This ARC approach can be both adaptive to the uncertain parameters and at the same time robust against the external disturbances, which is suitable for the systems with uncertainties and the external disturbances, for example, vehicle dynamic systems and robot manipulation systems [45].
The main target of yaw-moment 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 backstepping-type 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 2, and controller design is presented in Section 3, where both the traditional and improved ARC designs are presented.Section 4 provides a design example to illustrate the effectiveness of the proposed control laws and some concluding remarks are given in Section 5.
Nomenclature.The following nomenclature is used throughout the paper: • is used to denote the estimate of •, • is used to denote the parameter estimation error of •, and • max , • min are the maximum and minimum values of •() for all , respectively.

Problem Formulation
In this paper, a "bicycle model" is used to model the dynamics of the car, as shown in Figure 1.In this figure,  represents the car chassis;   is the moment of inertia about the yaw axis through the center of gravity (CG).The front and rear axles are located at distances   and   , respectively, from the vehicle CG.The front and rear lateral tire forces   and   depend on slip angles   and   , respectively, and the steering angle  changes the heading of the front tires. and  stand for the sideslip angle and yaw rate, respectively, and   is the external yaw moment.In this paper, it is assumed that the vehicle velocity V is a constant and the steering angle is small.The equations of the vehicle's handling dynamics in the yaw plane are given as The front and rear lateral tire forces in (1) can be given as where are the lateral forces at the front and rear wheels in case that the tires operate in the linear region and  1 (),  2 () are the additional nonlinear terms and are bounded.  and   are the cornering stiffness for the front and rear tires, respectively, and the front and rear slip angles are defined as Define the nonlinear disturbance uncertainty Δ 1 () = (( Shock and Vibration 3 where and  = 1/  ∈ [ min  max ] with  min = 1/  max ,  max = 1/  min being the uncertain parameter. Problem 1.For the lateral dynamics systems in (1), synthesize a control input  to control the sideslip angle as small as possible, and keep the real yaw rate tracking the desired trajectory, in the presence of parametric uncertainties and external disturbances.

Control Law Synthesis
3.1.Adaptive Robust Controller Design.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.
Step 1. Choose () as the virtual control and design a desired function (), such that if () = (), then the tracking error  1 =  −   is guaranteed to converge to zero or be bounded, where   is the reference trajectory.
Starting with the equation of tracking error, we have Let  2 be an error variable representing the difference between the actual and virtual control of (7); that is,  2 =  − .Thus we can rewrite (7) as Then, the desired virtual control  can be proposed as where is used to achieve a model compensation, is the stabilizing feedback term,  1 is a positive design parameter, and   is a robust control law designed to satisfy the following conditions: where  1 is a design parameter which can be arbitrarily small.Basically, condition 1 of (12) shows that   is synthesized to dominate the model uncertainties coming from uncertain nonlinearity Δ 1 , and condition 2 is to make sure that   is dissipating in nature so that it does not interfere with nominal process of control part   and   .Then, the robust control part   can be chosen as where Then, we will show how   in (13) guarantees conditions 1 and 2 in (12).
Substituting   into condition 1 in ( 12), we have Substituting ( 9)-( 11) and ( 13) into ( 8) results in Step 2. Synthesize an adaptive robust control law for , so that the error  2 is bounded in the presence of uncertain parameter  and external disturbances Δ  ,  = 1, 2.
Differentiating the error dynamics for  2 =  −  results in where () =  2  +  2  +  2  + .Design the control input where  2 is a positive design parameter and α  , α  respect the certain and uncertain parts of α   = − 2  2 is a robust control law designed to satisfy the following conditions: where  2 is a design parameter which can be arbitrarily small.To satisfy the conditions above, one can choose the nonlinear control gain  2 as where ℎ 2 () is a function which satisfies θ is the estimation of , which is chosen as the projection type with the following form: and   > 0 is a tunable gain and  = () 2 .The standard projection mapping Proj θ( −1  ) is introduced as This projection mapping Proj θ1 () guarantees that the parameter estimate is always within the known bounds, that is,  min <  <  max , and θ( −1  Proj θ() − ) ≤ 0, for all , which enables one to show that the use of projection modification to the traditional integral type adaptation law does not interfere with the perfect learning capability of the original integral type adaptation law.
Then, substituting ( 17)-( 23) into ( 16) results in The structure diagram of the adaptive robust controller design is given in Figure 2, and based on the above processing of controller design, we have the following theorem.
Theorem 2. If  is designed as (17)- (19), and the adaptive law is chosen as (23), then the following results hold.
(a) In general (i.e., the system is subjected to parametric uncertainties, unmodelled uncertainties, and external disturbances), both the tracking errors  1 and  2 are bounded and, specifically, defining  0 = min( 1 ,  2 ), the steady-state output tracking errors  1 and  2 are bounded by (b) 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  1 and  2 will asymptotically converge to zero.
Proof.Firstly, the proof of statement (a) is given.Choose a positive definite function as whose derivative is given as After defining  0 , we have which can further result in Inequality (29) implies that (∞) ≤ ( 1 +  2 )/2 0 , which guarantees Therefore, the proof of statement (a) is completed.If, after a finite time, the system is only subjected to parametric uncertainties, the dynamic systems can be written as Choose a positive definite function as and then we have Noticing the property of the projection mapping we have Integrating both sides of inequality V  ≤ 0 from 0 to  results in which implies  1 ,  2 , θ ∈  ∞ , and thus , ,  ∈  ∞ .Therefore, and thus By using Lyapunov-like lemma, we have V  → 0 as  → ∞, which means that the tracking errors  1 and  2 converge to zero asymptotically.The proof is completed.Remark 3. 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: where and   is a positive constant.Using this trajectory in (39) to replace the desired trajectory zero, we can guarantee that the virtual input  converges to the desired yaw rate   = , where in which   is a stability factor.
Next, we will give the proof that  will converge to the desired yaw rate   , using the designed   .Defining  =  −   (42) and choosing a positive function as we have Shock and Vibration Substituting (39) into (44) results in which means that  will converge to zero as  → ∞, and it further implies that  coverages to the desired yaw rate   .

Remark 4.
Actuator saturation appears frequently in engineering systems, which is also a source of performance degradation and the closed-loop 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 ( 1 ,  2 ,  1 ,  2 ) are adjusted properly.

Adaptive Robust Controller Design via DSC Technique.
From the design process above, it can been seen that it is quite complicated to obtain α , especially to split α into the known part α  and the unknown part α  .To overcome the problem of "explosion of complexity" caused by backstepping method in the traditional ARC design, the dynamic surface control technique is used to estimate the derivative of the virtual control α .Therefore, the design process for Step 2 is modified as follows.

Modified
Step 2. Let  pass through a first-order filter with time constant  2 , which means Defining the estimate errors as  2 =  − ,  2 =  − , the derivative of  1 can be rewritten as Differentiating the error dynamics for  2 =  −  results in After choosing the adaptive robust controller  as we can obtain the error dynamic systems as where   = − 2  2 is a robust control law designed to satisfy the following conditions: where  2 is a design parameter which can be arbitrarily small.Similarly, we can choose the nonlinear control gain  2 as  2 = (ℎ which implies Before the main result is given, the following definitions are given.For any  > 0, define where Obviously, Π is a compact subset; hence there must be a point corresponding to the supreme value of  2 in Π.We denote this supreme value as  2 ; that is, Noting that we have Based on conditions (58), we have where which implies that  1 ,  2 , and  2 are bounded as Next, we will show that the steady-state tracking errors  1 ,  2 , and  2 can be made arbitrarily small.Define a positive definite function   () satisfying In order to make a contradiction, we assume that there exists  > 0 so that, when  > , where  =  1 + 2 +(1/4) 2 2 and  is a positive constant which can be set arbitrarily small.On the other hand, the following inequality is true: Integrating both sides of the above inequality from zero to any  > 0, we obtain Because   () is bounded, we have is bounded as well.It is obvious that is bounded, and then, based on Barbalat's lemma, we have lim which implies Therefore, the tracking errors  1 ,  2 , and  2 are uniformly ultimately bounded and the steady-state tracking error can be made arbitrarily small by properly choosing tuning parameters.

A Design Example
In this section, we provide an example to illustrate the effectiveness of the proposed approach.The vehicle model parameters are listed in Table 1.
The initial state values are assumed as zeros, and (0) = 1/1500.The controller parameters are given in Table 2.
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: (S1) closed-loop systems with the vehicle mass  =  4.1.Periodic Demand Signal.In order to illustrate the effectiveness of the proposed control law, in this paper, it is assumed that the steering angle  = sin(), and the lumped nonlinearities The sideslip angle responses of the open-loop system, closed-loop systems with designed ARC controller for different vehicle masses are compared in Figure 3, and Figure 4 is the local enlargement of Figure 3 from −0.5 to 0.5 at yaxis.From these two figures, we can see that the response peak has been reduced substantially, by using the proposed controller, compared with the open-loop system without controller.Also, we can get the conclusion that the sideslip angle performance is still guaranteed at a high level, although there are changing vehicle masses, external disturbances, and other uncertainties.between the real yaw rate and the desired yaw rate.It can be seen that the closed-loop systems with the proposed ARC controllers can track the desired trajectories perfectly.By contrast, the open-loop systems without controller cannot do this work, and the tracking errors of the open-loop systems are very large.From Figures 5 and 6, we observe that the yaw rate response of the controlled system is better than the uncontrolled system, regardless of the change of vehicle mass.In particular, from Figures 7 and 8, the tracking errors are close to zero, and the yaw rates are all smaller than the corresponding uncontrolled system responses.The simulation result in Figure 9 characterizes the adaptation of the uncertain parameter   , and from this figure we can see that the moment of inertia about the yaw axis   does converge to the true value (1600 kgm 2 ), which achieves the estimation of the uncertain parameters.
The control inputs (the yaw moment,   ) for different cases plotted in Figure 10.The time histories of yaw rate for both open-loop systems and closed-loop systems with ARC controller in the case of different vehicle masses are shown in Figure 12, and Figure 13 is the corresponding responses of tracking errors between the real yaw rate and the desired yaw rate.From these figures, we can see that the proposed controller yields the least value of the maximum tracking error, compared with the open-loop system.Figure 14 shows the yaw moment,   , for different cases (changing masses).

Concluding Remarks
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.

Figure 2 :
Figure 2: Structure diagram of the adaptive robust controller design.

Figure 3 :
Figure 3: The sideslip angle responses of the open-loop system, closed-loop systems with designed ARC controller for different vehicle masses.

Figure 12 :Figure 13 :
Figure 12: The yaw rate responses of the open-loop system, closedloop systems with designed ARC controller for different vehicle masses.

Theorem 5 .
If the virtual input, control input, and adaptive law are designed as (9), (49), and (23), respectively, then, for any initial states in Π, there exist positive parameters  1 ,  2 ,  2 ,   ,  1 , and  2 satisfying Define a positive function as shown in (56).The derivative of   () is such that the tracking errors  1 ,  2 , and  2 are uniformly ultimately bounded and the steady-state tracking error can be made arbitrarily small.Proof.

Table 1 :
The model parameters of vehicle lateral systems.

Table 2 :
The controller parameters of adaptive robust controller.