2. Bosch Automotive Products(Suzhou) Co., Ltd, Suzhou 215021, China;
3. Huachen Auto Group, Shenyang 110141, China
With the rapid development of automobile industry, handling stability related to vehicle safety has attracted more attention. The four wheel steering (4WS) vehicles have been studied in order to improve the maneuverability of vehicle at low speed and enhance the stability at high speed. As rear wheels participate in steering process, the transient response of a vehicle can be improved [1], [2].
Traditional 4WS vehicles mainly refer to automobiles with active rear wheel steering systems, and their control types include both feedforward control and feedback control [3]. Early 4WS vehicles intend to achieve zero sideslip angle to reduce the phase difference between lateral acceleration and yaw rate. This kind of 4WS vehicles usually use a feedforward control method. Sano proposes a ratio between rear wheels and front wheels, which can guarantee a zero sideslip angle during steady steering [4]. This method is easy to implement, but it increases the time it takes for yaw rate and lateral acceleration to reach steady states. Thus, it does not perform well during transient motion. For 4WS vehicles using the feedback control method, two control strategies are presented. One also focuses on reaching a zero sideslip angle, the other focuses on tracking the reference yaw rate. The feedback control method has the characteristic of fast response, which can effectively reduce the effect of external disturbances. Quite a few modern 4WS control methods are established based on it. In [5], a set of linear maneuvering equations representing 4WS vehicle dynamics is used, which formulates the multiobjective vehicle controllers as a mixed
The development of the steerbywire (SBW) technique has made it possible for front and rear wheels to actively steer, simultaneously [11]. In this paper, motivated by this idea, the strategy of model reference control is proposed in order to achieve both the desired sideslip angle and the yaw rate. With the combination of feedforward control and feedback control, front wheel active steering and rear wheel active steering are used to achieve the desired performance of a reference vehicle model. In [12], a similar model reference control strategy has also been used, where a
The rest of this paper is organized as follows. 4WS vehicle models are built up in Section Ⅱ. Section Ⅲ describes the main results for 4WS control strategy. Disturbance observer based control scheme is illustrated in Section Ⅳ. The simulation results are carried out in Section Ⅴ. Section Ⅵ concludes the paper with a short summary.
Ⅱ. 4WS VEHICLE MODELSIn this section, 4WS vehicle models will be of interest. First a 3DOF 4WS vehicle model with a nonlinear tyre model is set up, which also takes the crosswind disturbance into consideration. Such a model captures the most relevant characteristics associated to lateral and longitudinal stabilization of 4WS vehicle. For simplicity and easy implementation, a linear 4WS vehicle model is used for the design of controllers.
A. Nonlinear 4WS Vehicle ModelA nonlinear 3DOF 4WS vehicle model is shown in Fig. 1. The body frame
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Fig. 1 The diagram of 3DOF vehicle model. 
Considering lateral, yaw and roll motions, the 3DOF vehicle model of the 4WS system is given by [5], [13]
$ \begin{equation*} mv(\dot{\beta}+r)+m_sh_s\ddot{\varphi}=\Sigma F_y \end{equation*} $  (1) 
$ \begin{equation*} I_z\dot{r}I_{xz}\ddot{\varphi}=\Sigma M_z \end{equation*} $  (2) 
$ \begin{equation*} I_x\ddot{\varphi}I_{xz}\dot{r}=\Sigma L_x \end{equation*} $  (3) 
where
As crosswind has some influences on dynamics of the 4WS system, the following equations are
$ \begin{equation*} \Sigma F_y =F_f+F_r+F_w \end{equation*} $  (4) 
$ \begin{equation*} \Sigma M_z =aF_fbF_r+F_wl_w \end{equation*} $  (5) 
$ \begin{array}{c} \Sigma L_x =m_sh_sg{\text{sin}}\varphi+m_sh_sv(\dot{\beta}+r){\text{cos}}\varphi\\ C_\varphi\dot{\varphi}k_\varphi\varphiF_wh_w \end{array} $  (6) 
where
The magic formula tyre model [14] is adopted here to determine the lateral force of front and rear wheels
$ \begin{equation}\label{magic} \begin{aligned} F_i=D{\text{sin}}\{C{\text{arctan}}[B(1E)\alpha_i+E{\text{arctan}}(B\alpha_i)]\} \end{aligned} \end{equation} $  (7) 
where
Assume that
$ \alpha_f = \beta\frac{ar}{v}+\delta_f\\ \alpha_r = \beta+\frac{br}{v}+\delta_r $  (8) 
where
The vertical load of each tyre includes both the static load and the lateral load transfer caused by the vehicle rolling motion
$ \left\{ \begin{array}{l} {F_{zfl}} = \frac{{mgb}}{{2L}} + \triangle{F_{zf}}\\ {F_{zfr}} = \frac{{mgb}}{{2L}}  \triangle{F_{zf}}\\ {F_{zrl}} = \frac{{mga}}{{2L}} + \triangle{F_{zr}}\\ {F_{zrr}} = \frac{{mga}}{{2L}}  \triangle{F_{zr}}. \end{array} \right. $  (9) 
The lateral load transfer is given by [5]
$ \left\{ \begin{array}{l} \triangle {F_{zf}} = \frac{{{a_y}}}{{{E_f}}}(\frac{{{m_s}{l_{rs}}{h_f}}}{L} + {m_{usf}}{h_{uf}}) + \frac{1}{{{E_f}}}(  {k_{\varphi f}}\varphi  {C_{\varphi f}}\dot \varphi )\\ \triangle {F_{zr}} = \frac{{{a_y}}}{{{E_r}}}(\frac{{{m_s}{l_{fs}}{h_f}}}{L} + {m_{usr}}{h_{ur}}) + \frac{1}{{{E_r}}}(  {k_{\varphi r}}\varphi  {C_{\varphi r}}\dot \varphi ) \end{array} \right. $  (10) 
where
In order to implement easily, it is reasonable to use a relatively simple vehicle model in the stage of controller design. Two degreeoffreedom vehicle model contains sideslip and yaw rate, which captures the most relevant dynamics of a 4WS vehicle. As is shown in Fig. 2, the coordinate frame is fixed at the vehicle's CG. According to Newton's second law, the vehicle dynamic equations can be written as
$ \begin{equation}\label{2dof1} \begin{aligned} &mv(\dot{\beta}+r)=F_f+F_r\\ &I_z\dot{r}=aF_fbF_r. \end{aligned} \end{equation} $  (11) 
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Fig. 2 The diagram of 2DOF vehicle model. 
While the tyre slip angle is small, the front and rear lateral tyre forces vary linearly with their slip angles
$ \begin{equation}\label{2dof2} \begin{aligned} F_f=k_f\alpha_f\\ F_r=k_r\alpha_r \end{aligned} \end{equation} $  (12) 
where
$ \begin{equation}\label{2dof4} \begin{aligned} mv(\dot{\beta}+r)=(k_f+k_r)\beta\frac{ak_fbk_r}{v}r+k_f\delta_f+k_r\delta_r\\ I_z\dot{r}=(ak_fbk_r)\beta\frac{a^2k_f+b^2k_r}{v}r+ak_f\delta_fbk_r\delta_r. \end{aligned} \end{equation} $  (13) 
From (13), the state space representation of the linear 4WS vehicle model can be expressed as
$ \begin{equation}\label{2dofstate} \dot{x}=Ax+Bu \end{equation} $  (14) 
where
$ \begin{equation*} \begin{aligned} &A=\begin{bmatrix}\frac{k_f+k_r}{mv}&\frac{bk_rak_f}{mv^2}1\\[2mm] \frac{bk_rak_f}{I_z}&\frac{a^2k_f+b^2k_r}{I_zv}\end{bmatrix}\\ &B=\begin{bmatrix}\frac{k_f}{mv}&\frac{k_r}{mv}\\ \frac{ak_f}{Iz}&\frac{bk_r}{I_z}\end{bmatrix}. \end{aligned} \end{equation*} $ 
Taking the external disturbances caused by the crosswind into consideration, (11) can be rewritten as
$ \begin{equation}\label{2dofdis} \begin{aligned} &mv(\dot{\beta}+r)=F_f+F_r+F_w\\ &I_z\dot{r}=aF_fbF_r+F_wl_w. \end{aligned} \end{equation} $  (15) 
In accordance with (14), the state space representation of (15) can be written as
$ \begin{equation}\label{2dofstate2} \begin{aligned} \dot{x}=Ax+B_{f}w_f+Bu \end{aligned} \end{equation} $  (16) 
where
The performance requirement of a 4WS vehicle is to improve the vehicle maneuverability at low speed and to enhance vehicle handling stability at high speed. The strategy of 4WS control is proposed in Fig. 3, where the 4WS control problem is simplified as a yaw rate and a sideslip angle tracking problem.
$ \begin{equation}\label{uinput} \begin{aligned} u=\begin{bmatrix}\delta_f\\\delta_r\end{bmatrix} =u_f+u_e. \end{aligned} \end{equation} $  (17) 
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Fig. 3 Model reference control of 4WS vehicle. 
Note that crosswind disturbances here are not considered. In the next section, a disturbance observer will be introduced so as to attenuate the effects of crosswind disturbances.
A. Reference ModelIn order not to change the driver's perception of vehicle handling, the steady state value of desired reference yaw rate should be the same as that of the front wheel steering (FWS) vehicle with the same parameters. Based on vehicle theory, the desired yaw rate can be described as follows [15]:
$ \begin{equation}\label{gamma} \begin{aligned} r^\ast&=\frac{v}{L(1+kv^2)}\frac{1}{1+\tau_rs}\delta^\ast_f\\ k&=\frac{m}{L^2}\left(\frac{a}{k_r}\frac{b}{k_f}\right) \end{aligned} \end{equation} $  (18) 
where
$ \begin{equation}\label{beta} \begin{aligned} \beta^\ast=\frac{k_\beta}{1+\tau_\beta s}\delta^\ast_f \end{aligned} \end{equation} $  (19) 
where
Define the state vector
$ \begin{equation}\label{refmodle} \begin{aligned} \dot{x}_d=A_dx_d+B_du_d \end{aligned} \end{equation} $  (20) 
where
$ A_d\!=\!\begin{bmatrix} \frac{1}{\tau_\beta}&0\\ 0 & \frac{1}{\tau_\beta}\end{bmatrix}, B_d\!=\!\begin{bmatrix} \frac{k_\beta}{\tau_\beta} \frac{k_r}{\tau_r}\end{bmatrix}, k_\beta\!=\!0, k_r\!=\!\frac{1}{1+kv^2}\frac{v}{L}. $ 
Applying Laplace transform on the linear 4WS vehicle model (14), the transfer function matrix can be obtained
$ \begin{equation}\label{gtf} \begin{aligned} G=(sIA)^{1}B. \end{aligned} \end{equation} $  (21) 
The role of feedforward controller
$ \begin{equation}\label{cfequ} \begin{aligned} GC_f\cdot\delta_f^\ast=\begin{bmatrix} \beta^\ast\\ r^\ast\end{bmatrix}. \end{aligned} \end{equation} $  (22) 
Substituting (18), (19) and (21) into (22), the feedforward controller can be obtained
$ \begin{equation} \begin{aligned} C_f=\begin{bmatrix}\frac{I_zvs+ak_fL+bmv^2}{vk_fL}\cdot\frac{v}{L(1+kv^2)}\cdot\frac{1}{1+\tau_rs}\\ \\ \frac{I_zvsbk_rL+amv^2}{vk_rL}\cdot\frac{v}{L(1+kv^2)}\cdot\frac{1}{1+\tau_rs}\end{bmatrix}. \end{aligned} \end{equation} $  (23) 
Then the feedforward control input can be expressed as
$ \begin{equation}\label{uf} \begin{aligned} u_f=C_f\cdot\delta_f^{\ast}. \end{aligned} \end{equation} $  (24) 
A state feedback controller will be proposed in this part. Define the following error vector as
$ \begin{equation}\label{error1} \begin{aligned} x_e=xx_d=\begin{bmatrix}\beta\beta^\ast\\ rr^\ast\end{bmatrix}. \end{aligned} \end{equation} $  (25) 
According to (16), (17) and (20), we get
$ \begin{equation}\label{errorstate1} \begin{aligned} \dot{x}_e\!=\!Ax_e\!+\!Bu_e\!+\!B_f\!w_f\!+\!(Ax_d\!+\!Bu_f)\!\!(A_dx_d\!+\!B_du_d) \end{aligned} \end{equation} $  (26) 
Substituting (24) into (22), we have
$ \begin{equation}\label{Guf} \begin{aligned} G\cdot u_f=x_d. \end{aligned} \end{equation} $  (27) 
Substituting (21) into (27) and then applying Laplace inverse transform, we get
$ \begin{equation}\label{error2} \begin{aligned} \dot{x}_d=Ax_d+Bu_f. \end{aligned} \end{equation} $  (28) 
Substituting (28) and (20) into (26), the last two terms of (26) are eliminated. Thus a simple state space description of the error system is achieved
$ \dot{x}_e=Ax_e+B_fw_f+Bu_e. $  (29) 
Here we design a linear quadratic regular (LQR) controller without considering disturbances that will be dealt with in the next section. The performance index
$ \begin{equation} J=\int_0^\infty(x_e^TQx_e+u_e^TRu_e)dt \end{equation} $  (30) 
where
$ \begin{equation*}\label{errorstate} Q= \begin{bmatrix} 400&0 \\ 0&180 \end{bmatrix}, ~~~~~~~ R= \begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix}. \end{equation*} $ 
The obtained LQR feedback control gain is
$ \begin{equation*} K_e= \begin{bmatrix} 14.06&7.96 \\ 12.22&10.67 \end{bmatrix}. \end{equation*} $ 
In this section, a disturbance observer is designed to deal with the crosswind. Furthermore, this disturbance observer also takes the ignored nonlinear dynamics into account. As the linear vehicle model inevitably neglects some high order nonlinear terms compared with nonlinear vehicle model, a nonlinear function
$ \begin{equation}\label{dobcstate} \begin{aligned} \dot{x}_e=Ax_e+B_fw_f+Bu_e+O(x_e, u_e, w_f). \end{aligned} \end{equation} $  (31) 
The DOBC approach [16] is used to deal with both disturbances and uncertainties. For simplicity, the effect caused by nonlinearities is merged into the disturbance term, i.e.,
$ \begin{equation}\label{dis} \begin{aligned} w_d=B_fw_f+O(x_e, u_e, w_f). \end{aligned} \end{equation} $  (32) 
Substituting (32) into (31), the full dynamic model of the nonlinear 4WS vehicle is described as
$ \begin{equation}\label{flstate} \begin{aligned} \dot{x}_e=Ax_e+Bu_e+B_dw_d \end{aligned} \end{equation} $  (33) 
where
$ \begin{equation}\label{disobserver} \left\{ \begin{aligned} &\dot{p}=LB_d(p+Lx_e)L(Ax_e+Bu_e)\\ &\hat{w}_d=p+Lx_e \end{aligned} \right. \end{equation} $  (34) 
where
$ \begin{equation}\label{kd} \begin{aligned} K_d=[C(A+BK_e)^{1}B]^{1}\times C(A+BK_e)^{1}Bd. \end{aligned} \end{equation} $  (35) 
The disturbance observer control law of system (33) is designed as [16]
$ \begin{equation} u_e=K_ex_e+K_d\hat{w}_d \end{equation} $  (36) 
where
$ \begin{equation} L= \begin{bmatrix} 0.1&0 \\ 0&0.1 \end{bmatrix}. \end{equation} $  (37) 
The control structure of the nonlinear 4WS vehicle with the proposed control strategy is shown in Fig. 4. The disturbance compensation gain vector can be obtained based on (35), where
$ \begin{equation} K_d= \begin{bmatrix} 0.93&0.10 \\ 0.67&0.10 \end{bmatrix}. \end{equation} $  (38) 
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Fig. 4 4WS vehicle with the proposed control strategy. 
In this section, some simulation results are given to show the effectiveness of the proposed control strategies for 4WS systems. Choose the vehicle forward velocity as
$ \begin{equation} K=\frac{\delta_f}{\delta_r}=\frac{b+\frac{mav^2}{k_rL}}{a+\frac{mbv^2}{k_fL}}. \end{equation} $  (39) 
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Fig. 5 Front wheel angle input. 
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Fig. 6 Crosswind disturbance input. 
As shown in Fig. 7, the steadystate gain of sideslip angle of 4WS vehicles with or without disturbance observer is much smaller than the one of the proportional control, but the control strategy with disturbance observer is slightly better. In cases where crosswind disturbances occur, both proposed strategies show great advantages. However, only the yaw rate of the 4WS vehicle with disturbance observer can achieve zero tracking error in the steadystate without overshoot Compared with the proportional control for 4WS vehicle, the proposed 4WS control strategies can provide drivers a familiar driving sensation. In addition, the proposed controller both work very well against the crosswind disturbance that occurred from 2 s to 4 s, while the 4WS vehicle based on DOBC is clearly superior.
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Fig. 7 The responses of step steering angle input: 4WS vehicle with disturbance control (cf. Fig. 4), 4WS vehicle without disturbance control (cf. Fig. 3), classic proportional control 4WS vehicle, FWS vehicle. 
On the basis of analyses, it is shown that the proposed disturbance observer based control strategy not only copes with the ignored nonlinear dynamics, but also presents strong robustness against external disturbances. Since there are ignored nonlinear dynamics, it can not achieve desired tracking performance. Although the feedback structure effectively reduces the effect of crosswind disturbances, the additional DOBC can have a better control effect.
Ⅵ. CONCLUSIONSIn this paper, a 4WS control strategy based on DOBC approach was presented. First, a nonlinear 3DOF 4WS vehicle model involving dynamic effects of crosswind was introduced. In the stage of controller design, a relatively simple vehicle model, the linear 2DOF 4WS vehicle model, was used. Taking the ignored nonlinear dynamics and crosswind disturbances into consideration, a disturbance observer was designed. Finally, step response experiments were carried out, which showed that the proposed 4WS vehicle with disturbance observer is superior to the 4WS vehicle without the disturbance observer, FWS vehicle, and 4WS vehicle with the classical proportional control. The proposed scheme not only copes with ignored nonlinear dynamics, but also presents strong robustness against external disturbances.
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