Model predictive control of multi-objective adaptive cruise system based on extension theory

2.1. Variable time headway model

In the theory of variable time headway (VTH), it is believed that the time headway $$ \tau $$ is influenced by environmental variables. In this paper, it is established that when the relative speed between two vehicles is constant, the time headway should be increased when the preceding vehicle decelerates and decreased when the preceding vehicle accelerates^[9].

Therefore, the calculation formula is designed as follows:

In the formula, $$ a_p $$ stands for acceleration of the preceding vehicle, $$ v_p $$ stands for the speed of the preceding vehicle, $$ v $$ stands for the speed of the host vehicle, $$ \tau_0 $$ represents the default time headway, set to 1.5s; $$ a $$ and $$ b $$ are parameters set to 0.3 and 1.5, respectively. Meanwhile, the upper and lower limits of the time headway are considered, with a range of 1.4-2.2s.

2.2. Longitudinal dynamic model

The corresponding mechanical balance equation can be established for the vehicle in the longitudinal direction:

Where $$ F_d $$ represents the longitudinal force of the tire, $$ m $$ represents the mass of the vehicle, $$ g $$ represents the gravitational acceleration, $$ f $$ represents the coefficient of friction of the road, $$ \theta $$ represents the road slope, $$ a_x $$ represents the longitudinal acceleration of the vehicle, $$ \rho $$ represents the air density, $$ C_D $$ represents the air resistance coefficient, $$ A $$ represents the frontal area, and $$ v_x $$ represents the longitudinal velocity of the vehicle^[10].

3.1. Model predictive control

3.1.1. Prediction model

The kinematic models for the host vehicle and the preceding vehicle can be expressed as follows^[11]:

(3) $$ \begin{equation} \left\{ \begin{aligned} \Delta x(k) = x_p(k)-x(k)\\ \Delta x_{\text des}(k) = \Delta x_0(k)+\tau v(k)\\ \delta(k) = \Delta x(k)-\Delta x_{\text des}(k) \end{aligned} \right. \end{equation} $$

Here, $$ \Delta x $$ represents the actual following distance, $$ \Delta x_{\text des} $$ represents the desired following distance, $$ x_p $$ represents the position of the preceding vehicle, $$ x $$ represents the position of the host vehicle, $$ k $$ represents the discrete time index, and $$ \delta $$ represents the error between the actual following distance and the desired following distance^[12]. The dynamic model is shown in Figure 1.

(4) $$ \begin{equation} \left\{ \begin{aligned} a(k) = \frac{v(k)-v(k-1)}{T_s}\\ j(k) = \frac{a(k)-a(k-1)}{T_s} \end{aligned} \right. \end{equation} $$

Figure 1. Longitudinal following diagram.

where $$ a(k) $$ represents the acceleration of the host vehicle at time $$ k $$, $$ j(k) $$ represents the acceleration change rate at time $$ k $$, and $$ T_s $$ stands for the discrete sampling period^[13], which is represented by the calculation step size in Matlab.

Under discrete conditions, the state variables based on kinematics can be selected as distance, relative velocity, host speed, host acceleration, and the rate of change of host acceleration, which can be expressed as follows^[14]:

(5) $$ \begin{equation} \Delta x(k+1) = \Delta x(k)-v_{\text rel}(k)*T_s+\frac{1}{2}*a(k)*T_s^2 \end{equation} $$

(6) $$ \begin{equation} v_{\text rel}(k+1) = v_{\text rel}(k)+a_p(k)*T_s-a(k)*T_s \end{equation} $$

(7) $$ \begin{equation} v(k+1) = v(k)+a(k)*T_s \end{equation} $$

(8) $$ \begin{equation} a(k+1) = (1-\frac{T_s}{t})*a(k)+\frac{T_s}{t}*u(k) \end{equation} $$

(9) $$ \begin{equation} j(k+1) = \frac{-1}{t}*a(k)+\frac{1}{t}*u(k) \end{equation} $$

$$ v_{\text rel} $$represents the relative velocity between two vehicles, $$ a_p $$ represents the acceleration of the preceding vehicle, $$ t $$ represents the control time constant, and $$ u $$ represents the desired acceleration.

In this paper, the disturbance variable is chosen as the acceleration of the preceding vehicle, and the state variables are selected as relative distance, host vehicle speed, relative speed, acceleration, and acceleration change rate^[15]: $$ x(k)=[\Delta x(k), v(k), v_{\text rel}(k), a(k), j(k)]^{\rm T} $$. The system output variables are the relative distance error, relative velocity, acceleration, and acceleration change rate: $$ y(k)=[\delta(k), v_{\text rel}(k), a(k), j(k)]^{\rm T} $$. Based on the selected state and output variables, the predictive model can be established as follows:

(10) $$ \begin{equation} x(k+1) = Ax(k)+Bu(k)+Gw(k) \end{equation} $$

(11) $$ \begin{equation} y(k) = Cx(k)-Z \end{equation} $$

where $$ A $$, $$ B $$, $$ G $$, $$ C $$, and $$ Z $$ are system state-space matrices.

$$ A= \begin{bmatrix} 1 & 0 & T_s & -\frac{1}{2}T_s^2 & 0\\ 0 & 1 & 0 & T_s & 0\\ 0 & 0 & 1 & -T_s & 0\\ 0 & 0 & 0 & 1-\frac{T_s}{t} & 1\\ 0 & 0 & 0 & -\frac{1}{t} & 0 \end{bmatrix}, B= \begin{bmatrix} 0 \\ 0\\ 0 \\ \frac{T_s}{t}\\ \frac{1}{t} \end{bmatrix}, G= \begin{bmatrix} \frac{1}{2}T_s^2\\ 0\\ T_s\\ 0\\ 0 \end{bmatrix}, C= \begin{bmatrix} 1 & -\tau & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 & 0 \end{bmatrix}, Z= \begin{bmatrix} \Delta x_0\\ 0\\ 0\\ 0 \end{bmatrix} $$

In this paper, the reference trajectory^[16] is defined as: $$ y_r(k)=0.95*I_id $$, where $$ I_id $$ stands for identity matrix, $$ y_r(k)=[\delta_{\text ref}(k), v_{\text rel_ref}(k), a_{\text ref}(k), j_{\text ref}(k)]^{\rm T} $$, where $$ \delta_{\text ref}(k) $$ represents the reference value of relative distance error, $$ v_{\text rel_ref} $$ represents the reference value of relative velocity, $$ a_{\text ref}(k) $$ represents the reference value of acceleration, and $$ j_{\text ref}(k) $$ represents the reference value of acceleration change rate. The standard form of the discrete linear MPC prediction model is:

(12) $$ \begin{equation} \widehat X_p(k+p|k) = \overline{A}x(k)+\overline{B}U(k+m)+\overline{G}W(k+p)+\overline{H}e_x(k) \end{equation} $$

(13) $$ \begin{equation} \widehat Y_p(k+p|k) = \overline{C}x(k)+\overline{D}U(k+m)+\overline{E}W(k+p)+\overline{F}e_x(k)-\overline{Z} \end{equation} $$

In these equations, p represents the prediction time horizon, and m represents the control time horizon. $$ \widehat X_p(k+p|k) $$ and $$ \widehat Y_p(k+p|k) $$ represent the predicted values of the state and output variables at future time $$ k+p $$, based on the information available at time $$ k $$, respectively. $$ e_x $$ represents the set of outputs of the system, that is, the expected acceleration. $$ W(k+p) $$ represents the error between the actual and predicted values of the state variables. represents the disturbance matrix^[17].

$$ \widehat X_p(k+p|k)= \begin{bmatrix} \widehat x_p(k+1|k)\\ \widehat x_p(k+2|k)\\ \vdots\\ \widehat x_p(k+p|k) \end{bmatrix}, \widehat Y_p(k+p|k)= \begin{bmatrix} \widehat y_p(k+1|k)\\ \widehat y_p(k+2|k)\\ \vdots\\ \widehat y_p(k+p|k) \end{bmatrix}, U(k+m)= \begin{bmatrix} u(k)\\ u(k+1)\\ \vdots\\ u(k+c-1) \end{bmatrix}, W(k+p)= \begin{bmatrix} w(k)\\ w(k+1)\\ \vdots\\ w(k+p-1) \end{bmatrix} $$

$$ e_x=x(k)-x(k-1) $$

$$ \overline{A} $$, $$ \overline{B} $$, $$ \overline{G} $$, $$ \overline{H} $$, $$ \overline{C} $$, $$ \overline{D} $$, $$ \overline{E} $$, $$ \overline{F} $$, and $$ \overline{Z} $$ are the discrete matrices, and the correlation coefficient matrices are given as follows^[18]:

$$ \overline{A}= \begin{bmatrix} A\\ A^2\\ \vdots\\ A^{p-1}\\ \end{bmatrix}_{p*1}, \overline{B}= \begin{bmatrix} B & 0 & \dots & 0\\ AB & B & \dots & \vdots\\ \dots & \dots & \vdots & 0\\ A^{p-1}B & A^{p-2}B & \dots& \sum\limits_{l=0}^{p-c}A^lB \\ \end{bmatrix}_{p*c}, \overline{G}= \begin{bmatrix} G & 0 & \dots & 0\\ AG & G & \dots & \vdots\\ \dots & \dots & \vdots & 0\\ A^{p-1}G & A^{p-2}G & \dots & G \\ \end{bmatrix}_{p*p} $$

$$ \overline{H}= \begin{bmatrix} H_1\\ H_2\\ \vdots\\ H_P\\ \end{bmatrix}_{p*1}, \overline{C}= \begin{bmatrix} CA\\ CA^2\\ \vdots\\ CA^{p-1}\\ \end{bmatrix}_{p*1}, \overline{D}= \begin{bmatrix} CB & 0 & \dots & 0\\ CAB & CB & \dots & \vdots\\ \dots & \dots & \vdots & 0\\ CA^{p-1}B & CA^{p-2}B & \dots& \sum\limits_{l=0}^{p-c}CA^lB \\ \end{bmatrix}_{p*c} $$

$$ \overline{E}= \begin{bmatrix} CG & 0 & \dots & 0\\ CAG & CG & \dots & \vdots\\ \dots & \dots & \vdots & 0\\ CA^{p-1}G & CA^{p-2}G & \dots& GCG \\ \end{bmatrix}_{p*p}, \overline{F}= \begin{bmatrix} CH_1\\ CH_2\\ \vdots\\ CH_P\\ \end{bmatrix}_{p*1}, \overline{Z}= \begin{bmatrix} Z\\ Z\\ \vdots\\ Z\\ \end{bmatrix}_{p*1} $$

3.2. Extension control

3.2.1. Partition of characteristic variables and extension sets

It is essential to ensure the longitudinal performance stability of the ACC system mentioned in this paper before considering ride comfort, fuel economy, and other factors. Therefore, when the longitudinal following state is not within an acceptable range, it is necessary to adjust the weight matrix of MPC. Based on actual driving experience, it is known that the impact on passengers caused by the relative distance between two vehicles is more significant than the relative vehicle speed^[22]. Therefore, in this paper, the longitudinal following error is chosen as the characteristic variable, and the weight $$ q_{\delta} $$ is adjusted further while setting $$ q_v, q_a, q_j $$ to constant values. As a result, the evaluation criteria are that when the longitudinal following distance error is small, it indicates that the ACC system's following ability is strong; when the following distance error is large, it indicates that the ACC system's following ability is weak.

As shown in Figure 2, a one-dimensional extension set is established, and the boundary values of the classical domain and the extension domain are defined as $$ d_1 $$ and $$ d_2 $$, respectively. According to the definition of the extension domain, $$ d_2 $$ is specified as the maximum allowable value of the following distance error. The boundary value of the classical domain is usually the smaller value, so their calculation formulas are as follows:

(21) $$ \begin{equation} d_2 = d_{max}*SDE^{-1} \end{equation} $$

(22) $$ \begin{equation} d_1 = 0.1*d_2 \end{equation} $$

(23) $$ \begin{equation} SDE^{-1} = k_{SDE}v_x+d_{SDE} \end{equation} $$

Figure 2. One-dimensional extension set of following distance error.

where $$ SDE^{-1} $$ represents the sensitivity of the intelligent vehicle to the following distance error, and the values of the relevant parameters are selected based on driver test data^[23]: $$ d_{max}=7.2m $$, $$ k_{SDE}=0.06 $$, $$ d_{SDE}=0.12m $$.

3.2.2. Calculate correlation degree

In the theory of extension control, it is believed that the correlation degree $$ K(S)=0 $$ and $$ K(S)=-1 $$ reflect qualified and unqualified responses to whether a feature variable can be accepted^[24]. In the case of the longitudinal following distance error, the point of zero indicates the absence of the following distance error and is also the ideal point of this feature variable. Additionally, the extensible distance is defined as the distance from a point to a set. As shown in Figure 3, the classical domain is $$ =X_c $$, and the extensible domain is $$ =X_e $$. Therefore, the distance from point Q to the classical domain is defined as $$ \rho(Q, X_c) $$, and the distance from point Q to the extensible domain is defined as $$ \rho(Q, X_e) $$:

(24) $$ \begin{equation} \rho(Q, X_c)= \left\{ \begin{aligned} -|Od_1|, Q\in \\ |Od_1|, Q\in \end{aligned} \right. \end{equation} $$

(25) $$ \begin{equation} \rho(Q, X_e)= \left\{ \begin{aligned} -|Od_2|, Q\in \\ |Od_2|, Q\in \end{aligned} \right. \end{equation} $$

(26) $$ \begin{equation} \left\{ \begin{aligned} K(S)= \frac{\rho (Q, X_e)}{D(Q, X_e, X_c)}\\ D(Q, X_e, X_c)=\rho (Q, X_e)-\rho (Q, X_c) \end{aligned} \right. \end{equation} $$

Figure 3. Measurement mode recognition flowchart.

3.3. Lower controller

First, an inverse engine model is established, assuming that the vehicle is traveling on a straight road with a zero slope^[26]:

The vehicle driving equation is as follows:

Where $$ F_t $$ represents driving force, $$ F_f $$ represents rolling resistance, $$ F_w $$ represents air resistance, and $$ F_j $$ represents acceleration resistance. Furthermore, $$ T_{des}=F_t*\frac{r_{eff}}{i_gi_0\eta_T} $$, which can be obtained after unfolding:

In the equation, $$ T_{des} $$ stands for desired engine torque, $$ m $$ stands for the total the vehicle mass, $$ g $$ stands for the gravitational acceleration, $$ f $$ stands for the road friction coefficient, $$ C_D $$ stands for the air resistance coefficient, $$ A $$ stands for the frontal area, $$ \delta $$ stands for the rotational mass conversion coefficient, $$ r_{eff} $$ stands for the effective radius of the wheel, $$ i_g $$ stands for the transmission ratio of the gearbox^[27], $$ i_0 $$ stands for the transmission ratio of the main reducer, $$ \eta_T $$ stands for the transmission efficiency, $$ I_f $$ stands for the rotational inertia of the flywheel, and $$ I_w $$ stands for rotational inertia of the wheel^[28].

Based on the throttle opening curve shown in Figure 4, the current expected throttle opening can be obtained from the expected torque and engine speed and output to the vehicle model.

Figure 4. Throttle opening curve surface.

Secondly, an inverse brake model is established, assuming a linear relationship between the brake master cylinder pressure and the braking torque^[29]: The magnitude of braking torque can be derived from the resistance torque function:

$$ F_{br} $$represents the sum of the braking forces acting on the wheels, and there is $$ T_{br}=F_{br}*r_{eff} $$. Therefore:

Where $$ T_{br} $$ stands for braking torque, $$ P_{br} $$ stands for brake pressure, and $$ K_1 $$ stands for the linearity coefficient. The next step is to establish the logic for switching between the accelerator and brake. Based on experimental data, the baseline for the switching logic can be obtained. In order to avoid frequent switching between the accelerator and brake and ensure the smoothness and fuel economy of the vehicle, we set threshold values for the accelerator and brake by shifting the curve up and down. The switching curve is shown in Figure 5.

Figure 5. Throttle/brake switching logic diagram.

4.1. Simulation condition settings

The E-type car in Carsim was selected for use in this paper, and the setting of vehicle parameters can be seen in Figure 6.

Figure 6. Parameters of vehicle model.

At the same time, the input variables of the Carsim interface are set as brake master cylinder pressure signals and open loop throttle control signals; output variables are relative distance, relative speed, host vehicle longitudinal speed, host vehicle acceleration, engine crankshaft spin, transmission gear ratio. The simulated road conditions are set to a straight track with a length of 500 meters, and the road friction coefficient is set to a constant value of 0.9. As shown in Figure 7, in the driver control section, the initial speed is set to 30 km/h, and the brake control signal is provided by the upper controller, so it is set to a constant value of 0; the transmission is set to 6-speed transmission.

Figure 7. Settings of the host vehicle model.

4.2. Analysis of simulation results

According to the previous introduction, the overall control structure is built, as shown in Figure 8.

Figure 8. Model control structure.

Three distinct working conditions were set to verify the effectiveness of the proposed control method, and the control results under the traditional MPC were taken as the control. The simulation results are shown in Figures 9-11.

Figure 9. Simulation results of working condition No.1. a: condition No.1 vehicle speed; b: condition No.1 relative speed; c: condition No.1 acceleration.

Figure 10. Simulation results of working condition No.2. a: condition No.2 vehicle speed; b: condition No.2 relative speed; c: condition No.2 acceleration.

Figure 11. Simulation results of working condition No.3.a: condition No.3 vehicle speed; b: condition No.3 relative speed; c: condition No.3 acceleration.

In working condition 1, the target speed is set as the sine curve between 30 km/h and 50 km/h. (A), (B), and (C) in Figure 9 show the variation curves of vehicle speed, the relative speed of the preceding and host vehicles, and the acceleration of the host vehicle, respectively. It can be seen from the speed curve that in the whole simulation process, the speed of the host vehicle controlled by an extension theory is closer to the speed curve of the target vehicle, with a maximum error of 0.6km/h. Furthermore, the relative speed of the two vehicles is smaller than that controlled under traditional MPC. The overall amplitude variation range is reduced by 50%, which indicates that the tracking effect of the host vehicle is better. In addition, it can be seen from the acceleration curve that the variation amplitude of vehicle acceleration under the extension control is smaller. Compared with the traditional MPC, the oscillation amplitude of acceleration is reduced by 8%; therefore, the ride comfort and fuel economy of the vehicle are also optimized. In the second condition, the speed of the preceding vehicle is set to accelerate to 50 km/h and maintained for 5 s, and then it changes between 30 km/h and 50 km/h. It can be seen from the speed curve and relative speed curve that extension control has more advantages in tracking effect in the acceleration stage. The control method has a certain optimization effect, and the maximum error of tracking speed is 0.66km/h. However, in the deceleration stage, for example, the range of relative speed variation of 35–40 s is larger than that in the traditional MPC, and the range increases by 43%. This indicates that the control effect of extension control on relative distance error is not ideal under the current working conditions, so the extension control on relative speed can be considered to generate a new real-time weight matrix for further optimization.

In the third condition, the speed of the preceding vehicle was set to accelerate from 30 km/h to 50 km/h, and it remained unchanged in the following simulation process. It can be found that in the entire simulation process, the vehicle speed under the extension control is closer to the target speed, the maximum relative speed error is 0.64 km/h, and the variation range of relative speed and acceleration is reduced by 40% and 17%, respectively. The results show that when compared with the traditional MPC, there is an optimization effect, and the goal of multi-objective ACC is achieved. However, in the 35–50 s range, that is, the constant speed stage, the relative speed curve has a small amplitude of oscillation, which also appears in the acceleration curve. The amplitude increases by 20% compared with that controlled by the traditional MPC, indicating that the current control method can improve the vehicle performance to a greater extent in the acceleration stage. However, under the condition of constant speed, the weight setting corresponding to the acceleration change rate needs to be updated. Therefore, an extension control theory should be applied to the corresponding state variable in future research to optimize the control results.

Acknowledgments

This work was funded by the Natural Science Foundation of China (Grant No. 52275082) and the Fundamental Research Funds for the Central Universities QNXM20220029 and FRF-TP-20-037A2.

Authors' contributions

Made substantial contributions to the conception and design of the study and performed data analysis and interpretation: Li Z, Zhao X, Yang J Performed article typography: Liu M

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Conflicts of interest

All authors declared that there are no conflicts of interest.

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Copyright

© The Author(s) 2023.