Frontiers in Computing and Intelligent Systems 
ISSN: 2832-6024 | Vol. 1, No. 3, 2022 

 

42 

Research on Vehicle Longitudinal Control Method 
Based on Model Predictive Control 
Xingbang Gao*, Jiaojiao Li, Ruiyuan Liu, Shuai Zhang, Pengcheng Ma 

School of Transportation and Vehicle Engineering, Shandong University of Technology, Zibo 255049, China. 
* Corresponding author: Xingbang Gao. 

 
Abstract: In order to improve the safety of autonomous vehicles during driving and the comfort of drivers and passengers, the 
longitudinal dynamics model of the vehicle is first established. Secondly, the longitudinal motion control strategy is designed 
considering the influence of vehicle driving safety and driver comfort. Based on this, a longitudinal motion controller based on 
model predictive control is established. The upper controller uses the model predictive control algorithm to calculate the expected 
acceleration, and the lower controller uses the vehicle inverse longitudinal dynamics model to convert the expected acceleration 
calculated by the upper controller into throttle opening and braking pressure. Finally, the effectiveness of the longitudinal motion 
controller is verified by MATLAB / Simulink under different working conditions. The simulation results show that the 
longitudinal motion controller designed in this paper improves the comfort of drivers and passengers under the premise of 
ensuring the safety of vehicles. 

Keywords: Autonomous vehicle; Longitudinal control; Model predictive control; Occupant comfort; Vehicle longitudinal 
dynamics model. 

 

1. Introduction 
Autonomous driving technology has become a research 

hotspot in the field of automotive engineering due to its 
significant advantages in traffic safety and traffic efficiency. 
As one of the three major technical fields of autonomous 
driving, vehicle motion control can complete the lateral 
control, longitudinal control and lateral longitudinal coupling 
control of autonomous vehicles according to the target 
planning path and speed. Stable and efficient longitudinal 
control is the basis for ensuring safe and comfortable speed 
tracking of vehicles, and is the key technology to improve 
driving safety and road traffic efficiency. 

In terms of control methods, there are two main types of 
longitudinal motion control: traditional control algorithms 
and artificial intelligence algorithms. Zhang et al. used double 
PID controller to control the speed and position error of 
vehicle longitudinal motion. Zheng used sliding mode control 
algorithm to design longitudinal controller and compared it 
with traditional PID algorithm. Zhu et al.designed 
longitudinal acceleration and braking controller based on 
fuzzy control method. Zheng et al.designed a vehicle 
longitudinal motion controller based on particle swarm 
optimization PID algorithm. Li et al.adopted a model 
predictive control method based on improved particle swarm 
optimization to realize the longitudinal motion control of 
vehicle. There are many control methods for vehicle 
longitudinal motion control. The traditional PID control is 
popular in the market due to its maturity of research and 
convenience of use, but its control accuracy is poor and its 
ability to deal with delay problems is not strong. Therefore, 
this paper selects the model predictive control algorithm with 
higher control precision and better processing delay ability to 
design the longitudinal motion controller. 

In the control structure, longitudinal motion control is 
divided into direct control and hierarchical control structure. 
Direct control has the characteristics of high integration and 
strong accuracy, but its design is complex and flexible. Zheng 

designed a longitudinal controller based on fuzzy control by 
direct control. Hierarchical control is simpler and more 
flexible than direct control. Jian designed a longitudinal 
controller with hierarchical control. The upper layer uses 
sliding mode variable structure control to obtain the desired 
acceleration, and the lower layer uses fuzzy PID control and 
longitudinal dynamics model to calculate the throttle opening 
and braking pressure. In order to study the longitudinal 
control system, this paper adopts hierarchical control 
structure. 

In summary, few existing studies have considered the 
comfort of the occupants to design the longitudinal motion 
controller. Therefore, this paper builds a vehicle longitudinal 
dynamics model, comprehensively considers the influence of 
the motion state of the front vehicle in the same lane on the 
vehicle and the comfort of the occupants to design the vehicle 
longitudinal control strategy. Based on this, the hierarchical 
control structure is used to design the longitudinal motion 
controller based on model predictive control. Finally, under 
different working conditions, simulation experiments are 
carried out based on MATLAB / Simulink. This paper lays a 
theoretical foundation for improving vehicle driving safety 
and diversity of control scenarios. 

2. Longitudinal vehicle dynamic model 
The vehicle completes a series of actions such as 

acceleration, uniform speed and deceleration through the 
force and reaction force between the ground and the tire. The 
design of the control algorithm needs to be based on the 
vehicle dynamics model. Therefore, this paper analyzes the 
force of the vehicle under acceleration and deceleration 
conditions during driving, and establishes the vehicle driving 
dynamics model and braking dynamics model: 



 

43 

 
Fig.1 Longitudinal force diagram of uphill vehicle 

It can be seen from Fig.1 that taking the vehicle with front 
wheel drive as an example, the vehicle mainly needs to be 
subjected to its own gravity during driving G ,air resistance

wF
,hill climbing resistance iF

,rolling resistance fF
,and 

the driving force acting on the wheel tF or braking force bF . 
According to Newton s second law, the dynamic 

equilibrium expression of the vehicle uphill can be 
established as: 

𝛿𝛿𝛿𝛿𝑣𝑣𝑥𝑥 ′ = 𝐹𝐹𝑡𝑡 − 𝐹𝐹𝑤𝑤 − 𝐹𝐹𝑖𝑖 − 𝐹𝐹𝑓𝑓 (1) 

Among them, m  is the vehicle mass ;δ  is the rotational 

mass conversion coefficient xv is longitudinal speed. 
The expression of air resistance is : 

𝐹𝐹𝑤𝑤 =
1
2
𝐶𝐶𝐷𝐷𝐴𝐴𝐴𝐴𝑢𝑢𝑟𝑟2 (2) 

Among them, DC is the air resistance coefficient ; A is the 

windward area of the vehicle ; ρ  is air density ; ru  is the 
relative speed of speed and wind speed. 

The expression of slope resistance is : 
𝐹𝐹𝑖𝑖 = 𝛿𝛿𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 (3) 

Among them,θ is the slope angle of the road. 
The expression of rolling resistance is : 

𝐹𝐹𝑓𝑓 = 𝛿𝛿𝑚𝑚𝑚𝑚 𝑐𝑐𝑐𝑐𝑠𝑠 𝜃𝜃 (4) 

Among them, f is the rolling resistance coefficient. 
During the driving process of the vehicle, the expression of 

the driving force tF of the engine acting on the tire is: 

𝐹𝐹𝑡𝑡 =
𝑇𝑇𝑡𝑡
𝑟𝑟

=
𝑇𝑇𝑡𝑡𝑡𝑡𝑠𝑠𝑔𝑔𝑠𝑠0𝜂𝜂𝑡𝑡

𝑟𝑟
 (5) 

Among them, tqT
 is engine torque; gi  , 0i  are the 

transmission ratio of transmission and main reducer 

respectively; tη  is the mechanical efficiency of the 
transmission; r is the wheel radius. 

Vehicle braking process, the ground on the tire braking 

force bF  to replace the driving force engine driving force 
acting on the tire, the expression is : 

𝐹𝐹𝑏𝑏 =
𝑇𝑇𝑏𝑏
𝑟𝑟
≤ 𝐹𝐹𝑍𝑍𝜑𝜑 (6) 

Among them, bT is the friction torque of the brake; ZF is the 
normal reaction force of the ground to the tire;ϕ is the ground 
adhesion coefficient. 

The direction of the force of the vehicle is different in the 
uphill, downhill, acceleration and braking. It is necessary to 
change the direction of the force in the model. The overall 
idea is similar to the uphill driving of the vehicle, and it is not 
described too much in the text. 

3. Longitudinal controller design based 
on model predictive control 

3.1. Longitudinal Control Strategy 
Considering Motion State of Other 
Vehicles 

In addition to the road environment and regulations, the 
vehicle speed is also affected by the motion state of other 
participants in the traffic system. In the normal driving 
process, when the speed of the front vehicle in the same lane 
is low, or the front vehicle is braked from a higher speed to a 
lower speed, a traffic conflict will occur to the normal driving 
of the vehicle. In order to avoid accidents, the vehicle should 
take braking to control longitudinal motion; when the current 
vehicle speed is greater than the self-vehicle or away from the 
current lane of the self-vehicle, the self-vehicle should drive 
stably at the set speed. In the process of driving, ensuring the 
safety of vehicle driving is the bottom line of autonomous 
driving technology. If the acceleration can be constrained on 
this basis to achieve predictive braking, the comfort of drivers 
and passengers can be greatly improved. Therefore, aiming at 
the safety and comfort of autonomous vehicles, this paper 
designs a longitudinal control strategy considering the motion 
state of the preceding vehicle under the following traffic 
conflict conditions. 

(1) Condition 1: the same lane in front of the car or 
stationary speed 

When the front vehicle in the same lane runs at a constant 
speed and the speed of the front vehicle is less than that of the 
self-vehicle, if the self-vehicle cannot complete the braking 
within a safe distance, the two vehicles will have traffic 
conflicts. In order to avoid accidents, the longitudinal speed 
of the vehicle should be controlled. Since the vehicle does not 
need the driver s control in the autonomous driving state, the 
driver s reaction time is not included in the braking process, 
and replaced by the reaction time of the autonomous driving 
system. Let the system reaction time be rt ,The braking force 

growth time is it , the continuous braking time is ct , the initial 

speed of the self-vehicle is m0v , and the initial speed of the 

front vehicle is 0qv  ,Under this condition, the minimum 

safety distance min1S from the self-vehicle to the two vehicles 
without collision is : 

𝑆𝑆𝑚𝑚𝑖𝑖𝑚𝑚1 = �𝑣𝑣𝑚𝑚0 − 𝑣𝑣𝑡𝑡0�𝑡𝑡𝑟𝑟 + 1
2
�𝑣𝑣𝑚𝑚0 − 𝑣𝑣𝑡𝑡0�𝑡𝑡𝑖𝑖 −

�𝑣𝑣𝑚𝑚0−𝑣𝑣𝑞𝑞0�
2

2𝑎𝑎𝑏𝑏𝑚𝑚𝑚𝑚𝑚𝑚
+

𝑎𝑎𝑏𝑏𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑖𝑖2

24
 

Where maxba is the maximum braking deceleration? 
In the actual driving process, in order to improve the safety, 

the critical state of no collision should not be considered only 
when braking. When the braking process is completed, a 
certain distance should be retained between the two vehicles, 
and the longitudinal control of the vehicle still needs to 
improve the comfort as much as possible under the premise 
of ensuring safety. In order to improve comfort, experienced 
drivers will gradually increase the braking pedal to a certain 
appropriate braking force and maintain it for a period of time 
when performing braking operations under safe conditions. 
When the vehicle brakes to a safe speed or stops, the driver 
will gradually reduce the braking force to ensure smooth and 
comfortable braking. Based on this, this paper constrains the 



 

44 

braking deceleration, and uses the trapezoidal braking 
deceleration curve to represent the change of acceleration 
with time in the braking stage, as shown in Fig.2. 

 
Fig.2 Vehicle speed change with time 

After constraint, the expected safety distance 𝑆𝑆𝑒𝑒1 under 
condition 1 is: 

𝑆𝑆𝑒𝑒1 = �𝑣𝑣𝑚𝑚0 − 𝑣𝑣𝑡𝑡0� �𝑡𝑡𝑟𝑟 +
𝑡𝑡𝑖𝑖
2
� −

�𝑣𝑣𝑚𝑚0 − 𝑣𝑣𝑡𝑡0�
2

2𝑎𝑎𝑒𝑒
+
𝑎𝑎𝑒𝑒𝑡𝑡𝑖𝑖2

24 −
𝑎𝑎𝑒𝑒𝑡𝑡𝑑𝑑2

6
+ 𝑆𝑆0 

 

In the formula, ea  is the expected acceleration, dt  is the 

time required for the braking force slow-down stage, 0S is the 
distance between the two cars when the two cars brake to the 
same speed, referring to the Mazda safety distance model, 
here take 5m. 

When the preceding vehicle in the same lane is stationary, 
it is only necessary to set the preceding vehicle speed𝑣𝑣𝑡𝑡0 = 0 
in Formulas (1) and (2). 

Therefore, under condition 1, after setting the comfortable 
expected acceleration ea , the braking force increase time it , 

and the braking force slow-down phase time dt ,the expected 
safety distance that takes into account both safety and comfort 
can be calculated. When the distance between the two 
vehicles is less than the minimum safety distance min1S  , 
emergency braking is taken; when the distance between the 
two cars is in[𝑆𝑆𝑒𝑒1 , 𝑆𝑆𝑒𝑒1 + 5], comfortable braking is adopted. 

(2) Condition 2: uniform deceleration in the same lane 
When the vehicle in front of the same lane slows down, and 

the current speed is less than the speed of the vehicle, the 
vehicle needs to control the longitudinal speed to avoid traffic 
conflicts. Under condition 2, the minimum safety distance 

min 2S from the car to take emergency braking to the two cars 
without collision is: 

𝑆𝑆𝑚𝑚𝑖𝑖𝑚𝑚 2 = 𝑣𝑣𝑚𝑚0𝑡𝑡𝑟𝑟 +
𝑣𝑣𝑚𝑚0𝑡𝑡𝑖𝑖

2 +
𝑣𝑣𝑡𝑡2 − 𝑣𝑣𝑚𝑚02

2𝑎𝑎𝑏𝑏𝑚𝑚𝑎𝑎𝑥𝑥
+
𝑎𝑎𝑏𝑏𝑚𝑚𝑎𝑎𝑥𝑥 𝑡𝑡𝑖𝑖2

24 −
𝑣𝑣𝑡𝑡2 − 𝑣𝑣𝑡𝑡02

2𝑎𝑎𝑏𝑏𝑡𝑡
  

In the formula, qv  is the speed of the front vehicle after 

braking, and bqa  is the braking deceleration of the front 
vehicle. 

After the driver s comfort is taken into account to impose 
constraints on the self-driving brake deceleration, the 
expected safety distance 2eS under Condition 1 is: 

𝑆𝑆𝑒𝑒2 = 𝑣𝑣𝑚𝑚0𝑡𝑡𝑟𝑟 +
𝑣𝑣𝑚𝑚0𝑡𝑡𝑖𝑖

2 +
𝑣𝑣𝑡𝑡2 − 𝑣𝑣𝑚𝑚02

2𝑎𝑎𝑒𝑒
+
𝑎𝑎𝑒𝑒𝑡𝑡𝑖𝑖2

24 + 𝑣𝑣𝑡𝑡𝑡𝑡𝑑𝑑 −
𝑎𝑎𝑒𝑒𝑡𝑡𝑑𝑑2

6

−
𝑣𝑣𝑡𝑡2 − 𝑣𝑣𝑡𝑡02

2𝑎𝑎𝑏𝑏𝑡𝑡
+ 𝑆𝑆0 

 

Under condition 2, when the distance between the two 
vehicles is less than the minimum safety distance𝑆𝑆𝑚𝑚𝑖𝑖𝑚𝑚 2 , 
emergency braking is taken; when the distance between the 
two cars is in[𝑆𝑆𝑒𝑒2 , 𝑆𝑆𝑒𝑒2 + 5], comfortable braking is adopted. 

3.2. Longitudinal Controller Design Based on 
Model Predictive Control 

The model predictive control algorithm is used to design 
the hierarchical longitudinal motion controller. The upper 
controller is based on the model predictive control algorithm 
to obtain the expected acceleration𝑎𝑎𝑒𝑒according to the relative 
distance 𝑑𝑑𝑟𝑟𝑒𝑒 between the two vehicles, the front vehicle 
speed𝑣𝑣𝑡𝑡  , the initial vehicle speed𝑣𝑣𝑚𝑚0 , and the real-time 
vehicle speed𝑣𝑣𝑚𝑚. The lower controller is based on the vehicle 
inverse longitudinal dynamics model, and the control 
acceleration output by the upper layer is converted into 
throttle opening and braking pressure, and finally the vehicle 
can stably complete the speed tracking. The structure design 
of the longitudinal motion controller is shown in Fig.3: 

 
Fig. 3 Structure diagram of longitudinal motion controller 

(1) Upper controller design 
In this paper, only the longitudinal motion of the vehicle is 

studied, and the influence of yaw motion on the longitudinal 
motion control of the vehicle is neglected. Therefore, the 
acceleration of the vehicle is approximately equal to the first 
derivative of the longitudinal speed of the vehicle to time, and 
the longitudinal control of the vehicle can be expressed by the 
first order inertial system. 

𝑎𝑎′ =
𝐾𝐾(𝑎𝑎𝑒𝑒 − 𝑎𝑎𝑡𝑡)

𝜏𝜏
  

𝐾𝐾 is the system gain; 𝑎𝑎𝑡𝑡 is the actual longitudinal 
acceleration; τ is the time constant. 

The continuous state equation of the vehicle can be 
expressed as: 

𝑥𝑥 ′ = �
0 1

0 −
1
𝜏𝜏
� 𝑥𝑥 + �

0
𝐾𝐾
𝜏𝜏
� 𝑢𝑢 

In the formula, x  is a continuous state quantity; u  is the 
system control input, 𝑢𝑢 = 𝑎𝑎𝑒𝑒 . 

The discrete state equation of the system is obtained by 
discretizing Eq. (11): 

𝑥𝑥(𝜆𝜆 + 1) = 𝐴𝐴𝜆𝜆𝑥𝑥(𝜆𝜆) + 𝐵𝐵𝜆𝜆𝑢𝑢(𝜆𝜆) 

where Aλ is the state matrix, 𝐴𝐴𝜆𝜆 = �
1 𝑇𝑇𝑐𝑐
0 1 − 𝑇𝑇𝑐𝑐

𝜏𝜏
�; Bλ is the 

control input matrix, 𝐵𝐵𝜆𝜆 = �
0
𝐾𝐾𝑇𝑇𝑐𝑐
𝜏𝜏
�; λ is the current sampling 

time ; 𝜆𝜆 + 1 is the next sampling time ; cT  is the sampling 
period. 

Speed as the output of the system, the output equation is: 
𝑣𝑣(𝜆𝜆) = 𝑐𝑐𝑥𝑥(𝜆𝜆), 𝑐𝑐 = [0 1] 

According to the control strategy designed by 3.1, the 
control goal of the automatic driving vehicle in the actual 



 

45 

driving process is to improve the driving comfort as much as 
possible under the premise of ensuring the safety of the 
vehicle speed tracking accuracy, that is, not to occur violent 
acceleration and deceleration and acceleration and 
deceleration rate of change. Based on this, the system 
performance evaluation index (objective function) is: 

𝐽𝐽�𝑥𝑥(𝜆𝜆),𝑢𝑢(𝜆𝜆 − 1),𝛥𝛥𝑢𝑢(𝜆𝜆)� 

��𝑌𝑌𝑝𝑝(𝜆𝜆 + 𝑠𝑠|𝜆𝜆) − 𝑌𝑌𝑟𝑟𝑒𝑒(𝜆𝜆 + 𝑠𝑠|𝜆𝜆)�

𝑁𝑁𝑝𝑝

𝑖𝑖=1 𝑄𝑄

2

+�‖𝛥𝛥𝑢𝑢(𝜆𝜆 + 𝑠𝑠|𝜆𝜆)‖
𝑁𝑁𝑐𝑐

𝑖𝑖=1 𝑅𝑅

2

+ 𝜀𝜀𝜎𝜎2  

In the formula, pN is the prediction time domain ; cN is the 
control time domain ;𝜆𝜆 + 𝑠𝑠|𝜆𝜆 indicates that the state value at 
time 1λ +  is predicted according to the state value at time
λ ;𝑌𝑌𝑝𝑝(𝜆𝜆 + 𝑠𝑠|𝜆𝜆)is the output prediction value ;𝑌𝑌𝑟𝑟𝑒𝑒(𝜆𝜆 + 𝑠𝑠|𝜆𝜆)is 
the output reference value ; Q and R  are weight matrix ;ε is 
the weight coefficient ;σ is the relaxation factor. 

In order to ensure that the system output is within a 
reasonable range, it is necessary to add active constraints, 
namely acceleration constraint and acceleration change rate 
constraint: 

𝑢𝑢𝑒𝑒𝑚𝑚𝑖𝑖𝑚𝑚 ≤ 𝑢𝑢(𝜆𝜆 + 𝑠𝑠) ≤ 𝑢𝑢𝑒𝑒𝑚𝑚𝑎𝑎𝑥𝑥          (𝑠𝑠 = 1,2, …𝑁𝑁) 
𝛥𝛥𝑢𝑢𝑒𝑒𝑚𝑚𝑖𝑖𝑚𝑚 ≤ 𝛥𝛥𝑢𝑢(𝜆𝜆 + 𝑠𝑠) ≤ 𝛥𝛥𝑢𝑢𝑒𝑒𝑚𝑚𝑎𝑎𝑥𝑥  (𝑠𝑠 = 1,2, . . .𝑁𝑁) 

where𝑢𝑢(𝜆𝜆 + 1)and𝛥𝛥𝑢𝑢(𝜆𝜆 + 1)are the longitudinal expected 
acceleration constraint and its incremental constraint ; mineu
and maxeu are the minimum and maximum of the longitudinal 
expected acceleration constraint, 
respectively ;𝛥𝛥𝑢𝑢𝑒𝑒𝑚𝑚𝑖𝑖𝑚𝑚  and𝛥𝛥𝑢𝑢𝑒𝑒𝑚𝑚𝑎𝑎𝑥𝑥  are the minimum and 
maximum values of the longitudinal expected acceleration 
increment constraint, respectively. 

The system solves the optimization problem at each 
moment, obtains the optimal input control quantity, and takes 
the first control quantity as the output quantity. The system 
will re-predict the control state of the next moment according 
to the current state, and continuously scroll the optimization 
until the control is completed. The optimization problem of 
model predictive control can be transformed into a quadratic 
programming problem: 

𝛿𝛿𝑠𝑠𝑠𝑠
Δ𝑈𝑈

1
2
𝑈𝑈𝑇𝑇𝐻𝐻Δ𝑈𝑈 + 𝐺𝐺𝑇𝑇Δ𝑈𝑈 

𝑠𝑠. 𝑡𝑡.     𝑈𝑈𝑒𝑒𝑚𝑚𝑖𝑖𝑚𝑚 ≤ 𝐴𝐴1Δ𝑈𝑈 + 𝑈𝑈𝜏𝜏 ≤ 𝑈𝑈𝑒𝑒𝑚𝑚𝑎𝑎𝑥𝑥           
Δ𝑈𝑈𝑒𝑒𝑚𝑚𝑖𝑖𝑚𝑚 ≤ Δ𝑈𝑈𝜏𝜏 ≤ Δ𝑈𝑈𝑒𝑒𝑚𝑚𝑎𝑎𝑥𝑥   

 

In the formula,𝐻𝐻 ∈ 𝑅𝑅2×2;𝐺𝐺 ∈ 𝑅𝑅2;Δ𝑈𝑈 = [Δ𝑢𝑢(𝜆𝜆), Δ𝑢𝑢(𝜆𝜆 +
1), . . . , Δ𝑢𝑢(𝜆𝜆 + 𝑠𝑠)]𝑇𝑇;𝑈𝑈𝜏𝜏 = 1𝑁𝑁𝑒𝑒 ⊗ 𝑢𝑢(𝜆𝜆 − 1) , 1

eN  is the unit 

vector of eN  line, ( )1u λ −  is the control quantity of the 

previous moment ; mineU  and maxeU  are the minimum and 
maximum sets of the control quantity in the control time 
domain respectively. mineU∆  and maxeU∆  are the minimum 
and maximum sets of control increment in control time 
domain respectively ; 

𝐴𝐴1 =

⎝

⎜
⎛

1 0 0 ⋯ 0
1 1 0 ⋯ 0
1 1 1 ⋯ 0
⋮ ⋮ ⋮ ⋮
1 1 1 ⋯ 1⎠

⎟
⎞

𝑁𝑁𝑒𝑒×𝑁𝑁𝑒𝑒

. 

The quadratic programming problem can be solved by the 
quadprog function provided in MATLAB. 

(2) Lower layer controller design 
In order to avoid damage to the vehicle system, the brake 

pedal and the accelerator pedal cannot be operated at the same 
time during the driving process of the vehicle. At the same 
time, considering the comfort of the occupants, frequent 
switching between braking and driving is avoided. Based on 
this, the switching logic is designed according to the expected 
acceleration of the vehicle. 

Mod = �
𝑎𝑎𝑗𝑗 , 𝑎𝑎𝑒𝑒 ≥ 𝑎𝑎𝑚𝑚𝑎𝑎𝑥𝑥 + 𝛥𝛥𝑎𝑎

0 , 𝑎𝑎𝑚𝑚𝑎𝑎𝑥𝑥 − 𝛥𝛥𝑎𝑎 ≤ 𝑎𝑎𝑒𝑒 ≤ 𝑎𝑎𝑚𝑚𝑎𝑎𝑥𝑥 + 𝛥𝛥𝑎𝑎
𝑎𝑎𝑏𝑏 , 𝑎𝑎𝑒𝑒 ≤ 𝑎𝑎𝑚𝑚𝑎𝑎𝑥𝑥 − 𝛥𝛥𝑎𝑎

  

Mod is the output mode ; ja represents the drive mode ;

ba  represents the braking mode ; maxa  is the maximum 
deceleration when the throttle opening is 0 ; a∆ is the dead 
zone buffer value. 

From Eqs.(1)-(5), the expected output torque of the engine 
is : 

𝑇𝑇𝑡𝑡𝑡𝑡

=
�𝛿𝛿𝛿𝛿𝑎𝑎𝑒𝑒 + 1

2𝐶𝐶𝐷𝐷𝐴𝐴𝐴𝐴𝑢𝑢𝑟𝑟
2 + 𝛿𝛿𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 + 𝛿𝛿𝑚𝑚𝑚𝑚 𝑐𝑐𝑐𝑐𝑠𝑠
𝑠𝑠𝑔𝑔𝑠𝑠0𝜂𝜂𝑡𝑡

 
 

The expected throttle opening tvpa  can be obtained 
according to the external characteristics of the engine and the 
power and torque curve: 

𝑎𝑎𝑡𝑡𝑣𝑣𝑝𝑝 = 𝑚𝑚�𝑠𝑠,𝑇𝑇𝑡𝑡𝑡𝑡� 
The longitudinal dynamic model of the vehicle established 

by 2.1 shows that the braking force required by the vehicle in 

the braking process is braF : 

𝐹𝐹𝑏𝑏𝑟𝑟𝑎𝑎 = 𝛿𝛿𝛿𝛿𝑎𝑎𝑒𝑒 −
1
2
𝐶𝐶𝐷𝐷𝐴𝐴𝐴𝐴𝑢𝑢𝑟𝑟2 − 𝛿𝛿𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃
− 𝛿𝛿𝑚𝑚𝑚𝑚 𝑐𝑐𝑐𝑐𝑠𝑠 𝜃𝜃 

 

Therefore, the expected brake master cylinder pressure
bP is: 

𝑃𝑃𝑏𝑏 =
𝐹𝐹𝑏𝑏𝑟𝑟𝑎𝑎
𝑘𝑘𝑏𝑏

  

4. Longitudinal control simulation 
analysis 

In order to verify the control effect of the longitudinal 
motion controller, MATLAB / Simulink is used to build the 
longitudinal controller simulation model based on model 
predictive control, including the self-vehicle control model 
and the other-vehicle control model, and the longitudinal 
control of the self-vehicle is simulated under three conditions : 
the same-lane vehicle driving at a constant speed or at rest, 
the same-lane vehicle decelerating, the same-lane vehicle 
accelerating or driving out of the current lane. 

4.1. Build simulation model 
Through MATLAB / Simulink, the front vehicle model, the 

longitudinal control model and the visual model are built 
respectively. The dynamic model between acceleration and 
speed is used to build the front vehicle model. The simulation 
model of the self-driving car mainly includes three driving 
modes: constant speed cruise, intelligent car-following and 
emergency braking. The constant speed cruise mode can track 
the set speed and drive steadily. The intelligent car-following 
mode can track the front vehicle speed and maintain a certain 
safe distance. The emergency braking mode can make the 
vehicle brake at the maximum deceleration when the safe 
distance cannot be guaranteed to avoid or reduce the damage 



 

46 

caused by the accident. In order to realize the switching of the 
three modes and realize the continuous and complete 
longitudinal control, the state machine is used to switch the 
three driving modes according to the safe distance set by 
section 3.1. The simulation structure of the vehicle 
longitudinal motion control model based on model predictive 
control is shown in Fig.4. 

 
Fig. 4 Simulation framework of longitudinal motion control 

4.2. Simulation Results and Analysis 
According to the longitudinal motion simulation model of 

the self-driving vehicle built above, the longitudinal motion 
control of the self-driving vehicle under the three conditions 
of the front vehicle driving at a constant speed below the self-
driving vehicle speed, the front vehicle decelerating and the 
front vehicle accelerating is simulated. 

(1) Condition 1: The preceding vehicle travels at a constant 
speed below its own speed 

The initial speed of the front vehicle is set to 20 m/s, the 
target cruise speed is 20 m/s, and the initial position is 40 m 
in front of the vehicle. The initial speed of the vehicle is set 
to 20 m/s, and the target cruise speed is set to 25 m/s. The 
simulated velocity and acceleration results are shown in Fig.5: 

 
Fig.5 Curves of self-driving speed and acceleration versus time 

under condition 1 

As shown in Fig.7, at the beginning of the simulation, the 
self-driving car first accelerates to track the set target cruise 
speed, but in the acceleration process, the relative distance 
between the self-driving car and the front car is less than the 
expected safe distance under this condition. Therefore, the 
self-driving car adjusts the speed and finally stabilizes the 
speed to about 20 m/s, that is, the speed of the front car is 
equal to that of the front car, and the acceleration range is [-
2,2], which is within the expected comfortable acceleration 
range. Finally, the safe and comfortable intelligent car-
following driving is realized. 

(2) Condition 2: Front car slows down 
The initial vehicle speed is set to 25 m/s, the target cruise 

speed is 15 m/s, and the initial position is 40 m in front of the 
vehicle. The initial speed of the vehicle is set to 20 m/s, and 
the target cruise speed is 20 m/s. The simulated velocity and 
acceleration results are shown in Fig.6: 

 
Fig.6 The curve of self-driving speed and acceleration with time 

under condition 2 
As shown in Fig.8, at the beginning of the simulation, the 

vehicle runs at a uniform speed of 20 m/s, and the front 
vehicle brakes from the initial 25 m/s to 15 m/s. At about the 
5th second of the simulation, the relative distance between the 
two vehicles is less than the expected safe distance, and the 
self-vehicle is converted from the fixed-speed cruise mode to 
the intelligent car-following mode, and the self-vehicle speed 
is adjusted to make the self-vehicle speed consistent with the 
front vehicle speed. In this process, the actual acceleration 
range is[-2,0.75], in line with the expected comfortable 
acceleration, and ultimately to achieve safe and comfortable 
car following. 

(3) Condition 3: Front vehicle accelerates 
The initial vehicle speed is set to 15 m/s, the target cruise 

speed is 30 m/s, and the initial position is 30 m in front of the 
vehicle. The initial speed of the vehicle is set to 20 m/s, and 
the target cruise speed is 25 m/s. The simulated velocity and 
acceleration results are shown in Fig.7: 

 
Fig.7 The curve of self-driving speed and acceleration with time 

under condition 3 

As shown in Fig.9, at the beginning of the simulation, the 
speed of the front vehicle is less than that of the self-vehicle, 
and the relative distance between the two vehicles is less than 
the expected safe distance. Therefore, the self-vehicle turns to 
the intelligent car-following mode to track the speed of the 
front vehicle. With the passage of time, the speed of the front 
vehicle is gradually higher than that of the self-vehicle, and 
when the relative distance between the two vehicles is greater 
than the expected safe distance, the self-vehicle changes from 
the intelligent following mode to the fixed-speed cruise mode, 
and accelerates with the expected comfortable acceleration 
until the target cruise speed set by the self-vehicle is reached. 
In this process, the acceleration threshold ranges from [−2, 2] 
to the desired comfortable acceleration range. 

5. Conclusion 
In this paper, a vehicle longitudinal motion controller based 

on model predictive control is established by considering the 
motion state of the preceding vehicle and the comfort of the 
driver and passenger. The effectiveness of the longitudinal 
motion controller is simulated by MATLAB / Simulink under 
three conditions: the constant speed of the preceding vehicle, 
the constant speed of the preceding vehicle lower than the 
speed of the vehicle, the deceleration of the preceding vehicle 
and the acceleration of the preceding vehicle. 



 

47 

(1) The vehicle longitudinal motion controller based on 
model predictive control can realize the longitudinal motion 
control of the controlled vehicle according to the motion state 
of the preceding vehicle. According to different working 
conditions, the controlled vehicle can be switched under three 
driving modes: constant speed cruise, intelligent following 
and emergency braking, which improves the driving safety. 

(2)A longitudinal motion control method is proposed. 
Under the premise of ensuring the safety of vehicle driving, 
the comfort of drivers and passengers is considered 
comprehensively, and the control of vehicle longitudinal 
motion state is completed by adopting a more moderate 
comfortable acceleration, which improves the driving 
comfort. 

References 
[1] Yin Zhishuai, He Jiaxiong, Nie Linzhen, butler. Longitudinal 

adaptive control of autonomous vehicles based on optimization 
algorithms [J]. Journal of system simulation,2021, 33(02): 409-
420. DOI:10.16182/j.issn1004731x.joss.19-0372. 

[2] Riley, Zhang Tong. Longitudinal following control of 
intelligent platoon based on fuzzy MPC[J] .Journal of Jiangsu 
University ( Natural Science Edition ),2022,43(04):394-
399+430. 

[3] Zhang Xuyuan, Li Jun. Lateral and longitudinal coordinated 
control of intelligent electric vehicle trajectory tracking based 
on LQR double PID[J].Journal of Automotive Safety and 
Energy Saving,2021,12(03):346-354. 

[4] Zheng Yannan. Longitudinal Study of Following Vehicle 
Based on Sliding Mode Control[J].Agricultural Equipment and 
Vehicle Engineering,2022,60(04):103-107. 

[5] Zhu Zenghui, Xu Youchun, Ma Yulin, Li Jianshi, Li Yongle. 
Design of intelligent vehicle longitudinal acceleration tracking 
controller based on fuzzy control[J].Journal of Military 
Transportation College,2014,16(12):31-35. 

[6] Zheng Shuhua, Qiao Yajun, Wang Xiangzhou, Li Shouxiang, 
Chen Dongqing. Research on Influence of Load Change on 
Mine Vehicle Trajectory Tracking[J].Journal of Beijing 
University of Technology,2021,41(04):403-
409.DOI:10.15918/j.tbit1001-0645.2019.320. 

[7] Li Guangnan, Ye Hongtao, Luo Wenguang. Longitudinal 
motion control of unmanned vehicle based on IPSO-
MPC[J].Journal of Guangxi University of Science and 
Technology,2022,33(01): 94-
100+109.DOI:10.16375/j.cnki.cn45-1395/t.2022.01.014. 

[8] Research on Automatic Driving Control Strategy of Intelligent 
Vehicle on Special Road[D].chongqing jiaotong 
university,2018. 

[9] Jiang Shuzhen.Research on Lateral and Longitudinal 
Integrated Control Strategy of Intelligent Vehicle Trajectory 
Tracking[D].Chang'an 
University,2021.DOI:10.26976/d.cnki.gchau.2021.001930. 

[10] Wu Hao, Wang Sishan, Zhou Haiying, Shang Rong. Design of 
intelligent vehicle lateral and longitudinal controller based on 
model predictive control[J].Journal of Hubei University of 
Automobile Industry,2021,35(01):11-16+20. 

[11] He Xianglin. Research on Lateral and Longitudinal Control 
Strategy of Unmanned Vehicle Path Tracking[D].Hunan 
University,2020.DOI:10.27135/d.cnki.ghudu.2020.003390. 

 


	Research on Vehicle Longitudinal Control Method Based on Model Predictive Control
	1. Introduction
	2. Longitudinal vehicle dynamic model
	3. Longitudinal controller design based on model predictive control
	3.1. Longitudinal Control Strategy Considering Motion State of Other Vehicles
	3.2. Longitudinal Controller Design Based on Model Predictive Control

	4. Longitudinal control simulation analysis
	4.1. Build simulation model
	4.2. Simulation Results and Analysis

	5. Conclusion
	References