Vehicle platoon has been demonstrated to be a promising driving pattern for its prominent advantages in enhancing traffic safety, improving highway capacity, and increasing fuel economy as well as reducing carbon emissions. However, the uncertain driving resistance and saturated actuator output decay the control performance and may even lead to the instability of a vehicle platoon. Therefore, a distributed adaptive sliding mode control algorithm for vehicle platoon with uncertain driving resistance and actuator saturation is proposed in this paper. First of all, sliding mode control technique, together with the coupled sliding surface (CSS) method, is adopted to design the vehicle platoon control algorithm and an adaptive updating law is proposed to estimate the unknown driving resistance coefficients. Then, for the problem of actuator saturation, an antiwindup compensation-based approach is utilized to attenuate the integral windup of the adaptive platoon control laws in the case of actuator saturation. In addition, considering the chattering problem inherent in sliding mode control, a sigmoid-like function
In recent years, the automated highway system (AHS) has gained considerable attentions from governments, automobile manufactures, and academia because of the increasing traffic congestion problem in large cities [
During the past few years, many achievements have been developed for vehicle platoon control, such as backstepping approach [
For the problem of unknown driving resistance in vehicle dynamics, Altmannshofer and Endisch [
On the other hand, the mechanical constraints, especially the actuator saturation [
In this paper, we are trying to investigate the vehicle-platoon problem with uncertain driving resistance and actuator saturation via adaptive sliding mode control approach. Firstly, coupled sliding surface (CSS) is deployed to link interconnected vehicles and an adaptive control method is adopted to identify and estimate the variations of resistance coefficients. After that, a distributed adaptive sliding mode control algorithm for vehicle platoon with uncertain driving resistance is proposed. Then, an antiwindup compensation based approach is utilized to attenuate the integral windup of the adaptive platoon control laws in case of actuator saturation. Moreover, the chattering phenomena inherent in sliding mode control are relieved by using a sigmoid-like function. Finally, various numerical simulations are performed to demonstrate the feasibility and effectiveness of the proposed control algorithm.
The rest of this paper is organized as follows. In Section
Assume a vehicle platoon, which consists of a string of autonomous vehicles, includes a leader vehicle and
Topological structure of vehicle platoon.
Consider a vehicle platoon moving in a string with the following longitudinal dynamics:
Generally,
The explicit form of Rolling resistance where where Air resistance where Grade resistance If the vehicle is running on the hill, the component of gravity along the slope is defined as the grade resistance where
As
In order to simplify the protocol design and stability analysis, we rewrite (
Generally speaking, the main purpose of vehicle platoon is to enhance the highway capacity and relieve traffic congestion by maintaining the desired safety distance between two consecutive vehicles and reaching the velocity consensus among vehicles [
In particular, the time-varying driving resistance will inevitably influence the vehicle dynamics and decay the vehicle platoon performance. Therefore, one needs to specifically design the distributed control input
In addition, due to the physical and mechanical limitations of actuators, the control input
Based on above description, the main control objective of this paper can be concluded as follows: The vehicle platoon is achieved such that the follower’s velocity can converge to the velocity of the leader and each vehicle can maintain a safe intervehicle distance to avoid collision with each other The unknown time-varying coefficients When the control input exceeds the maximum output of the vehicle actuator (servo motor for an electric vehicle or engine for a gasoline vehicle), the proposed algorithm can regulate the actuator output autonomously such that the actuator life as well as the vehicle platoon performance is guaranteed
Firstly, the position tracking error for the
We also denote the velocity error by
Here, the following assumptions are made for facilitating the control protocol design and theoretical analysis.
We first consider the case of vehicle platoon with uncertain driving resistance. For the dynamics of vehicles with error
Hence, sliding mode control technique is employed to develop the vehicle platoon controller; we choose each sliding surface as
The solution of (
From (
Taking the time derivative of (
It is worth noting that (
Since
In order to illustrate the same convergence of
Equivalence of the convergence of the CSS and each sliding surface toward zero:
Therefore, the problem of making
Accordingly, the novel adaptive platoon control law for the
The adaptive estimation law for unknown coefficients is determined by
Particularly, when
The adaptive platoon control law for the
Thus, the coefficients adaptation law can be designed as
Then, the following theorem, which guarantees the stability of each vehicle and string stability of the whole vehicle platoon, can be obtained.
Consider a vehicle platoon described by (
First, we define the estimation error of coefficients
Choose the following Lyapunov function candidates:
Then, the time derivative of
Because
According to Young’s inequality [
Substituting (
Thus,
From (
Taking the derivative of (
Cconstrued sliding surface (
The main objective of the designed controller is to make
In practice, the actuator saturation of vehicles has proved to be a source of performance degradation [
Let
By considering explicitly the actuator saturation of vehicles, the modified adaptive control input
The adaptive estimation laws for unknown coefficients are determined by
The adaptive platoon control law of the
The coefficients adaptation law is designed as
The signal
In addition, for the signal
For the signal
From (
The following theorem will provide our result on the vehicle platoon with actuator saturation.
Consider a vehicle platoon described by ( When the saturation does not occur, i.e., When the saturation occurs, i.e.,
For the case that the actuator is not saturated,
When saturation occurs, for the error dynamic (
Then, the time derivative of
Because
Thus,
Using the same analysis method as in Theorem
It is well known that the sliding mode control has inherently the phenomena of chattering, which is detrimental for the system performance [
Then, the control input
The adaptive estimation law of
Similar to the previous case,
The adaptive estimation law of the
To verify the feasibility and effectiveness of the proposed platoon control algorithm, numerical simulations are performed with 8 vehicles (1 leader and 7 followers).
Without loss of generality, we suppose the vehicle platoon drives in various driving conditions, such as acceleration, cruising, and braking. In addition, we also consider the influence of time-varying driving resistance as a disturbance on the vehicle velocity in the form of sine wave.
Therefore, the desired velocity of the leading vehicle is specifically designed as
Inspired by our previous work on adaptive control for vehicle platoon [
Initial states of each vehicle and initial estimated values of driving resistance coefficients.
Vehicle | Leader | V1 | V2 | V3 | V4 | V5 | V6 | V7 |
---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
68 | 63 | 46 | 38 | 30 | 21 | 8 | 0 | |
0 | 0.8 | 0.9 | 0.85 | 0.85 | 0.8 | 0.85 | 0.8 | |
0 | 0.004 | 0.004 | 0.0045 | 0.004 | 0.0045 | 0.004 | 0.0045 | |
0 | 0.00016 | 0.00016 | 0.00016 | 0.00016 | 0.00016 | 0.00016 | 0.00016 | |
0 | 0.3 | 0.4 | 0.4 | 0.4 | 0.5 | 0.4 | 0.5 | |
0 | 0.004 | 0.003 | 0.005 | 0.004 | 0.005 | 0.004 | 0.005 | |
0 | 0.00018 | 0.00016 | 0.00014 | 0.00017 | 0.00015 | 0.00017 | 0.00015 |
Control parameters of vehicles.
10 | 100 | 5 | 10−10 | 0.5 | 0.5 | −2 | 2 |
To better illustrate the effectiveness of the proposed vehicle platoon control algorithm, three simulation cases are performed.
Platoon with uncertain driving resistance.
In this case, we only consider the influence of uncertain driving resistance for vehicle platoon, where (
Figure
The simulation results of each vehicle with uncertain driving resistance: (a) the velocity curves; (b) the position curves; (c) the distances curves between two neighbor vehicles; (d) the control input curves.
Platoon with actuator saturation.
In this case, the control laws (
Figure
However, it triggers the chattering phenomenon, which may deteriorate the vehicle actuator and lead to the driving uncomfortableness for passengers.
To avoid chattering in practical implementation, the modified control algorithms (
The simulation results of each vehicle with uncertain driving resistance and actuator saturation: (a) the velocity curves; (b) the position curves; (c) the distances curves between two neighbor vehicles; (d) the control input curves.
Platoon with reduced chattering
In this case, the control laws (
Figure
The simulation results of each vehicle with uncertain driving resistance, actuator saturation, and reduced chattering: (a) the velocity curves; (b) the position curves; (c) the distances curves between two neighbor vehicles; (d) the control input curves.
In Case
In this paper, we discuss the distributed adaptive control problem for vehicle platoon with uncertain driving resistance and actuator saturation. Coupled sliding surface (CSS) is deployed to link interconnected vehicles and an adaptive control method is adopted to estimate the variations of resistance coefficients. An antiwindup compensation based approach is utilized to attenuate the integral windup of the adaptive platoon control laws in case of actuator saturation. Theoretical results are verified via numerical simulations, which demonstrate that the proposed control algorithm can make every vehicle keep the desired distance with the preceding vehicle and all followers’ velocity will gradually converge to the velocity of the leader even in the presence of uncertain driving resistance coefficients and actuator saturation.
The data used to support the findings of this paper are included within the article and are available from the corresponding author upon request.
Part of the content of this paper has been accepted for a podium presentation at the 36th Chinese Control Conference (CCC).
The authors declare that they have no conflicts of interest.
This work was supported by the Fundamental Research Funds for the Central University of China-Excellent Doctoral Dissertation Foundation of Chang’an University (no. 300102320720) and the Natural Science Basic Research Plan in Shaanxi Province of China (no. 2020JM-255).