Study on state feedback control strategy for car-following system

doi:10.1016/j.physa.2020.124938

Physica A: Statistical Mechanics and its Applications

Volume 558, 15 November 2020, 124938

https://doi.org/10.1016/j.physa.2020.124938 Get rights and content

Highlights

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A state feedback control strategy was presented to analyze the car-following system.
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The stability of car-following system with state feedback control strategy was analyzed.
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The control parameters’ effects was verified with the numerical simulation.

Abstract

A state feedback control strategy of modern control theory is proposed to analyze the classical car-following system. From the point of view of control theory, the influence of changing two important states velocity and headway on the stability of the car-following system is analyzed. The stability conditions of the control parameters are obtained by the conditions for roots of characteristic polynomials and the small gain theorem in the control theory. A dynamical car-following model is derived by the state feedback control method. Simulation experiments are conducted to verify the state feedback control strategy on traffic flow stability by the new model under the conditions of open boundary and periodic boundary respectively. The simulation results show that with the increase of the influence coefficient, the stability of the traffic flow system is enhanced, and the simulation results are consistent with the theoretical analysis results.

Introduction

In recent decades, the traffic flow problem has been widely investigated in the field of physics. Many theoretical results and dynamic models are obtained to explain the characteristics of traffic flow and the formation mechanism of traffic jams, such as fluid model, cellular automaton model and car-following model. Car-following model is a kind of microscopic model, which can describe the moving behavior of every car in traffic flow. The first classical car-following model was proposed by Pipes in 1953 [1]. In 1961, Newell [2] improved the classical car-following model and proposed a first-order optimal velocity model. In the next 30 years, this model did not improve until Bando put forward the optimal velocity model (OVM) in 1995 [3], which overcame the shortcomings of classical following model in describing continuous vehicle motion. Since then, the car -following theory based on Bando’s model has attracted many scholars’ interest and has been developing continuously. Lenz [4] proposed an improved optimal speed model in 1999 considering the relationship between the headway information of multiple vehicles. Konishi [5], [6] proposed decentralized feedback control and coupled map car-following model. Nagatani [7], [8], [9] improves car-following model considering a next-nearest-neighbor interaction. Sawada [10], [11] proposed a generalized optimal velocity model (GOVM) in 2002, which pointed out that the optimal speed function is not only related to the headway of each vehicle, but also to the headway of the front vehicle. Many domestic experts and scholars have done a lot of research on the development of vehicle following model, and have made a lot of achievements. Jiang [12] puts forward the full velocity difference model (FVDM), which introduces the speed difference between the current car and the front car into the model. In 2005, Zhao [13] proposed a full velocity difference and acceleration difference car-following model(FVAD). Considering the influence of safety distance and time delay, Zheng [14] proposed an improved car following model. Based on the control theory, Ge [15], [16] takes the headway of two adjacent vehicles into the optimal speed function and discusses the two lane traffic trend considering lane change based on the control theory. Tang [17], [18], [19], [20], [21], [22] not only considered the attribution of the driver, the communication between the vehicles and their reliability, but also considered the rationality of the limited driver to improve the car following model. Zhu and Zhang [23], [24], [25], [26], [27], [28] introduced the PD (proportional differential) term into the traffic flow model. Using the classical control theory, through the analysis of vehicle following, he obtained the transfer function of vehicle following model. The improved model is better than the original model in general. In recent years, Peng [29], [30], [31], [32], [33], [34], [35] et al. have made good progress in the study of traffic flow models and proposed a control signal from honk information for lattice hydrodynamic model. In the development of traffic flow model, the state analysis method of modern control theory has never been used to study the performance of car-following model. In this paper, the car-following model is improved based on the state analysis method, and the improved model is verified by numerical simulation to improve the stability of traffic flow.

The rest of this paper is organized as follows. In Section 2, the state analysis method is used to analyze the classical optimal velocity model, and a new optimal velocity model is obtained through the feedback regulation of the internal state-variables of the system. In Section 3, stability analysis of the new model using Routh criterion and small gain theorem. In Section 4, set up numerical simulation experiment to verify the improvement of system stability by the new model. In Section 5 the summary is given.

Section snippets

Model

Suppose that all vehicles are driving on an infinitely long single-lane road after the leading vehicle one by one without overtaking. Fig. 1 shows the car-following system, where $x_{n} (t)$ represents the position of the nth car at time t and $n = 1$ , 2, …, N, N is the total number of the vehicles and the Nth vehicle is the leading vehicle. Under periodic boundary conditions, the $N + 1$ th vehicle is the leading one. According to the optimal velocity model proposed by Bando et al. the equation of motion of

Stability analysis

Based on classical control theory, If all the roots of the characteristic polynomial $d (s)$ are in the left half of the phase plane and ${‖ Φ^{*} (j w) ‖}_{\infty} < 1$ , the traffic flow system is stable. The specific proof process is as follows.

Firstly, according to the stability conditions in classical control theory, if the characteristic polynomial $d (s)$ satisfies the following inequalities, the control system is stable. $\{\begin{matrix} a + m_{0} > 0 \\ a Ω - m_{1} > 0 \end{matrix}$

Here, $a > 0, Ω > 0, m_{0} > 0, m_{1} > 0$ , then the above inequality can be reduced to $Ω > \frac{m_{1}}{a}$

Secondly,

Numerical simulation

We set up three numerical simulation experiments to discuss the influence of coefficient $m_{0}$ , $m_{1}$ on the stability of traffic flow. The following describes the specific process and simulation results of the three simulation experiments.

Summary

In this paper, the state feedback control method of modern control theory is investigated to analyze car-following system, and a dynamical optimal velocity model is derived through the Laplace Inverse Transform. Based on the conditions for roots of characteristic polynomials and small gain theorem in control theory, the stability conditions of the system are obtained. By setting up the numerical simulation experiment, it is verified that the new control strategy improves the stability of the

CRediT authorship contribution statement

Tao Song: Conceptualization, Software, Data curation, Writing - original draft , Investigation. Wen-Xing Zhu: Supervision, Methodology, Writing - review & editing.

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (Grant No. 61773243), and Natural Science Foundation of Shandong Province, China (Grant No. ZR2017MF011).

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Study on state feedback control strategy for car-following system

Highlights

Abstract

Introduction

Section snippets

Model

Stability analysis

Numerical simulation

Summary

CRediT authorship contribution statement

Acknowledgments

Declaration of competing interest

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