String stability for vehicular platoon control: Definitions and analysis methods

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Abstract

The platooning of connected and automated vehicles (CAVs) is expected to have a transformative impact on road transportation, e.g, enhancing highway safety, improving traffic efficiency, and reducing fuel consumption. One critical task of platoon control is to achieve string stability, for which various models and methods had been proposed. However, different types of definitions and analysis methods for string stability were proposed over the years and were not thoroughly compared. To fill these gaps, this paper aims to clarify the relationship of ambiguous definitions and various analysis methods, providing a rigorous foundation for future studies. A series of equivalences are summarized and discussed. The pros and cons of different analysis methods and definitions are discussed, too. All these discussions provide insights for practical selection of analyzing methods for vehicle platoons.

Introduction

Vehicular platoon control, as an effective way of improving traffic efficiency, has attracted extensive interest (Guanetti, Kim, Borrelli, 2018, Horowitz, Varaiya, 2000, Ioannou, Chien, 1993, Li, Zheng, Li, Wang, 2015, Sheikholeslam, Desoer, 1990, Shladover, 1995). A system of vehicular platoon is a string of two or more closely vehicles traveling with desired cruising velocity and distance. Compared with human drivers, an automatic platoon control system has the advantages of decreasing intervehicle distance (i.e., tight formation), and is thus considered a promising solution for reducing traffic congestion, aerodynamic drag, and fuel consumption (Al Alam, Gattami, Johansson, 2010, Chien, Ioannou, 1992, Li, Chen, 2017).

The tight formation control of platoons has a particular difficulty known as “string instability”, i.e., disturbances of system states are amplified along the string of vehicles, as shown in Fig. 1(b) (from Peppard (1974)). As demonstrated by observations (Treiterer & Myers, 1974) and experiments (Sugiyama et al., 2008), the string instability of tight formation platoons can cause the emergence of a jam (e.g., stop and go) without bottleneck in both circuits and highways, which seriously compromises the benefits of platoon control.

To resolve this problem, the property of string stability has been widely studied for platoon control. Intuitively, a platoon is said to be string stable if it has the property, i.e., the disturbances are not amplified when propagating along the vehicle string (Peppard, 1974), as shown in Fig. 1(c). The basic process of string stability studies can be divided into three steps: (1) mathematically define the property of string stability; (2) derive sufficient conditions by analysis methods; (3) design controllers to satisfy the sufficient conditions.

A vast amount of literature has proposed various types of definitions and analysis methods. In many studies, although the simulation results show similar property to that in Fig. 1(c), the defined properties of string stability have confusing discrepancies. Various definitions have been described by different domains (e.g., frequency-domain and time-domain), norms (e.g., L2, Lp, and L), strength (e.g., weak and strict). The ambiguous definitions block the comparison among different studies. The rigorous analysis of their relations is required for further study. Moreover, a lot of analysis methods have been proposed and derived many alternative properties. Their relations, pros and cons, and solvable problems are not yet discussed much. The better understanding of these methods and properties is the foundation to further studies of troublesome issues.

This paper focuses on the definitions of string stability and the analysis methods which derive alternative properties of string stability. To better explain the related conceptions, we will briefly introduce the studies of vehicular platoon control. To make the research concise, however, we will not discuss other problems in the field of vehicular platoon control (Li, Zheng, Li, Wang, 2015, Li, Zheng, Li, Wu, Hedrick, Gao, Zhang, 2017) in this paper.

The aims of this paper are: (1) to clarify the relations of ambiguous definitions of string stability and recommend a unified definition; (2) to discuss the relations, pros and cons, and solvable problems of various analysis methods and recommend methods for existing troublesome issues; and (3) to dig into the relations of the derived alternative properties, which provide insights of solutions to combat string instability.

The major contributions of this paper are as follows:

First, the relations of ambiguous definitions are rigorously analyzed. The commonly used definitions are introduced and compared. Three essential properties are summarized for string stability, i.e., convergence, boundedness, and scalability. Analogous to stability definitions in control theory, three types of string stability definitions, i.e., Lyapunov, input-to-output, and input-to-state string stability, are proposed as the bridges of the relations. A rigorous analysis of these string stability definitions is elaborated in Theorem 1. Inspired by the theorem, the proposed definition, i.e., input-to-state string stability (ISSS), is recommended for future studies. The justifications of ISSS are also provided. This paper extends and deepens the discussions on definitions of the previous major survey in this field (Ploeg, Van De Wouw, Nijmeijer, 2014, Stüdli, Seron, Middleton, 2017).

Second, the various analysis methods are compared, and their derived alternative properties are rigorously analyzed. The methods are classified into three families, i.e., time-, z-, and s- domain analysis methods. The analysis tasks are divided into the temporal and spatial perspectives. The cons and pros of these methods are discussed by these perspectives, based on which we recommend methods for existing troublesome issues. Moreover, the derived alternative properties are rigorously analyzed in Theorem 2, which shows the relations among these properties and the recommended definition (i.e., ISSS) for the commonly studied platoon system. The common solutions to combat string instability are compared with the “weak coupling property”.

The remainder of this paper is arranged as follows. Section 2 describes the preliminaries of the study. Section 3 briefly introduces the vehicular platoon control that led to the studies of string stability. The commonly used definitions of string stability are introduced in Section 4 and their relations are analyzed in Section 5. Section 6 introduces the common analysis methods and their relations. The alternative properties derived by these methods are discussed in Section 7. The future directions are discussed in Sections 8 and 9 concludes the paper.

Section snippets

Preliminaries

The field of a real number is denoted by R, whereas N={1,2,}. For a vector xRn, its p-norm is given asxp=(i=1n|xi|p)1/p,p[1,)x=maxi|xi|.Given a Lebesgue measurable signal x(t):IRn, xLpI denotes its Lp norm defined asxLpI=(Ix(t)ppdt)1/p<,p[1,)xLI=suptIx,where the shorthand notation xL=xL[0,) is used when I=[0,). Given a transfer function G() of the system, the H norm of the system is defined asGH=supω|G(jω)|,(SISO)GH=supωσ¯(G(jω)),(MIMO)where SISO

Vehicular platoon control

This section briefly introduces the control problem for vehicular platoon that led to the studies of string stability. The platoon control problem, originally proposed by Levine and Athans (1966), studies how to design a controller to achieve control objectives of a platoon system. This includes platoon system description, control objectives, and controller design methods. The background of the two focuses of this paper, i.e., definition and analysis methods of string stability, is introduced.

A catalog of string stability definitions

This section introduces the commonly used definitions of string stability in the platoon control problem. The development of string stability definitions is closely related to the assumptions on platoon systems, and deeply influences the analysis methods. Therefore, the specific platoon system and analysis methods are briefly discussed when a new definition is introduced. The abbreviations of definitions are listed in Table A.2.

Relations of string stability definitions

This section provides major results of relations of above-mentioned definitions. From the perspective of control theory, three types of definitions, i.e., Lyapunov, input-to-output, and input-to-state string stability, are proposed as the bridges of the above-mentioned definitions. All equivalences of definitions are summarized. Based on the equivalences, the proposed input-to-state string stability is recommended as a formal definition of string stability for further related studies.

Analysis methods

This section introduces the commonly used analysis methods, which derive sufficient conditions of the string stability properties for controller design. To better understand their relations, the analysis methods are classified into three families, i.e., z-, s-, and time domain analysis methods. Although most methods were designed for specific definitions of string stability and systems, relations of these methods are discussed, which is insightful for future studies.

Alternative properties

This section digs into the commonly used alternative properties of string stability derived by the above-mentioned analysis methods. The properties are compared for the commonly studied platoon system (i.e., linear, identical, and with rPF topology), and their relations are identified. The relations provide insights of the common solutions to combat string instability.

Longitudinal and lateral

Most of the studies mentioned above focus on longitudinal platoon control. For lateral platoon control, lateral string stability was proposed (Alleleijn, Nijmeijer, Öncü, & Ploeg, Khatir, Davidson, 2005, Papadimitriou, Tomizuka, 2004, Solyom, Idelchi, Salamah, 2013). A related approach is to simultaneously consider the longitudinal and lateral string stability, which is also known as mesh stability (Pant, Seiler, & Hedrick, 2002).

There are many remaining problems for lateral string stability.

Conclusions

In this paper, we present a literature review of string stability studies from the perspectives of definitions and analysis methods. The relations of ambiguous definitions, analysis methods, and derived alternative properties are discussed rigorously. It provides a solid foundation for understanding the existing studies and resolving future research questions. The representative studies are summarized in Table 3.

Specifically, we outline the commonly used definitions, and propose three types of

Acknowledgement

This work was supported in part by National Natural Science Foundation of China Grant 61790565 and 71671100, Beijing Municipal Science and Technology Program under Grant D171100004917001/2, and the Beijing Municipal Commission of Transport Program under Grant ZC179074Z.

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