String-stable CACC design and experimental validationThe corresponding closed-loop system is...

String-stable CACC design andexperimental validation

R.P.A. Vugts

2009.131

TU/e Master’s ThesisApril 2008 - Januari 2010

Coaches: dr.ir. M.J.G. van de Molengraft (TU/e)ir. J. Ploeg (TNO)ir. G.J.L. Naus (TU/e)

Engineering thesis committee: prof.dr.ir. M. Steinbuch (TU/e)prof.dr. H. Nijmeijer (TU/e)dr.ir. M.J.G. van de Molengraft (TU/e)ir. J. Ploeg (Advisor, TNO)ir. G.J.L. Naus (Advisor, TU/e)

TNO Science & IndustryBusiness Unit AutomotiveDepartment of Integrated Safety

Technische Universiteit EindhovenDepartment of Mechanical EngineeringControl Systems Technology Group

Abstract

The design, analysis and experimental validation of a Cooperative Adaptive Cruise Control (CACC)system are presented. Focussing on the feasibility of the implementation, vehicle-to-vehicle com-munication with the directly preceding vehicle only is considered. This communication structureis known as semi-autonomous ACC and enables ad-hoc platooning and dealing with non-CACC-eqquipped vehicles in a straightforward way compared to other communication structures. Viawireless communication, the acceleration of the directly preceding vehicle is obtained. The sig-nal is used in a feedforward setting in combination with ACC feedback. Hence, in the case ofno feedforward, an ACC system instead of a CACC system results. Consequently, if the directlypreceding vehicle is not CACC-equipped, or if communication fails, ACC functionality is stillavailable. For ACC, the inter-vehicle distance and relative velocity of the preceding vehicle areobtained by on-board sensors, e.g. radar or lidar. To allow straightforward implementation ondifferent types of vehicles, the (C)ACC system is divided into a lower- and an upper-level con-troller. The lower-level controller is vehicle-dependent and controls the acceleration of the vehicleby actuation of the throttle and brake system. The corresponding closed-loop system is approx-imated by a simple process model, called the vehicle model. The vehicle model parameters areeasy to identify, e.g. using step response measurements. The desired acceleration is computedby the upper-level controllers, based on the information from on-board sensors and, in the caseof CACC, wireless communication. The design variables of the upper-level controller are theACC feedback controller, the CACC feedforward controller and the spacing policy dynamics. Thespacing policy is the desired inter-vehicle distance. Using a frequency domain approach, the pre-sented (C)ACC system is analyzed for stability, performance and string stability, where vehicledynamics and communication delay are taken into account. String stability considers the propa-gation of oscillations upstream a string of vehicles, i.e. a platoon. Input- output- and error stringstability functions are derived, which indicate the amplification of these signals upstream a pla-toon. For a homogeneous platoon, the error-, input- and output string stability transfer functionsare identical. For heterogeneous traffic, the output string stability function has to be considered.Considering a velocity-dependent spacing policy, the CACC system enables string stable behaviorat smaller inter-vehicle distances and smaller distance errors. Hence, a small inter-vehicle dis-tance can more safely be adopted in the case of CACC compared to ACC. Road experiments withtwo CACC-equipped vehicles validate this theoretical result.

Samenvatting

In dit verslag wordt het ontwerp van een Cooperative Adaptive Cruise Control (CACC) systeemgepresenteerd. CACC is een uitbreiding van ACC. ACC systemen kunnen een gewenste afs-tand houden tot een voorliggende auto door middel van radar en automatische aansturing vanhet gas- en remsysteem. Bij CACC wordt er ook gemaakt van draadloze communicatie. Er isgekozen voor een communicatie structuur waarbij alleen wordt gecommuniceerd met de directvoorliggende auto. Deze keuze is mede gebaseerd op de wens om tijdens het rijden treintjesvan auto’s te kunnen vormen. Bovendien kan deze communicatie structuur relatief eenvoudigomgaan met auto’s die niet kunnen communiceren. Namelijk, als het systeem voor het eerstop de markt wordt geïntroduceerd, zullen de meeste auto’s niet in staat zijn om te communi-caren. De draadloze verbinding wordt gebruikt om de acceleratie van de voorliggende auto doorte sturen. Deze acceleratie wordt, grof gezegd, opgeteld bij de gewenste acceleratie die het ACCsysteem bepaalt. Dit heeft als voordeel dat het CACC systeem automatisch degradeert tot eenACC systeem in het geval dat de voorliggende auto niet kan communiceren. Het is wenselijkom het CACC systeem zo te ontwerpen dat het eenvoudig op verschillende soorten voertuigengeïmplementeerd kan worden. Daarom is het CACC systeem opgedeeld in twee delen. Het enedeel stuurt het gas- en remsysteem aan. Het andere deel bepaalt de gewenste acceleratie op ba-sis van radar en gecommuniceerde informatie. De gesloten lus voertuigdynamica is benaderddoor een simpel model met drie parameters die makkelijk geïdentificeerd kunnen worden aande hand van stapresponsie metingen. Dit model is experimenteel gevalideerd en komt overeenmet meetresultaten. Het ontwerp van het CACC systeem bestaat uit drie variabelen: de ACC feed-back regelaar, de CACC feedforward regelaar en de gewenste volgafstand. Een frequentie-domeinaanpak is gebruikt voor het ontwerp en de analyse van het systeem, waarbij gekeken is naar sta-biliteit, performance en zogenaamde string stabiliteit. Hierbij wordt rekening gehouden met devoertuigdynamica en de vertraging in het gecommuniceerde signaal. String stabiliteit geeft aanof signalen versterkt dan wel afgezwakt worden in een treintje voertuigen. Er zijn zogenaamdestring stabiliteit overdrachten afgeleid voor verschillende signalen. Het is gebleken dat, voor eentreintje dat bestaat uit verschillende voertuigen, gekeken moet worden naar de overdracht vande snelheden of acceleraties tussen voertuigen. String stabiliteit betekent dan dat snelheden enacceleraties uitgedempt worden van voertuig naar voertuig. Het is aangetoond dat met een CACCsysteem, vergeleken met een ACC systeem, een kortere volgafstand behaald kan worden zonderstring onstabiel gedrag te vertonen. Bovendien kan een kleinere afstandsfout gerealiseerd wor-den, waardoor het ook veiliger is om dichter achter de voorligger te rijden. Experimenten opde weg met twee auto’s valideren deze theoretische bevinding. Gesloten lus simulaties komenovereen met de meetresultaten. Hieruit blijkt dat, op zijn minst voor deze experimenten, tijdenshet CACC ontwerp en de bijbehorende analyse rekening gehouden is met het grootste gedeeltevan de dynamica dat verantwoordelijk is voor het gedrag van de auto.

Contents

1 Introduction 5

1.1 Cooperative Adaptive Cruise Control . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 System specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Literature survey 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Practical issues and considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 On-board sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Wireless communication . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.3 Non-communicating vehicles . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Communication structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 String stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.1 Homogeneous string stability . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.2 Heterogeneous string stability . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Traffic flow stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6 Spacing policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Problem formulation 23

3.1 Control objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 System setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.1 Information structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.2 Design considerations and assumptions . . . . . . . . . . . . . . . . . . . 25

3.2.3 Control structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 String stability - a frequency-domain approach . . . . . . . . . . . . . . . . . . . . 31

3.3.1 Definition of a heterogeneous string stability condition . . . . . . . . . . 31

1

2 CONTENTS

3.3.2 Frequency-domain versus time-domain string stability . . . . . . . . . . . 35

3.4 Mathematical objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4.1 Vehicle stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4.2 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.3 String stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 System analysis 41

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Vehicle-stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.1 Constant inter-vehicle spacing . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.2 Velocity-dependent inter-vehicle spacing . . . . . . . . . . . . . . . . . . . 42

4.3 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3.1 ACC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3.2 CACC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.4 String stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.4.1 ACC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4.2 CACC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5 Experimental validation 59

5.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2 Vehicle model identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.3 Control design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.4 System analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.5.1 ACC, string-stable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.5.2 ACC, string-unstable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.5.3 CACC, string stable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.6 Conclusions and recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6 Conclusions and recommendations 77

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Appendix A Planning experiments I 87

CONTENTS 3

Appendix B Planning experiments II 89

Appendix C Implementation ACC feedback controller 91

Appendix D Simulation input 93

4 CONTENTS

Chapter 1

Introduction

Section 1.1 provides a general introduction to Cooperative Adaptive Cruise Control (CACC). In Sec-tion 1.2, background information with respect to associated developments at TNO Automotive and theTU/e is presented. CACC system specifications, provided by TNO Automotive, are presented in Sec-tion 1.3. Finally, the organization of this report is presented in Section 1.4.

1.1 Cooperative Adaptive Cruise Control

Adaptive Cruise Control (ACC) is an extension of Cruise Control (CC) functionality. Today, CCis a well-known and widespread functionality in modern vehicles. It automatically controls thethrottle to maintain a desired velocity, set by the driver. In addition, ACC automatically controlsthe throttle and brake to maintain a certain desired distance, in case there is preceding traffic.Commonly, on-board sensors such as radar and lidar are used to detect preceding traffic, that is,to measure the inter-vehicle distance and relative velocity between vehicles. Based on these mea-surements, ACC enables automatic vehicle following. As ACC is primarily intended as a comfortsystem, and, to a smaller degree, as a safety system, a relatively large inter-vehicle distance is typ-ically adopted [IC93] [KD04b]. Decreasing this distance is expected to yield an increase in trafficthroughput, and, specifically focusing on heavy-duty vehicles, a significant reduction of the dragforce, thus decreasing fuel consumption [ADV06] [Shl05].

In the case of ACC, string-unstable behavior may result if the adopted inter-vehicle distance

xr,3, xr,3

vehicle 3 vehicle 2 vehicle 1

x2 x1

xr,2, xr,2

radar beam

wireless

commu-

nication

Figure 1.1: Schematic representation of a string of CACC-equipped vehicles. where xr,i, xr,i and xi

represent the relative position, the relative velocity and the acceleration of vehicle i, respectively.

5

6 CHAPTER 1. INTRODUCTION

is too small. The so-called string stability of a string of vehicles, a platoon, indicates whetheroscillations are amplified upstream, i.e., from the leading vehicle i = 1 to vehicle i > 1 inthe platoon, see Figure 1.1. String-unstable behavior may result in traffic jams. No accidentor bottleneck need to be present, just too much traffic or erratic driving behavior may cause ashockwave of continuously increasing braking upstream a string of vehicles, until vehicles cometo a standstill and a traffic jam results 1. Hence, considering traffic throughput and fuel economy,as well as comfort and safety, string-unstable driving behavior is highly undesirable.

Extending ACC functionality with wireless (inter-vehicle) communication is called Coopera-tive ACC (CACC). In Figure 1.1, an example of a CACC system is shown. Using wireless vehicle-to-vehicle communication, more information can be obtained from neighboring vehicles then byusing on-board sensors only. For example, acceleration and braking commands, vehicle char-acteristics, etc. In literature, CACC is shown to enable string-stable driving behavior at smallerinter-vehicle distances compared to ACC.

1.2 Background

In 2005, an ACC Stop&Go system has been developed for a heavy-duty vehicle at TNO Automo-tive, Helmond. It was implemented and tested successfully. Stop&Go (S&G) indicates that theACC system may also be used at low velocities, e.g. stopping and starting in congested traffic.The basis of the ACC concept is a gain scheduled PD-controller. In order to obtain a good com-promise between comfort and safety for different traffic scenarios, the final tuning (by means ofsituation-dependency of the gains) of the ACC S&G system has become rather complicated andtime consuming. An easy-to-tune (re)design of the controller was desired.

In 2007, Roel van den Bleek, a M.Sc. student from the Eindhoven University of Technol-ogy (TU/e), has developed an explicit MPC ACC S&G system for TNO Automotive, which wasimplemented in an Audi S8 and tested successfully on the road [Ble07]. The Model PredictiveControl (MPC) framework enables an easy-to-tune ACC design, ensuring comfortable and de-sirable behavior for a broad envelope of working conditions. Furthermore, MPC is able to dealwith constraints in a straightforward way. For example, actuator saturation is taken into accountduring the approach of a preceding vehicle. The on-line computational effort is reduced by solv-ing the MPC optimization problem off-line, i.e. computing the explicit solution, such that thecontroller is available as a lookup table. A disadvantage of explicit MPC is the rapid growth ofsize in the explicit solution as the problem size, primarily the number of states and constraints,increases. Without going into much detail, the challenges that were considered are i) modelingof the vehicle dynamics and ii) the switching between throttle and brake. Furthermore, CACChas been no point of issue yet. Enhancement of the explicit MPC ACC S&G system is part ofthis project. The focus of this report, however, is on the development of a CACC system. Con-sequently, only the results regarding the development of a CACC system are presented in thisreport.

1YouTube video: “Shockwave traffic jams recreated for first time”http://www.youtube.com/watch?v=Suugn-p5C1M (January 6, 2010)

1.3. SYSTEM SPECIFICATIONS 7

1.3 System specifications

TNO Automotive aims to develop a CACC system that can be introduced to the market withinseveral years. Consequently, the main focus of the design is on the feasibility of implementation,or, practical realizability. TNO Automotive has provided the following system specifications:

a) Graceful degradation from CACC to ACC.Intuitively, the best performance with respect to a platoon is to be expected when all vehi-cles are CACC-equipped. Currently, CACC systems are not commercially available. Whenit does become available, most vehicles on the road will not be CACC-equipped. As thenumber of CACC-equipped vehicles decreases, it is desired that the performance degradesgradually and predictably to that of ACC.

b) Ad-hoc platooning.Platoons, i.e. strings of (CACC-equipped) vehicles, must be able to form and unform whiledriving, as opposed to driving in a fixed platoon with a designated platoon leader from startto finish. Vehicles must be able to join and leave a platoon at any time.

c) Add-on to existing ACC systems.ACC systems are already present in a number of commercially available vehicles. Hence,the chance of actual implementation of the CACC system is increased significantly if it isdesigned as an addition, or “add-on”, to existing ACC systems.

d) Robustness to communication delay.For the sake of reliability, the CACC system must be able to cope with communication delayup to a certain degree. At least, a certain level of performance must be guaranteed in thepresence of delay.

e) Vehicle independency.The CACC system must be implementable on different types of vehicles, e.g. on passengercars as well as on heavy-duty trucks.

Comparing CACC to ACC systems, the focus shifts from comfort towards safety, traffic through-put and fuel economy. Particularly considering heavy-duty vehicles, a significant reduction inaerodynamic drag force and, hence, a more fuel economic ride can be achieved if a small inter-vehicle distance is adopted. In order to increase traffic throughput and fuel economy, a smallinter-vehicle distance and string-stable behavior are demanded. Regarding safety, accurate track-ing of the preceding vehicle is required. The final goal of this project is to design, analyse andexperimentally validate a CACC system which meets these specifications.

8 CHAPTER 1. INTRODUCTION

1.4 Outline

This thesis is organized as follows. In Chapter 2, a categorized overview of available literature onautomatic vehicle following systems is presented. In Chapter 3, the control objectives and systemsetup are presented. For this setup, conditions for stability and performance are presented. Inaddition, necessary and sufficient conditions for heterogeneous string stability are derived. InChapter 4, the (C)ACC system is analyzed with respect to stability, performance and string sta-bility. The analysis focusses on the influence of the design variables and communication delay.The vehicle dynamics are assumed to be ideal. In Chapter 5, the theoretical results are validatedusing road experiments with two vehicles. In this chapter, the vehicle dynamics are not assumedideal. Stability, performance and string-stability are analyzed for the identified vehicle dynamicsand communication delay. Finally, in Chapter 6, the main conclusions and recommendations forfuture research and developments are presented.

Chapter 2

Literature survey

This chapter provides a categorized overview of the available literature on automatic vehicle follow-ing systems. First, in Section 2.1, an introduction and global overview of the literature on intelligenttransportation systems is presented. In the following sections, specific topics regarding automatic vehiclefollowing systems (AVHS) are addressed. The first topic is on practical issues and considerations, in Sec-tion 2.2. In Section 2.3, various communication structures found in literature are presented. Sections 2.4and 2.5, are on string stability and traffic flow stability, respectively. Various definitions of homogeneousand heterogeneous string stability are presented. Finally, in Section 2.6, various spacing policies arepresented.

2.1 Introduction

The invention of the automobile with internal combustion engine, i.e. the birth of the modernday passenger car, resulted in a technical revolution in the transportation of mankind. During thelast decades, there has been an overwhelming increase in the number of passenger cars world-wide [DG99]. Especially in rapid developing countries with a large population, such as Chinaand India, the number of vehicles on the road is expected to increase rapidly. Also in Westerncountries, i.e. Europe and North America, the number of vehicles on the road is expected to in-crease [DG99]. We are already confronted with problems caused by increasing road traffic: heavycongested traffic, increased number of traffic accidents, energy consumption, air pollution andglobal warming due to CO2 emissions. Hence, the expected increase of the amount of vehicleson the road is reason for concern.

The application of (information and communication) technology in vehicles and transportinfrastructure is believed to alleviate these problems. The collective term for a system that en-compasses such technology is an intelligent transportation system (ITS). As part of ITS, the fieldof intelligent vehicles (IV) is rapidly growing worldwide, both in the diversity of applicationsand in increasing interest in the automotive, truck, public transport, industrial, and military sec-tors [Bis00]. IV application areas can be segmented into

warning systems, which provide an advisory warning to the driver;

driver assistance systems which take partial control of the vehicle, either for steady-statedriver assistance or as an emergency intervention to avoid a collision;

9

10 CHAPTER 2. LITERATURE SURVEY

vehicle automation systems vehicle automation systems, which take full control of vehicleoperation.

Table 2.1: Examples of intelligent vehicle systems.

Warning systems Driver assistance systems Vehicle automationforward collision warning cruise control (CC) (low speed) vehicle automationblind spot warning adaptive CC (ACC) intersection managementlane departure warning cooperative ACC (CACC) merging managementlane change/merge warning lane-keepingintersection collision warning parking assistancepedestrian detection and warningbackup warningrear impact warningrollover warning for heavy vehiclesdriver monitoring

In Table 2.1, examples of IV systems are presented. The (advanced) driver assistance systems(DAS/ADAS) and vehicle automation systems can be divided into longitudinal control systems(“feet-off”), lateral control systems (“hands-off”), or a combination of both (“hands- and feet-off”).For example, CC, ACC and CACC systems are longitudinal control systems, while a lane-keepingsystem is a lateral control systems. Intersection- and merging management, for example, requiresboth longitudinal and lateral control. Although not entirely correct, ACC en CACC systems arealso commonly referred to as Automatic Vehicle Following Systems (AVFS). In the followingsections, and the rest of this report, ACC and CACC systems only are discussed.

An attempt is made here to categorize the literature on Intelligent Transportation systems(ITS), see Figure 2.1. An Automated Highway System (AHS) and an Intelligent Vehicle (High-way) System (IV/IVS/IVHS) are two clearly distinct proposed ITS’s. AHS uses vehicle-to-roadcommunication, while IVHS uses vehicle-to-vehicle communication.

AVFS

IVHS

AHSACC

ITS

CACC

Figure 2.1: Schematic representation of the organization in literature. Intelligent TransportationSystem (ITS), Automated Highway System (AHS), Intelligent Vehicle Highway System (IVHS),Automatic Vehicle Following Systems (AVFS), (Cooperative) Adaptive Cruise Control (C)ACC.

2.2. PRACTICAL ISSUES AND CONSIDERATIONS 11

2.2 Practical issues and considerations

In this section, several practical issues and considerations are discussed: on-board sensors, wire-less communication and non-communicating vehicles. These topics are considered to be impor-tant regarding the feasibility of the implementation. For each topic, a few examples from literateare presented to provide a feeling to the reader about some of the practical issues and limitationsregarding an AVFS

2.2.1 On-board sensors

If the longitudinal control system requires absolute positions, GPS/DGPS can be used . Alterna-tively, small magnets can be placed throughout the whole highway . In addition, an observer suchas an extended Kalman Filter (EKF) can be used, which is fed by additional on-board measure-ments such as the velocity measurement . GPS/DGPS cannot be used alone, because it has a lowupdate rate ∼ 1 Hz and is not sufficiently accurate. Absolute velocities are measured accuratelyat a sufficiently high rate, e.g. using wheel-encoders. The absolute acceleration may be estimatedfrom the velocity measurement, e.g. by differentiation and filtering or an Extended Kalman Filter(EKF). Alternatively, an accelerometer may be used.

Relative positions and velocities can be measured accurately at a sufficiently high rate usingon-board sensors, e.g. radar, lidar. Unfortunately, the noise on the relative position and velocitymeasurements is such that the relative acceleration cannot be estimated with an acceptable signal-to-noise ratio or phase lag. An option is to use wireless communication to obtain the accelerationof other vehicles.

2.2.2 Wireless communication

There are two main types of communication: vehicle-to-vehicle and vehicle-to-road communi-cation. Vehicle-to-road communication implies that there is a central communication stationwhich communicates with every vehicle on a particular section of a highway. Such a commu-nication structure requires significant investments to the existing infrastructure. Moreover, ifthe central communication station sends and receives wireless information at a single location,large communication delays and increased risk of communication failure may result . Vehicle-to-vehicle communication, on the other hand, does not require a central communication station andlarge investments. Each vehicles must be equipped with an inter-vehicle communication (IVC)protocol, which must be reliable and secure [HCL04] [KRM07]. Moreover, for safety at smallinter-vehicle distances, the communication delay must be acceptably small, i.e. at least (much)smaller than the brake perception-reaction time (PRT) of a human driver. In two field studies, thePRT is shown to be 1.5 seconds on average [Ler93] [MMB00]. The processing time of the humanvisual system, is shown to be much faster, about 150 ms [TFM96]. Since the communicationsystem is (part of) the “eyes” of the automatic vehicle following system, the latter value is used asa benchmark for the admissible communication delay. Further discussion on IVC is out of thescope of this project. Literature on IVC is abundant, see [LH06] for a survey.


2.2.3 Non-communicating vehicles

When a CACC system is first introduced to the market, the penetration level will be low. Most ve-hicles will not be equipped with CACC and will not be able to communicate. Non-communicatingvehicles may become a problem when they are undetectable. Vehicles are detectable by on-boardsensors, e.g. radar, or by communication. A non-communicating vehicle is undetectable if i) it isdriving beyond radar range, or ii) if it is driving between two other non-communicating vehicles.In a worst case scenario, the behavior of an undetectable vehicle may be completely opposite tothe behavior of surrounding (detectable) vehicles, for example in case of an accident. Hence, froma safety point of view, the CACC controller must be able to deal with non-communicating vehi-cles up to a certain level. This is a challenging task, depending on the communication structure.Discussions on how to deal with undetectable vehicles have not been found in literature.

2.3 Communication structures

The communication structure specifies the flow of information between vehicles and/or road.The communication structure is determined by the information that is used by the longitudinalcontroller, e.g. absolute and/or relative positions and/or velocities, etc. The information must beobtained by on-board sensors and/or communication. Hence, the communication structure is animportant design consideration, because it has a large influence on the (feasibility of the) imple-mentation of the longitudinal controller (see Section 2.2). This section provides an overview ofthe various communication structures found in literature. It must be noted that all communica-tion structures are based on the concept of platooning, i.e. the controllers are designed to controlthe inter-vehicle distances of a limited sized string of vehicles.

The communication structures can be divided into two main groups: centralized and decen-tralized controllers. Centralized controllers are also referred to as “smart road” systems [Var93].An Automated Highway System (AHS), enabling driverless cars on a dedicated highway, is anexample of a centralized control system [HR97]. Centralized control implies that the controlleris located at a central control location, which gathers and interprets data from all communicatingvehicles, determines an appropriate response for each vehicle (e.g. a driving action) and sendthis response to each vehicle. Centralized controllers use vehicle-to-road communication, seeFigure 2.2.

command

center

Figure 2.2: An example of a centralized communication structure (AHS approach).

A perfect example of a centralized controller is the optimal error regulation of a string ofmoving vehicles, which was first studied by Levine, Levis, and Athans [LA66]. In 1971, Melzer

2.3. COMMUNICATION STRUCTURES 13

presents a closed-form solution to the Riccati equation for this problem [MK71a] [MK71b]. Sim-ulation results for a string of three vehicles indicate that the suggested design method yields anexcellent control system [LA66]. The authors emphasize, however, that such a centralized controlsystem imposes serious economic considerations due to the size of the communication systemrequired. The optimal design requires that the force acting on each vehicle is a function of theabsolute positions and velocity deviations of all vehicles.

Besides AHS, a so-called Intelligent Vehicle/Highway System (IVHS) is another proposedITS. The IVHS is a decentralized control system, also referred to as a “smart car” system, whichcan be designed to use vehicle-to-vehicle communication only. In a decentralized control system,each vehicle gathers and interprets information from neighboring vehicles on its own, using com-munication and/or on-board sensors, and determines its own response. If on-board sensors onlyare used, the system is said to be “autonomous”, whereas it is referred to “non-autonomous” or“cooperative” if communication is involved. Most of the recent work on an ITS focusses on decen-tralized control instead of centralized control, because the requirements of the communicationsystem and investments on the existing infrastructure are less severe. The California Partners forAdvanced Transit and Highways (PATH) program of the University of California provided in amajor contribution to the development of an IVHS system. An overview of the developments inthe PATH program is presented in [SDH+91] [Shl92].

a) b)

c) d)

f)e)

Figure 2.3: Decentralized communication structures for platoonining (IVHS approach): a)“everybody knows everything”, b) designated platoon leader, c) bi-directional, d) N -vehicleslook-ahead, e) directly preceding vehicle only, f) mini-platoons with designated platoonleader.

On-board sensor signals are indicated by black lines, communication signals are indicated by bluelines.

Various communication structures are presented in literature for decentralized control sys-tems, see Figure 2.3. In 1974, Peppard suggests a decentralized bidirectional communicationstructure, where each vehicle obtains inter-vehicle separation measurements of the directly pre-ceding and directly following vehicle [Pep74] (Figure 2.3(c)). Bidirectional control means that thecontroller in each vehicle uses information from traffic in two directions, i.e. preceding as wellas following traffic (Figure 2.3(a,c)). As part of the PATH program in 1998, Yanakiev, Eyre andKanellakopoulos regard a platoon of vehicles as a mass-spring-damper system and analyze stringstability for unidirectional and bidirectional controllers [YEK98]. It is shown that the analysis ofstring stability for bidirectional is more complicated because disturbance propagation in both theforward and backward direction has to be considered.

Unidirectional control means that the controller uses information from preceding traffic only


(Figure 2.3(b,d,e,f)). For autonomous ACC systems, the inter-vehicle distance and, possibly, thevelocity of the directly preceding vehicle is used, obtained by on-board sensors [IC93]. Includ-ing communication of the acceleration of the preceding vehicle is shown to provides significantadvantages over ACC systems in terms of safety and traffic flow [YK98]. Communication ofthe velocity and/or acceleration of the lead vehicle, i.e. the platoon leader, is shown to have apositive effect on string stability, because all vehicles in the platoon follow the same referencevehicle (Figure 2.3(b)). In order to avoid communication over long distances, Swaroop suggestsa mini-platoon communication structure, where every platoon is divided into mini-platoons andthe last vehicle of a mini-platoon becomes the reference vehicle for the following mini-platoon(Figure 2.3(f)). Within each mini-platoon, every vehicle obtains information from the directly pre-ceding vehicle and the mini-platoon leader [SH99a]. Another suggested communication struc-ture is communication with a limited amount of directly preceding vehicles (Figure 2.3(d)). Anycombination of the presented communication structures is possible. For example, communica-tion with one following- and five preceding vehicles, while communicating with a platoon leaderin mini-platoons, may be considered.

A methodology for optimal control design of a string of vehicles within the IVHS approachis presented, called “decentralized overlapping control” [SSS00]. This methodology allows tosystematically take into account optimality in a predefined sense, similar to optimal error regula-tion [LA66]. For example, the controller may be designed to optimize safety in the platoon. Theavailable information on each vehicle is determined on beforehand and results a decentralizedcontroller. Therefore, this methodology somewhat closes the gap between centralized (optimal)control and decentralized control. First, a linearized state model for an entire string of vehiclesis derived, containing all the states, i.e. positions, velocities, of each vehicle. This platoon modelis then expanded and decomposed into overlapping subsystems by using the so-called inclusionprinciple [SMS98] [SXBMA99] [SSS05]. The subsystems are defined in such a way that theirstate vectors are composed of the available measurements in each vehicle. Using the inclusionprinciple, it is shown that the closed-loop (platoon) system preserves the main properties, e.g.stability and sub-optimality, obtained by the design done in the expanded space.

2.4 String stability

String stability is concerned with the stability of a platoon, i.e. a string of interconnected vehicles.Considering a platoon with a possibly infinite amount of vehicles, the behavior of each vehiclemust be such that oscillating behavior of the leading vehicle, i.e. the platoon leader, is not ampli-fied upstream the platoon. Else, the oscillating behavior might become so severe at some pointin the string, that the vehicles reach their accelerating or braking limits. Hence, a string unstableplatoon may result a “harmonica effect”, which, in turn, may result in traffic jams or even colli-sions. String stability must not be confused with “ordinary stability”, i.e. stability of solutions ofdifferential equations and of trajectories of dynamical systems under small perturbations of ini-tial conditions, as they evolve in time. String stability considers the propagation of disturbancesfrom vehicle to vehicle, i.e. as they evolve in vehicle index.

To my knowledge, the term “string stability” is first used by Peppard in 1973 [Pep74]. Inearly publications, the term “slinky-effect” is used to describe the same phenomenon. Accordingto Swaroop and Hedrick [SH95], Chu defined string stability in the context of vehicle followingsystem for the first time in 1974 [Chu74], while Chang is said to introduce a stronger version

2.4. STRING STABILITY 15

of string stability in 1980 [Cha80]. For string stability, according to Swaroop and Hedrick, “it isrequired to ensure that the distance errors do not amplify upstream from vehicle to vehicle in a platoon”[SH94] [SH95] [Swa97] [SHC01]. This notion of string stability is shared by other researchers[Shl91] [SD93] [WR94], [YK98] [RCL+00] [GS02] [RZ02].

A distinction is made between homogeneous and heterogeneous string stability. Homoge-neous refers to string stability of a platoon where all vehicle are identical. That means, identicalvehicle dynamics, identical spacing policy, identical longitudinal controller, etc. As a result, theclosed-loop transfer functions that describe the following behavior are identical for each vehicle.Heterogeneous string stability, on the other hand, refers to string stability of a platoon where eachvehicle may have different vehicle dynamics, which is more realistic regarding ad-hoc platooningin real traffic.

Literature on the effects of communication delay on string stability is limited. The effect of aconstant communication delay on string stability is considered in [LGMH01] [LG01] [LSH04]. Itis not considered in other references in this report.

As will soon become clear, literature on string stability is abundant. More importantly, andconfusingly, different notions and definitions of string stability are found in literature. Some ofthe notions and definitions are repeated in the following sections.

2.4.1 Homogeneous string stability

e (t)3e (t)2

x (t)3

x (t)1x (t)2

x (t)d,2x (t)

d,3

Figure 2.4: A homogeneous vehicle string, where xi(t) are absolute positions, xd,i(t) are desiredinter-vehicle distances and ei(t) are distance errors for i = 1, 2, 3.

Consider a homogeneous string of vehicles i = 1, . . . ,∞, see Figure 2.4, with distance, veloc-ity and acceleration error

ei(t) = xi(t)− xi−1(t)− xr,d,i(t) (2.1a)ei(t) = xi(t)− xi−1(t)− xr,d,i(t) (2.1b)ei(t) = xi(t)− xi−1(t)− xr,d,i(t) (2.1c)

where xi and xr,d,i(t) are the absolute position and desired inter-vehicle distance of vehicle i > 1.The vehicle index i = 1 refers to the leading vehicle and increases upstream. Note that, fora homogeneous platoon, xr,d,i(t) = xr,d(t) is identical for each vehicle. For a homogeneousplatoon, the transfer function

Ψi(s) =Ei(s)

Ei−1(s)(2.2)

is identical for each vehicle, i.e. Ψi(s) = Ψ(s) for all i. In order to assure that the distance errorsdo not amplify from vehicle to vehicle, the following condition is used [Shl91] [SD93] [WR94]


[GS02]||Ψi(jω)||∞ ≤ 1 (2.3)

where the ∞-norm indicates that the magnitude |Ψi(jω)| ≤ 1 for all ω. Condition (2.3) is nec-essary, in the sense that if |Ψ(jω0)| > 1 for some frequency ω0, an error with this frequency isamplified upstream the platoon. In addition, ω → |Ψi(jω)| is required to be a strictly decreas-ing function of ω [Shl91]. In [SPH04], homogeneous string stability (2.3) is analyzed for twocommunication structures, using a frequency-domain approach: i) predecessor following and ii)predecessor&leader following.

In [RCL+00], it is desired that the impulse response function ψi(t) = L−1(Ψi(s)) does notchange sign, where L−1(·) is the inverse Laplace transform. Since the step response is the in-tegrated impulse response, this condition corresponds to a non-overshooting step response ofΨi(s), which ensures that ||ei(t)||∞ ≤ ||ei−1(t)||∞ if ei(t) is a step signal. ||ei(t)||∞ is the maxi-mum amplitude of ei(t) for t ≥ 0.

Instead of considering the amplification of errors (2.2), the amplification of the inter-vehicledistance may be considered [CH99]

||Ψ′i(s)||∞ =

∣∣∣∣∣∣∣∣

Xi−1(s)−Xi(s)Xi−2(s)−Xi−1(s)

∣∣∣∣∣∣∣∣∞≤ 1 (2.4)

Using (2.1), it can be shown that condition (2.4) is similar to condition (2.3) if the desired distancesxr,d,i(t) = xr,d,i are constant.

String stability for a platoon of N vehicles is defined as [LSH04]

||e1(t)||∞ ≤ ||e2(t)||∞ ≤ . . . ≤ ||eN (t)||∞ (2.5)

From linear system theory [LSH04],

||ψi(t)||1 =

∞∫

0

|ψi(τ)|dτ (2.6a)

||ψi(t)⊗ ei(t)||∞ ≤ ||ψi(t)||1||e1(t)||∞ (2.6b)||Ψi(s)||∞ ≤ ||ψi(t)||1 (2.6c)

where ⊗ denotes a convolution. From (2.6), the authors conclude that the platoon is string stableif ||ψi(t)||1 < 1 and string unstable if ||ψi(t)||1 ≥ ||Ψi(s)||∞ > 1. The 1-norm of the impulseresponse is also used in [KD04b] [KD04a], which state that ||ei(t)||∞ ≤ ||ei−1(t)||∞ if ||ψi(t)||1 ≤1. The 1-norm of ψi(t) is the surface of the absolute impulse response.

Swaroop and Hedrick generalized the concept of string stability for a class of nonlinear sys-tems and seek sufficient conditions to guarantee string stability [SH95]. Quoting: “Intuitively,string stability implies uniform boundedness of all the states of the interconnected system for all time ifthe initial states of the interconnected system are uniformly bounded”. Applying this notion of stringstability on a platoon of vehicles, results the following definition [SH99a]

“A platoon is string stable if, given any γ > 0, there exists δ > 0 such that whenever

max

{||ei(0)||∞, ||ei(0)||∞,

∣∣∣∣∣

∣∣∣∣∣i∑

1

ej(0)

∣∣∣∣∣

∣∣∣∣∣∞

,

∣∣∣∣∣

∣∣∣∣∣i∑

1

ej(0)

∣∣∣∣∣

∣∣∣∣∣∞

}< δ ⇒ sup

i{||ei||∞, ||ei||∞} < γ.′′

(2.7)


The string stability condition (2.7) implies that the spacing and velocity errors of vehicles in aplatoon remain small at all instants of time, if the initial spacing and velocity errors are sufficientlysmall. They also present a definition of weak string stability:

“A platoon is string stable in the weak sense if, given any γ > 0, there exists δ > 0 such thatwhenever

max {||ei(0)||1, ||ei(0)||1} < δ ⇒ supi||ei||∞ < γ.′′ (2.8)

Weak string stability of a platoon (2.8) implies that the spacing errors of vehicles in a pla-toon remain small at all instants of time, if the initial spacing and velocity errors are absolutelysummable, i.e. if the sum of the absolute values of its summands converges, and if the sumof the absolute values of the initial spacing and velocity errors are sufficiently small. Somewhatcounterintuitive, weak string stability (2.8) is called “weak” while imposes a stronger conditionon the initial spacing and velocity errors compared to (2.7) [SH99a].

2.4.2 Heterogeneous string stability

e (t)3e (t)2

x (t)3

x (t)1x (t)2

x (t)d,2x (t)

d,3

Figure 2.5: A heterogeneous vehicle string, where xi(t) are absolute positions, xd,i(t) are desiredinter-vehicle distances and ei(t) are distance errors for i = 1, 2, 3.

Heterogeneous string stability refers to string stability of a platoon where each vehicle mayhave different dynamics, see Figure 2.5. In this case, instead of the distance error, absolute posi-tions [HLBM03] and velocities [Lia00] [BI99] are considered.

Φi(s) =Vi(s)

Vi−1(s)=

sXi(s)sXi−1(s)

=Xi(s)

Xi−1(s)(2.9)

where Xi(s) and Vi(s) are the Laplace transforms of the position xi(t) and velocity vi(t) = xi(t).The following definition of heterogeneous string stability for a platoon of N vehicles is presented[Lia00] ∣∣∣∣∣

∣∣∣∣∣N∏

1

Φi(jω)

∣∣∣∣∣

∣∣∣∣∣∞≤ 1 (2.10)

which ensures that oscillations in absolute position and velocity remain bounded for boundedoscillatory behavior of the leading vehicle i = 1. Condition (2.10) is automatically satisfied if||Φi(jω)|| ≤ 1 for all i = 1, . . . , N . Somewhat remarkable, condition (2.10) had already beenproposed by Peppard in 1974 [Pep74].

Another proposed definition of heterogeneous string stability is [BI99]


“A platoon of N vehicles, given in (2.9), is string stable if

||ei(t)||p ≤ ||ei−1(t)||p (2.11a)||ei(t)||p ≤ ||ei−1(t)||p (2.11b)||ei(t)||p ≤ ||ei−1(t)||p (2.11c)

for all p = 1, . . . ,∞ and for all i = 1, . . . , N”

It is shown that, for a constant headway time policy, the closed-loop transfer functions thatdescribe the propagation of errors ei(t), ei(t) and ei(t) is clearly different than that of the velocitiesxi(t), if different headway times are adopted in each vehicle. These transfer functions are shownto be identical in the homogeneous case [BI99].

String stability of a platoon with non-identical controllers is analyzed in [KD04b] [KD04a].The longitudinal controller of each vehicle is assumed to be PID, of which the gains are notnecessarily identical. In one example, the PID gains of a string of vehicles are derived suchthat condition (2.3) holds for every vehicle i. It turned out that the PID gains must increasewith vehicle index. Simulation results show that the resulting peaks of the distance error indeeddecreases with vehicle index. However, the velocity peaks are shown to increases with vehicleindex. In another example, PID gains are derived such that the distance error and velocity peaksare attenuated, i.e. such that condition (2.3) holds and ||Φi(s)||∞ ≤ 1 for every vehicle i. In thiscase, simulation results show that the distance error and velocity peaks are attenuated for a largestring of vehicles. However, the velocity peaks do increase for the first 500 vehicles up to 450%of the maximum velocity of the leading vehicle i = 1. [KD04b] [KD04a]

In 2007, Shaw and Hedrick analyzed heterogeneous string stability for vehicle strings ofarbitrary length and arbitrary vehicle type ordering, using a frequency-domain approach [SH07b].Three communication structures are considered: i) predecessor following, ii) leader following andiii) predecessor&leader following. They note that, for a homogeneous platoon, the distance errorsof a string stable platoon attenuate uniformly down the vehicle chain, while this is not the casefor a heterogeneous platoon. Simulation results showing the distance error are presented for aplatoon with three types of vehicles (slow, medium and fast), corresponding to transfer functions(2.2) with a low, intermediate and high bandwidth. Separate controllers are designed for eachvehicle type, such that a homogeneous vehicle string of each vehicle type is string stable. Whenthe three vehicle types are put in the same string, the distance errors depend on the ordering ofthe vehicles. Hence, it is not obvious from their simulations whether the system is string stableor not. Therefore, they propose the following definition a heterogeneous string stability:

“A heterogeneous vehicle string is string stable if the propagating errors stay uniformly bounded forall string lengths and vehicle type orderings.”

A condition is derived, which results in heterogeneous string stability according to this def-inition. However, bode diagrams and simulation results regarding the propagating of distanceerrors only, are presented. Simulation results regarding the propagating of velocities and/or ac-celeration, are not shown.

2.5 Traffic flow stability

Traffic flow stability is not the same as string stability. String stability refers to a property of aplatoon, see Section 2.4. Traffic flow stability, on the other hand, deals with the evolution of

2.6. SPACING POLICIES 19

traffic velocity and density in response to the addition and/or removal of vehicles from the flow,see [SR99] [SH99b] [SR98] [SR00] [SRT08]. The distinction between string stability and trafficflow stability has first been made in [SR99]. The goal of this section is to provide some insight inthe analysis of traffic flow stability and to show the difference with respect to string stability.

In the mentioned publications, e.g. [SR99], traffic flow is modeled as a continuum. In directanalogy to fluid flow, most macroscopic models of the traffic flow assume that traffic throughput(or volume, e.g. in number of vehicles per hour) Q is equal to the product of the aggregate trafficdensity ρ (in number of vehicles per unit length of the highway) and the aggregate traffic velocity v(in unit length of the highway per hour). Assume the following steady-state relationship betweentraffic density ρ and speed v:

v = vf

(1− ρ

ρmax

)⇒ Q = ρv = ρvf

(1− ρ

ρmax

)(2.12)

where vf is the free velocity of traffic and ρmax is the maximum traffic density, or jam density.The latter equation, i.e. Q = Q(ρ), is a constitutive relation for traffic flow, commonly referredto as the fundamental traffic characteristic. Note that the traffic throughput is either zero if thereare no vehicles at al, i.e. Q(ρ = 0) = 0, or if the maximum capacity is reached, i.e. Q(ρ = ρmax).It can be shown that the throughput Q increases with increasing density up to a critical densityρcrit, with corresponding maximum traffic throughput Qmax. For densities ρ > ρcrit, the trafficthroughput decreases. Moreover, it can be shown that traffic flow is stable as long as δQ/δρ > 0,i.e. up to the critical density, and unstable for δQ/δρ < 0 [SH99b] [SR00].

A literature survey on the impact of intelligent vehicles on traffic flow characteristics is pre-sented in [ADV06]. Based on the literature study, Arem concludes that: “Extensive research intothe traffic flow impact of CACC in terms of traffic flow stability and throughput is lacking. Thelimited CACC effect studies that have been performed emphasize that CACC is able to increasethe capacity of a highway significantly. CACC can potentially double the highway capacity at ahigh penetration level.” The penetration level is the ratio of CACC equipped vehicles to all ve-hicles. Simulation results are presented for a highway merging scenario from 4 to 3 lanes, forequipped and non-equipped CACC vehicles for a varying penetration level. A traffic flow simula-tion model MIXIC is used that was specially designed to study the impact of intelligent vehicleson traffic flow [ADV06].

2.6 Spacing policies

The term “spacing policy” is commonly used to classify the desired inter-vehicles distance. Aconstant spacing policy, for example, refers to a constant desired distance. The spacing policyhas a large influence on, for example, the driving behavior, the string stability characteristicsand traffic flow stability [SR99] [SH99b]. Besides that, the spacing policy influences the safety-and comfort perception of the human driver. If, for example, the desired inter-vehicle distanceis decreased when the vehicle is moving faster, the driver will likely not feel comfortable norsafe. Hence, it is worth to investigate what kind of spacing policies are considered in literature.This section does prove a complete set of suggested spacing policies in literature. A few spacingpolicies are selected and discussed in more detail.

The two most encountered spacing policies in literature are


1. a constant headway time policy, e.g. see [CJ07] [Dre94] [FGSB04] [GF97] [HLBM03] [Lia00][CH99] [LHD02] [ZI04] [RZ02] [SMS98].

2. a constant spacing policy, e.g. see [BH05]] [GL94] [HR97] [HY06] [KD04a] [GS02] [LSH04][RCL+00] [RZD07] [SH07a] [SD93] [LGMH01] [WR94] [SHC01]

For a constant headway time policy, the desired inter-vehicle distance xr,d(t) is linearly depen-dent on the velocity, i.e.

xr,d,i(t) = ri + hd,ivi(t) (2.13)

where ri is the desired distance at standstill, hd,i is the so-called desired headway time, and vi(t)is the velocity of vehicle i. The headway time is the time it takes for vehicle i to reach the cur-rent position of the corresponding preceding vehicle i− 1 when continuing to drive at a constantvelocity. Presumably, the constant headway time spacing policy is popular due to its simplic-ity. It includes one of the most basic elements of human driving behavior, i.e. the distance isincreased with increasing velocities, while it is still linear. Human driver models are typicallynon-linear, e.g. see [Win99]. Regarding platooning, it has been shown that a constant headwaytime spacing policy enables string-stable behavior for a certain minimum desired headway timeusing inter-vehicle distance and relative velocity measurements only. Hence, communicationwith the preceding- and/or leading vehicle is not necessary to achieve string stability for a contantheadway time spacing policy. On the other hand, it must be noted that a constant headway timepolicy is shown to yield traffic flow instability over the whole density region [SR99].

At high velocities, constant headway time spacing policies (2.13), may result in large inter-vehicle distances (depending on the adopted headway time, of course). This is disadvantageous inthe case where traffic throughput and road capacity are considered to be important, or, particularlyregarding heavy-duty vehicles, in the case that a reduction in aerodynamic drag force is desiredfor the sake of fuel consumption. In these cases, a constant spacing policy

xr,d,i(t) = ri (2.14)

is preferred, which is an exception of a constant headway time policy (2.13) with hd,i = 0 s.Regarding platooning, it has been shown that string-stable behavior cannot be achieved for aconstant spacing policy (2.14) without communication [SH99a]. However, string stability canbe achieved if communication with the platoon leader is enabled [SH99a]. Only marginal stringstability can be achieved if communication with the directly preceding vehicle is enabled [SH99a].

Several non-linear spacing policies are presented to achieve small distance errors without in-creasing the inter-vehicle distances [YEK98] [YK98]. One example of such a non-linear spacingpolicy is presented here: the variable headway time spacing policy. This spacing policy is identicalto (2.13), except that the desired headway time hd,i is made dependent on the relative velocity. Theidea behind this spacing policy is as follows. If the relative velocity between two vehicles is posi-tive, that is, if the preceding vehicle is moving faster, then it is safe to reduce the desired headwaytime. If the preceding vehicle is moving slower, however, it would be advisable to increase thedesired headway time. This led to the following spacing policy

xr,d,i(t) = ri + hd,i(t)vi(t), with (2.15a)hd,i(t) = sat (hd,0,i − ch,ivr,i(t)) (2.15b)

where hd,0,i and ch,i are constants and vr,i(t) = vi−1(t)− vi(t) is the relative velocity. The satura-tion limits are chosen as 0 and 1. For safety reasons, the desired headway time cannot be allowed

2.6. SPACING POLICIES 21

to become negative, while large values are undesirable for reasons of traffic throughput. Notethat hd,0,i is the desired headway time if the relative velocity is zero, i.e. if vr,i(t) = 0.

In [IC93], a spacing policy is derived based on a safety requirement. For safe operation, theyrequire the following vehicle to maintain a sufficiently safe distance throughout all maneuvers, inorder to avoid a collision. An extreme stopping situation is considered, i.e. when the precedingvehicle is assumed to be at full negative acceleration while the following vehicle is at full positiveacceleration at the instant the stop maneuver commences. This led to the following spacingpolicy:

xr,d,i(t) = λ1

(vi(t)2 − v2

i−1(t))

+ λ2vi(t) + λ3 (2.16)

where λ1,λ2 and λ3 are constants which depend on the maximum acceleration and jerk levelsof the two subsequent vehicles. The spacing policy (2.16) is also referred to as a “constant safetyfactor policy” [SH99b]. For a constant safety factor policy, the desired inter-vehicle distance isproportional to the controlled vehicle’s stopping distance and hence, varies quadratically with thecontrolled vehicle’s speed. Note that, if λ1 = 0, or if both vehicles are driving at the same velocity,i.e. vi(t) = vi−1(t), (2.16) is similar to (2.13).

A desired inter-vehicle distance based on a curve fitting of human driver behavior is presented[XS03]

xr,d,i(t) = ri + hd,ivi(t)ki (2.17)

The constant 0 < ki < 1 in (2.17) ensures that the desired inter-vehicle distance at high velocitiesis smaller than in case of a constant headway time policy (2.13).

A spacing policy based on a traffic flow requirement is presented [SR98], yielding

xr,d,i(t) =L0,i

1− vi(t)vf

− Lv,i (2.18)

where Lv,i is the vehicle length, L0,i − Lv,i the desired distance at standstill. Without going intodetail, the spacing policy (2.18) yields a desirable fundamental traffic characteristic, i.e. the trafficthroughput as a function of the traffic density. The spacing policy (2.18) ensures that the trafficflow is stable up to the critical traffic density, at which the traffic throughput is at its maximum,and unstable beyond the critical density. Stable traffic flow over the whole density range is notpossible [SH99b] [SR98]. Intuitively, the spacing policy (2.18) does not seem very attractive, sincethe desired distance xr,d,i(t) approaches infinity as the velocity vi(t) approaches the free trafficvelocity vf .

Chapter 3

Problem formulation

This chapter focusses on a clear formulation of the control problem for a particular system setup. First,the control objectives are formulated in Section 3.1. Then, in Section 3.2, the system setup is presentedand motivated. In Section 3.3, necessary and sufficient conditions for heterogeneous string stability aredefined. Finally, the control objectives from Section 3.1 are translated to mathematical objectives inSection 3.4.

3.1 Control objectives

The control objectives are separated in local and global objectives. The local objectives are relatedto individual vehicles

Vehicle stability, the (C)ACC controller must yield a stable closed-loop system;Safety, collisions must be avoided;Performance, small distance errors are desired;Comfort, a low acceleration and jerk profile is considered to be comfortable;Fuel economy, for a reduction in travel costs.

whereas the global objectives are related to traffic as a whole.

Traffic throughput, string-stable behavior at small inter-vehicle distances;Road capacity, small inter-vehicle distances are desired;Fuel economy, for a reduction in CO2-emmissions and its effect on global warming.

Safety is considered to be similar to performance, i.e. collisions are avoid by achieving smalldistance errors. Vehicle stability and performance are considered to be the most important objec-tives, closely followed by traffic throughput. Some objectives are in agreement with each other,while others are in conflict. For example, vehicle stability is a prerequisite for safety and perfor-mance. Comfort and fuel economy automatically results by demanding string-stable behavior.As will soon become clear, string-stable behavior implies that velocity, acceleration and jerk peaksare attenuated from vehicle to vehicle. As a result, considering a string of vehicles, the driv-ing behavior automatically becomes increasingly comfortable and fuel-economic from vehicle tovehicle. Moreover, small distance errors are more easily achieved if the preceding vehicle has

23

24 CHAPTER 3. PROBLEM FORMULATION

a low acceleration profile. Hence, performance is increased from vehicle to vehicle if the pla-toon is string-stable. String-stable driving behavior is also believed to increase traffic throughputbecause it prevents the occurrence of shock-waves, which may result in traffic jams or even colli-sions. Small inter-vehicle distances are desired to increase road capacity, traffic throughput and,particularly considering heavy-duty trucks, a reduction in fuel consumption due to a reductionin aerodynamic drag force. Besides that, human drivers typically prefer small inter-vehicle dis-tances. On the other hand, small inter-vehicle distances require small distance errors, whichrequire large control inputs and, consequently, high acceleration profiles. Taking the objectivesand corresponding priorities as a whole, the general goal is to design a stable CACC system,which guarantees string-stable behavior at small inter-vehicle distances.

3.2 System setup

In the previous chapter, a research overview is presented. Based on this overview and the objec-tives for the CACC system, the information structure and control structure are chosen.

3.2.1 Information structure

The information structure specifies the flow of wireless information between vehicles and/orroad. The decision making process of the information structure is schematically depicted inFigure 3.1.

Centralized Decentralized

Semi-autonomous adaptive cruise control

bi-

direc-

tional

uni-

direc-

tional

with

platoon

leader

single

preceding

vehicle

multiple

preceding

vehicles

without

platoon

leader

Information structures

Figure 3.1: Schematic representation of the decision making process of the information structure.

Focussing on the feasibility of implementation of the CACC system, communication with asingle, directly-preceding vehicle only is considered. In literature, this is often referred to as semi-autonomous adaptive cruise control. First of all, a centralized information structure requires amajor investment to the existing infrastructure, e.g. due to the need of vehicle-to-road communi-cation and/or a central control station. A decentralized information structure, however, involving

3.2. SYSTEM SETUP 25

vehicle-to-vehicle communication, does not require large investments or adaptations to the exist-ing infrastructure. Each vehicle gathers information and determines its driving behavior on itsown. A bi-directional information structure, i.e. when vehicles communicate with preceding aswell as following traffic, is thought to highly complicate the control design. Conflicting situationsregarding preceding and following traffic occur on a daily basis. For example, the risk of a deadlyfront-end collision may increase while attempting to avoid a rear-end collision. Such traffic sce-nario’s require a complex decision making algorithm, which is not considered here as it does notfacilitate the control design. Moreover, by law, the following vehicle is normally held responsiblein case of an accident. Hence, a uni-directional information structure is adopted, where commu-nication with preceding traffic only is considered. In literature, communication with a platoonleader is shown to have a positive effect on the so-called string stability of a platoon. However, thepractical problems of a platoon leader are often ignored. Firstly, a platoon with a large amountof vehicles would involve communication over long distances, which presumably increases com-munication delay and the risk of communication failure. Secondly, for ad-hoc platooning, it isdifficult to define the platoon leader. Imagine an extremely large string of CACC-equipped ve-hicles from the beginning to the end of the highway, with inter-vehicle distances ranging fromseveral meters to several hundreds of meters. Which vehicle is assigned as the platoon leader?What happens if multiple non-CACC-equipped vehicles merge into the lane? These problemsdo not exist for a limited-sized platoon which arrives and departs in the same formation and forwhich the inter-vehicle distances are so small that non-CACC-equipped vehicles are unable topenetrate the platoon. For ad-hoc platooning, however, these problems do exist if communica-tion with a platoon leader is considered. To avoid these problems, platooning without a platoonleader is considered.

When communicating with a single preceding vehicle, there are five possible scenario’s thatmay occur, see Figure 3.2. In scenario’s 4 and 5, there is a possibility that there is one, or more, un-detectable vehicle(s) present between the two CACC-equipped vehicles. A vehicle is undetectable,if it is not CACC-equipped and out of radar range, or driving in front of another vehicle withoutradar. For a platoon with an increasing amount of vehicles, the possibility of undetectable vehiclesmerging into the platoon increases. The behavior of an undetectable vehicle is not known and itis thought to be unsafe to make any assumption with respect to its behavior. Therefore, commu-nicated information beyond non-CACC-equipped vehicles must be discarded, leaving standardCC or ACC functionality (in scenario’s 4 and 5 of Figure 3.2). It is not hard to imagine that theproblem of undetectable vehicles becomes a lot more cumbersome when communicating withmultiple preceding vehicles is enabled. Moreover, it would increase the controller complexityand impose higher demands on the wireless communication due to the increased amount ofinformation and increased communication distances. Therefore, communication with a single,directly preceding vehicle only is considered (scenario 3 of Figure 3.2). In some specific situations,however, it is desirable to use communicated information beyond non-CACC-equipped vehicles.For example, a traffic jam may be detected at an earlier time such that the vehicle is able to startslowing down earlier. These kind of traffic scenario’s will not be discussed here.

3.2.2 Design considerations and assumptions

All systems are described by single-input single-output (SISO), linear time-invariant (LTI) transferfunctions. The inter-vehicle distance and relative velocity are assumed to be measured by radarand/or lidar. The absolute velocity and acceleration are assumed to be measured or estimated


vehicle i

vehicle i

vehicle i − 1

vehicle i vehicle i − 1

vehicle i vehicle i−?

vehicle i vehicle i − 1 vehicle i−?

radar rangeradar range

communication rangecommunication range

1.

2.

3.

4.

5.

Figure 3.2: Five possible scenario’s during communication with a single preceding vehicle only: 1)there is no preceding traffic, 2) the directly preceding vehicle is non-CACC-equipped, 3) the directlypreceding vehicle is CACC-equipped, 4) the CACC-equipped vehicle is out of radar range and 5) theCACC-equipped vehicle is driving beyond a non-CACC-equipped vehicle. The communication range

is assumed to be larger than the radar range. Radar beams are indicated by black curved lines.Wireless communication signals are indicated by blue lines.

locally on each vehicle using on-board sensors, i.e. wheel encoders and/or accelerometers. Allthese measurements are assumed to be corrupted by measurement noise. The time delay onthe radar/lidar measurements is assumed to be negligible and therefore neglected. In case thedirectly preceding vehicle is CACC-equipped, the acceleration of this vehicle is assumed to beavailable by wireless communication. The communication delay is approximated by a constanttime delay, such that it can be described by a linear system. The focus is on the development of aCACC system concept instead of wireless communication modeling.

Although the communication delay is generally not constant, a “communication-buffer” maybe implemented which receives communicated packages with a variable delay, and releases pack-ages at a constant delay. The constant delay must be equal to a certain maximum (measured orallowed) communication delay. If the actual, variable delay is smaller than the maximum delay,additional delay is added. This is somewhat justified by the fact that stability and a certain levelof performance can still be guaranteed, or predicted, for a constant delay using linear analysis.System analysis for a variable communication delay is suggested for future work.

3.2.3 Control structure

Vehicle-to-vehicle communication allows to transmit any relevant information, for example thevehicle weight, engine and braking power, number of occupants, etc. Any of this information canbe used to adapt the behavior of the following vehicle. This research, however, focusses merelyon communication of the acceleration, which is difficult to estimate from radar/lidar data due tothe amplification of measurement noise. In Figure 3.3, a schematic representation of a platoon isshown, where the acceleration of the preceding vehicles is available via wireless communicationin addition to the relative position (inter-vehicle distance) and velocity obtained by radar and/orlidar.


xr,3, xr,3

vehicle 3 vehicle 2 vehicle 1

x2 x1

xr,2, xr,2

radar beam

wireless

commu-

nication

Figure 3.3: Schematic representation of a platoon of CACC-equipped vehicles, where xr,i, xr,i andxi represent the relative position, the relative velocity and the acceleration of vehicle i, respectively.

If the communicated acceleration of the directly preceding vehicle is used in a feedforwardsetting, ACC functionality is still available when communication fails, or when the preceding ve-hicle is not able to communicate at all. Another advantage is that it enables the design of a CACCsystem as an “add-on” for existing ACC systems. ACC functionality is already commercially avail-able. Consequently, the probability of actual implementation of the CACC system is significantlyincreased.

The primary control objective is to follow the preceding vehicle at a desired distance xr,d,i(t).Consider a velocity-dependent spacing policy, where the desired relative position, or desired inter-vehicle distance, is given by

xr,d,i(t) = ri + hd,ixi(t), for i > 1 (3.1)

where ri is the desired distance at standstill, hd,i is the so-called desired headway time, and xi(t)is the velocity of vehicle i. In literature, (3.1) is widely used and is called a constant headway timespacing policy. The headway time is the time it takes for vehicle i to reach the current positionof the preceding vehicle i − 1 when continuing to drive with a constant velocity. The relativedistance, or inter-vehicle distance,

xr,i(t) = xi−1(t)− xi(t)− Li−1, for i > 1 (3.2)

is obtained by radar/lidar and used in a feedback setting by a standard ACC controller. Li−1 is thelength of vehicle i − 1. The acceleration of the preceding vehicle xi−1(t) is available via wirelesscommunication and is used in a feedforward setting.

The length of the vehicle Li−1 and the desired distance at standstill ri are omitted from nowon, as these constant values only influence the initial conditions of the closed-loop system. Thisproject focusses on the scenario where vehicles are driving within a platoon and not on the sce-nario where a vehicle approaches a platoon. In the former scenario, the vehicle length Li−1 andconstant distance ri do not matter. Here, only the forced response of vehicle i to an external input,i.e. the behavior of the directly preceding vehicle i− 1, is considered.

The resulting control setup, for a constant headway time spacing policy, is depicted schemat-ically in Figure 3.4. Because the desired inter-vehicle distance (3.1) as well as the actual inter-vehicle distance (3.2) depend on the output of the vehicle model Gi(s), the control structurecomprises a double feedback loop. The inputs of the inner control loop are xr,d,i and nr,i and theoutput is xr,i, see Figure 3.4. Concerning the inner loop, the feedforward signal and the positionof the preceding vehicle xi−1 are considered as disturbances. Here, a time signal is denoted by


−

xiei uiKi

xr,d,i

Gi1

s1

s

xi xi

−

xr,i

+

xi−1

xi−1

1

s2

hd,i

DiFi

na,i−1

nv,i nr,i

Figure 3.4: CACC control structure of a single vehicle i, adopting a velocity-dependent spacingpolicy (3.1). The inner control loop is shown in black. Gi = Gis

−2 represents the vehicle model, Ki

the ACC feedback controller, Fi the feedforward controller and Di the communication delay model.The signals nr,i, nv,i and na,i represent noise on the relative distance, velocity and acceleration

measurements, respectively.

a lower-case letter, e.g. the error ei(t), and the corresponding Laplace transforms L(·) are de-noted by upper-case letters, e.g. Ei(s) = L(ei(t)). The following relations follow directly fromFigure 3.4.

Ei(s) = −Si {Xr,d,i(s)−Nr,i(s)}+ . . . (3.3a)Si

(1−GiFiDis

2)Xi−1(s)− SiGiFiDiNa,i−1(s)

Xr,i(s) = +Ti {Xr,d,i(s)−Nr,i(s)}+ . . . (3.3b)Si

(1−GiFiDis

2)Xi−1(s)− SiGiFiDiNa,i−1(s)

for i > 1, where Si = (1 + GiKi)−1 and Ti = GiKi(1 + GiKi)−1 = 1− Si represent the closed-loop sensitivity and complementary sensitivity of the inner control loop, respectively. Withoutfeedforward, i.e. Fi(s) = 0, the transfer functions (3.3) reduce to

Ei(s) = −Si {Xr,d,i(s)−Nr,i(s) + Xi−1(s)} (3.4a)Xr,i(s) = +Ti {Xr,d,i(s)−Nr,i(s) + Xi−1(s)} (3.4b)

for i > 1. Perfect tracking of the desired inter-vehicle distance requires Ti = 1 and Si = 0.However, a high-frequent roll-off Ti(s) is generally desired to ensure attenuation of radar noisenr,i and high-frequent driving behavior xi−1. Consequently, and inevitably, the sensitivity Si =1− Ti approaches one at high-frequencies.

For ease of representation, the block diagram in Figure 3.4 is simplified to the block diagramin Figure 3.5. Besides re-arranging the summation points, the spacing policy dynamics Hi(s)is introduced to get rid of the extra feedback loop. The input and output of the spacing policydynamics Hi(s) is xi and xd,i−1, respectively. From vehicle i’s point of view, xd,i−1 is regarded asthe desired position of vehicle i− 1, see Figure 3.6. The inputs of the outer control loop are xi−1,nr,i, nv,i = hd,inv,i and na,i−1. The output of the outer loop is xi, see Figure 3.5.

The following relations follow directly from Figure 3.5.

Ei(s) = S′i

{Nr,i(s)− Nv,i(s)

}+ S

′i

(1−HiGiFiDis

2)Xi−1(s) (3.5a)

Xd,i−1(s) = T′i

{Nr,i(s)− Nv,i(s)

}+ S

′i

(HiGi(FiDis

2 + Ki))Xi−1(s) (3.5b)


xiei ui

Di

Gi

s2 Fi

Ki

Hi

−

+

xd,i−1

nv,i

nr,i

xi−1

na,i−1

Figure 3.5: Simplified and generalized CACC control structure of a single vehicle i, where Hi

represents the spacing policy dynamics, Gi the vehicle model, Ki the ACC feedback controller, Fi thefeedforward controller and Di the communication delay model. The signals nr,i, nv,i = hd,inv,i andna,i represent noise on the relative distance, velocity and acceleration measurements, respectively.

for i > 1, where S′i = (1 + HiGiKi)−1 and T

′i = HiGiKi(1 + HiGiKi)−1 = 1 − S

′i represent

the closed-loop sensitivity and complementary sensitivity of the outer control loop, respectively.Without feedforward, i.e. Fi(s) = 0, the equations (3.5) reduce to

Ei(s) = S′i

{Xi−1(s) + Nr(s)− Nv(s)

}(3.6a)

Xd,i(s) = T′i

{Xi−1(s) + Nr(s)− Nv(s)

}(3.6b)

for i > 1. A high-frequent roll-off of T′i is desired to ensure attenuation of measurement noise

nr,i and nv,i. Consequently, the sensitivity S′i = 1− T

′i approaches one at high-frequencies.

The vehicle model Gi(s) = Gis−2 includes a controller Kl,i for the longitudinal vehicle dy-

namics, see Figure 3.7. Hence, the input ui(t) of Gi(s) can be regarded as a desired accelerationxi,d(t). The controller Kl,i ensures tracking of this desired acceleration via actuation of the throt-tle and the brake system. The corresponding closed-loop system is approximated by

Gi(s) =kG,i

s2(τis + 1)e−φis (3.7)

where τ−1i = ωG,i the closed-loop bandwidth of Gi = Gis

2, and kG,i the loop gain, which equals1 for an appropriately designed controller Kl,i.

xi(t)

xi−1(t)

vehicle i − 1vehicle i

hd,i

xi(t)

ei(t)

xd,i−1(t)

xr,i

(t)

xr,d,i

(t)

ri

Li−1

Figure 3.6: Clarification of previously mentioned signals regarding the inter-vehicle spacing. Thevehicle length Li−1 and the constant distance ri are only included for the sake of illustration, they

are not included in the analysis.


Kl,i s−2

ui,aui = xd,i x

i xi

Gi(s)di

Figure 3.7: Schematic representation of the closed-loop vehicle dynamics, approximated by Gi

(3.7), where Kl,i is the controller for the longitudinal vehicle dynamics, ui,a the throttle and brakeactuators control signals, ui = xi,d the desired acceleration, di disturbances such as aerodynamic

drag, xi the actual acceleration and xi the absolute position of vehicle i.

Considering Figure 3.4, the distance error between the desired distance xr,d,i(t) (3.1) and theactual distance xr,i(t) (3.2) equals

ei(t) = xr,i(t)− xr,d,i(t) (3.8)= (xi−1(t)− xi(t))− hd,ixi(t)= xi−1 − (xi + hd,ixi(t))

Defining ei(t) in this manner implies that positive control action, i.e., acceleration, is requiredwhen the inter-vehicle distance xr,i(t) is too large with respect to the desired distance xr,d,i(t),which is intuitive. The Laplace transform of ei(t) (3.8), equals

L (ei(t)) = Ei(s) = Xi−1(s)−Hi(s)Xi(s) (3.9)

whereHi(s) = 1 + hd,is (3.10)

represents the so-called spacing policy dynamics for a constant headway time. Note that a con-stant spacing policy is a special case of a constant headway time policy with hd,i = 0, such thatxr,d,i(t) = ri and

Hi(s) = 1, (3.11)

see (3.1) and (3.10)).

Given the vehicle dynamics Gi(s) (3.7), a feedback controller with PD action provides freedomto choose the bandwidth of the inner closed-loop system Ti(s) (3.4). Correspondingly, the ACCfeedback controller Ki(s) is defined as

Ki(s) = k−1G,i (kP,i + kD,is) (3.12)

= k−1G,iωK,i(ωK,i + s) (3.13)

where kP,i = k−1G,iω

2K,i and kD,i = k−1

G,iωK,i are the proportional and derivative gains, respectively,and ωK,i is the breakpoint of the controller. This particular choice of Ki(s) (3.12) stabilizes theinner control loop Ti(s) (3.4), at least in the case of ideal vehicle dynamics, Gi(s) = s−2, whichwill be discussed in more detail in the next chapter. Moreover, in case of ideal vehicle dynamics,the breakpoint always equals ωK,i ≈ 1.8ωbw, where ωbw,i is the bandwidth of Ti(s). Hence,the breakpoint ωK,i is used as a measure for the aggressiveness of the controller Ki(s). Thebandwidth ωbw,i is defined as the frequency where the gain |Ti(jω)| drops below −3 dB. Note

3.3. STRING STABILITY - A FREQUENCY-DOMAIN APPROACH 31

that, for an appropriately designed lower-level controller Kl,i, the gain kG,i equals one, such thatthe ACC feedback controller Ki(s) is independent of all vehicle model parameters.

The wireless communication includes delay, which is represented by a constant delay time θi,yielding L (xi−1(t− θi)) = Di(s)s2Xi−1(s), where

Di(s) = e−θis (3.14)

The acceleration of the preceding vehicle is used as a feedforward control signal via a feedfor-ward filter Fi(s). The design of this feedforward filter is based on a zero-error condition, wherethe error is defined as in (3.5a). Demanding Ei(s) = 0 for Xi−1(s) 6= 0 and accounting for thefact that time delays (φi and θi) can not be compensated by a causal feedforward filter, yields

Fi =1

HiGis2with Gi =

kG,i

s2(τis + 1), (3.15)

The spacing policy dynamics and vehicle dynamics are compensated by Fi(s). Note that Gi issimply the vehicle model Gi(s) = Gi(s)e−φis (3.7) without actuator delay time, i.e. φi = 0.

3.3 String stability - a frequency-domain approach

In literature, different notions and definitions of string stability are found, see Section 2.4. Thegoal of this section is to obtain a better understanding of string stability and to arrive at a generalcondition for heterogeneous string stability, using a frequency-domain approach.

3.3.1 Definition of a heterogeneous string stability condition

The so-called string stability of a platoon indicates whether oscillations are amplified upstreamthe platoon, i.e., from the leading vehicle i = 1 to vehicle i > 1 in the platoon. An exampleof string-unstable behavior are traffic jams that appear for no apparent reason. No accident orbottleneck needs to be present, just too much traffic or erratic driving behavior may cause ashockwave of increased braking, upstream a string of vehicles, until vehicles come to a standstilland a traffic jam appears. The main ambiguity in literature concerning string stability, is thesignal(s) to consider. Either the control input ui(t), control error ei(t), vehicle output (or state)xi(t), or a combination of these are considered. In Figure 3.8, these signals are indicated in acontrol structure of a three-vehicle platoon. The leading vehicle i = 1 is assumed to follows acertain reference signal x0(t), which may be another preceding vehicle.

The transfer functions from the reference position L(x0(t)) = X0(s) to the Laplace trans-forms of the input ui(t), error ei(t) and output xi(t), i.e. L(ui(t)) = Ui(s), L(Ei(t)) = Ei(s) and


L(xi(t)) = Xi(s), are derived from the block diagram in Figure 3.8 and are given by

Ui

X0=

K1S′1, for i = 1

K1S′1

{i∏

k=2

S′k

(FkDks

2 + Kk

)Gk−1

}, for i > 1

(3.16a)

Ei

X0=

S′1, for i = 1

S′i

(1−HiGiFiDis

2) Xi−1

X0, for i > 1

(3.16b)

Xi

X0=

G1K1S′1, for i = 1

G1K1S′1

{i∏

k=2

S′k

(FkDks

2 + Kk

)Gk−1

}, for i > 1

(3.16c)

(3.16d)

whereS′i = (1 + HiGiKi)−1 (3.17)

represents the closed-loop sensitivity corresponding to vehicle i, see (3.6) in Section 3.2.3. Asstring stability concerns amplification of signals upstream the platoon, focus lies on the magni-tude of the so-called string stability transfer functions SS∗Λ,i(s), Λ ∈ {U,X, E}

SS∗Λ,i =Λi

Λ1=

Λi

X0

(Λ1

X0

)−1

, for i > 1, Λ ∈ {U,X, E} (3.18)

The magnitude of these transfer functions is a measure for the amplification of signals upstreamthe platoon. A necessary condition for string stability thus is

∣∣SS∗Λ,i(jω)∣∣ ≤ 1, for i > 1, ∀ω, Λ ∈ {U,X, E} (3.19)

To fulfill condition (3.19) for vehicle i > 2, the dynamics of all preceding vehicles k ∈ {1, . . . , i−1}have to be known. Considering heterogeneous traffic, this requires an extensive communicationstructure. As communication with the directly preceding vehicle only is considered, a moreconservative condition for string stability is defined as

|SSΛ,i(jω)| ≤ 1, for i > 1, ∀ω, Λ ∈ {U,X,E} (3.20)

F3

G3K3 H3

D3

x3u3

vehicle 3

F2

G2K2 H2

D2s2

x2u2

x2

x2

vehicle 2

G1Ki H1

s2

x1u1

x0x1

x1

vehicle 1

e3 e2 e1

− − −

Figure 3.8: Control structure of a three-vehicle platoon, where Gi represents the dynamics ofvehicle i, Ki the corresponding feedback controller, Fi the feedforward controller, Di the

communication delay model and Hi the spacing policy dynamics, for i=1,2,3.


where

SSΛ,i =Λi

Λi−1=

Λi

X0

(Λi−1

X0

), for i > 1, Λ ∈ {U,X, E} (3.21)

As it holds that

SS∗Λ,i =i∏

k=2

SSΛ,k, for i > 1, Λ ∈ {U,X, E} (3.22)

condition (3.19) is automatically satisfied if (3.20) is satisfied. String stability condition (3.19)considers the platoon as a whole: if compensated somewhere else, local string-unstable behaviorcan be allowed in the platoon. Hence, condition (3.19) is considered as a weak string stabilitycondition. The string stability condition (3.20), on the other hand, requires string-stable behaviorof every vehicle in the platoon and is considered as a strong string stability condition.

Combining (3.16) and (3.21) yields the input, the error and the output string stability transferfunctions, SSU,i(s), SSE,i(s) and SSX,i(s), respectively.

SSU,i =Ui

Ui−1= S

′i

(FiDis

2 + Ki

)Gi−1, for i > 1 (3.23a)

SSE,i =Ei

Ei−1=

S′i

(1−HiGiFiDis

2)Gi−1Ki−1, for i = 2

S′i

(1−HiGiFiDis

2)

S′i−1 (1−Hi−1Gi−1Fi−1Di−1s2)

Xi−1

Xi−2, for i > 2

(3.23b)

SSX,i =Xi

Xi−1= S

′i

(FiDis

2 + Ki

)Gi, for i > 1 (3.23c)

For homogeneous traffic, i.e., Gi = G, Ki = K, etc., the string stability transfer functions(3.23a,b,c) are equal; SSU,i(s) = SSX,i(s) = SSE,i(s)|i>2. In literature, often, homogeneoustraffic is considered, see Section 2.4. Focusing on heterogeneous traffic, however, the stringstability transfer functions (3.23a,b,c) are clearly different.

In Figure 3.9, the Bode magnitude plots of these transfer functions for a heterogeneous pla-toon of three vehicles, all three equipped with a CACC system, are shown. For the sake of clarity,constant inter-vehicle distances hd,i = 0 are adopted and no delay is assumed, i.e. θi = 0, φi = 0and τG,i = 0. Consequently, Hi(s) = 1, Di(s) = 1,

Gi(s) = kG,is−2, (3.24a)

Ki(s) = k−1G,iωK,i(ωK,i + s) and (3.24b)

Fi(s) = k−1G,ikF,i (3.24c)

where k−1G,i is the vehicle mass, ωK,i is the breakpoint of the PD feedback controller and kF,i

is the feedforward gain. Moreover, a constant inter-vehicle distance Hi(s) = 1 is adopted andno communication delay Di(s) = 1 is assumed. As a result, the differences between the threevehicles come down to differences in the breakpoint frequency ωK,i, the vehicle mass k−1

G,i andthe feedforward gain kF,i.

As Figure 3.9 shows, the low-frequent asymptotic values of SSU,i(s) differ. They equal

limω→0

|SSU,i(jω)| = kG,i−1

kG,i, for i > 1 (3.25)


10-1

100

101

-10

-5

0

5

10

Frequency (rad/s)

Ma

gn

itu

de

(d

B)

Input string stability SSU,i

10-1

100

101

-10

-5

0

5

10

Frequency (rad/s)M

ag

nit

ud

e (

dB

)

Error string stability SSE,i

10-1

100

101

-10

-5

0

5

10

Frequency (rad/s)

Ma

gn

itu

de

(d

B)

Output string stability SSX,i

Figure 3.9: Bode magnitude of the input, error and output string stability transfer functions (3.23)corresponding to a platoon of four heterogeneous vehicles i = 1, 2, 3, 4 with

k−1G,i ∈ {1.0, 1.5, 1.0, 1.1} 103 kg, ωK,i ∈ {1.0, 1.2, 1.4, 1.1} rads−1 and kF,i ∈ {0.0, 0.1, 0.2, 0.3}. As

vehicle 1 follows a reference signal x0(t), the string stability functions are not defined for this vehicle.

Hence, with the adopted setup, input string stability |SSU,i(jω)| ≤ 1, i > 1, ∀ω is achievedonly if the preceding vehicle is heavier kG,i > kG,i−1. This makes perfect sense, since heaviervehicles simply require larger forces in order to keep up with their predecessors. Analogously, thelow-frequent asymptotic values of the error string stability transfer function, for heterogeneoustraffic, equal

limω→0

|SSE,i(jω)| = ω2K,i−1

ω2K,i

1− kF,i

1− kF,i−1, for i > 2 (3.26)

where it is assumed that kF,i−1 6= 1, for example because the mass k−1G,i−1 is not known exactly.

From (3.26) it follows that error string stability is achieved if, for example, ωK,i > ωK,i−1 andkF,i = kF,i−1 = 0. In other words, error string stability is achieved only if the following vehiclehas a more aggressive feedback controller.

Considering the magnitude of the output string stability transfer function |SSX,i(jω)|, thecorresponding steady-state value lim

ω→0|SSX,i(jω)| equals 1 for all i > 1, see Figure 3.9. Indepen-

dent of the vehicle dynamics and CACC controller, the same condition for string stability can beused. Hence, for heterogeneous traffic, the condition (3.20) for Λ = X has to be adopted, andthe output string stability transfer function SSX,i(s) has to be considered. In practice, this im-plies that velocity and acceleration peaks may not be amplified upstream the platoon. A sufficientcondition for string stability of a platoon of heterogeneous vehicles is thus defined as

|SSX,i(jω)| =∣∣∣∣

Xi(jω)Xi−1(jω)

∣∣∣∣ ≤ 1, for i > 1, ∀ω (3.27)

or, equivalently,||SSX,i(jω)||∞ ≤ 1, for i > 1 (3.28)

whereXi

Xi−1=

GiFiDis2 + GiKi

1 + HiGiKi, for i > 1 (3.29)

Condition (3.27) guarantees that the magnitude of oscillations in absolute position, velocityand acceleration do not amplify continuously upstream, as the vehicle index i increases. The


amplitudes of the control input and error, on the other hand, do not decrease uniformly, unlessthe vehicle mass decreases and the control gains increases upstream. As this is not the case inreal (heterogeneous) traffic, error and input string stability are not considered to be necessaryand/or desirable. However, it is desired that the errors and inputs do not amplify continuouslyupstream. To illustrate that this will not happen if output string stability is achieved, consider astring of vehicles with uniformly decreasing magnitudes in output oscillations. Vehicles with lowacceleration profiles at the end of the platoon are easier to follow, generally resulting in smallercontrol errors of their respective followers. In turn, small control errors result in smaller controlinputs. Hence, although the errors and inputs do not decrease uniformly, they do not becomeunbounded for an output string stable platoon, using the adopted control setup.

The input, error and output string stability functions can also be derived for a control setupwhere feedforward of the control input is used instead of feedforward of the measured accelera-tion. An expected benefit of such a setup is that the control input, i.e. the desired acceleration,precedes the actual acceleration. In other words, the actual acceleration has phase lag and timedelay compared to the actual acceleration, due to vehicle dynamics (3.7). Secondly, the controlinput is not corrupted by measurement noise. It turned out, however, that the resulting outputstring stability function includes the vehicle model Gi−1(s). Consequently, this alternative setuprequires modeling of Gi−1(s), which introduces additional model uncertainty. Note that (3.29)contains information about vehicle i only, which is convenient.

3.3.2 Frequency-domain versus time-domain string stability

From linear system theory, it is clear that the frequency-domain string stability condition (3.27) isnot the same as the time-domain condition

||xi(t)||∞||xi−1(t)||∞ ≤ 1, for i ≥ 1 (3.30)

see (2.6) in Section 2.4. In order to satisfy condition (3.30), the 1-norm of the impulse responsessX,i(t) = L−1(SSX,i(s)) of the output string stability transfer function must be smaller than orequal to one, i.e. ||ssX,i(t)||1 ≤ 1. As ||SSX,i(s)||∞ ≤ ||ssX,i(t)||1, the time-domain condition isconsidered stronger. In this section, the frequency-domain string condition (3.27) will be testedfor a 2nd order system, by analyzing the corresponding behavior in the time-domain. Moreover,a spacing policy will be derived which guarantees time-domain string stability (3.30) in case ofideal vehicle dynamics and no feedforward, i.e. in the case of ACC.

String-stability analysis of a 2nd order linear system

Imagine a homogeneous platoon with an infinite number of vehicles i = 1, . . . ,∞ where thedynamics and controllers of each vehicle are such that its output string stability transfer functioncan be described by a 2nd order linear system

SSX,i =Xi

Xi−1=

ω2n

s2 + 2ξωns + ω2n

, for i > 1 (3.31)

with undamped natural frequency ωn and damping ratio ξ. It is well-known that the 2nd orderlinear system (3.31) has step response overshoot if 0 < ξ < 1, and no overshoot if ξ ≥ 1. Consider


a reference vehicle i = 0 with a unit step in the acceleration x0(t). Then, the percentage overshootof a following vehicle i with respect to the reference vehicle is defined as

||xi(t)||∞ − ||x0(t)||∞||x0(t)||∞ × 100% = (||xi(t)||∞ − 1)× 100%, for i ≥ 1 (3.32)

as ||x0(t)||∞ = 1 for a unit step. In (3.32), the∞-norm is used to denote the maximum amplitudeof the acceleration

||xi(t)||∞ = maxt|xi(t)|, for i ≥ 0 (3.33)

Combining condition (3.27) and (3.31) shows that string stability is guaranteed if

ω2n√

(ω2n − ω2)2 + 4ξ2ω2

nω2

≤ 1 (3.34)

holds, or, after squaring and rearranging

2ω2n

(1− 2ξ2

) ≤ ω2 (3.35)

With min(ω2) = 0 and ωn > 0, condition (3.35) is satisfied if

ξ ≥ 2−1/2 ≈ 0.71 (3.36)

Now, the characteristics of the 2nd order linear system can be divided into three regions, seeTable 3.1. Overshoot in the frequency-domain refers to condition (3.27) and overshoot in thetime-domain refers to condition (3.30).

Table 3.1: Properties of the 2nd order system (3.31) for different damping ratios ξ.Region Overshoot in Overshoot in

frequency-domain time-domain0 ≤ ξ < 2−1/2 yes yes2−1/2 ≤ ξ < 1 no yes

ξ ≥ 1 no no

Figure 3.10(b) shows that for 0 ≤ ξ < 2−1/2, a platoon of vehicles is not string stable, as theovershoot of the responses xi(t) to a step in the reference trajectory x0(t) continues to increasefor increasing i. For ξ = 2−1/2, the overshoot slowly converges to a constant value of approx-imately 24%. This value has been computed numerically by simulating ten-thousand vehicles.For 2−1/2 < ξ < 1, the overshoot reaches a maximum value (<24%) after which it slowly dimin-ishes to zero. For ξ ≥ 1, there is no overshoot at all. As the magnitude of the acceleration is notamplified to infinity if the number of vehicles in the platoon goes to infinity for 2−1/2 ≤ ξ < 1,this behavior is still considered to be string-stable, which is in accordance with the string stabilitycondition (3.27).

In general, time-domain string stability (3.30) is considered to be more desirable than frequency-domain string stability (3.27). It is shown that in the case of frequency-domain string stability,there may still be overshoot on a step response, which corresponds with results from linear sys-tem theory. Imagining a homogeneous platoon, where the first vehicle suddenly decelerates at


maximum braking capacity, there is a probability that the vehicles will collide, as every follow-ing vehicle reaches its deceleration limit. Nonetheless, the frequency-domain condition is usedinstead of the time-domain condition, because it is harder to derive analytical results for the time-domain condition.

0

1

1.24

Time (s)

Acc

eler

atio

n (m

s−2 )

(a) Responses xi(t) of a ten-vehicle platooni = 1, . . . , 10 (dark to light) with damping ratio

ξ = 2−1/2.

i=0 i = 10000

5

10

15

20

25

Vehicle index (−)O

vers

hoot

(%

)

ξ = 21/2

≈ 0.707

ξ = 0.697

ξ = 0.717

ξ = 0.75

ξ = 1

(b) Percentage overshoot max(xi(t))× 100% for athousand-vehicle platoon i = 1, . . . , 103, with

damping ratio ξ ∈ {0.697, 0.707, 0.717, 0.75, 1.00}(light to dark).

Figure 3.10: Response to a step in the reference acceleration x0(t) for a platoon of vehicles forwhich the string stability transfer function SSX,i = Xi X−1

i−1 equals a 2nd order system (3.31).

Derivation of a velocity-dependent spacing policy

The idea of the velocity-dependent spacing policy (3.1) presumably originates from human drivingbehavior and logical reasoning. In literature, it is shown to achieve string-stable driving behaviorfor a certain minimum headway time. Here, by reversed reasoning, a spacing policy is derived bydemanding string-stable behavior without overshoot in the time-domain for the ACC case.

For a constant inter-vehicle spacing policy (Hi(s) = 1), without feedforward (Fi(s) = 0),ideal vehicle dynamics (Gi(s) = s−2) and ACC feedback controller Ki(s) as defined in (3.12), theoutput string stability transfer function (3.29) equals

SSX,i =GiKi

1 + HiGiKi=

ωK,is + ω2K,i

s2 + ωK,is + ω2K,i

(3.37)

for which the limits lims→0

SSX,i = 1 and lims→∞SSX,i = lim

s→∞ωK,is−1. Thus, a low-frequent slope

of 0 dB/decade and a high-frequent slope of −20 dB/decade. By demanding that

SSX,i =GiKi

1 + HiGiKi= Ri (3.38)

withRi =

ωK,i

s + ωK,i⇒ ||Ri||∞ ≤ 1 (3.39)


the spacing policy dynamics Hi(s) can be derived which guarantees to yield a step responsewithout overshoot. Note that ||Ri(jω)||∞ = 1, such that the string stability condition (3.27) isautomatically satisfied if SSX,i(s) = Ri(s). Moreover, it has a low-frequent slope of 0 dB/decadeand a high-frequent slope of −20 dB/decade. The step response of a first-order system does notovershoot. Damping is not defined for a first-order-system Ri(s). However, it can be shown thatR2

i (s) equals the 2nd order system (3.31) with ξ = 1 and ωn = ωK,i. Solving the spacing policydynamics Hi(s) in (3.38) yields

Hi(s) =2s + ωK,i

s + ωK,i= 1 +

1s + ωK,i

s (3.40)

which is interpreted asHi(s) = 1 + hd,ifi(s)s (3.41)

with desired headway time hd,i = ω−1K,i and low-pass velocity filter

fi(s) =ωf,i

s + ωf,i(3.42)

with cut-off frequency ωf,i = ωK,i. The spacing policy dynamics (3.41) corresponds to the velocity-dependent spacing policy

xr,d,i(t) = ri + hd,iνi(t) (3.43)

where νi(t) is the velocity xi(t) filtered by fi(s). Comparing spacing policy dynamics (3.10) and(3.41), the only difference is the low-pass filter fi(s), which introduces additional freedom in thecontrol design. The constant headway time spacing policy (3.10) is considered a special case of(3.43), as it can be shown that

limωf,i→0

fi(jω) = 0, limωf,i→∞

fi(jω) = 1 (3.44)

andlim

ωf,i→0Hi(s) = 1, lim

ωf,i→∞Hi(s) = 1 + hd,is (3.45)

assuming that 0 < ω < ∞ in the frequency range of interest. The influence of the spacing policy,i.e. the desired headway time hd,i and the cut-off frequency ωf,i, on the closed-loop system willbe thoroughly analyzed in the next Chapter.

3.4 Mathematical objectives

In this section, the control objectives with respect to vehicle stability, performance and stringstability are formulated (or repeated) mathematically. At the end, the objectives are summarized.

3.4.1 Vehicle stability

For vehicle stability, both the inner- and outer control loop are required to be stable, see Figures3.4 and 3.5. Stability is determined by the roots of the characteristic equations 1 + Li = 0 and1 + L

′i = 0, with open-loop

Li = GiKi andL′i = HiGiKi

3.4. MATHEMATICAL OBJECTIVES 39

Note that an accent (′) is used to differentiate between the inner- from the outer loop. The feed-forward part does not affect stability. Assuming that Hi(s), Gi(s) and Ki(s) do not have zerosor poles in the right-half of the complex plane (RHP), the critical −1 + 0j point must be on theleft-hand side of the open-loop paths Li and L

′i with no encirclements, according to the Nyquist

Stability Criterion. For robust stability, the phase margin and modulus margin are evaluated.

3.4.2 Performance

For performance, the inter-vehicle distance error ei(t) with respect to manoeuvres xi−1(t) is ofimportance. Hence, the closed-loop sensitivity

S′′i = Ei X

−1i−1 = S

′i

(1−HiGiFiDis

2)

(3.46)

is evaluated, see (3.5). Note that a double accent (′′) is used to differentiate between cases with andwithout feedforward, i.e. S

′′i = S

′i for Fi(s) = 0. Using the final value theorem, the steady-state

error is computed as

ess,i = limt→∞ ei(t) = lim

s→∞ sEi(s) = lims→∞ sS

′′i (s)Xi−1(s) (3.47)

The Laplace transform of a constant equal s−1 times the amplitude of the constant. Hence, fora constant acceleration ai−1 = xi−1, Xi−1(s) = s−2Ai−1(s) = s−3|Ai−1|, where Ai−1(s) =L(ai(t)). Hence, the steady-state error for a constant acceleration with amplitude |Ai−1| = |ai−1|becomes

ess,i = lims→∞ sS

′′i (s)Xi−1(s) = lim

s→∞ s−2S′′i (s)|ai−1| (3.48)

Substituting ideal vehicle dynamics Gi(s) = s−2, feedback controller Ki(s) (3.12), spacing policydynamics Hi(s) (3.10) and no feedforward Fi(s) = 0 in (3.46) and (3.48) yields

ess,i = lims→∞ s−2S

′′i (s)|ai−1| = ω−2

K,i|ai−1| (3.49)

i.e. a constant value, which is conveniently used as a performance measure.

3.4.3 String stability

For string stability, condition (3.27) must be satisfied, see Section 3.3

3.4.4 Summary

The control objectives are summarized as follows

(Gi,Ki) and (Hi, Gi, Ki) stable; (vehicle stability)|S′′i | and |ess,i| small; (performance, safety)|SSX,i| = |Xi X−1

i−1| ≤ 1; (string stability)small headway time hd,i. (road capacity, traffic throughput, human acceptance)

The notion of “small” will be evaluated from case to case in the following chapters. As mentionedearlier, string-stable behavior is believed to have a positive effect on traffic throughput. More-over, as velocity, acceleration and jerk peaks are attenuated from vehicle to vehicle, string-stablebehavior is expected to have a positive effect on performance, safety, comfort and fuel economy.

Chapter 4

System analysis

This chapter focusses on the influence of the control design parameters on stability, performance andstring stability.

4.1 Introduction

In the following sections, stability, performance and string stability are analyzed for differentcontroller settings. Consider the CACC system setup as presented in Section 3.2.3. The designvariables are the ACC feedback controller Ki(s) (3.12), the feedforward controller Fi(s) (3.15) andthe spacing policy dynamics Hi(s) (3.41). In this chapter, ideal vehicle dynamics are assumed,i.e. Gi(s) = s−2, as the focus of this chapter will be lost out of sight if the effect of parametervariations in Gi(s) (3.7) are taken into account. In the next chapter, stability, performance andstring stability will be thoroughly analyzed for a particular set of vehicle parameters.

4.2 Vehicle-stability

In this section, closed-loop stability will be analyzed for three spacing policy settings: a con-stant spacing policy, a velocity-dependent spacing policy without low-pass filter and a velocity-dependent spacing policy with low-pass filter. The feedforward part does not influence stability.For stability, both the inner and outer feedback loops are required to be stable (see Figure 3.4 inSection 3.1). Hence, the roots of the characteristic equations 1 + Li as well as 1 + L

′i must be in

the left-half of the complex plane (LHP), with

Li = GiKi andL′i = HiGiKi

Equivalently, the critical −1 + 0j point must be on the left-hand side of the open-loop path’sLi(jω) and L

′i(jω).

41

42 CHAPTER 4. SYSTEM ANALYSIS

4.2.1 Constant inter-vehicle spacing

A constant spacing policy is a special case of the general velocity-dependent spacing policy Hi(s)(3.41) with hd,i = 0. In this case, Hi(s) = 1 and L

′i = Li. The open-loop transfer functions equal

L′i(s) = Li(s) =

ωK,i(s + ωK,i)s2

(4.1)

and the roots of the characteristic equations Li + 1 = 0 and L′i + 1 = 0 are

p1,2 = −ωK,i

2

(1± j

√3)

(4.2)

which are in the LHP of the complex plane for any ωK,i > 0. Correspondingly, the critical−1+0jpoint is on the left-hand side of the path L

′i(jω) = Li(jω), see Figure 4.1. A phase margin (PM) of

about 52◦ and a minimum distance to the−1+0j point of about 1.15 is achieved. Consequently,both the inner and outer control loops are stable, i.e. vehicle-stability is achieved, for a constantinter-vehicle spacing policy.

10-2

10-1

100

101

102

-50

0

50

100

Ma

gn

itu

de

(d

B)

10-2

10-1

100

101

102

-180

-90

PM

Frequency (rad)

Ph

ase

(d

eg

)

(a) Bode diagram.

-2 -1 0 1-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

Real axis

Ima

gin

air

y a

xis PM

MM

(b) Nyquist diagram.

Figure 4.1: Bode and Nyquist diagram of L′i(jω) = Li(jω) = GiKi for a constant spacing policy

Hi(s) (3.11), ideal vehicle dynamics Gi = s−2 and feedback controller Ki (3.12) withωK,i = 1 rad s−1.

4.2.2 Velocity-dependent inter-vehicle spacing

Two types of velocity-dependent spacing policies are considered, without and with low-pass filterfi(s), corresponding to the spacing policy dynamics Hi(s) (3.10) and (3.41), respectively.

4.2. VEHICLE-STABILITY 43

Without low-pass filter

The open-loop L′i(s) with spacing policy dynamics Hi(s) = 1 + hd,is equals

L′i(s) =

hd,iωK,is2 + ωK,i(hd,iωK,i + 1)s + ω2

K,i

s2(4.3)

and the roots of the characteristic equation L′i + 1 = 0 are

p1,2 = −ωK,i

2

1±

√(hd,iωK,i)

2 − 2hd,iωK,i − 3

hd,iωK,i + 1

(4.4)

With ωK,i > 0 and hd,i ≥ 0, the square root√· in (4.4) is imaginary for 0 ≤ hd,iωK,i ≤ 3,

and positive real for hd,iωK,i > 3. In both cases, the (real parts of the) roots (4.4) are negativefor any ωK,i > 0 and hd,i ≥ 0. Hence, vehicle-stability is guaranteed for any ωK,i > 0 andhd,i ≥ 0. By increasing the desired headway time hd,i, the path L

′i(jω) moves away from the

critical −1 + 0j point, see Figure 4.2(b). Hence, the stability margins are increased. For hd,i 6= 0,additional phase ∠L

′i is introduced due to the differentiator in Hi(s), see Figure 4.2(a). Another

consequence of increasing hd,i is that the high-frequent open-loop gain lims→∞L(s) = hd,iωK,i

increases, see Figure 4.2(a). This will have a negative side-effect on the closed-loop performance,which is discussed in the next section.

10-2

10-1

100

101

102

-50

0

50

100

Ma

gn

itu

de

(d

B)

10-2

10-1

100

101

102

-180

-90

0

Frequency (rad)

Ph

ase

(d

eg

)

(a) Bode diagram.

-2 -1 0 1-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

Real axis

Ima

gin

air

y a

xis


Figure 4.2: Bode and Nyquist diagram of L′i(jω) = HiGiKi for a velocity-dependent inter-vehicle

spacing policy Hi(s) without low-pass filter (3.10), with hd,i ∈ {0, 0.5, 1, 1.5, 2} s (black to lightgrey). Feedback controller Ki(s) (3.12) with ωK,i = 1 rad s−1 and ideal vehicle dynamics Gi(s) = s−2

are assumed.

With low-pass filter

By filtering the measured velocity in the velocity-dependent spacing policy (3.43) with a first-orderlow-pass filter fi(s) = ωf,i(s+ωf,i)−1 (3.42), the corresponding spacing policy dynamics becomes


Hi(s) = 1 + hd,ifi(s)s (3.41) with open-loop transfer function

L′i(s) =

ωK,i(1 + hd,iωf,i)s2 + ωK,i(ωf,i + ωK,i + hd,iωf,iωK,i)s + (ωf,iω2K,i)

s3 + ωf,is2(4.5)

The corresponding characteristic equation is a 3rd order equation, for which the roots are elab-orate and hard to analyze. Therefore, in this case, stability is analyzed using the following twoapproaches: 1) by simplifying and analyzing the roots (4.5) for a special case where ωf,i = ωK,i

and 2) by analyzing the open-loop path L′i(jω) in the complex plane for the general case.

Substituting ωf,i = ωK,i in (4.5) yields

L′i(s) =

ωK,i(1 + hd,iωK,i)s + ω2K,i

s2(4.6)

for which the roots of the characteristic equation L′i + 1 = 0 are

p1,2 = −ωK,i

2

(1 + hd,iωK,i ±

√(hd,iωK,i)

2 + 2hd,iωK,i − 3)

(4.7)

With ωK,i > 0 and hd,i ≥ 0, the square root√· in (4.7) is imaginary for 0 ≤ hd,iωK,i < 1, and

positive real for hd,iωK,i ≥ 1. In both cases, the real parts of (4.7) are negative for any ωK,i > 0and hd,i ≥ 0 and, hence, vehicle-stability is guaranteed for ωf,i = ωK,i. By increasing the desiredheadway time hd,i, the path L

′i(jω) moves away from the critical−1+0j point, see Figure 4.3(b).

Comparing Figures 4.2 and 4.3 shows that the low-pass filter fi(s) ensures that the open-loopgain |L′i| goes to zero for high frequencies.

10-2

10-1

100

101

102

-50

0

50

100

Ma

gn

itu

de

(d

B)

10-2

10-1

100

101

102

-180

-90

0

Frequency (rad)

Ph

ase

(d

eg

)

(a) Bode diagram.

-2 -1 0 1-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

Real axis

Ima

gin

air

y a

xis


Figure 4.3: Bode and Nyquist diagram of L′i(jω) = HiGiKi for a velocity-dependent spacing policy

Hi with low-pass filter (3.41) with ωf,i = 1 rad s−1 and hd,i ∈ {0, 0.5, 1, 1.5, 2} s (black to light grey).Feedback controller Ki(s) (3.12) with ωK,i = 1 rad s−1 and ideal vehicle dynamics Gi(s) = s−2 are

assumed.

Stability for any cut-off frequency ωf,i > 0 is proven by showing that the critical−1+0j pointis always on the left-hand side of the path L

′i(jω) in the complex plane. The roots of 1+Li = 0 are

4.2. VEHICLE-STABILITY 45

always in the LHP (see (4.2)) and, correspondingly, the critical −1 + 0j point is on the left-handside of the path Li(jω), see Figure 4.1(b). As L

′i = HiLi, phase lead ∠Hi > 0 causes the path

L′i(jω) to move away from the critical −1 + 0j point such that stability is guaranteed. On the

other hand, phase lag ∠Hi < 0 may result in an unstable system. Now, consider

Hi(s) = 1 + hd,ifi(s)s =(1 + hd,iωf,i)s + ωf,i

s + ωf,i= kH,i

s− zH,i

s− pH,i(4.8)

with gain kH,i = 1 + hd,iωK,i, zero zH,i = −ωf,i(1 + hd,iωf,i)−1 and pole pH,i = −ωf,i. Now, if|pH,i| < |zH,i|, Hi(s) is a lag filter with a minimum phase lag ∠Hi of -90◦ within the frequencyrange |pH,i| < ω < |zH,i|. On the other hand, if |pH,i| > |zH,i|, Hi(s) is a lead filter witha maximum phase lead ∠Hi of 90◦ within the frequency range |zH,i| < ω < |pH,i|. Hence,stability is guaranteed if

ωf,i ≥ ωf,i(1 + hd,iωf,i)−1

1 + hd,iωf,i ≥ 1hd,iωf,i ≥ 0

which holds for any hd,i ≥ 0 and ωf,i > 0. Therefore, vehicle-stability is guaranteed. Figure 4.4shows that L

′i(jω) with Hi(s) = 1+hd,ifi(s)s for all ωf,i > 0 is enclosed by L

′i(jω) for Hi(s) = 1

and Hi(s) = 1 + hd,is. Since both of the enclosing paths are proven to be on the right-hand sideof the critical −1 + 0j point, the enclosed path must also be on the right-hand side of the critical−1 + 0j point, see Figure 4.4(b).

10-2

10-1

100

101

102

-50

0

50

100

Ma

gn

itu

de

(d

B)

10-2

10-1

100

101

102

-180

-90

0

Frequency (rad)

Ph

ase

(d

eg

)

(a) Bode diagram.

-2 -1 0 1-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

Real axis

Ima

gin

air

y a

xis


Figure 4.4: Bode and Nyquist diagram of L′i(jω) = HiGiKi for a velocity-dependent inter-vehicle

spacing policy Hi(s) = 1 + hd,ifi(s)s (dark to light grey) with hd,i = 1 s andωf,i ∈ {10−1, 100, 101, 102} rad s−1 (dark to light grey). The black dashed line corresponds toωf,i →∞, i.e Hi(s) = 1 + hd,is and the black solid line corresponds to hd,i = 0, i.e Hi(s) = 1.

Feedback controller Ki(s) (3.12) with ωK,i = 1 rad s−1 and ideal vehicle dynamics Gi(s) = s−2 areassumed.


4.3 Performance

For performance, the inter-vehicle distance error with respect to manoeuvres of the preceding ve-hicle, i.e. the closed-loop sensitivity S

′′i = Ei X

−1i−1 (3.46), and the steady-state error for a constant

acceleration (3.48) are evaluated. Again, ideal vehicle dynamics Gi(s) = s−2 are assumed to keepfocus on the influence of the design parameters on performance. First, performance is analyzedin the case of no feedforward, i.e. in case of ACC. The influence of the different spacing policieson performance is analyzed. Next, performance is analyzed in the case of CACC.

4.3.1 ACC

In the case of ACC, i.e. Fi(s) = 0, the closed-loop sensitivity (3.46) reduces to

S′′i = Ei X

−1i−1 = (1 + HiGiKi)−1 = (1 + L

′i)−1 = S

′i (4.9)

Using Hi(s) = 1, the steady-state error for a constant acceleration |ai−1| (3.48) equals

ess,i = lims→∞ s−2S

′′i (s)|ai−1| = lim

s→0(s2 + ωK,is + ω2

K,i)−1|ai−1| = ω−2

K,i|ai−1| = k−1P,i|ai−1|

Thus, for a constant acceleration |ai−1| = 1 ms−2, the steady state-distance error is 1 m withkP,i = 1 (rad s−1)2. By increasing the proportional gain of the feedback controller kP,i, thesteady-state error |ess,i| resulting from a constant acceleration |ai−1| is reduced linearly. As

lims→0

Hi(s) = 1 (4.10)

for a velocity-dependent spacing policy dynamics, (3.10) or (3.41), the steady-state error for a con-stant acceleration (3.48) does not change by using a velocity-dependent spacing policy instead ofa constant spacing policy..

The influence of the desired headway time hd,i and cut-off frequency ωf,i on the closed-loopsensitivity (4.9) for a velocity-dependent spacing policy, without and with low-pass filter, is de-picted in Figures 4.5(a) and 4.5(b), respectively. Recall that a constant spacing policy is a specialcase with hd,i = 0. Figure 4.5(a) shows that the the closed-loop sensitivity (4.9) always approachesa constant value at high frequencies in case no low-pass filter is used, i.e. Hi(s) as defined in(3.10). This value is computed as

lims→∞S

′i(s) = lim

s→∞s2

(1 + hd,iωK,i)s2 + ωK,i(1 + hd,iωK,i)s + ω2K,i

=1

1 + hd,iωK,i

This result corresponds with Figure 4.2(a), as the open-loop gain |L′i| does not approach zero athigh frequencies. Correspondingly, the complimentary sensitivity, T

′i = 1 − S

′i = L

′i(1 + L

′i)−1

does not roll-off at high frequencies. Instead, it approaches the value

lims→∞T

′′i (s) = 1− 1

1 + hd,iωK,i=

hd,iωK,i

hd,iωK,i + 1

In practise, a high-frequent roll-off of T′i is desired to ensure attenuation of measurement noise

nr,i and nv,i (see Figure 3.5 in Section 3.2.3). Therefore, in case of ACC, a low-pass filter fi(s)(3.42) is desired to ensure a high-frequent roll-off T

′i , see Figure 4.5(b).


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0

10

Frequency (rad)

Ma

gn

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(d

B)

(a) without low-pass filterHi = 1 + hd,is.

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100

101

-40

-30

-20

-10

0

10

Frequency (rad)

Ma

gn

itu

de

(d

B)

(b) with low-pass filterHi = 1 + hd,ifi(s)swith ωf,i = ωK,i.

Figure 4.5: Bode magnitude plot of the closed-loop sensitivity S′′i (jω) (4.9) for different spacing

policies Hi, with hd,i ∈ {0, 0.5, 1, 1.5, 2} s (black to light grey). Feedback controller Ki(s) (3.12) withωK,i = 1 rad s−1 and ideal vehicle dynamics Gi(s) = s−2 are assumed.

4.3.2 CACC

In case of CACC, the feedforward controller equals Fi(s) = H−1i (s) (3.15), where ideal vehicle

dynamics Gi(s) = s−2 are assumed. The closed-loop sensitivity (3.46) equals

S′′i = Ei X

−1i−1 = (1−Di)(1 + HiGiKi)−1 = (1−Di)S

′i (4.11)

which equals zero if communication delay is not taken into account, i.e. Di(s) = 1. Using

lims→0

Di(s) = lims→0

e−θis = 1, (4.12)

it can be shown that the steady-state error for a constant acceleration |ai−1| (3.48) always equalszero. Hence, performance is clearly improved by including communication and feedforwardof the acceleration. The influence of the parameters in Di, Hi, Gi and Ki on the closed-loopsensitivity (4.11) is not discussed here. Instead, in the next chapter, the closed-loop sensitivity isanalyzed for a particular set of parameters.

4.4 String stability

For string stability, i.e. attenuation of velocity and acceleration peaks, condition (3.27) must besatisfied, see Section 3.3. The condition is repeated here for convenience:

|SSX,i(jω)| =∣∣∣∣

Xi(jω)Xi−1(jω)

∣∣∣∣ ≤ 1 (4.13)

whereXi

Xi−1=

GiFiDis2 + GiKi

1 + HiGiKi, for i > 1 (4.14)


First, string stability is analyzed for the ACC case. Moreover, three different spacing policy dy-namics Hi(s) are considered: a constant spacing policy (3.11), a velocity dependent spacing policywithout low-pass filter (3.10) and a velocity dependent spacing policy with low-pass filter (3.41).For the velocity-dependent spacing policies, a minimum headway to guarantee string stability isderived. The analysis includes frequency and time responses. Then, string stability is analyzedfor the CACC case in a similar way.

4.4.1 ACC

In the case of no feedforward, i.e. Fi(s) = 0, an ACC system instead of a CACC system results.Then, the output string stability transfer function (4.14) reduces to

SSX,i =GiKi

1 + HiGiKi(4.15)

Constant inter-vehicle spacing policy

Considering a constant spacing policy, i.e. Hi(s) = 1 (3.11). Then, the output string stabilitytransfer function (4.15) reduces to

SSX,i =GiKi

1 + GiKi= Ti (4.16)

In practice, a high-frequent roll-off of Ti(s) is desired to ensure attenuation of radar noise nr,i (seeFigure 3.5 in Section 3.2.3). As a result, the magnitude |Ti(jω)| is increased at other frequencies,such that ||Ti(s)||∞ = ||SSX,i(s)||∞ > 1. This is the result of a fundamental limitation of linearfeedback systems, quantified by the well-known Bode’s sensitivity integral. Consequently, in caseof an ACC system, string stability can not be guaranteed for a constant inter-vehicle spacing byany linear controller.

This general conclusion will now by verified for this case by evaluating the string stabilitycondition (4.13). Substituting Hi(s) = 1, ideal vehicle dynamics Gi(s) = s−2 and ACC feedbackcontroller Ki(s) (3.12) in the output string stability function (4.15) yields

SSX,i(s) =ωK,i(s + ωK,i)

s2 + ωK,is + ω2K,i

(4.17)

Substitution of (4.17) in the string stability condition (4.15) and rearranging yields

2ω2K,i ≤ ω2, ∀ω (4.18)

As min{ω2} = 0, it follows directly that string stability can be guaranteed if and only if ωK,i =0, i.e. if there is no feedback controller at all. In Figure 4.6, frequency and time responsesof SSX,i(s) (4.17) are shown. The peak in the output string stability function is about 1.48 ≈3.33 dB, see Figure 4.6(a). Figure 4.6(b) shows the acceleration responses of a homogeneous six-vehicle platoon to a unit step in the acceleration of the reference vehicle x0(t). The accelerationpeak ||x3(t)||∞ of the third following vehicle is almost twice as large as the unit step and continuesto grow rapidly upstream, see Figure 4.6(b).


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Frequency (rad/s)

Ma

gn

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(d

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(a) Bode magnitude |SSX,i|.

0 5 10 15 20

0

0.5

1

1.5

2

2.5

3

Time (s)

Acc

ele

rati

on

(m

s-2)

(b) Time responses xi(t) of a string of vehiclesi = 0, . . . , 5 (black to light grey) with a unit step in

the acceleration x0(t) (dashed).

Figure 4.6: ACC, constant inter-vehicle spacing policy. Bode magnitude (left) and time (right)responses of the string stability transfer function SSX,i (4.17) with ωK,i = 1 rad s−1.

Velocity-dependent spacing policy, without low-pass filter

Consider the velocity dependent spacing policy without low-pass filter, i.e. Hi(s) = 1+hd,is (3.10).Substituting Hi(s), ideal vehicle dynamics Gi(s) = s−2 and ACC feedback controller Ki(s) (3.12)in the output string stability function (4.15) yields


(hd,iωK,i + 1)s2 + (hd,iω2K,i + ωK,i)s + ω2

K,i

(4.19)

Now, string stability is guaranteed for

2− h2d,iω

2K,i ≤

ω2

ω2K,i

(hd,iω

2K,i + 1

)2, ∀ω (4.20)

As min{ω2} = 0,2− h2

d,iω2K,i ≤ 0 (4.21)

it follows directly that string stability can be guaranteed for hd,i ≥ hd,i,min, where hd,i,min =√2ω−1

K,i ≈ 1.41ω−1K,i is the minimum headway time to guarantee string stability. Figure 4.7(a)

clearly shows that ||SSX,i(s)||∞ ≤ 1 for hd,i ≥ hd,i,min. Figure 4.7(b) shows time responses ofSSX,i(s) for hd,i = hd,i,min

Velocity-dependent spacing policy, with low-pass filter

Consider the velocity dependent spacing policy with low-pass filter, i.e. Hi(s) = 1 + hd,ifi(s)s(3.41). First, the influence of the cut-off frequency ωf,i of the first-order low-pass filter fi(s) (3.42)is evaluated. It has been shown that a constant spacing policy (3.11) and a velocity-dependentspacing policy without low-pass filter (3.10) are special cases with ωf,i → 0 (or hd,i = 0) and


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101

102-20

-15

-10

-5

0

5

10

Frequency (rad/s)

Ma

gn

itu

de

(d

B)

(a) Bode magnitudes with hd,i ∈ {0, 0.5, 1, 1.5, 2} s(solid, dark to light grey) and hd,i = 21/2 ≈ 1.41

(dashed, black).

0 5 10 15 20

0

0.2

0.4

0.6

0.8

1

Time (s)

Acc

ele

rati

on

(m

s-2)


the acceleration x0(t) (dashed) withhd,i = 21/2 ≈ 1.41 s for i > 1.

Figure 4.7: ACC, velocity-dependent spacing policy without low-pass filter. Bode magnitude (left)and time (right) responses of the string stability transfer function SSX,i (4.19) with ωK,i = 1 rad s−1.

ωf,i → ∞, respectively. Hence, using previous result, it can be concluded directly that no stringstability can be guaranteed for ωf,i → 0, while string stability can be guaranteed for hd,i ≥21/2ω−1

K,i (4.21) for ωf,i →∞. In Section 3.3.2, however, it has been shown that string stability in

the ACC case can be guaranteed for hd,i = ω−1K,i and ωf,i = ωK,i. In other words, this particular

choice for ωf,i improves the string stability characteristics of the system.

Substituting Hi(s) (3.41) with ωf,i = ωK,i, ideal vehicle dynamics Gi(s) = s−2 and ACCfeedback controller Ki(s) (3.12) in the output string stability function (4.15) yields


s2 + ωK,i(hd,iωK,i + 1)s + ω2K,i

(4.22)

for which string stability can be guaranteed for

(2− 2hd,iωK,i − h2d,iω

2K,i) ≤

ω2

ω2K,i

, ∀ω (4.23)

As min{ω2} = 0,

2− 2hd,iωK,i − h2d,iω

2K,i ≤ 0 (4.24)

it follows directly that string stability can be guaranteed for hd,i ≥ hd,i,min where hd,i,min =(√

3 − 1)ω−1K,i ≈ 0.73ω−1

K,i. The low-pass filter with ωf,i = ωK,i reduces the minimum head-

way time, for which string stability can be guaranteed, by a factor (√

3 − 1)/√

2 ≈ 1.93. Com-paring Figures 4.7(a) and Figure 4.8(a), shows that, due to the low-pass filter, the amplitude of|SSX,i(jω)| changes only around the breakpoint frequency ωK,i. Comparing Figures 4.7(b) andFigure 4.8(b) shows that the “speed” of the responses clearly increases, which is the result of thereduction in headway time.


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101

102-20

-15

-10

-5

0

5

10

Frequency (rad/s)

Ma

gn

itu

de

(d

B)

(a) Bode magnitudes with hd,i ∈ {0, 0.5, 1, 1.5, 2} s(solid, dark to light grey) and

hd,i = (31/2 − 1) ≈ 0.73 (dashed, black).

0 5 10 15 20

0

0.2

0.4

0.6

0.8

1

Time (s)

Acc

ele

rati

on

(m

s-2)


the acceleration x0(t) (dashed) withhd,i = (31/2 − 1) ≈ 0.73 s for i > 1.

Figure 4.8: ACC, velocity-dependent spacing policy with low-pass filter. Bode magnitude (left) andtime responses (right) of the string stability transfer function SSX,i (4.22) with

ωK,i = ωf,i = 1 rad s−1.

4.4.2 CACC

For the CACC case, with Fi(s) = H−1i (s) (3.15) for ideal vehicle dynamics Gi(s) = s−2, the string

stability transfer function (4.14) equals

SSX,i =Di + HiGiKi

Hi(1 + HiGiKi)(4.25)

If, for the sake of clarity, communication delay is not taken into account, i.e., Di(s) = 1, thestring stability transfer function (4.14) reduces to

SSX,i = H−1i = Fi (4.26)

Constant inter-vehicle spacing policy

If communication delay is not taken into account, the string stability transfer function (4.26)equals SSX,i = H−1

i = 1, for a constant inter-vehicle spacing policy. Hence, only marginal stringstability |SSX,i(jω)| = 1, ∀ω, can be guaranteed. Marginal in this case indicates that the designis not robust for uncertainties or modeling errors. For example, taking into account time delay aswell would mean that no string stability can be guaranteed.

Velocity-dependent spacing policy, without low-pass filter

For a velocity-dependent spacing policy Hi(s) without low-pass filter (3.10) and no communica-tion delay Di(s) = 1, the string stability transfer function (4.26) equals

SSX,i = H−1i = (hd,is + 1)−1 (4.27)


for which string stability can be guaranteed for hd,i ≥ hd,min,i = 0. Hence, the headway timemay be chosen arbitrarily small in the ideal case. For hd,i = 0, only marginal string stability canbe guaranteed. In the corresponding ACC case, a minimum headway time of hd,i,min =

√2ω−1

K,i

(4.21) is required for a velocity-dependent spacing policy without low-pass filter, see Section 4.4.1.

Consider the Bode magnitude plots of the string stability function in Figure 4.9(a). Reducingthe desired headway time hd,i causes |SSX,i(jω)| to roll-off at a higher frequency, which indi-cates ‘tighter’ control. For hd,i = 0, |SSX,i(jω)| = 1 for all frequencies ω. The accelerationresponses xi(t) of a six-vehicle platoon, adopting a desired headway time of hd,i = 0.1 s, ∀i, areshown in Figure 4.9(b). Comparing Figures 4.7 (ACC) and 4.9 (CACC) clearly shows the effectof feedforward.

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Frequency (rad/s)

Ma

gn

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de

(d

B)

(a) Bode magnitudes withhd,i ∈ {0, 0.1, 0.2, 0.3, 0.4} s

(dark to light grey).

0 1 2 3 4 5

0

0.2

0.4

0.6

0.8

1

Time (s)

Acc

ele

rati

on

(m

s-2)

(b) Time responses xi(t) of a string of vehiclesi = 0, . . . , 5 (black to light grey) with a unit step inthe acceleration x0(t) (dashed) with hd,i = 0.1 s for

i > 1.

Figure 4.9: CACC, velocity-dependent spacing policy without low-pass filter. Bode magnitude(left) and time (right) responses of the string stability transfer function SSX,i (4.27).

Taking into account communication delay Di(s) (3.14), a minimum headway time hd,i,min forhd,i is required to guarantee string stability. Depending on the breakpoint ωK,i and on the size ofthe delay θi, string stability can be guaranteed for hd,i ≥ hd,i,min(ωK,i, θi). In Figure 4.10, the re-sults of a numerical approximation of hd,i,min(ωK,i, θi) are shown. Assuming that the breakpointωK,i and the communication delay θi are known, the minimum value hd,i,min(ωK,i, θi) for hd,i

follows from these results. If, for example, the communication delay equals θi = 100 ms, and thebreakpoint ωK,i of the feedback controller equals ωK,i = 1 rad s−1, string stability is guaranteedfor hd,i & 0.4 s (see Figure 4.10). This value is clearly smaller than the value required to achievestring stability in case of ACC, see Section 4.4.1.

A Bode magnitude plot of the string stability transfer function, with hd,i ∈ [0, 0.4]s and acommunication delay of θi = 100 ms is shown in Figure 4.11(a). In accordance with Figure 4.10, adesired headway time of hd,i = 0.4 s results in a string-stable transfer function, i.e ||SSX,i||∞ ≤ 1.Comparing Figures 4.9 and 4.11 shows the effect of the communication delay.


0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0.5

0.60.7

0.80.91

1.1

1.2

1.3

1.4

1.5

1.6

ωK,i

(rad/s)

θi(m

s)

10-1

100

101

0

50

100

150

200

hd

,min

,i(s

)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Figure 4.10: CACC, velocity-dependent spacing policy without low-pass filter. Contour plot ofωK,i vs θi, indicating the corresponding minimal value for hd,i,min = hd,i,min(ωK,i, θi) for which

string stability can be guaranteed.

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Ma

gn

itu

de

(d

B)



0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

Time (s)

Acc

ele

rati

on

(m

s-2)


i > 1.

Figure 4.11: CACC, velocity-dependent spacing policy without low-pass filter. Bode magnitude(left) and time (right) responses of the string stability transfer function SSX,i (4.25) with idealvehicle dynamics Gi(s) = s−2, ωK,i = 1 rad s−1 and a communication delay of θi = 100 ms.

Velocity-dependent spacing policy, with low-pass filter

Taking the low-pass filter fi(s) (3.42) into account with a velocity-dependent spacing policy (3.41),the string stability transfer function (4.26) equals

SSX,i =s + ωf,i

(1 + hd,iωf,i)s + ωf,i(4.28)


for which string stability can be guaranteed for hd,iωf,i ≥ 0, i.e. for any hd,i ≥ hd,min,i = 0with ωf,i > 0. Hence, the headway time may be chosen arbitrarily small, whereas a minimumheadway time of approximately 0.73ω−1

K,i in the corresponding ACC case, see Section 4.4.1. Dueto the low-pass filter, the string stability transfer function (4.28) is no longer strictly-proper, i.e.

limω→∞ |SSX,i(jω)| = 1

1 + hd,iωf,i6= 0 (4.29)

see Figure 4.12(a). As a result, the time responses show a direct feed-through effect, see Fig-ure 4.12(b). Moreover, the settling time of the time responses increases significantly (compareFigures 4.9(a) and Figure 4.12(a)), which is caused by the relative small cut-off frequency of thelow-pass filter, i.e. ωf,i = ωK,i = 1 rad s−1. Decreasing the cut-off frequency causes the stringstability transfer function to roll off at a lower frequency (not shown). Without low-pass filter,which corresponds to ωf,i → ∞, the string stability transfer function rolls-off at approximately1 rad s−1, see Figure 4.9(a). With low-pass filter and ωf,i = 1 rad s−1, the string stability transferfunction starts to roll of at approximately 0.1 rad s−1, see Figure 4.12(a). Generally, the lowerthe frequency at which the string stability function start to rolls off, the slower its time response.Hence, in the case of CACC, the cut-off frequency ωf,i is preferred to be larger to reduce thesettling time. Another benefit of increasing the cut-off frequency is that the high-frequent gain(4.29) decreases. As the feedforward controller equals Fi = H−1

i (3.15), this result in more atten-uation of high-frequent measurement noise of the communicated acceleration xi(t).

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101

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-15

-10

-5

0

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10

Frequency (rad/s)

Ma

gn

itu

de

(d

B)



0 1 2 3 4 5

0

0.2

0.4

0.6

0.8

1

Time (s)

Acc

ele

rati

on

(m

s-2)


i > 1.

Figure 4.12: CACC, velocity-dependent spacing policy with low-pass filter. Bode magnitude (left)and time (right) responses of the string stability transfer function SSX,i (4.28) with ωf,i = 1 rads−1.

Taking into account communication delay Di(s) (3.14), while assuming ideal vehicle dynam-ics Gi(s) = s−2, a minimum value hd,i,min(ωK,i, ωf,i, θ) for hd,i is required to guarantee stringstability. In case ωf,i = ωK,i, the minimum headway time depends on the breakpoint and com-munication delay only, i.e. hd,i,min = hd,i,min(ωK,i, ωf,i, θ), see Figure 4.13. Due to the low-passfilter, the minimum headway time hd,i,min for which string stability can be guaranteed is clearlysmaller compared to the CACC case without low-pass filter (compare Figures 4.10 and 4.13). If,

4.5. CONCLUSIONS 55

for example, the communication delay equals θi = 100 ms, string stability is guaranteed forhd,i > 0.2 s (see Figure 4.13) with ωf,i = ωK,i = 1 rad s−1, while the minimum headway time isabout 0.4 s in case no low-pass filter is used (see Figure 4.10). Hence, it can be concluded thatthe low-pass filter also has a positive effect on the minimum headway time in the CACC case.

0.1

0.1

0.2

0.3

ωK,i

(rad/s)

θi(m

s)

10-1

100

101

0

50

100

150

200

hd

,min

,i(s

)

0

0.1

0.2

0.3

0.4

Figure 4.13: CACC, velocity-dependent spacing policy with low-pass filter and ωf,i = ωK,i.Contour plot of ωK,i vs θi, indicating the corresponding minimal value forhd,i,min = hd,i,min(ωK,i, θi) for which string stability can be guaranteed.

A Bode magnitude plot of the string stability transfer function, with hd,i ∈ [0, 0.4] s and acommunication delay of θi = 100 ms is shown in Figure 4.14(a). In accordance with Figure 4.13,a desired headway time of hd,i = 0.2 s results in a string-stable transfer function, i.e ||SSX,i||∞ ≤1. Comparing Figures 4.12(b) 4.14 shows the effect of the communication delay on the time-responses.

4.5 Conclusions

In this chapter, ideal vehicle dynamics are assumed. In that case, the PD breakpoint frequency ofthe ACC feedback controller may be chosen arbitrarily large, while guaranteeing stability of theinner loop. In general, the outer loop may still be unstable due to the spacing policy dynamics. Fora constant spacing policy, however, there is no outer loop and stability is guaranteed automaticallyif the inner loop is stable. Moreover, for the velocity-dependent spacing policies, stability of theouter loop is guaranteed for any desired headway time and any cut-off frequency of the first orderlow-pass velocity filter. Feedforward does not affect vehicle-stability.

Without feedforward, an ACC system instead of a CACC system results. For ACC, thesteady-state distance error for a constant acceleration of the preceding vehicle, reduces linearlywith increasing proportional gain, irrespective of the adopted spacing policy. In case a velocity-dependent spacing policy without low-pass filter is adopted, the high-frequent distance error be-


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5

10

Frequency (rad/s)

Ma

gn

itu

de

(d

B)



0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

Time (s)

Acc

ele

rati

on

(m

s-2)


i > 1.

Figure 4.14: CACC, velocity-dependent spacing policy with low-pass filter. Bode magnitude (left)and time (right) responses of the string stability transfer function SSX,i (4.25) with ideal vehicle

dynamics Gi(s) = s−2, ωf,i = ωK,i = 1 rad s−1 and a communication delay of θi = 100 ms.

comes smaller with increasing desired headway time. On the other hand, high-frequent noise onthe velocity measurement is amplified correspondingly. Including a low-pass velocity filter in thevelocity-dependent spacing policy, prevents the amplification of high-frequent velocity noise and,hence, yields better performance. In the case of CACC, assuming ideal vehicle dynamics and nocommunication delay, the distance error equals zero. Hence, feedforward of the acceleration ofthe preceding vehicle enables a significant improvement in performance.

In the case of ACC, string stability cannot be achieved by any linear controller if a constantspacing policy is adopted. In case of a velocity-dependent spacing policy, the minimum headwaytime for which string stability can be guaranteed is inversively proportional to the PD breakpointωK,i. Without a low-pass velocity filter, the minimum headway time equals hd,min,i =

√2ω−1

K,i.

With low-pass filter, the minimum headway time equals hd,min,i = (√

3 − 1)ω−1K,i if the cut-off

frequency of the low-pass filter is chosen equal to the PD breakpoint. Hence, in case of ACC, thelow-pass filter and this particular choice of cut-off frequency has a strong positive effect on stringstability. In the case of CACC, assuming ideal vehicle dynamics and no communication delay, theheadway time may be chosen arbitrarily small while string stability is guaranteed. Only marginalstring stability can be achieved if a constant inter-vehicle spacing policy is adopted, i.e. if theheadway time equals zero. If communication delay is taken into account, a certain minimumheadway time larger than zero is required to guarantee string stability, which is dependent onthe size of the delay and the control design. The low-pass velocity filter is also shown to have apositive effect on string stability in the CACC case.

Overall, it may be concluded that the proportional gain, and correspondingly the PD break-point, must be chosen as large as possible in order to achieve small distance errors and a smallheadway time while guaranteeing string-stable behavior in the ACC case. Moreover, a first orderlow-pass filter on the velocity should be included to prevent amplification of velocity noise whenthe headway time is increased. In the ACC case, the cut-off frequency should be chosen equal to

4.5. CONCLUSIONS 57

the PD breakpoint. In the CACC case, however, it should be chosen larger in order to speed upthe time response of the vehicle. It should be chosen based on the amount of noise on the veloc-ity measurement without considering the PD breakpoint. For both ACC and CACC, the headwaytime must be chosen larger than the theoretical minimum headway time in order to guaranteestring-stable driving behavior. Increasing the headway time is shown to decrease the amplitudeof the string stability transfer function at high frequencies and, hence, increase robustness tohigh-frequent modeling uncertainties.

Chapter 5

Experimental validation

This chapter focusses on validation of the theoretical results by road experiments. In Section 5.1, theexperimental setup is presented. Then, in Section 5.2, the vehicle dynamics are identified and validated.The design variables are chosen and motivated in Section 5.3 after which the system is analyzed inSection 5.4. Experimental and simulation results are presented in Section 5.5. Finally, conclusions andrecommendations are summarized/formulated in Section 5.6.

5.1 Experimental setup

Two Citroën C4’s are used as a testing platform, see Figure 5.1. For the wireless inter-vehiclecommunication, the standard Wi-Fi protocol IEEE 802.11g is used, with an update rate of 10 Hz.The acceleration of both vehicles are derived from the built-in Electronic Stability Program (ESP).The acceleration of vehicle 1 is transmitted to vehicle 2 by wireless communication. A zero-order-hold approach is adopted for the communicated signal, introducing a corresponding delayof about 50 ms. Based on identification measurements, an additional communication delay ofabout 10 ms is identified. Combination of these values yields θ2 = 60 ms as a total delay for themodel D2(s) (3.14).

Figure 5.1: Experimental setup with two Citroën C4Šs.

Vehicle 2 is equipped with an Electro-Hydraulic Braking (EHB) system, i.e. a brake-by-wiresystem. Vehicle 1 is driven manually. Implementation of the controller for the longitudinal

59

60 CHAPTER 5. EXPERIMENTAL VALIDATION

dynamics of the vehicle as well as actuation of the throttle and EHB system are covered by theTNO Modular Automotive Control System (MACS) [Oud03]. An OMRON laser radar, i.e. a lidar,with 150 m range is built-in, see Figure 5.2(a). Using rapid control prototyping, the CACC systemis implemented on a dSpace AutoBox with a sample rate of 100 Hz, see Figure 5.2(b). Finally, alaptop is used for monitoring and logging data. A schematic overview of the instrumentation isshown in Figure 5.2(c).

Two sets of experiments have been executed. The experiment plan of the first set of experi-ments is shown in Appendix A1. The experiment plan of the second set of experiments is shownin Appendix B. Only the results of the second set of experiments are discussed here.

5.2 Vehicle model identification

Vehicle 2 includes a lower level controller Kl,2 for the longitudinal dynamics, see Figure 3.7 inSection 3.2.3. The corresponding closed-loop dynamics are approximated by the vehicle modelG2(s) (3.7), which includes the loop gain kG,2 ≈ m2m

−12 , an actuator time constant τG,2 and a

pure time delay φ2. To identify these vehicle model parameters, step-response measurements areperformed. The measurements and corresponding simulation results are shown in Figure 5.3.The simulation results correspond to the model

G2(s) = G2(s)s2 =kG,2

τG,2s + 1e−φ2s (5.1)

with kG,2 = 0.72, τG,2 = 0.38 s and φ2 = 0.18 s. These values have been computed by averagingidentified parameters for separate acceleration and deceleration step-response measurements.Four of these measurements are shown in Figure 5.3. Comparing the measurement and thesimulation results shows that the model covers the main longitudinal vehicle dynamics.

The average identified vehicle model is validated in open-loop using the desired and measuredacceleration during actual ACC and CACC experiments. One of these validation measurementsis shown in Figure 5.4. Again, the simulation results are plotted on top of the experimentalresults, showing that the model G2(s) accurately represents the vehicle dynamics.

5.3 Control design

In this section, the control settings during the experiments are presented and motivated. Thedesign variables are the ACC feedback controller K2(s) (3.12), the spacing policy dynamics H2(s)(3.41) and the feedforward controller F2(s) (3.15). For each design variable, the control parameter-s/settings are motivated. Finally, an overview of the experiments and the corresponding controllersettings is presented.

ACC feedback controller

The first step is to choose the PD breakpoint frequency ωK,2 of the ACC feedback controller K2(s)(3.12), which should be as large as possible in order to yield i) a small distance error and good

1YouTube video: “CACC string stability tests”http://www.youtube.com/watch?v=aNYh4MjeSpU (January 6, 2010)

5.3. CONTROL DESIGN 61

Lidar

(a) Citroën C4 and OMRON lidar.

1

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23

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5

(b) Hardware: 1) dSpace AutoBox, 2) RTK-GPS,3) GPS time pule, 4) Power supply 230 V and 5)

Ethernet Gateway

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autobox

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MACS

EHB

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uth

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(c) Schematic overview of the instrumentation of the vehicles. The main communication channels andcorresponding signals are indicated, where uth and ubr the throttle and brake system control signals, xd andx the desired and actual acceleration, xr and xr the relative position and velocity, xi−1 the communicatedacceleration of the preceding vehicle, and t the time stamping signal. For clarity, the index i, indicating

vehicle i, is omitted.

Figure 5.2: Overview of vehicle instrumentation.

tracking (see Section 4.3) and ii) string stability at a small headway time in the case of ACC (seeSection 4.4). The time constant τG,2 and time delay φ2 in G2(s) (5.1) introduce additional phaselag at high frequencies. Consequently, as the breakpoint frequency is increased, the closed-loopT2 = L2(1 + L2)−1 with L2 = G2K2, eventually becomes unstable as the PD controller does notprovide sufficient phase lead at high frequencies. For robustness, the breakpoint frequency ωK,2

is chosen such that |L2jω + 1| > 0.5 for all frequencies ω. The final choice is ωK,2 = 0.5 rad s−1.A Bode and Nyquist diagram of the open-loop L(jω) is shown in Figure 5.5. The phase margin(PM) is about 31.5◦ .


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Figure 5.3: Identification step-response measurement results for accelerating (a,c), and braking(b,d): desired acceleration (dashed black), measured acceleration (solid black) and correspondingsimulation results (solid grey) with the model G2(s) = G2(s)s2, kG,2 = 0.72, τG,2 = 0.38 s and

φ2 = 0.18 s

Spacing policy dynamics

During the experiments, a velocity-dependent spacing policy is adopted. An additional low-passfilter f2(s) (3.42) is used to filter the measured velocity signal. The resulting spacing policydynamics are H2(s) = 1+hd,2f2(s)s (3.41). The design parameters are the cut-off frequency ωf,2

and the desired headway time hd,2 .

In the case of ACC, the cut-off frequency ωf,2 is chosen equal to the breakpoint, i.e. ωf,2 =ωK,2 = 0.5 rads−1. Results from the previous chapter showed that this particular choice has astrong positive effect on string stability. In the case of CACC, a large value for the cut-off fre-quency ωf,2 has been used. In the previous chapter, it is shown that a small cut-off frequencyresults in a large setting time for a step response. Hence, in the case of CACC, the cut-off fre-quency is chosen as ωf,2 = 10ωK,i = 5 rad s−1 to speed up the response.

One of the goals of the experiments is to validate wether string-stable behavior can be pre-

5.3. CONTROL DESIGN 63

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G2(s) = G2(s)s2

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Figure 5.5: Bode and Nyquist diagram of the open-loop L2(jω) = G2K2, where G2 (5.1) theidentified vehicle model with kG,2 = 0.72, τG,2 = 0.38 s and φ2 = 0.18 s and feedback controller K2

(3.12) with ωK,2 = 0.5 rad s−1.

dicted from theoretical results. Hence, the desired headway time hd,2 is either chosen smalleror larger than the minimum headway time hd,2,min to guarantee string stability. As hd,2,min hasbeen derived analytically for ideal vehicle dynamics only, it is computed numerically for eachexperiment using the identified vehicle dynamics G2(s) (5.1) and, in the case of CACC, the com-munication delay model D2(s) (3.14) with θ2 = 60 ms. During the experiments, a headway timeof hd,2 = 3 and 1 s has been used for ACC. A headway time of hd,2 = 1 s is used for CACC. Abode magnitude of the various spacing policy dynamics H2 (3.41) is shown in Figure 5.6


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Figure 5.6: Bode magnitude of the spacing policy dynamics H2(s) (3.41) for a cut-off frequency ofωf,2 = 0.5 rad s−1 and headway time of hd,2 = 3 s (dashed, black) and hd,2 = 1 s (solid, black), and a

cut-off frequency of ωf,2 = 5.0 rad s−1 and headway time of hd,2 = 1 s (grey).

Feedforward controller

The feedforward controller F2(s) (3.15) is based on a zero-error condition and compensates forthe vehicle dynamics G2(s) (5.1) and the spacing policy dynamics H2(s) (3.10). Hence, no newcontrol parameters are introduced. The feedforward equals

F2(s) =1

H2G2s2=

1kG,2

τG,2s2 + (τG,2ωf,2 + 1)s + ωf,2

(hd,2ωf,2 + 1)s + ωf,2(5.2)

For τG,2 6= 0, or ωf,i 6= ∞, the feedforward controller F2(s) (5.2) becomes non-proper.2 Imple-mentation of a non-proper feedforward controller involves differentiation of the communicatedacceleration, resulting in amplification of measurement noise. Hence, such a feedforward con-troller is considered as undesirable. To resolve this problem, the filter f2(s) (3.42) has beenremoved from H2(s) = 1 + hd,2f2(s)s in F2(s) (5.2), yielding the implemented controller

F2(s) =1

kG,2

τG,2s + 1hd,2s + 1

(5.3)

The implemented feedforward controller (5.3) does not appropriately compensate for the actualspacing policy dynamics H2(s) (3.41) with filter f2(s) (3.42). The effect of the implemented feed-forward (5.3) is evaluated later on in this section. A Bode magnitude plot of both feedforwardcontrollers is shown in Figure 5.7.

Experiment overview

The experimental results of three (C)ACC experiments will be discussed: (a) an ACC experimentwith a large headway time, (b) an ACC experiment with a small headway time and (c) a CACC

2A transfer function f(s) with limit

limω→∞

|f(jω)| =

0 is strictly-proper ;constant is proper ;∞ is non-proper.

5.4. SYSTEM ANALYSIS 65

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Figure 5.7: Bode magnitude of the feedforward controller Fi(s) (5.2) (grey) and implementedfeedforward controller (5.3) (black) F2(s), with kG,2 = 0.72, τG,2 = 0.38 s, ωf,2 = 5 rad s−1 and

hd,i = 1 s.

experiment with a small headway time. The control design parameters for each experiment arepresented in Table 5.1. Note that for experiments (a) and (c), string-stable behavior is expected ashd,2 > hd,2,min, while string-unstable behavior is expected for experiment (b) as hd,2 < hd,2,min.

Table 5.1: Overview of the experiments.Experiment ACC/CACC F2(s) ωf,2 (rad s−1) hd,2 (s) hd,2,min (s)

(a) ACC - 0.5 3.0 1.46(b) ACC - 0.5 1.0 1.46(c) CACC (H2G2s

2)−1 5 1.0 0.84

5.4 System analysis

For each experiment, see Table 5.1, the system is analyzed for performance and string stability.Performance and string stability are analyzed by evaluating the magnitudes of the closed-loopsensitivity S

′′2 = E2 X−1

1 (3.46) and string stability SSX,2 = X2 X−11 (3.29) functions for the

identified models G2(s) (5.1) and D2(s) (3.14) and design variables (See Table 5.1). A small am-plitude |S′′2 (jω)| at low frequencies ω is desired to yield good tracking, i.e. small inter-vehicledistance errors, for low-frequent driving behavior of vehicle 1. For string stability, the amplitude|SSX,2(jω)|must be smaller or equal to one for all frequencies ω. The magnitude of both transferfunction is compared with the case of ideal vehicle dynamics G2(s) = s−2 and no communicationdelay D2(s) = 1.

a) ACC, string-stable

Without feedforward F2(s) = 0, the sensitivity equals S′′2 = (1 + L

′2)−1 = S

′2 and the string

stability function equals SSX,2 = L2(1 + L′2) = L2S

′2, with L2 = G2K2 and L

′2 = H2G2K2 =


H2L2. The vehicle gain kG,2 in G2(s) (5.1) does not influence these transfer functions as it iscanceled out by the feedback controller K2(s) (3.12).

Figure 5.8(a) shows that there is a peak on the sensitivity |S′′2 (jω)| for identified vehicle dy-namics (the identified vehicle model), whereas there is no peak for ideal vehicle dynamics. Thispeak is caused by the time constant τG,2 and, for a smaller amount, by the time delay φ2. Theseparameters introduce additional phase lag such that, in the complex plane, the open-loop L

′2 is

shifted towards the critical −1 + 0j point. The spacing policy dynamics H2(s) does not providemuch phase lead in this frequency range. The distance of L

′2(jω) to−1+0j is equal to the ampli-

tude |L′2(jω)+1|. Hence, as this distance decreases, a peak in the sensitivity |S′2| = |L′2(jω)+1|−1

appears.

The peak in |S′′2 (jω)| = |S′2(jω)| causes a little ‘bump’ at ω ≈ 1 rad s−1 in the amplitudeof the string stability function |SSX,2| = |L2S

′2|, see Figure 5.8(b). At frequencies beyond this

bump, the slope decreases from −20 to −40 dB/decade, which is a result of the time constantτG,2. At frequencies before the bump, the amplitude |SSX,2(jω)| is smaller than 0 dB, due to thelarge headway time hd,2 = 3 s. It can be shown that, for an increasing headway time hd,2, themagnitude |H2(jω)| increases around the breakpoint ωK,2. The magnitude |L2(jω)| ≈ 1 aroundthe breakpoint frequency ωK,2, see Figure 5.5. Hence, increasing the magnitude of |H2(jω)|in the frequency range around the breakpoint ωK,2 has a large influence on the amplitude of|SSX,2|, because |SSX,2| ≈ |(1 + H2)−1| at frequencies where |L2(jω)| ≈ 1. For hd,2 = 3 s,the amplitude of the string stability function |SSX,2| drops below 0 dB at low frequencies, suchthat string stability is guaranteed. The minimum headway time to guarantee string stability ishd,2,min ≈ 1.46, see Table 5.1.

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Figure 5.8: Experiment (a) Bode magnitude of the sensitivity (left) and string stability (right)functions for ideal vehicle dynamics (dashed, grey) and identified vehicle dynamics.

b) ACC, string-unstable

The sensitivity S′′2 (s) (3.46) and string stability SSX,2(s) (3.29) functions are similar to those

in the previous ACC experiment. The changes with respect to the smaller headway time arediscussed here. The influence on the amplitude |S′′2 (jω)| = |S′2(jω)| is not dramatic, see Fig-

5.4. SYSTEM ANALYSIS 67

ures 5.9(a) and 5.8(a). For a smaller headway time, the peak in |S′2(jω)| = |H2(jω)L2(jω)| issomewhat smaller due to the smaller amplitude of |H2(jω)| at that frequency, see Figure 5.6.

By reducing the headway time hd,2, the amplitude in |H2(jω)| decreases in the frequencyrange where |L2(jω)| ≈ 1, i.e. at frequencies around the breakpoint frequency ωK,2. As a result,the magnitude |SSX,2(jω)| = |L2(1 + H2L

−1)−1| increases in this frequency range, such thatthere is now a peak ||SSX,2||∞ > 1 = 0 dB. Hence, the system in string-unstable.

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Figure 5.9: Experiment (b). Bode magnitude of the sensitivity (left) and string stability (right)functions for ideal vehicle dynamics (dashed, grey) and identified vehicle dynamics.

c) CACC, string-stable

With feedforward (5.3), the sensitivity S′′2 (s) (3.46) and string stability SSX,2(s) (3.29) functions

are clearly different, see Figure 5.10. The magnitudes of these functions are shown for

1. ideal dynamics (G2(s) = s−2, D2(s) = 1) and feedforward F2(s) (5.2)

2. identified dynamics (G2(s) (5.1), D2(s) (3.14)) and feedforward F2(s) (5.2).

3. identified dynamics (G2(s) (5.1), D2(s) (3.14)) and implemented feedforward F2(s) (5.3).

For the ideal dynamics and feedforward controller F2(s) (5.2), the sensitivity S′′2 (s) equals

zero and, hence, its magnitude is not visible in Figure 5.10(a). For the identified dynamics andfeedforward controller F2(s), the amplitude |S′′2 (jω)| is increased at high frequencies, which isthe results of the communication delay θ2 = 60 ms and the time delay φ2 = 180 ms, which are notcompensated by the feedforward controller. The exact shape of |S′′2 (jω)| at high frequencies, seeFigure 5.10(a), has not been explained yet. The feedforward controller F2(s) (5.2) compensatesfor the remaining vehicle dynamics G2(s) (5.1) and spacing policy dynamics H2(s) (3.41). For theimplemented feedforward controller, the magnitude |S′′2 (jω)| somewhat increases at frequenciesbetween about 0.3 and 8 rad s−1, which is the result of not compensating for the velocity filterf2(s). By increasing the cut-off frequency ωf,2 in the spacing policy dynamics H2(s) (3.41), the in-crease in magnitude may be reduced. Comparing string-stable ACC and CACC, in Figures 5.8(a)


and 5.10(a), shows that the frequency at which |S′′2 (jω)| = 1 has moved from about 0.6 rad s−1

to 2 rad s−1, which is the result of feedforward. Hence, the distance error resulting from ma-noeuvres of vehicle 1 is expected to be much smaller compared in the case of CACC, comparedto ACC.

The string stability functions SSX,2(s) are shown in Figure 5.10(b). The identified dynamicscause the magnitude |SSX,2(jω)| to increase in the frequency range between 0.5 and 10 rad s−1.At lower frequencies, ω < 0.5 rad s−1, the magnitude is hardly influenced, because the ACC feed-back controller K2(s) is active in this frequency range. An additional effect of the implemented,feedforward filter is that the amplitude |SSX,2(jω)| rolls-off at the cut-off frequency ωf,2. Com-paring string-stable ACC and CACC, in Figures 5.9(b) and 5.10(b), shows that the magnitude|SSX,2(jω)| starts to roll-off at a larger frequency. This indicates that the CACC system enablestracking of high-frequent driving behavior of vehicle 1.

Looking at the differences between the feedforward controllers, in Figures 5.10(a) and (b), itcan be concluded that the implemented feedforward controller has a deteriorating effect, as ex-pected. The amplitude |S′′2 (jω)| increases between 0.5 and 5 rad s−1. However, the amplitude|S′′2 (jω)| is still considered small, at least compared to the ACC case, e.g. compare Figures 5.8(a)and 5.9(a). More importantly, the amplitude |SSX,2(jω)| does not increase in the critical fre-quency range, i.e. where it starts to roll-off. Hence, string stability is still guaranteed in this case.The influence of the implemented controller on the magnitudes |S′′2 (jω)| and |SSX,2(jω)| can bereduced by increasing the cut-off frequency ωf,2 of the low-pass filter f2(s) (3.42).

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Figure 5.10: Experiment (c). Bode magnitude of the sensitivity (left) and string stability (right)functions for ideal dynamics (G2(s) = s−2, D2(S) = 1) with feedforward controller F2(s) (5.2)

(dashed, grey), for identified dynamics (5.1), D2(s) (3.14)) with feedforward controller F2(s) (5.2)(grey, solid) and for identified dynamics (5.1), D2(s) (3.14)) and implemented feedforward controller.

5.5 Results

In this section, the measurement results of the experiments, see Table 5.1, are analyzed andcompared with closed-loop simulation results of a five-vehicle platoon. Vehicle 1 is the preceding

5.5. RESULTS 69

vehicle and vehicle 2 is the following vehicle, see Figure 5.1. The simulation has two purposes:

1. to validate whether the following behavior of the vehicle 2 can be described using linearmodels;

2. to emphasize the effect of string-(un) stable behavior of the experiments.

If the results from the closed-loop simulation results are in good correspondence with the exper-imental results, it may be concluded that there are no unmodeled dynamics which are of largeinfluence, e.g. on-board sensor delay, a variable communication delay, and/or high-order/non-linear vehicle dynamics. The dynamics of the simulated platoon are equal to the (identified)dynamics of vehicle 2, i.e. a homogeneous platoon. Hence, if the simulation results are accurate,the string-(un)stable of vehicle 2 is emphasized.

Vehicle 1 is driving in front of the simulated string of vehicles i = 2, 3, 4, 5, 6. Correspond-ingly, the input of the closed-loop simulation is the absolute position x1(t), velocity x1(t) and theacceleration x1(t) of vehicle 1. The communicated velocity and acceleration are considered thebest available measurements and are used as input for the simulation. The absolute position isobtained by integrating the velocity, see Appendices C and D for more detailed information.

For each experiment, see Table 5.1, measurement and simulation results are compared forstring-stability and performance. Here, for ease of discussion, string-stability is discussed first.For string stability, the velocities vi(t) = xi(t) and accelerations ai(t) = vi(t) = xi(t) are evalu-ated (i = 1, 2 for experimental results, i = 2, 3, 4, 5, 6 for simulation results). For performance,the desired and measured inter-vehicle distances (xr,d,2(t), xr,2(t)) and relative velocities (vr,d,2(t),vr,2(t)) are evaluated, along with the corresponding distance and relative velocity errors (ex,2(t),ev,2(t)). During the experiments, a desired distance at standstill of r2 = 10 m has been used forsafety, allowing testing of string unstable behavior.

5.5.1 ACC, string-stable

String stability

In Figure 5.11, the velocities and accelerations of the measured and simulated vehicles are shown.As the amplitude of the oscillations in velocity and acceleration are smaller for vehicle 2 comparedto vehicle 1, vehicle 2 behaves string-stable, as expected. The results of the simulated vehicle 2are in correspondence with the measured vehicle 2. A possible explanation for the differences ison-board sensor delay. On-board sensor delay has not been modeled. If the vehicle dynamics aresimulated in open-loop, the rapid changes in the acceleration are covered by the model. Hence,unmodeled high-order vehicle dynamics is believed not to be the reason for the differences be-tween the closed-loop simulation and experimental results. The ‘shocky’ acceleration of vehicle 2between 70 and 80 s, see Figure 5.11(b), was perceptible during the tests and is yet unexplained.There are two ‘dips’ in the acceleration a1(t) between 60 and 70 s, resulting from gear changes.This is not the cause, however, as the acceleration of vehicle 1 continues to oscillate rapidly whilethe acceleration a2(t) does not. Overall, the simulation results cover the main characteristics ofthe experimental results pretty well. The simulated platoon shows the effect of a homogeneousplatoon with vehicles that behave as vehicle 2. The string-stable behavior damps out oscillationsin velocity and acceleration, resulting in a more comfortable and fuel economic ride at the end ofthe platoon.


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Figure 5.11: The velocities (top) and accelerations (bottom) of vehicle 1 (black), vehicle 2 (blue)and five simulated vehicles i = 2, 3, 4, 5, 6 (dark to light green) for the string-stable ACC experiment.

Performance

In Figure 5.12(a) and (b), the desired and measured distances are shown. As a result from theheadway time hd,2 = 3 s and an average velocity of about 60 km h−1 ≈ 17 m s−1 (see Fig-ure 5.11(a)), the desired inter-vehicle distance is about 10 + 3 × 17 ≈ 60 m on average with adesired distance at standstill of r2 = 10 m. For ACC, the minimum headway time to guaran-tee string stability equals hd,2,min ≈ 1.46 s. Hence, the headway time hd,2 may be reduced,thereby reducing the desired inter-vehicle distance. However, even for hd,2 = hd,2,min, the result-ing distance is still considered large. Especially if the velocity v2(t) would increase to, lets say,120 km h−1. The inter-vehicle distance error varies between −4 and 2 m, which is the result oflarger deceleration values compared to acceleration values of vehicle 1, see Figure 5.11.

Comparing the acceleration profiles of vehicle 2 in Figures 5.4 and 5.11, it can be seen that theacceleration profile is significantly smoother during validation of the vehicle model. Overall, thesimulation results cover the main characteristics of the following behavior of vehicle 2.

5.5. RESULTS 71

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Figure 5.12: The desired and measured inter-vehicle distance and relative velocity andcorresponding errors for the string-stable ACC experiment (blue) in comparison with simulation

results (green).

5.5.2 ACC, string-unstable

String stability

In Figure 5.13, the velocities and accelerations of the measured and simulated vehicles are shown.The amplitude of the oscillations in velocity and acceleration are slightly larger for vehicle 2 com-pared to vehicle 1. Hence, the behavior is string-unstable as expected. Again, the simulationresults cover the main characteristics of the experimental results reasonably well, except that thesimulation results show stronger string-unstable behavior. Again, a possible explanation is on-board sensor delay, which has not been modeled. Despite the difference between the experimentand simulation results, the difference between the experimental results for a different headwaytime, in Figures 5.11 and 5.13, is obvious. By trusting the simulation results, it can be concludedthat a string of vehicles which all behave like vehicle 2, is string-unstable as the magnitude in theoscillations in velocity and acceleration are increased rapidly upstream. As a result, vehicles at theend of the platoon may reach their acceleration or braking limits, resulting in possibly dangeroussituations.

Performance

In Figure 5.14(a) and (b), the desired and measured distances are shown. As a result fromthe headway time hd,2 = 1 s and an average velocity of about 60 km h−1 ≈ 17 m s−1 (see Fig-ure 5.13(a)), the desired inter-vehicle distance is about 10 + 17 ≈ 27 m on average with a desireddistance at standstill of r2 = 10 m. The inter-vehicle distance error varies between −5 and 4 m,


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experiment.

resulting from acceleration/deceleration manoeuvres of vehicle 1, see Figure 5.13. During theexperiment, a desired distance at standstill of r2 = 10 m has been adopted for safety. In practise,the desired distance at standstill r2 is typically only several meters. Then, with a distance error of−5 m during decelerations, the driver may not feel safe in these kind of situations.

5.5.3 CACC, string stable

String stability

In Figure 5.15, the velocities and accelerations of the measured and simulated vehicles are shown.Comparing ACC and CACC for an identical headway time hd,2 = 1 s, in Figures 5.13 and 5.15,the effect of the feedforward controller is obvious. Although the magnitudes in the oscillationsin velocity and acceleration do not decrease drastically, the behavior of vehicle 2 is string-stable inthe case of CACC, whereas it is string-unstable in the case of ACC. Hence, it can be concludedthat wireless communication and feedforward of the acceleration a1(t) = x1(t), i.e. CACC,enables string-stable driving behavior at a smaller headway time compared to the case of nocommunication and feedforward, i.e. ACC.

In this case, the simulation results almost perfectly correspond with the experimental results,meaning that the behavior of vehicle 2 is accurately described by using a simple linear vehicle

5.6. CONCLUSIONS AND RECOMMENDATIONS 73

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results (green).

model G2(s) (5.1. Figure 5.15(b) shows that even rapid changes in the acceleration a2(t) arecovered well by the simulation. As the identified communication delay is small, i.e. θ2 = 60 ms,it does not have much influence and not much can be said about the validity of the linear modelD2(s) (3.14). Based on these results, it is concluded that the on-board sensor delay is the maincause for that differences between the experiment and simulation results for the case of ACC,see Figure C.1. In the case of CACC, the feedback action is much smaller due to the feedforwardaction and, hence, the effect of the on-board sensor delay is reduced significantly.

Performance

In Figure 5.16(a) and (b), the desired and measured distances are shown. Similar to the ACC case,the desired inter-vehicle distance is about 27 m on average for a headway time for hd,2 = 1 s andan average velocity of about 60 kmh−1. Comparing ACC and CACC, however, in Figures 5.14(c)and 5.16(a), the distance error is only about ±1 m in the case of CACC. Also the relative velocityerror is reduced significantly, which contributes to the safety perception of the driver. Hence, itcan be concluded that performance and, hence, safety is greatly improved by wireless communi-cation and feedforward of the acceleration a2(t), i.e. in the case of CACC.

5.6 Conclusions and recommendations

The vehicle dynamics have been identified using step-response measurements. Comparing themeasurement and simulation results showed that the identified model, a simple process model,


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accurately represents the longitudinal vehicle dynamics. The control variables have been chosenbased on the identified vehicle model and a small identified communication delay. Three exper-iments have been designed: a string-stable ACC experiment, a string-unstable ACC experimentand a string stable CACC experiment. The system has been analyzed for each experiment. De-pending on the experiment, the experimental results show that the vehicle exhibits string-stableor -unstable behavior as expected. In the case of CACC, string-stable behavior is achieved at aheadway time of 1 s. In the case of ACC, the behavior is string-unstable at a headway time of 1 s.The distance errors are reduced from−6, 4 m in the case of ACC, to−1, 1 m in the case of CACC,while the acceleration profile of the preceding vehicle is similar.

The experimental results have been compared with closed-loop simulation results. In the caseof CACC, the experimental and simulation results are almost identical. Hence, it is concludedthat, in this case, all the relevant dynamics are taken into account. In the case of ACC, the ex-perimental and simulation results show a good comparison. However, the differences betweenexperimental and simulation results are larger for ACC compared to CACC. The differences arethought to be caused by delay on the on-board sensor, i.e. lidar, measurements. Hence, it isrecommended that delay and/or phase lag of the on-board sensors is taken into account in futureanalysis. The same can be said about possibly present filters for the velocity and accelerationmeasurements. During these experiments, a low-pass filter is used to filter the on-board velocityand acceleration measurements. This filter is taken into account in the control design and systemanalysis. As the cut-off frequency of this filter is chosen relatively small, the effect of other pos-

5.6. CONCLUSIONS AND RECOMMENDATIONS 75

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sibly present filters is reduced, yielding predictable results. This, however, may not be the caseduring future experiments if the cut-off frequency is increased or if the low-pass filter is removed.

Chapter 6

Conclusions and recommendations

6.1 Conclusions

A theoretical framework for the frequency-domain analysis and design of a CACC system is pre-sented. The framework is based on a system setup focusing on feasibility of implementation.Vehicle-to-vehicle communication with the directly preceding vehicle only is considered. In addi-tion to on-board sensor information for the feedback ACC controller, the wireless communicationis used to obtain the acceleration of the directly preceding vehicle, in the case of CACC, which isused in a feedforward setting.

Using a frequency domain approach, control objectives for the (C)ACC system have beenformulated regarding stability, performance and string stability. For heterogeneous traffic, theoutput string stability function must be evaluated. Input- and error string stability can only beachieved by increasing the control gains as well as decreasing the vehicle mass upstream a pla-toon. For a homogeneous platoon, the error-, input- and output string stability transfer functionsare identical.

Stability, performance and string stability have been thoroughly analyzed for the case of idealvehicle dynamics, focussing on the effect of the design variables. The design variables are theACC feedback controller, the CACC feedforward controller and the so-called spacing policy dy-namics. The ACC feedback controller is a simple PD controller. The CACC feedforward con-troller is based on a zero-error condition and compensates for the vehicle dynamics and the spac-ing policy dynamics. A constant and a velocity-dependent spacing policy are considered.

Considering a velocity-dependent spacing policy, the control structure includes an additionalfeedback loop. Both feedback loops must be stable. Stability of the outer-loop is shown to beguaranteed for a velocity-dependent spacing policy, provided that the inner-loop is stable. TheCACC feedforward controller is shown to improve performance. This result is experimentallyvalidated.

In the case of ACC, string stability cannot be guaranteed for a constant spacing policy. Fora velocity-dependent spacing policy, a minimum headway time is required to guarantee stringstability, which is shown to be inversely proportional to the proportional gain of the ACC feedbackcontroller. Considering a velocity-dependent spacing policy, a first-order low-pass velocity filter isshown to reduce the minimum headway time to guarantee string stability by half, if the cut-

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78 CHAPTER 6. CONCLUSIONS AND RECOMMENDATIONS

off frequency of the low-pass filter is chosen equal to the breakpoint frequency of the ACC PDcontroller.

In the case of CACC, a minimum headway time is required to guarantee string stability.The minimum headway time depends on the vehicle dynamics, communication delay and ACCfeedback controller. The minimum headway time is shown to increase with increasing commu-nication delay in the case of ideal vehicle dynamics. Only marginal string stability can be achievedfor a constant spacing policy, meaning that string stability cannot be achieved e.g. in the presenceof communication delay.

To validate the theoretical results, experiments are performed with two CACC-equipped ve-hicles. The closed-loop longitudinal vehicle dynamics have been identified using step-responsemeasurements. Comparing the measurement and simulation results showed that the identifiedmodel, a simple process model, accurately represents these vehicle dynamics. The design vari-ables are based on the identified vehicle model and, to a smaller amount, the identified communi-cation delay. Hence, time- and money consuming modeling of vehicle dynamics is not necessaryfor the design of the (C)ACC system.

During an ACC experiment, string stability is not achieved if a headway time of 1 second isadopted, as predicted by theory. During acceleration manoeuvres of the preceding vehicle rangingbetween −1.5 and 1 ms−2, the distance errors range between −6 and 4 m, respectively. During aCACC experiment, the driving behavior is string-stable if a headway time of 1 second is adopted,as predicted by theory. In the case of CACC, the distance errors ranges between −1 and 1 m.Hence, the experimental results validate the theoretical analysis results. Comparing CACC toACC, CACC enables string-stable driving behavior at a smaller headway time while achievingsmaller distance errors. As a result, a smaller headway time can more safely be adopted in thecase of CACC.

6.2 Recommendations

Systematic controller design approach. The ACC feedback controller is a simple PD con-troller and a constant headway time spacing policy has been used because it is the mostcommonly found spacing policy in literature. The minimum headway time to guaranteestring stability has been derived for this particular setup. No attempt has been made to opti-mize the design to, for example, minimize the inter-vehicle distance or achieve the smallestdistance errors. Perhaps, much smaller inter-vehicle distances (or errors) can be achievedfor different control designs or spacing policies.

Other communication structures. As the focus is on the feasibility of the implementation,communication with the directly preceding vehicle only, is considered. Intuitively, im-proved performance can be achieved if communication with multiple vehicles is enabled.How should the CACC system react to communicated information from multiple vehicles?Does the string stability condition still hold? And how can string stability be defined fora bi-directional communication structure? Research is recommended to find answers tothese questions.

On-board sensor delay. The delay of the on-board sensor, i.e. radar or lidar, has been as-sumed to be negligible. In practice, however, this assumption may not be valid. For exam-

6.2. RECOMMENDATIONS 79

ple, the sensor may incorporate a low-pass filter to reduce measurement noise. Such a filtermay introduces a non-negligible delay and influence the results. It is recommended thaton-board sensor delay is taken into account during future analysis.

Longitudinal vehicle dynamics. Currently, disturbances such as aerodynamic drag forceand road incline are not compensated for by the lower-level controller. Hence, unpre-dictable behavior may result in the case of, for example, an increased road slope. Duringthe experiments, the vehicle dynamics are identified under a controlled environment, i.e. aflat road and no severe wind. Moreover, the vehicle dynamics are identified just before theactual string stability experiments. This is not the case in real traffic. Two recommendedsolutions are: 1) The lower-level is improved such that a certain level of performance isguaranteed under any circumstance. Then, the CACC system can rely on the lower-levelcontroller such that it can be designed completely vehicle-independent. A certain levelof performance of the CACC system can be guaranteed only if the lower-level controllermeets certain (pre-determined) performance requirements. 2) The lower-level controlleris not improved. In that case, better performance can be achieved if the CACC systemis designed vehicle-dependent. A possible option would be to identify the vehicle dynam-ics on-line while driving. Either way, the CACC system must know how the vehicle willrespond, up to a certain degree.

Saturations and constraints. During some of the experiments, not shown in this report, ac-tuator saturation occurred. The saturation occurred simply because the preceding vehiclewas accelerating at maximum capacity. In a platoon, the probability of actuator saturation ismore likely if the preceding vehicle is able to accelerate and decelerate harder. Saturation isa hard non-linear effect and, hence, is not covered by the linear analysis in this report. Onthe short term, it is recommended that future research first focuses on avoiding actuatorsaturations for reasons of safety, instead of focussing on the effects of actuator saturations.In my opinion, it is not useful to, for example, study string stability in the case of actuatorsaturation. Considering braking, it is more important not to collide and string stability isirrelevant. Collisions must be avoided at all times and, hence, is a hard constraint. A sug-gestion to avoid collisions and actuator saturation at the same time is to enable platooningof vehicles with similar characteristics only. It is not logical to form a platoon with both rac-ing cars and heavy-duty vehicles. I personally envision a highway where groups of similarCACC-equipped vehicles, i.e. with similar weight, accelerating and decelerating capabili-ties, group together to form platoons. The maximum acceleration and deceleration limitsof the platoon are determined by the weakest link in the chain. Considering safety at smallinter-vehicle distances, no vehicle is allowed to accelerate or decelerate harder than the pla-toon’s limits. If the platoon is too ‘slow’, the vehicle should find another, more suitable,platoon. If the platoon leader collides with preceding traffic, the whole platoon is likely tocollide. Hence, the platoon leader is responsible for adopting a safe distance to other pre-ceding traffic, such that the platoon is always able to come to a standstill without reachingthe braking limits and without colliding. In this case, actuator saturation and the constraint(of not colliding) is avoided. On the long term, focus may shift towards the effects of actua-tor saturations.

Approaching of a platoon. The approach of a platoon has not been considered. Currently,the CACC system is designed to drive in a platoon where the initial distance error is small.During the approach of a platoon, the initial distance error is large. The ACC feedback

80 CHAPTER 6. CONCLUSIONS AND RECOMMENDATIONS

controller is a simple PD controller and, hence, the vehicle will accelerate at maximumcapacity once the preceding platoon is detected by on-board sensors. An option is to designa reference path generator, which guarantees a smooth approach of the platoon. Anotheroption is to extend the explicit ACC MPC S&G system [Ble07] with CACC functionality, asit is able to deal with large initial errors in a straightforward way.

CC/ACC/CACC Switch. Switching between CC, ACC and CACC has not been considered.The switch between CC and (C)ACC is similar to the scenario of the approach of a platoon.A switch from CACC to ACC is required if the wireless communication fails. In that case,the headway time must be increased to guarantee string stability. Research to the effectsof switching between CC, ACC and CACC is recommended. A smooth and (string-)stabletransition between these modes is desired.

Communication modeling. To enable linear analysis, the communication delay has been as-sumed to be constant. Moreover, package loss has not been considered. In practise, thecommunication delay is certainly not constant and package loss may occur. More researchis recommended, for example, to study the effect of a varying communication delay.

Mixed traffic simulations. The effect of ACC and CACC on traffic flow has not been ana-lyzed. Mixed traffic simulation with a varying degree of penetration of ACC and/or CACCvehicles are recommended, as it will gain more insight in the effect of these systems intraffic.

2DOF control theory. The ACC system is, in fact, a two degree-of-freedom (2DOF) con-troller, with controllers Ki and Hi. 2DOF controllers are known to enable a non-overshootingstep response. In the process industry, for example, 2DOF controller may be used to avoidan overshooting temperature. For string stability, overshoot on the acceleration must beavoided. Hence, theory on 2DOF controllers may provide valuable insight.

Human Machine Interface (HMI). The design of an HMI is recommended to provide in-formation of the CACC system to provide the driver. For example, the rear-end of thepreceding vehicle may be shown on a display, where the braking lights become red to indi-cate that the CACC system knows that the preceding vehicle is braking. On the other hand,the driver must not be flooded with information.

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Appendix A

Planning experiments I

88 APPENDIX A. PLANNING EXPERIMENTS I

Testplan Autotuning

Customer: Autotuning

Customer project leader: Rene Vugts

TNO project leader: Jeroen Ploeg

Test period: July 2009

Authors: Thijs van den Broek, Rene Vugts

Date: 1juli 2009

Version: 2.3

scenario title location * date

1 Time-delay & package loss test TNO 10 July

2a Acceleration measurement test TNO 10 July

2b Acceleration measurement test DAF 20-24 July

3 Validation Ax-controller TNO VeHIL 10 July

4 Validation CACC algoritm VeHIL TNO VeHIL 14-15 July

5 Validation CACC algoritm road Test track 20-24 July

Test Summary

VeHIL Test Report customer name 1-7-2009

General information

Planning

What? Where? When? Timespan?

1

2aTNO road

3 TNO VeHIL

4 TNO VeHIL 14/15 July 1 morning

1 day

1 day

These experiments are part of the Autotuning project, which includes development of a CACC system. The main goals of these

experiments are i) to check wether the CACC algorithm works in practice, ii) to collect data in order to validate various

assumptions made during the design of a CACC algorithm and iii) to record movies for promotional purposes.

The experiments are split up into 5 scenarios. The goals of these scenario's are:

1) to measure the communication delay and package loss during vehicle-to-vehicle (V2V) communication,

2) to measure the acceleration of the leading vehicle with on-board sensors and with radar/lidar

3) to validate the Ax-controller by comparing the desired acceleration with the measured acceleration,

4) to validate the CACC algorithm and to compare it with ACC in VeHIL and

5) to validate the CACC algorithm and to compare it with ACC on the road.

Notes:

- Scenario 2 will be executed twice, once on a road near TNO (2a) and once on the test track (2b).

- Scenario 1 and 2a may be executed simultaneously.

Test track 20-24 July2b

5

10 July

Scenario title: Time-delay & Package loss testScenario no.: 1

Location: TNO - road

Date: 10 July 2009

# variants 1

# runs per variant 1

total # tests 1

Scenario description

Scenario hypothesis

test no. / run data file observations

parameter description

ID package identification number

t_GPS_t GPS time at which the package is sent by the target

t_GPS_h GPS time at which the package is received on the host


t_t time of the target

ID package identification numbert_GPS_t GPS time at which the packages are sent by the target

Logging following vehicle (host)

Preparation

Logging leading vehicle (target)

Thijs & Thijs XLPrepare both C4's (wireless communication + time-

synchronized GPS systems)

Software Thijs & Thijs XL

Software (intermediate test results) René

Off-line analysis René

Test Results

Scenario summaryMeasure the communication delay and package loss during vehicle-to-vehicle communication.Scenario objective

GPS time stamping information. Both Citroën C4's will be equipped with time-synchronized GPS systems. The target vehicle will

send packages to the host vehicle. The packages will at least include a GPS time stamp and an ID number. The communication

delay is measured by comparing the time at which the package is sent and received. The percentage package loss is measured by

dividing the total number of sent messages by the total number of received messages.

There is no specific pre-described trajectory, however, it is important to drive at a varying distance.

The expected amount of package loss is 0%.

The expected average communication delay is 20 ms.

The communication delay is expected to increase with increasing distance.

Drivers (2)

What? Who?


Scenario title: Acceleration measurement testScenario no.: 2a 2b

Location: TNO - road Test track

Date: 10 July 2009 between 20-24 July 2009

# variants 1


total # tests 1

Extra information

Scenario description

Leading C4

Required sensors Measurement

wheel-encoders velocity

HQ accelerometer acceleration

Following C4

Required sensors Measurement

lidar/radar relative position and velocity

wheel-encoders velocity

HQ accelerometer acceleration (optional)

Both C4's will be equipped with wireless communication. The host vehicle will be equipped with radar/lidar. The accelation of the target

vehicle can now be obtained in 2 ways:

1) on the host vehicle, by differentiating/filtering the radar/lidar measurements. This gives a relative acceleration and must be added to

the host acceleration to obtain the target acceleration.

2) on the target vehicle, using on-board sensors and communicate it to the host vehicle via a wireless communication link.

The on-board sensors are: i) wheel-encoder to measure velocity and ii) high-quality accelerometers to measure acceleration. Both

sensors will be used to estimate the actual acceleration of the host/target vehicle as accurately as possible.

There is no pre-described trajectory. However, take the following into account:

- wheel-encoder precision results in a poor velocity measurement when driving at low velocities.

- small accelerations/decelerations result in a poor signal-to-noise ratio.

What? Who?

Prepare both C4's (wireless communication + radar/lidar +

HQ accelerometers)

Thijs & Thijs XL



Preparation

Drivers (2)

Software (intermediate test results) René

Scenario summaryScenario objective Measure/estimate the acceleration of the host vehicle with on-board sensors and via radar/lidar on the

target vehicle.

Communication of the acceleration of the target vehicle, measured by on-board sensors, will result in a better measurement than by

estimating the acceleration from radar/lidar measurements.

Scenario hypothesis

The test described below will be executed twice (a & b), on a different location and time. The TNO road is not very long and velocities

cannot be very large, for safety. A test track is much larger and smoother. However, the test on the TNO road is important to see if

everything works properly (for example the wireless connection, GPS, accelerometers, etc.).

C4 (target)

C4 + lidar/radar

(host)

wireless

comm.

link



t_h time of the hostv_h velocity of the hosta_h acceleration of the hostx_r relative positionv_r relative velocityID package identification numbert_GPS_t GPS time at which the packages are sent by the targett_GPS_h GPS time at which the packages are received by the hosta_t_COM acceleration of the target via wireless communication



ID package identification numbert_GPS_t GPS time at which the packages are sent by the targeta_t acceleration of the target

Note: this test is all about data logging. The will be no on-line calculations, this will be done off-line.

Logging leading vehicle (target)

Logging following vehicle (host)

Test Results

Scenario title: Validation Ax-controllerScenario no.: 3

Location: TNO - VeHIL

Date: 10 July 2009

# variants 2


total # tests 2

Extra information

Scenario description variant 1


Acceleration and deceleration with various magnitudes between the velocity range of certain gears. Supply the Ax-controller with

the desired acceleration trajectory (an example of the trajectory is shown below), and measure the actual acceleration of the

vehicle. The acceleration and deceleration levels, as well as the velocity ranges, are dependent on the results of variant 1. There

will be three tests, one in three different gears, 3rd 4th and 5th gear.

This test will give more detailed information about the performance of the Ax-controller, because acceleration trajectories are

defined within gear changes. The goal is to determine the steady state error, time-delay and closed-loop bandwidth.

During the design of the CACC controller, the closed-loop dynamics of the vehicle and Ax-controller is approximated by a first-order

system with a certain bandwidth. More importantly, it is assumed that the bandwidth of the closed-loop is much larger than the

bandwidth of the ACC controller, which is typically of order 0.01-0.1 Hz. Moreover, the steady state error is assumed to be zero.

Scenario summaryScenario objective Validate the Ax-controller by comparing the desired acceleration with the measured acceleration.

Determine the bandwidth, steady state error and delay during gear changes.

Acceleration and deceleration with various magnitudes from 0 to 30 m/s. Supply the Ax-controller with the desired acceleration

trajectory (shown in the figure below), and measure the actual acceleration of the vehicle. The acceleration can be estimated by

differentiating the velocity, measured by wheel-encoder. When accelerating/decelerating, the vehicle will change gears

automatically .

Besides the performance of the Ax-controller, this test will give information about i) the velocity range of each gear and ii) gear

changes itself, e.g. the time it takes to change gears. This information is usefull for variant 2 of this scenario.

0 20 40 60 80-4

-2

0

2

4

t (s)

ah,d

(m

s-2

)

Exp 3, Var 1, Test A

0 20 40 60 80

0

50

100

t (s)

v (km

h-1

)

0 20 40-4

-2

0

2

4

t (s)

ah,d

(m

s-2

)

Exp 3, Var 1, Test B

0 20 40

0

50

100

t (s)

v (km

h-1

)

0 10 20 30 40-4

-2

0

2

4

t (s)

ah,d

(m

s-2

)Exp 3, Var 1, Test C

0 10 20 30 40

0

50

100

t (s)

v (km

h-1

)

Scenario description variant 3 (suggestion, dependent on results of variant 1 and 2)

Scenario hypothesis



t_h time of the host

a_h_d desired acceleration

v_h velocity of the host

a_h acceleration of the host

The Ax-controller has a zero steady-state error and the first-order approximation of the closed-loop system will have a bandwidth of

order 10 Hz.

What?

Logging (host)

Test Results

Software (desired trajectories, intermediate test results) René


Fix C4 on rollerbench Thijs & Thijs XL

Preparation


Who?

Prepare C4 Thijs & Thijs XL

If the performance of the Ax controller is "good enough", the Ax controller may be provided with sinusoidal inputs in order to

identify a Frequency Reponse Model (FRM). The gear switches, torque-speed curves, etc. make the closed-loop system highly

nonlinear. However, if the vehicle operates close to a specific operation point, i.e. a specific gear and velocity, the behavior might

be described well by a linear model.

First, the vehicle has to drive at a certain speed in a certain gear. Then, the Ax controller may be provided with a sinusoid with a

certain frequency and amplitude. By comparing the amplitude and phase of the measured acceleration (output) with the desired

acceleration (input) in steady state, a single measurement point of the FRM is obtained.

In order to draw a Bode Diagram of the FRM, at least 10 frequencies should be tested, which makes this scenario very time

consuming. The result is a linear frequency response model in a single operation point.

0 50 100-4

-2

0

2

4

t (s)

ah,d

(m

s-2

)

Exp 3, Var 2, Test A

0 50 100

0

50

100

t (s)

v (km

h-1

)

0 50 100-4

-2

0

2

4

t (s)

ah,d

(m

s-2

)

Exp 3, Var 2, Test B

0 50 100

0

50

100

t (s)

v (km

h-1

)

0 50 100 150-4

-2

0

2

4

t (s)

ah,d

(m

s-2

)

Exp 3, Var 2, Test C

0 50 100 150

0

50

100

t (s)

v (km

h-1

)

Scenario title: Validation CACC algoritm VeHILScenario no.: 4

Location: TNO - VeHIL

Date: 14-15 July 2009

# variants 2

# runs per variant xtotal # tests 2x

Extra information

min max

acc (m/s2): -5 5

vel (km/h) -50 50

pos (m) 0 100

General scenario description

Trajectories

Scenario I: reference tests

Ia ACC, constant distance (h=0), block trajectory

Ib ACC, constant distance (h=0), sine trajectory

Ic CACC, constant distance (h=0), block trajectory

Id CACC, constant distance (h=0), sine trajectory

Scenario summaryConfirm that CACC works and show the difference between ACC and CACC in an experiment.Scenario objective

The host (Citroën C4 with EHB, lidar/radar and wireless communication) will be on the rollerbench. The radar/lidar will be used to

measure the distance and relative velocity to the target vehicle, in this case the moving base (MB). In case of CACC, wireless

communication will be used to transmit the measured acceleration from the MB to the C4.

Two absolute target vehicle trajectories are designed. For both trajectories, the target vehicle accelerates to a certain desired set

speed. Then, for one trajectory, the target vehicle suddenly slows down to a certain speed, after which it accelerates to the set

speed again. This trajectory mainly focusses on the performance of ACC and CACC with respect to safety. For the other trajectory,

the target vehicle exhibits three sinusoidal accelerations, with three different frequencies close to the bandwidth frequency of the

ACC controller. This trajectory mainly focusses on string stability. For string stability, the amplitude of the sinusoids should be

attenuated.

On the host vehicle, different tuning sets are examined. A combination of the following settings is examined:

- ACC or CACC

- Headway filter on/of

- Headway time (h)

The relative motions of the MB will depend on the absolute trajectory of the target vehicle as well the tuning set on the

host vehicle.

The trajectories must be designed such that the moving base (MB) trajectories are within the following limits:

0 50 100

0

50

100

Time (s)

Vel. (km

h-1

)

Block - tracking & safety

0 50 100

-2

0

2

Time (s)

Acc. (m

s-2

)

0 100 200 300

0

50

100

Time (s)

Vel. (km

h-1

)

Sine - string stability

0 100 200 300

-2

0

2

Time (s)

Acc. (m

s-2

)


Scenario II: filter tests

IIa ACC, h2a = hmin, filter on, sine trajectory (string stable)

IIb ACC, h2b = h2a, filter off, sine trajectory (string unstable)

Scenario III: ACC/CACC string stability tests

IIIa CACC, h3a = hmin, filter on, sine trajectory (string stable)

IIIb ACC, h3b = h3a, filter on, sine trajectory (string unstable)

Scenario IV: ACC/CACC safety tests

IVa CACC, h4a = h3a, filter on, block trajectory (safe)

IVb ACC, h4b = h4a, filter on, block trajectory (unsafe)

Note: tests IIa and IIIb require a few tests to find the minimum headway time in ACC and CACC mode.

Scenario hypothesis

*String stable behavior roughly means: no overshoot and attenuation of sinusoidal trajectories.

PreparationWhat? Who?

Prepare C4

Ayse

Controller implementation Thijs & Thijs XL

Implementaiton in VeHIL

Software (control design, desired trajectories, intermediate

test results)

René

Moving base operator Hanno?

Cameraman


Thijs & Thijs XL


CACC enables driving at a shorter distance, while i) keeping a saver distance to the target vehicle, i.e. faster reaction time, smaller

error and ii) showing string stable behavior*.




t_h time of the hostv_h velocity of the hosta_h acceleration of the hosx_r relative positionv_r relative velocitya_t acceleration of the target, via radar/lidara_t_COM acceleration of the target, via wireless communicationw_bw bandwidth ACC controllerh_d desired headway timex_r_d desired relative positionv_r_d desired relative velocity

e_x distance error (e_x = x_r - x_r_d)e_v velocity error (e_v = xdot_r - xdot_r_d)a_h_d_fb output of the feedback (ACC) controller: desired feedback accelerationa_h_d_ff output of the feedforward (CACC) controller: desired feedforward accelerationa_h_d desired acceleration (a_h_d = a_h_d_fb + a_h_d_ff)


Standard MB log file

Logging leading vehicle (MB | target)

Logging following vehicle (C4 | host)

Test Results


Scenario title: Validation CACC algoritm roadScenario no.: 5

Location: Test track

Date: between 20-24 July 2009

# variants

# runs per variant

total # tests

Extra information



Scenario hypothesis

*String stable behavior roughly means: no overshoot and attenuation of sinusoidal trajectories.

In order to get a reasonably comparison between ACC and CACC, it is probably best/safest to perform a certain manouvre in ACC

mode, for example a start and stop, and then try to perform the same manouvre in CACC mode.


Software (control design, desired trajectories, intermediate

test results)

PreparationWhat?


CACC. See variant 1. The only difference is that feedforward is enabled. The wireless communication will be used to transmit the

measured acceleration from the target to the host.

Cameraman

Scenario summaryConfirm that CACC works and show the difference between ACC and CACC in an experiment.

Record movies for promotional purpouses.

Scenario objective

ACC. See scenario 4. A major difference with this experiment is that the target vehicle is not the moving base, but another C4. This

C4 does not have EHB, unfortunately, such that it will have to be driven manually. To make a comparison between ACC and CACC,

at least some of the trajectories should be easily replicable.

The trajectories are:

i) stops & starts to simulate traffic jam,

ii) sudden braking from and to a certain speed to simulate merging,

iii) slow to fast speed variations around a constant speed to simulate (annoying) human driving behavior.

Who?

Prepare both C4's Thijs & Thijs XL

Drivers (2)

René

CACC enables driving at a shorter distance, while i) keeping a saver distance to the target vehicle, i.e. faster reaction time, smaller

error and ii) showing string stable behavior*.




t_h time of the hostv_h velocity of the hosta_h acceleration of the hosx_r relative positionv_r relative velocitya_t acceleration of the target, via radar/lidarID package identification numbert_GPS_t GPS time at which the packages are sent by the targett_GPS_h GPS time at which the packages are received by the hosta_t_COM acceleration of the target via wireless communicationw_bw bandwidth ACC controllerh_d desired headway timex_r_d desired relative positionv_r_d desired relative velocity

e_x distance error (e_x = x_r - x_r_d)e_v velocity error (e_v = xdot_r - xdot_r_d)a_h_d_fb output of the feedback (ACC) controller: desired feedback accelerationa_h_d_ff output of the feedforward (CACC) controller: desired feedforward accelerationa_h_d desired acceleration (a_h_d = a_h_d_fb + a_h_d_ff)



ID package identification numbert_GPS_t GPS time at which the packages are sent by the targeta_t acceleration of the target

Logging leading vehicle (target | C4)

Logging following vehicle (host | C4+EHB)

Test Results


Appendix B

Planning experiments II

Experimental setup

date: 4 December, 2010, 13:00 – 17:00u (preparation: 11:00u)

location: Ford Proving Grounds, Lommel

people: Jeroen Ploeg, Thijs Versteegh, Thijs van den Broek, René Vugts, Bart

Scheepers, Marieke Posthumus, Gerrit Naus

Objectives / test definition

CACC string stability tests / platooning

1. Compare human-driver car following with ACC / CACC: it is assumed that a

human driver behaves like an ACC system (or the other way around).

o driving at small and large inter-vehicle distances

2. Measure string-stable and string-instable driving behavior in the linear operating

range of the vehicle, i.e., without saturation of the acceleration (@ 1.8 m/s2).

name:

(C)ACC_hd_not/yes*

headway time

hd [s]**

additional low-pass

filter cut-off ωf [rad/s]

feedforward filter

F(s)

ACC_hd_not Large ∞ 0

ACC_hd_not Small ∞ 0

ACC_hd_yes Large ωk**

0

ACC_hd_yes Small ωk 0

CACC_hd_not Small ∞ (H(s)G(s)s2)

-1

CACC_hd_yes Small ωk H(s)-1

*replace hd with the actual used headway time, e.g., 20 for 2.0s, or 05 for 0.5s

**the values for ‘large’, ‘small’ and ωk are determined on the basis of the on-site vehicle

model identification

3. Switch between ACC and CACC, i.e., switch between string-stable headway

times for ACC and CACC driving, by faking a communication error.

o gradual switching

o instantaneous switching

4. While driving in CACC (or ACC), vary the headway time (think of merging or

exiting vehicles changing lanes).

o focus on CACC

5. Perform tests with reasonable values for the standstill distance to make a (more)

convincing movie.

MOVE vehicle state estimator / GPS quality tests

1. Check GPS accuracy (with RTK-GPS as “ground truth”) (measurements while

performing other experiments)

2. Cruise Control operated vehicle 2, large acceleration and deceleration values with

preceding vehicle 1.

3. Cruise Control operated vehicle 1, large acceleration and deceleration values with

following vehicle 2.

4. Slowly (20 km/h) moving vehicle 1, (slowly 50 km/h) approaching vehicle 2,

starting at a (very) large distance (outside radar/lidar range).

Settings / setup

- communicatie-update rate @ 10 Hz (worldwide adopted as a realistic and feasible

value for actual implementation) (Thijs V.)

- communication delay / synchronization of measurement data (René, Thijs V.)

- adjust Simulink model with (C)ACC tuning possibilities (Jeroen, René, Gerrit)

o vehicle model parameters

o ACC feedback controller bandwidth

o spacing policy dynamics with(out) additional low-pass filter f(s)

o feed forward filter Fi(s)=(s2Gi(s)Hi(s))

-1, with(out) Gi(s)

- evaluation scripts (Marieke, René, Gerrit)

Approach

- vehicle model identification

s

i

iG

iie

ss

ksG

*

)1()(

2

, ϕ

τ

−

+=

o normal driving and desired-acceleration step response measurements

during CCC tests in the morning

- appropriate tuning of longitudinal acceleration controller ilK ,

- design feedback controller Ki(s), design feedforward filter and compute minimal

headway time for string stability, based on identified vehicle model

- use a horizontal track (for MOVE)

System layout

In Figure 1, the system layout of the CACC-equipped vehicle (vehicle 2) for the CACC

experiments is shown. The vehicle driving in front of the CACC-equipped vehicle

(vehicle 1) is operated manually. The same system setup is used in vehicle 1, although

without the automatic actuation of the EHB and throttle, and without the lidar and

WLAN input. For the MOVE tests, an additional TomTom GPS is used.

Figure 1: Schematic overview of the instrumentation of the vehicles.

The main communication channels and corresponding signals are indicated, where uth

and ubr the throttle and brake system control signals, dx&& and x&& the desired and actual

acceleration, xr and rx& the relative position and velocity, 1−ix&& the communicated

acceleration of the preceding vehicle, and t the time stamping signal. For clarity, the

index i, indicating vehicle i, is omitted.

Logging list

CACC

- front wheel acceleration (used in feedforward)

- ESP acceleration

- … (Jeroen, René, Gerrit)

MOVE

Both vehicles

Inputs estimator:

- Tomtom GPS (position data)

- Ax

- Ay

- Vx (data van de auto)

- Wheel speed sensors

- Steering wheel angle

- Yaw rate sensor

Reference inputs

- RTK-GPS (long&lat position & velocity)

- ESP sensor Bosch:

o Yawrate

o Ax

o Ay

Only second vehicle

- Radar (only mio)

- Lidar (only mio)

- Received data over wifi

Logging with RTK time & TomTom GPS time

100 Hz data logging

Appendix C

Implementation ACC feedbackcontroller

During the experiments, both the inter-vehicle distance xr,2(t) and relative velocity vr,2(t) =xr,2(t) are provided by the on-board lidar sensor. Correspondingly, the ACC controller, a simplePD controller, is implemented as a multi-input single-output system (MISO) instead of a single-input single-output (SISO) system. The inputs of the MISO PD controller are the distance errorex,2(t) = e2(t) and relative velocity error ev,2(t) = e2(t). For a velocity-dependent spacing policy(3.1), the desired relative velocity equals

vr,d,2(t) = xr,d,2(t) (C.1)

=d

dt(r2 + hd,2xt(t)) (C.2)

= hd,2x2(t) (C.3)

as r2 and hd,2 are constants. The desired relative velocity is based on the measured accelerationa2(t) = x2(t). Typically, there is more noise on an acceleration measurement compared to avelocity measurement. Hence, a velocity-dependent spacing policy with low-pass filter fi(s) (3.43)is desirable. Now, the velocity error equals

ev,2(t) = vr,2(t)− vr,d,2(t) (C.4)

For the sake of clarity, assume kG,i = 1 for a moment. Then, the MISO feedback controller equalsK2 = ωK,2[ωK,2, 1], such that the control input equals U2 = K2[Ex,2, Ev,2]T = ωK,2(ωK,2Ex,2 +Ev,2). For the SISO implementation, Ki(s) = ωK,i(ωK,i +s) (3.12), the control input equals U2 =K2E2 = ωK,2(ωK,2E2 + sE2). As L(ex,2(t)) = sEx,2(s) = Ev,2, i.e. ev,2(t) = ex,2(t), the MISOimplementation yields similar results. It can be shown that the string stability transfer functionSSX,2(s) is identical for both the SISO and MISO implementation. Hence, the string stabilityanalysis in the previous chapter is still valid. For performance, two sensitivity functions can bedefined for the MISO implementation, S

′′x,2 = Ex,2 X−1

1 and S′′v,2 = Ev,2 X−1

1 = sEx,2 X−11 ,

while only a single sensitivity can be defined for the SISO implementation: S′′2 (s) = E2 X−1

1 . Inorder to be consistent with the theory in the previous chapters, the system will be analyzed forthe SISO implementation.

The relative velocity vr,2(t) is estimated by combining data from various on-board sensors,called sensor-fusion, to yield the best available measurement. In Chapter 4, it is assumed that the

0 10 20 30 40 50 60 70 80 90 10015

20

25

30

35

Time (s)

xR

,2(m

)

75 75.5 76

24

24.5

25

25.5

(a) Inter-vehicle distance

0 10 20 30 40 50 60 70 80 90 100-10

-5

0

5

10

Time (s)

vR

,2(k

mh

-1)

14 15 16 17

-4-2

02

(b) Relative velocity

Figure C.1: Inter-vehicle distance and relative velocity using the on-board sensor (black) andwireless communication (grey). The distance and relative velocity from the on-board sensor have

been used during the experiments.

on-board sensors have a negligible delay. In order to validate this assumption, the distance andrelative velocity have been determined off-line using wireless communication, see Figure C.1.The distance and relative vehicle using wireless communication have been determined off-linefor comparison only, they have not been used during testing. The relative velocity is determinedas vr,2(t) = v1(t) − v2(t), where v2(t) is measured locally and v1(t) is the communicated veloc-ity of vehicle 1. The inter-vehicle distance is determined by integrating the relative velocity, i.e.xr,2(t) =

∫(vr,2(t) + ∆v)dt. A small offset of ∆v = 0.02 m s−1 = 0.72 kmh−1 has been intro-

duced to remove a trend in the resulting xr,2(t), for comparison with the on-board measurement.Figure C.1(b) shows that the on-board velocity measurement has more noise and more (phase) de-lay. Moreover, the distance measured by the on-board sensor, see Figure C.1(a), also shows somedelay with respect to the off-line determined distance using wireless communication. Hence, theassumption that the on-board sensor delay is negligible is not valid.

Appendix D

Simulation input

The input of the closed-loop simulation is the absolute position x1(t), velocity x1(t) and the ac-celeration x1(t) of vehicle 1. The on-board measurements on the vehicles 1 and 2 have not beentime-synchronized and, hence, the on-board measurements on vehicle 1 cannot be used as it doesnot allow comparison of simulation and measurement results. The velocity x1(t) is available onvehicle 2 by i) on-board sensors, i.e. x1(t) = x2(t) + xr,2(t) and ii) wireless communicationx1(t− θ2). Based on a comparison of both measurements and integrating them to absolute posi-tion, it is concluded that the communicated velocity is the best available measurement. Therefore,the communicated velocity is used as input for the simulation, along with the communicated ac-celeration x1(t−θ2). The absolute position is obtained by integrating the communicated velocity,i.e. x1(t) =

∫x1(t− θ2)dt. As the communication delay is identified as θ2 = 60 ms, the effect of

the communication delay on the simulation results, with respect to the experimental results, isexpected to be small. Note that for an ACC simulation, the acceleration of the leading vehicle isnot necessarily required, although it is useful to compare it with the acceleration of the followingvehicle for string stability.

String-stable CACC design and experimental validationThe corresponding closed-loop system is...

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