Ecological Cooperative Adaptive Cruise Control for Autonomous Electric

Vehicles

BY

LORENZO BERTONIPolitecnico di Torino, Turin, Italy, 2016

THESIS

Submitted as partial fulfillment of the requirementsfor the degree of Mechanical Engineering

in the Graduate College of theUniversity of Illinois at Chicago, 2017

Chicago, Illinois

Defense Committee:

Sabri Cetinkunt, Chair and Advisor

Arunkumar Subramanian

Marco Masoero, Politecnico di Torino

ACKNOWLEDGMENTS

Firstly, I express my gratitude to Jacopo Guanetti. This work would not have been possible

without his guidance, support and patience. I am also very grateful to Francesco Borrelli for

welcoming me in his lab and for his precious advices.

Furthermore, I would like to thanks my advisors Sabri Cetinkunt and Marco Masoero for

their cooperation and for providing me the possibility to conduct my research at the University

of California, Berkeley.

Another thanks goes to my colleagues in the Model Predictive Laboratory for being very

helpful. A special thanks goes to Valerio Turri, from KTH, for his valuable inputs.

I am very grateful to my parents for all the support the always gave me, that means a lot

for me.

Last but certainly not least, thank you, Maria.

LB

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TABLE OF CONTENTS

CHAPTER PAGE

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Ecological Cooperative Adaptive Cruise Control . . . . . . . . 42.2 Electric Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Concept of Platooning . . . . . . . . . . . . . . . . . . . . . . . . 62.3.1 Spontaneous Platooning . . . . . . . . . . . . . . . . . . . . . . . 72.4 Hierarchical Control System . . . . . . . . . . . . . . . . . . . . 72.5 Vehicle to Vehicle Communication . . . . . . . . . . . . . . . . . 8

3 OPTIMAL CONTROL THEORY . . . . . . . . . . . . . . . . . . . . . 103.1 Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . 123.3 Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . 163.3.1 Advantages and Limits . . . . . . . . . . . . . . . . . . . . . . . 18

4 PROBLEM STATEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . 204.1 Non-linear Vehicle Dynamics . . . . . . . . . . . . . . . . . . . . 204.2 Powertrain Model . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 OFFLINE OPTIMIZATION . . . . . . . . . . . . . . . . . . . . . . . . . 375.1 Pontryagin Minimum Principle . . . . . . . . . . . . . . . . . . . 385.2 Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 405.2.1 Control Law Derivation . . . . . . . . . . . . . . . . . . . . . . . 435.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.3 Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . 525.4 Numerical Optimization: YALMIP Toolbox . . . . . . . . . . . 57

6 ONLINE OPTIMIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . 596.1 Optimal Control Problem . . . . . . . . . . . . . . . . . . . . . . 596.2 Suboptimality of the Solution . . . . . . . . . . . . . . . . . . . 606.3 Penalization of the Terminal Kinetic Energy . . . . . . . . . . 626.4 Penalization of the Terminal Distance . . . . . . . . . . . . . . 646.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7 CLOSED LOOP SIMULATIONS WITH EXPERIMENTAL FORE-CAST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

iii

TABLE OF CONTENTS (continued)

CHAPTER PAGE

7.1 QuasiStatic Simulation Toolbox . . . . . . . . . . . . . . . . . . 697.1.1 QSS Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707.1.2 QSS Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717.2 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.2.1 Forecast Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.2.2 Energy Savings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737.2.3 MPC Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 737.3 Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.3.1 Highway Scenario 1 . . . . . . . . . . . . . . . . . . . . . . . . . 757.3.1.1 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.3.1.2 Energy Savings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787.3.2 Highway Scenario 2 . . . . . . . . . . . . . . . . . . . . . . . . . 807.3.3 Urban Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

8 EMBEDDED IMPLEMENTATION . . . . . . . . . . . . . . . . . . . 848.1 Time Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . 848.2 FORCES Pro Code Generator . . . . . . . . . . . . . . . . . . . 868.3 Hardware in the Loop Simulation . . . . . . . . . . . . . . . . . 90

9 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

CITED LITERATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

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LIST OF TABLES

TABLE PAGEI VEHICLE AND ENVIRONMENTAL PARAMETERS . . . . . . . . . . . 28II MPC PARAMETERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

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LIST OF FIGURES

FIGURE PAGE1 Linear approximation and regression curve of experimental data (1) . . . . 262 Powertrain scheme in an electric vehicle . . . . . . . . . . . . . . . . . . . . 293 Nominal torque for an electric motor of 12 kW . . . . . . . . . . . . . . . . 304 Powertrain model of an electric motor: experimental data (blue dots) and

fitting surface Pm(t) = b1Tm(t)ω(t) + b2T2m(t). . . . . . . . . . . . . . . . . 34

5 Analytical optimal motor torque in its six different phases . . . . . . . . . 516 Analytical optimal speed (first plot) and position (second plot) profiles . . 527 Torque comparison using analytical solution and dynamic programming

approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 Optimal speed (first plot) and position (second plot)+ of the ego vehicle

using analytical solution and dynamic programming approach . . . . . . . 559 Relative speed between the ego and the target vehicle with and without

penalization of the kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . 6310 Relative speed (first plot) and distance (second plot) between the ego and

the target vehicle with and without the terminal costs . . . . . . . . . . . . 6811 QSS outline and elements not included in the model adopted in the Eco-

CACC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7012 Experimental speed profile collected driving in the Highway 101 around

the Silicon Valley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7213 Extract of speed profile of target and ego vehicles (first plot) and their

relative distance (second plot) of “Highway Scenario 1” . . . . . . . . . . . 7614 Speed profile of target and ego vehicles (first plot) and their relative dis-

tance (second plot) of “Highway Scenario 1” . . . . . . . . . . . . . . . . . . 7715 Comparison between the controller optimal torque and the QSS validation

using the same speed profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7816 Energy consumption comparison using the Eco-CACC and a traditional

ACC resulting in energy saving of 15.6 % . . . . . . . . . . . . . . . . . . . 7917 Speed profile of target and ego vehicles (first plot) and their relative dis-

tance (second plot) in “Highway Scenario 2” . . . . . . . . . . . . . . . . . . 8018 Speed profile of target ego vehicles (first plot) and their relative distance

(second plot) in an urban scenario . . . . . . . . . . . . . . . . . . . . . . . . 8219 Computational time in Matlab environment of ”Highway Scenario 2” with

N=10s, Ts=0.1s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8620 Optimal torque, speed and relative distance of the ego vehicle in ”Highway

Scenario 3” are shown in the first second and last plot respectively . . . . 8721 Computational time comparison between IPOPT and FORCES solvers. . 8922 Computational time during a Hardware in the Loop simulation with Ts=0.1

s and N=8 s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

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LIST OF ABBREVIATIONS

ACC Adaptive Cruise Control

CACC Cooperative Adaptive Cruise Control

CN Cellular Network

DP Dynamic Programming

DSRC Dedicated Short Range Communications

Eco-CACC Ecological Cooperative Adaptive Cruise Control

GPS Global Positioning System

HIL Hardware In the Loop

ITS Intelligent Transportation System

MPC Model Predictive Control

PMP Pontryagin Minimum Principle

QSS TB QuasiStatic Toolbox

V2I Vehicle to Infrastructure

V2V Vehicle to Vehicle

VANET Vehicle Ad hoc NETwork

vii

SUMMARY

This thesis develops an ecological cooperative adaptive cruise control, which exploits look-

ahead information coming from a preceding vehicle in order to minimize energy consumption.

The controller enables substantial fuel savings, combining an optimal eco-driving and an adap-

tive cruise control that exploits the reduction of aerodynamic resistance. The proposed control

approach is tailored for electric vehicles as it leverages a realistic powertrain model, validated

with experiments in the literature. Different optimization techniques are analyzed and com-

pared to solve the optimal control problem offline. For real-time implementation, a nonlinear

model predictive control framework is proposed. Look-ahead information is simulated using

experimental data collected driving in the Silicon Valley around the city of San Francisco. Sim-

ulations in real-world driving conditions suggest substantial energy savings when the proposed

Eco-CACC is compared to a standard adaptive cruise controller. Finally, hardware-in the-loop

tests have been conducted to show the effectiveness of the approach for vehicle implementations.

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CHAPTER 1

INTRODUCTION

In the European Union, the transportation sector is still responsible for 23.2% of the green-

house gas emissions. A substantial increase of 9% of share from 1990 well represents the

increasing importance of the this sector in the world economy [2]. The situation in the rest of

the world is even worse: in ten years (from 2000 to 2010) China doubled C02 emissions due

to the transportation sector and a further increase is expected. Furthermore, car accidents

lead to more than 120 000 victims every year in the EU and almost 1.3 million victims in the

world [3]. Thanks to the incessant progression of technology, in particular on Electronic Con-

trol Units (ECU), vehicle-to-vehicle (V2V) and vehicle-to-infrastructure (V2I) communication

and vehicle sensors, the concept of driving is rapidly changing. Cars are becoming more and

more autonomous every year, with a consequent decrease of emissions. In addition, they are

equipped with complex safety systems that permits a strong decrease of the mortality rate in

the transportation sector.

In this thesis we focus on electric vehicles, which are responsible for global warming emissions

through the electricity supply chain of the charging stations. Furthermore, energy savings are

even more important for this category, since the battery capacity appears to be their main

limitation.

There exists at least two effective ways to reduce energy consumption of vehicles. The first

one is the eco-driving concept. There are various ways of driving the same journey from an

1

2

energy point of view. Several studies already investigated how to optimize the speed trajectory

of a vehicle in order to minimize the energy consumption (e.g. [4], [5], [6]). The other one is

the platooning concept, which defines a group of vehicles traveling at a close inter-vehicular

distance. The main advantage of a platoon formation is the reduction of the overall aerodynamic

drag, which implies a reduction of fuel consumption. Starting from the sixties, many studies

have investigated this concept (e.g. [7]). However, the energy consumption aspect of platooning

has been analyzed in the literature especially regarding heavy-duty vehicles ([8] and [9]). The

reason is that freight transportation often requires more vehicles to travel the same journey at

the same time. Alternatively, departure and arrival time of heavy-duty vehicles can be flexible,

making easy to arrange a common route for the convoy.

The thesis proposes a different approach of platooning, suitable for passenger cars, and it

focuses on optimal eco-driving control in order to maximize energy savings. The ego vehicle

during its journey receives trajectory information from other cars around it. If it finds a vehicle

close enough with a common route (target vehicle), it may decide to reach the other car and to

travel behind it at a close inter vehicular distance until the end of the common route. In any

case, no surrounding vehicle will modify its trajectory, it will just continue its journey. For this

fact, we refer to the term“spontaneous” platoon formation.

This thesis is part of a work in which a two-layer hierarchical control architecture is proposed.

The high level decides whether or not joining a target vehicle in a platoon formation. In the

current work this decision has already been taken and the focus is on the low-level powertrain

controller.

3

An ecological cooperative adaptive cruise control (Eco-CACC) is therefore presented. The

controller relays on vehicle-to-vehicle communication (V2V) receiving real-time information

and forecast of the future states of the target vehicle. Adopting a predictive control strategy, it

generates the energy-optimal speed profile independently from the preceding vehicle trajectory.

The effectiveness of the controller lies in combining an eco-driving approach with the aerody-

namic reductions typical of platoon formations. A nonlinear model predictive controller has

been chosen due to its capacity of handle input and state constraints. The latter has been used

for safety purposes, guaranteeing collision avoidance between vehicles.

The thesis is organized as follows. In chapter 2 a background of autonomous vehicle tech-

nologies is provided. In chapter 3 the main concepts of the optimal control theory are recalled.

In chapter 4 vehicle and powertrain model for an electric car are defined. In chapter 5 various

choices for the optimization procedure are analyzed and compared. In chapter 6 the optimal

control formulation is illustrated. Then, in chapter 7 simulations are performed using experi-

mental data. Finally, in chapter 8 real-time solutions are revisited to prove whether it is fast

enough for vehicle implementations. A conclusion provides final remarks.

CHAPTER 2

BACKGROUND

2.1 Ecological Cooperative Adaptive Cruise Control

The aim of this work is to develop a real-time ecological cooperative adaptive cruise control

(Eco-CACC) for electric vehicles, which leads to significant energy savings with respect to a

traditional adaptive cruise control (ACC).

ACC is a technology fully developed and implemented in the new vehicles currently on the

market. During navigation, it automatically adjusts the speed in order to maintain a fixed

distance from the preceding vehicle. It is a more advanced form of control with respect to the

traditional cruise control (CC) which simply maintains the vehicle at a fixed speed.

Cooperative adaptive cruise control (CACC) is the last extension in the automation of

navigation. It receives real-time information from the vehicle ahead to perform the optimal

speed profile. It takes into account also the behavior of the other vehicles and the conditions

of the traffic. For example, the vehicle ahead can provide some information regarding a traffic

jam to the following vehicle which starts to slow down sooner thanks to this information.

The main objective of the Eco-CACC developed in this report is to use the real-time infor-

mation available from the preceding vehicle in order to save as much energy as possible over

the journey. The ego vehicle receives from the target vehicle (in front of it) its actual speed

profile and a forecast of the next seconds. The Eco-CACC uses that information to create an

4

5

optimal speed profile (for the ego vehicle). The length of the horizon of the forecast received

from the target vehicle is extremely variable and it affects the performances of the Eco-CACC.

The longer the horizon of the forecast available (realistically no more than 10-12 seconds), the

better the performances of the Eco-CACC. The trajectory obtained from the controller is the

solution of an optimization problem that minimizes fuel consumption within the horizon. In

order to accomplish that, a powertrain model of the vehicle has been considered. In this work,

the focus is on electric vehicles. Therefore, the electrochemical energy drained from the battery

to the electric motor has to be minimized.

2.2 Electric Vehicles

This thesis focuses on electric vehicles. They still represent a small share on the market,

but they are characterized by a sharp growth. In 2015, plug-in electric vehicles sales increased

of 80% with respect to the previous year, according with the trend of the past years [10].

This kind of vehicles has always been attractive for the absence of greenhouse gas emissions

and for the lack of dependence from oil. Nowadays, the escalation of renewable energy makes

this technology even more attractive. If the electric energy used in an electric vehicle is produced

by a power plant that burns oil or gas, this technology is still responsible for emissions: the

ones produced by the power plant in order to provide to the vehicle its energy. Instead, if

the electric energy is directly produced by a renewable energy source, the electric car can be

considered effectively a zero-emission technology.

Electric vehicles make use of the energy that comes from an external source. This energy is

stored in an electrochemical accumulator. The battery provides currents and constant tension

6

to an inverter that transforms direct current (DC) into alternating current (AC) and it fuels

an electric motor. Generally, the most used electric motors are asynchronous motor (induction

motor) or permanent magnet machines. One peculiar aspect of the electric vehicles is their

regenerative braking. It is possible to slow down the vehicle by converting the kinetic energy

into a form that can be stored into the battery. Therefore, the energy that would be wasted

into heat during a normal braking can be partially recovered.

2.3 Concept of Platooning

Platoon is the term for a group of vehicles coupled that travels at close inter-vehicular

distances in order to save energy. Platoon concept is mainly studied and applied to heavy-duty

vehicles which are electronic coupled. They usually accelerate and brake simultaneously in

order to save energy but also to reduce congestions of the roads.

Vehicles that travels at a close inter-vehicular distance experience a decreasing in the fuel

consumption. In fact, when a vehicle is traveling few meters behind another one, the resistance

force caused by the air on the vehicle behind is reduced up to the 50%. The reduced pressure

of the air creates the phenomenon called slipstream effect, extensively used in every car or bike

competition. The dependence between the distance of the two vehicles and the reduction of the

drag coefficients is not linear and it is function of various factors such as the frontal area of the

vehicles and its speed.

The contribution of the air resistance for the fuel consumption is anything but negligible,

especially at high speeds, and the main idea behind the platooning concept is to take advantage

of this aspect.

7

2.3.1 Spontaneous Platooning

Platooning concept, until now, has been extensively adopted for vehicles which do an entire

journey together, mainly for freight transportation. But, thanks to the spread of communication

technologies, is now interesting to study a more spontaneous way of creating platoons.

The ego vehicle, during his journey, can receives information of the trajectory from other

cars around it. If it finds a vehicle close enough with a common route (target vehicle), it can

decide to reach this other car and to travel behind him at a close inter-vehicular distance until

the end of the common route. Spontaneous approach means that the target vehicle ahead not

need to adapt its trajectory or to receive information from the ego vehicle. It will just continue

its journey, while the ego vehicle will take advantage of the air drag reduction.

2.4 Hierarchical Control System

The ecological driving proposed is characterized by two different phases. During the first one,

the ego vehicle receives information from all the vehicles around it and decides if it is convenient

for it to reach another vehicle and follow this vehicle at a close inter-vehicular distance. The

second phase starts when the other vehicle is reached. The Eco-CACC should guarantee in real

time that the best trajectory is performed. In fact, the behavior of the target vehicle can be

non-optimal in terms of fuel consumption and it is not efficient just following it at a close inter-

vehicular distance. Instead, the Eco-CACC receives information regarding the speed profile of

the target vehicle and calculate the optimal speed of the ego vehicle accordingly. These two

phases result in a control system articulated in two different layers, that is called hierarchical

control system. The higher layer takes into account the spontaneous platoon formation, i.e.

8

the decision to join another vehicle and to follow it at a close distance. The lower level (the

Eco-CACC) controls the longitudinal dynamics of the vehicle (powertrain control) in order to

perform the best speed profile in terms of energy savings.

This thesis consists in the development of the lower layer, the powertrain control of the

vehicle. The decision of joining another vehicle in order to form a spontaneous platoon is

assumed to be already taken [11].

2.5 Vehicle to Vehicle Communication

Connectivity is essential for autonomous driving and its development is the key for the

process of growth of the autonomous vehicles in the market. The integration of communication

technologies to the transportation systems leads to so-called intelligent transportation systems

(ITS).

Nowadays, extensive researches have been conducted on vehicular communication systems,

both vehicle to vehicle (V2V) and vehicle to infrastructure (V2I). These kind of communications

are possible using dedicated short range communication (DSRC) protocols. In particular, the

European Telecommunications Standard Institute (2008) as well as the US Federal Commu-

nications Commission (1999) decided to dedicate to the automotive use parts of the 5.9 GHz

band (respectively 30 MHz and 75 MHz). [12]

Those Vehicle ad hoc network (VANET) are the most used technology and a topic of ongoing

research. There are already standards that defines the data format and the message dictionary.

They are also characterized by a low latency (less than 200 µs within 1000m of distance), to-

gether with a fast network acquisition. Reliability of information is very important, peaks of

9

delay during communications can easily lead to safety problems between vehicles, especially

if they are traveling at a close distance. This dedicated network is able to work in high ve-

hicle speed mobility conditions and during extreme weather conditions, and improvement are

constantly made.

The main limitation is the distance between vehicle, since the latency of the data is directly

proportional to the distance. For this reason, a complimentary cellular network (CN) is often

adopted. It is used when vehicles has to communicate at a longer distance but this technology

is characterized by higher latency.

In particular, CN is characterized by peaks of latency of 4-5 seconds, not acceptable for fast

communication but it is meant as a complement to DSRC network for longer distances. [13]

Together with the DSRC radio, a GPS has to be part of the in-vehicle equipment, as well as

a radar that measures the relative speed and the distance with respect to the preceding vehicle.

GPS provides absolute position and time to the DSRC. Fusing the data from the radar and

from the GPS, together with acceleration measurements, it is possible to know the state of the

vehicle with a precision of centimeters [14]. On the other hand, it is not possible to rely just on

the radar. Its limited precisions and the fact that he can measure just relative distances imply

errors and delays in the measurement.

CHAPTER 3

OPTIMAL CONTROL THEORY

This chapter wants to provide a brief introduction to the optimal control theory and its

approaches. All the work developed has its basis on this theory. First, a generic definition

of optimal control problem is given. Then, dynamic programming (DP) and model predictive

control (MPC) approaches to solve an optimal problem are presented. Both approaches are

extensively used in this report, in particular the MPC one.

3.1 Optimization Problem

An optimization problem is formulated as [15]

minz f(z)

subj.to z ∈ S ⊆ Z

where:

z is a vector that contains all the decision variables.

Z is the domain of the optimization problem.

S defines the constraint set: all the feasible decisions admissible in the problem.

f : Z → R is the function that attributes a cost f(z) to each decision z.

10

11

The solution of the problem is the least possible cost f∗:

f∗ = infz∈S f(z)

In other words the number f∗ is the value such that:

f∗ ≤ f(z) ∀z ∈ S

If f∗ = −∞ the problem is unbounded.

If S contains to values the problem is considered infeasible.

If S=Z the problem is unconstrained.

Normally, one is also interested in finding the optimal solution, i.e. the decision whose cost

is the minimum cost. In other words, finding the z∗ ∈ S/f(z∗) = f∗.

z∗ is called optimal solution.

Therefore, the set of the solutions can be expressed by:

argminz∈Sf(z) = {z ∈ S : f(z) = f∗}

If Z is finite the optimization problem is called discrete. Optimization problems in the field of

autonomous vehicles are usually in the form of discrete problems.

12

3.2 Dynamic Programming

Dynamic programming (DP) is a technique that permits to unravel a generic optimal prob-

lem. The idea is to break the original problem down in a series of smaller optimal control

problems. This theory has been formulated by Bellman (1957) [16].

The Dynamic programming technique is based on a simple idea: the Bellman’s principle

of optimality. The principle simply specifies that for x0, x∗1, x∗2, ..., x

∗N to be optimal, also the

trajectory that starts from any intermediate point 0 ≤ j ≤ N − 1, (x∗j , x∗j+1, x

∗N ) must be

optimal.

The following example should clarify the concept: “suppose that the fastest route from Los

Angeles to Boston passes through Chicago. Then the principle of optimality formalizes the

obvious fact that the Chicago to Boston portion of the route is also the fastest route for a

trip that starts from Chicago and ends in Boston” [17]. This principle of optimality is used in

this thesis to solve optimal control problems: in order to explain how, we consider a generic

constrained finite time (discrete) optimal control problem [15].

13

J0→N (x0, U0→N ) = p(xN ) +N−1∑k=0

q(xk, uk)

subj.to xk+1 = g(xk, uk), k = 0, ...N − 1

h(xk, uk) ≤ 0, k = 0, ...N − 1

uk ∈ U

x0 = x(0)

in which N denotes the horizon length over time and xk represents the vector of the states

obtained at time k beginning at x0 = x(0) and applying to xk+1 = g(xk, uk) the input sequence

u0, u1, ..., uk−1.

q(xk, uk) is called stage cost while p(xN ) is called terminal cost. To give a practical clar-

ification, in a powertrain control problem the series of inputs u0, ..., uN usually represent the

torque and the brake, i.e. the variables that are used to control the system from the instant 0

to the instant N. The state vector of the system at time k includes the position of the car at

time k and its speed at that time.

14

The solution of the control problem is:

J∗0→N (x0) = minU0→N J0→N (x0, U0→N )

subj.to xk+1 = g(xk, uk), k = 0, ...N − 1

h(xk, uk) ≤ 0, k = 0, ...N − 1

uk ∈ U

x0 = x(0)

where J∗0→N is the optimal cost. We can now use the principle of optimality in order to solve

this control problem: a cost over a shorter horizon (from time j to time N) is here defined

Jj→N (xj , uj , ..., uN−1) = p(xN ) +

N−1∑k=0

q(xk, uk)

that is called cost to-go.

15

Consequently:

J∗j→N (xj) = minuj ,...uN−1 Jj→N (xj , uj , ..., uN − 1)

subj.to xk+1 = g(xk, uk), k = j, ...N − 1

h(xk, uk) ≤ 0, k = j, ...N − 1

uk ∈ U

x0 = x(0)

The principle of optimality specifies that this cost to-go J∗j→N (xj) can be found minimizing

the stage cost q(xk, uk) and the cost to-go J∗j+1→N (xj+1). Therefore, the only decision variable

that has to be optimized is the input at time j, uj . In fact, optimal inputs u∗j+1, ..., u∗N−1 have

been previously chosen to create J∗j+1→N (xj). This also explains why the optimal cost to-go

J∗j+1→N (xj) depends just on the state xj .

As it is possible to see, this algorithm works recursively backwards, the first step is to

determine the terminal cost:

J∗N→N (xN ) = p(xN )

16

and then proceed backwards

J∗N−1→N (xN−1) = minuN−1 q(xN−1, uN−1) + JN→N (g(xN−1, uN−1)

subj.to g(xN−1, uN−1) ∈ XN→N

h(xk, uk) ≤ 0

until the last step

J∗0→N (x0) = minu0 q(x0, u0) + J1→N (g(x0, u0)

subj.to g(x0, u0) ∈ X1→N

h(x0, u0) ≤ 0

x0 = x(0)

The main advantage of the dynamic programming technique is that the optimization takes

place one element uj at a time. However, the construction of the optimal cost to go J∗j→N (xj)

and the optimization can be rather complex [15].

3.3 Model Predictive Control

Model predictive control (MPC), also called Receding Horizon Control, is a control tech-

nique. It was invented in the seventies as a way to control chemical plants and oil refineries

[18]. Since solving real-time optimization problems requires an enormous computational power,

at the beginning it could only be applied to systems with very slow dynamics, such as chemical

17

plants. However, thanks to the computational power reached, nowadays it can be effectively

applied to various field including the automotive one.

This technique is based on the following concept: at the time step k an optimization problem

is solved until the time k+HMPC . Consequently, a series of optimal input from time k to time

k+HMPC are obtained. Then, just the first optimal input is applied. At the following time

step a different optimization problem from k+1 to k+1+HMPC is solved and, again, just the

first optimal input will be applied.

The following example should give a clear idea of the concept: if we want a car to do a lap

in a racetrack in the minimum possible time, the trivial solution would be to optimize the entire

trajectory so that the car can drive along the racetrack in the best possible way. Realistically,

though, this is not a safe solution. The car could meet some obstacle along the path and a new

optimization would be necessary at that point in order to take into account also the obstacle.

The main concept of the MPC is to optimize just a small portion of the path (until the temporal

horizon HMPC) and to apply just the first control input to the car. At the new time step, a

new optimization which takes into account the new conditions (an obstacle for example) can

be done and the entire procedure will be repeated until the lap is finished.

Note that just a single input will be applied to the system, but this first input takes into

account what will happen in the future thanks to the optimization until the horizon HMPC .

The technique is called receding horizon since, at every time step, the horizon seems to move:

from HMPC to HMPC + 1 etc.

The finite time optimal control problem is written as:

18

J∗k (x(k)) = minUk→k+HMPC |k

k+HMPC−1∑j=k

q(xj|k, uj|k + p(xk+HMPC |k)

subj.to xj+1|k = f(j, xj|k, uj|k)

xj|k ∈ X

uj|k ∈ U

xk = x(k)

and it is solved at time k. The terminology xj+k|k indicates the vector of the states at the

time j+k that is predicted at time k beginning from the initial value xk = x(k) applying to the

system xj+1|k = f(j, xj|k, uj|k) the input sequence Uk→k+HMPC |k = uk|k, ..., uk+HMPC−1|k.

As said before, thanks to the optimization at time k, an entire sequence of optimal input

Uk→k+HMPC |k is obtained, but just the first element of these inputs uk|k is applied. the entire

procedure of optimization is then repeated at time k+1, but this time until HMPC + 1.

3.3.1 Advantages and Limits

The spread of this technique has been slowed down with respect to its potentiality until the

last years. The main reason is that the various optimization procedures in real-time obviously

require high computational demand. At the same time this technique is very powerful and

it permits to address some issues of the classical feedback control. The possibility to include

control constraints in the input and process constraints in the future state are the most signifi-

cant advantages. A classical feedback control just tracks a reference trajectory and its action is

19

based on the error of the actual trajectory with respect to the reference one. It is not possible

to include input constraints, just a saturation block on the inputs. The saturation ,though, can

easily lead to instability.

Furthermore, the fact that at each time step a new optimization is provided permits to have

a feedback and this characteristic makes the controller robust to disturbances and uncertainties

of the model.

It is important to underline that using MPC it is not always possible to assure the stability

of the problem or its recursive feasibility, while some approaches can be adopted in order to

prevent these issues.

CHAPTER 4

PROBLEM STATEMENT

In this chapter, we describe the longitudinal dynamics and the powertrain model of an

electric vehicle. Both formulations will be used in all the control approaches presented in this

work.

4.1 Non-linear Vehicle Dynamics

A vehicle is a very complex non-linear system with a very high number of interacting

dynamics. However, the aim of this thesis is to develop an ecological cooperative adaptive

cruise control. Therefore, we are interested in the longitudinal dynamics of the vehicle, in

particular on the forces that act on a vehicle in motion. The energy provided to the vehicle has

to compensate three main energy losses due to:

- Air resistance

- Rolling resistance

- Brakes dissipation

The equations that describe the longitudinal dynamics of a car can be obtained using the

Newton’s second law and they have the following form [19]:

20

21

md

dtv(t) = Ft(t)− Fb(t)− Fa(v(t))− Fr − Fg(α(s))− Fd(t) (4.1)

ds

dt= v(t) (4.2)

where Ft is the traction force, Fb is the brakes resistance force, Fa is the aerodynamic force

due to the air density, Fr is the rolling friction and Fg is the longitudinal component of the

gravity force that acts when the road is not flat. Finally, Fd is a generic term that includes all

the non specified effects. This last term will be generally neglected.

Traction force

The traction force is the force that generates motion at the wheels and it can be modeled

as follow:

Ft(t) =Tm(t) gr η

sign Tm(t)t

rw(4.3)

where Tm(t) is the motor torque that will be treated as an input for our system, rw is

the radius of the wheels and gr is the gear ratio. Since we are considering an electric vehicle,

assuming a constant gear ratio is realistic; hence, gr is a constant value in this work.

22

The operator ηt is the transmission efficiency. The operator sign in Equation 4.3 catches the

fact that the electric motor also works as a generator (see below), in which case the direction

of the flow of energy is inverted. The same happens for the transmission losses.

Due to this aspect, the system the assumes two different behaviors depending on the value

of the motor torque Tm. Systems with this peculiarity are called hybrid systems and the

optimization procedure in these cases is much more complex. In order to deal with this issue,

we take into account the transmission efficiency in the powertrain model, while we will neglect

this term in the vehicle dynamics.

The electric motor provides the torque for the traction, but it acts as a generator as well.

It produces a regenerative braking, an energy recovery mechanism that improves the overall

efficiency of the vehicle. Indeed, the energy of the brakes can be chemically stored into the

battery. Therefore, the motor torque has two limits: the maximum power that can be provided

to the wheels and the maximum regenerative braking force, both function of the power of the

electric motor.

Tmin ≤ Tm ≤ Tmax (4.4)

It is reasonable to consider the maximum regenerative braking force equal in absolute value

to the maximum traction force, obtaining:

23

−Tmax ≤ Tm ≤ Tmax (4.5)

It is important to underline that the regenerative braking force cannot substitute the

friction-based braking since it drops off at lower speed. Bringing a vehicle to full stop gen-

erally requires the aid of friction brakes.

Braking force

The braking system for a car is made by different actuators, but regarding our purposes we

are interested in the maximum force that the braking actuators can generate:

Fbmax ≤ Fb ≤ 0 (4.6)

Aerodynamic losses

The air resistance force that acts on a vehicle is caused by two different components. The

first one is the viscous friction due to the air that surrounds the vehicle in motion. The second

one is the difference of pressure between the front and the rear part of the vehicle created by

the air flow separation. This aerodynamic drag force is usually approximated as:

24

Fa(v(t)) =1

2ρ cd Af v(t)2 (4.7)

where ρ is the density of the air, cd is the air drag coefficient, a parameter estimated using

Computational Fluid Dynamics (CFD) techniques, Af is the frontal area of the vehicle and v

is the speed of the vehicle in motion.

As it is possible to note, this force is a non-linear term due to the term v2 and this is the

reason why the longitudinal model of the vehicle is not linear.

As mentioned before, when a vehicle is traveling behind another one, the air resistance is

reduced due to the slipstream effect.

This fact can be formally considered including a reduced air drag coefficient. In particular,

the reduction will be a function of the relative distance between the two cars, i.e. cd = cd(d).

The effect of the distance in the reduction is not easily defined, as it can vary with several factors

including weather and vehicles shape. From experimental data [1] it is possible to model the

air drag reduction as a function of the relative distance between the two cars.

In this work, we will adopt two different models of drag reduction: a linear approximation

for the powertrain controller and a nonlinear function in order to better estimate fuel savings

of the Eco-CACC with respect to a traditional ACC.

In particular, the nonlinear function used has the following form [9], [8]:

25

cd(d) = cd,0

(1−

cd,1cd,2 + d

)(4.8)

where d is the distance between the two vehicles, cd,0 is the nominal (d → ∞) air drag

coefficient and cd,1 and cd,2 are parameters obtained from a regression curve created using

experimental data. A graphical representation is shown in Figure 1.

Rolling friction

The rolling friction is the resistance force that is generated when a generic body rolls onto

a surface; in this case it is generated by the wheels onto the road.

This force can be expressed as:

Fr = cr m g (4.9)

where cr is the rolling friction coefficient, a parameter function of different variables such as

speed of the vehicle and pressure of the air. However, it will be always considered as a constant

coefficient, since the influence of those variables is not critical.

Gravitational force

It is the force induced by gravity when the vehicle is not driving on a flat road.

It can be expressed as:

26

Figure 1: Linear approximation and regression curve of experimental data [1]

Fg(α(s)) = m g sin(α) (4.10)

However, in this thesis we will consider just flat roads. The evaluation of downhill and

uphill roads is very different and for this reason it will be included in future works.

27

Vehicle model final setup

In conclusion, the longitudinal dynamics of the vehicle, including all the hypothesis pre-

sented, can be written in the following form:

dv

dt=Tm(t) grrw m

− cr g −1

2mρ cd Af v(t)2 − Fb

m(4.11)

ds

dt= v(t) (4.12)

The state of our car is expressed by its position and its speed at each time. These variables

can be collected in the state vector

x = [v s]T (4.13)

while the control inputs considered are the motor torque (that provides traction to the

wheels and regenerative brakes) and the braking friction-based force. Again, these variables

can be collected in the input vector

u = [Tm Fb]T (4.14)

Numerical parameters for all the simulations presented in this work are presented in Table I.

28

TABLE I: VEHICLE AND ENVIRONMENTAL PARAMETERS

Name Symbol Value

Total mass of the vehicle [kg] m 1200

Vehicle cross-section [m2] Af 2

Wheel radius [m] rw 0.3

Drag coefficient [-] cd 0.22

Rolling friction coefficient [-] cr 0.008

Air density [kg/m3] ρ 1.22

4.2 Powertrain Model

The aim of this thesis is to develop a powertrain controller which maximizes the energy

savings over the journey. In order to do so, it is necessary to model the powertrain of the electric

vehicle considered. The main idea is to capture the relation that exists between the energy

provided by the electric motor and the traction force at the wheels. The energy provided by

the electric motor is directly proportional to the electrochemical energy drained from or supplied

to the battery. The battery is recharged from the grid, hence battery energy consumption is

related to grid energy costs and possibly to pollutant emissions. Therefore, the aim is to

minimize it.

29

Figure 2: Powertrain scheme in an electric vehicle

A powertrain model of an electric vehicle should catch the efficiencies of all the main com-

ponents. Then, the transmission efficiency has to be considered. Finally, a relation between

the motor power and the wheels traction has to be found.

The power provided by the electric motor is given by:

Pm =Tm ωm

ηm(Tm ωm)(4.15)

with Tm ≥ 0

Instead, when the motor works as a generator we have:

Pm = Tm ωm ηm(Tm ωm) (4.16)

where Tm is the motor torque, ωm is the rotational speed of the motor and ηm is the

efficiency.

30

Figure 3: Nominal torque for an electric motor of 12 kW

The motor efficiency is a function of both the motor torque and its speed has to be charac-

terized for each motor.

We consider the characteristic curve of an electric motor of 12 kW obtained from exper-

imental data [20]. As shown in Figure 3, the curve that represents the behavior of a typical

electric motor can be divided in two parts: the constant torque phase and the constant tension

phase. An electric motor can distribute its maximum torque until it reaches its nominal speed.

31

Above the nominal speed, the maximum torque starts to decrease as the hyperbole 1ωm

, such

that power is kept constant. The maximum speed is linked to the resistance of the mechanical

parts. The behavior as a generator in the second quadrant can be considered specular.

From the nominal curve, knowing the power of the electric motor from experimental data,

it is possible to build the efficiency map.

However, a motor of 12 kW is not suitable for an electric vehicle. Indeed, this types of

motor are mostly adopted in hybrid vehicles.

The solution adopted is to consider a more powerful motor with scaled values. The efficiency

map, function of torque and speed, can be opportunely scaled (without significantly changing

the efficiency values) simply modifying accordingly the torque scale on the map [21].

In this work, an electric motor of 60 kW has been considered.

Then, the map needs to be further modified in order to include the transmission efficiency,

neglected in the vehicle dynamics.

The efficiency is expressed as ηsign Tm(t)t and the operator sign models the double behavior

of the electric motor which acts both as a motor and as a generator during the regenerative

braking. As a consequence, the system becomes a hybrid system. The optimization of these

kind of systems is much more complex and, for this reason, another solution has been adopted.

ηt is generally considered as constant and in this thesis the value

η = 0.9

32

has been assumed.

The transmission efficiency has been removed from the vehicle dynamics and it has been

included in the efficiency map simply multiplying the first quadrant of the motor efficiency map

for 1η and the second quadrant for η.

Overall, the map now represents a decreased “global” efficiency which takes into account

both motor and transmission efficiencies.

The efficiency map is ready. However, for the powertrain model to be useful for control

design, it is necessary to build a closed form expression for the power of the motor or alterna-

tively, for the efficiency. Note that the power and the efficiency of the motor are connected by

the simple relation:

Pm(t) = Pm(Tm(t) ωm(t)) = ηm(Tm(t) ωm(t)) · Tm(t) · ωm(t) (4.17)

The electric power generated by or supplied to the electric motor can be approximated in

the form [4], [5]:

Pm(t) = b1Tm(t)ω(t) + b2T2m(t) + b′1Tm(t) + b0ω(t) + b′0 (4.18)

where b1, b2, b′1, b0 and b′0 are tunable parameters.

However, a simpler expression is generally adopted:

33

Pm(t) = b1Tm(t)ω(t) + b2T2m(t) (4.19)

where the term b1Tm(t)v(t) represents the mechanical power provided by the motor while

the term b2T2m(t) models the thermal losses of the motor.

Equation 4.18 is tunable and it should fit the electric motor considered. In order to do so,

the correct values of the parameters b1 and b2 need to be chosen.

A fitting procedure of Equation 4.19 (which graphically represents a surface) into the power

map of the motor has been adopted, obtaining as a result the values of the tunable parameters

(see Figure 4):

b1 = 1.005 (4.20)

b2 = 0.18 (4.21)

The result of the fitting is well represented in a surface graph that shows how the experi-

mental data have been fitted into the powertrain model.

34

Figure 4: Powertrain model of an electric motor: experimental data (blue dots) and fitting

surface Pm(t) = b1Tm(t)ω(t) + b2T2m(t).

Analyzing the confidence interval of this fitting, we obtain a 95% of the confidence bounds

in the following intervals:

35

b1 = (0.981, 1.03)

b2 = (0.165, 0.21)

The width of the intervals indicates the amount of uncertainty about the fitted coefficients.

In this case, the width is acceptable, and this a proof of the validity of the expression used to

model the powertrain.

Trying to fit the data into more accurate expressions, such as Equation 4.18, the following

values have been obtained:

b1 = 0.972 (0.945, 09978)

b2 = 0.18 (0.132, 0.164)

b′1 = 10.41 (4.75, 16.06)

b0 = 13.8 (10.85, 16.75)

b′0 = −41.79 (−1124, 1040)

The results show a good confidence interval on coefficients b1 and b2, comparable to the one

of the previous fitting. However, the confidence bounds on the other coefficients is much wider.

Coefficients b′1, b0 and b′0 have a minor impact in the fitting and they are characterized by more

36

uncertainties. For this reason, the approximation of considering Equation 4.18 as a model of

the powertrain is acceptable and a more precise expression (and therefore more complex to

optimize) is not necessary.

CHAPTER 5

OFFLINE OPTIMIZATION

In order to develop the Eco-CACC, the choice of the optimization approach to adopt is

fundamental. The main goal of this chapter is to build an energy optimal drive control problem

and to study different offline approaches in order to solve it.

With the term “offline” we assume that all position and road constraints are known in

advance and there are no needs of prediction or estimations of any sort. Furthermore, compu-

tational time is not an issue to take into account.

The problem can be written in the following form:

minu∈U(x(t) J =

∫ tf

0g(u, x(τ))dτ (5.1)

subj.todx

dt= f(u(t), x(t)) (5.2)

v(0) = vi, s(0) = 0 (5.3)

v(tf ) = vf s(tf ) = sf (5.4)

where g(u, x(τ)) is the cost function of the problem i.e. the motor power of the vehicle given

by Equation 4.19 and f(u(t), x(t)) represents the non-linear vehicle dynamics.

37

38

We consider a car in a flat road that has to travel a certain distance sf in a certain time tf

with an initial and a final speed fixed. In particular, for the problem chosen we consider the

following data:

vi = 0

vf = 0

s0 = 0

sf = 200m

tf = 24s

Tmax = 700Nm

In other words, the optimization has the aim to find the best speed profile (in terms of

energy-savings) in order to compute 200 meters in 24 seconds starting and arriving at zero

speed. Vehicle parameters are described in Table I.

5.1 Pontryagin Minimum Principle

The first approach adopted consists in finding the optimal solution using analytical expres-

sions instead of numerical ones. In order to do so, we refers to Pontryagin’s Minimum Principle

(PMP) approach. [22], [23]

PMP has been invented in optimal control theory in order to find the best possible solution

for taking a dynamic system from one state to another.

39

The first step of this approach consists in defining the Hamiltonian function of the prob-

lem[4]:

H(u, x, λ) = g(u, x) + λT f(u, x) (5.5)

where g(u, x) is the cost function, f(u, x) the state dynamic function, λ the adjoint vec-

tor, u∗(t) the optimal control strategy, x∗(t) and λ∗(t) the optimal state and adjoint state

trajectories. Then according to the PMP principle:

x(t) =∂H

∂λ(x∗(t), u∗(t), λ∗(t)) (5.6)

x∗(0) = x0 x∗(tf ) = xf (5.7)

λ(t) = −∂H∂x

(x∗(t), u∗(t), λ∗(t)) (5.8)

u∗(t) = argminu∈UH(x∗(t), u, λ∗(t)) (5.9)

The formulation of the PMP principle is given by Equation 5.9. It states that, thanks to the

Hamiltonian function H, is possible to solve a different problem obtaining the optimal control

input policy u∗ of the original one.

40

A consequence of the principle is that, when the Hamiltonian function is linear with respect

to the control variables u(t), the optimal solution is at the boundaries of the control region U.

This condition takes the name of ”bang-bang” control and the function σ is introduced

σ =∂H

∂u(5.10)

This function is called switching function since when the sign changes, the variable switches

from one point at the side of the region to the opposite one.

The main difficult is to solve the differential equation system given by Equation 5.8 since

the starting conditions of the adjoint vector are unknown, while the initial conditions for the

differential equations system of the state vector (Equation 5.6) are given by Equation 5.7.

5.2 Analytical Solution

Analytical solution for the problem described in the previous section is provided. The vehicle

parameters used are described in Table I.

The state function can be written as [4]:

f(u(t), x(t)) =

(v

s

)=

(h1Tmη

sign(Tm(t))t − h0 − w

v

)(5.11)

41

where

h1 =grrwm

, (5.12)

h0 = crg, (5.13)

w =Fbm

(5.14)

and the cost function, as usual, is given by Equation 4.19.

It is possible to note that in this case the transmission efficiency is included in the state

function; for this reason the coefficients b1 and b2 does not keep into account ηt this time. Note

also that the air resistance term has been neglected in Equation 5.11. This assumption can be

justified just in those scenarios where the speed is relatively low, such as urban scenarios. In

the problem chosen, the car starts from standstill and it arrives at the same condition, so the

assumption is acceptable.

The Hamiltonian function is:

H = b1Tmv + b2T2m + λv(h1Tmη

sign(Tm(t)t − w) + λsv (5.15)

where λv and λs represent the adjoint states.

Knowing that

{T ∗m(t), w∗(t)} = argmin H(Tm, w, λ)

42

It is possible to obtain the expression of the optimal motor torque using the following:

T ∗m(t) =

{Tm(t) :

∂(H)

∂(Tm)= 0

}(5.16)

Note that the operator sign in the transmission efficiency leads to two expressions of the torque:

T+m(t) = − 1

2b2(b1v(t) + h1ηtλv(t)− h0) (5.17)

T−m(t) = − 1

2b2(b1v(t) +

h1λv(t)− h0ηt

) (5.18)

The optimal braking force is given by:

w∗(t) =

0 if λv(t) < 0

W if λv(t) > 0

(5.19)

with

W =Fb,maxm

(5.20)

While the dynamics of the problem related to the adjoint states, λv and λs, can be obtained

applying Equation 5.8:

43

λv(t) = −b1T ∗m(t)− λs(t) λs(t) = 0 (5.21)

Thus, we obtain a set of 4 differential equations, given by Equation 5.6 and Equation 5.21

and 4 unknowns (x, v, λv, λs) to solve in order to obtain the optimal control law given by

(T ∗m(t), w∗(t)).

5.2.1 Control Law Derivation

The following rules can be applied to the control law:

• if T+m(t) > Tmax then T ∗m(t) = Tmax (5.22)

• if T−m(t) < Tmin then T ∗m(t) = Tmin (5.23)

• if T+m(t) < 0, T−m(t) > 0 then T ∗m(t) = 0 (5.24)

The first two assumptions come from the fact that the maximum and minimum torque are

fixed and constant. The third one appears due to the term ηsign(Tm)t . In this case, neither T ∗m

nor T−m can be the optimal solution. Therefore, the optimal solution is given by T ∗m = 0, i.e.

coasting condition.

The evolution of the control law can be generally divided into 6 different phases [5], [4].

44

First phase

The initial condition of the car is v = 0, therefore the vehicle needs a positive torque in order

to take off. For long distances and short time the conditon T+m(t)|t=t0 will certainly appear. So,

we can define the first control law:

T ∗m = Tmax

w∗ = 0

(5.25)

for t ∈ [0, t1] where the conditions t = t1 is given by T+m(t1) = Tmax.

Knowing the control law it is possible to obtain the expression of the optimal velocity and

optimal distance in this first phase. We simply need to integrate Equation 5.6:

v =

∫ tfin

tin

v dt+ v0

and

s =

∫ tfin

tin

v dt+ s0

45

obtaining:

v = (h1Tmax − h0)t with t ∈ [0, t1] (5.26)

s = (h1Tmax − h0)t212

with t ∈ [0, t1] (5.27)

The time t1 is given by the condition T+m = Tmax, which is:

1

2b2((b1h0 + h1λs)t− h1λv,0 − b1v0) = Tmax

→ t1 =2b2Tmax + h1λv,0 + b1v0

b1h0 + h1λs(5.28)

In order to know the time t1 in our problem (as well as the optimal speeds and distances

in the next phases) we need to know the initial values of the adjoint states λv,0 and λs. These

values can be find using the terminal conditions on the states, as it will be explained later on.

Second Phase

At the time t1, when T ∗m ≤ Tmax a second phase starts with the following condition: T ∗m =

T+m .

46

Substituting the vehicle model and the adjoint state expressions (Equation 5.11 and Equa-

tion 5.21) in the expression of T+m (Equation 5.17), we obtain the control law of the second

phase:

T ∗m = 1

2b2[(b1h0 + h1λsηt)t− h1λv,0ηt − b1v0]

w∗ = 0

(5.29)

For t ∈ [t1, t2] where the time t2, that represents the end of the second phase, can be find

applying the condition T+m = 0.

Again, in order to find the optimal speed and the optimal distance of this second phase it

is necessary to integrate Equation 5.26 and Equation 5.27 and to use v|t=t1 and s|t=t1 as initial

conditions of the integrals .

The expressions of the speed and the distance of this and of the following phases are not

reported from now on, since they are very long expressions and just a matter of straightforward

substitutions.

Third Phase

The third phase starts when T+m < 0. At this point, though, T−m can be still positive.

(T+m 6= T−m due to the operator sign in the transmission efficiency). So, neither T+

m nor T−m is

the optimal solution and therefore:

47

T ∗m = 0

w∗ = 0

(5.30)

for t ∈ [t2, t3]. t3 can be found applying the condition:T−m = 0.

Again, the optimal speed and distance expressions can be found by substitution, function

of λv,0, λs, v|t=t2 and s|t=t2 . Therefore, going forward with the phases we always obtain a more

complex expressions of v and s, functions of all the previous phases.

Fourth Phase

The control law in this phase is given by the condition T ∗m = T−m .

Therefore:

T ∗m =

(12b2

(b1h0 + h1λsηt

)(t− t3)− b1v3 − h1λv,3ηt

)w∗ = 0

(5.31)

for t ∈ [t3, t4]

where v3 = v|t=t3 and λv,3 = λv|t=t3 . λv,3 can be found simply integrating Equation 5.21.

Time t4 is obtained thanks to the condition T−m = Tmin.

Fifth Phase

The control law of this phase is defined by a constant control T−m :

48

T ∗m = Tmin

w∗ = 0

(5.32)

for t ∈ [t4, t5]

This phase ends when the switching function ∂H∂w changes sign becoming positive. It is

possible to find t5 applying the following: λv(t) = 0.

Sixth Phase

Once λv > 0 the braking force is active at its maximum value. Therefore the control law is:

T ∗m = Tmin

w∗ = W

(5.33)

for t ∈ [t5, tf ] with final time tf known in advance as a parameter of the problem.

5.2.2 Results

Now, all the control laws have been developed and an analytical expression of the optimal

speed and distance for each phase can be written. However, these formulations are functions

of the initial conditions of the adjoint states λv,0 and λs, the only remaining unknowns of the

problem.

49

It can be proved [5] that the optimal speed v(t) and distance s(t) for each phase depend

only on the unknown initial conditions of the adjoint states λv,0 and λs and on the known initial

condition of the states v(0) = v0, s(0) = s0.

In order to find λv,0 and λs we use the terminal condition on the states v(tf ) = vf , s(tf ) = sf .

In this way, we can obtain two equations and solve them in order to find λv,0 and λs:

v(λv,0, λs)|t=tf = vf

s(λv,0, λs)|t=tf = sf

(5.34)

where the explicit algebraic expressions of v(λv,0, λs)|t=tf and s(λv,0, λs)|t=tf are functions

of all the control laws in each phase.

To be more explicit, we can write:

v(λv,0, λs)|t=tf =

∫ tf

t5

(h1T

∗m

ηt− h0) dt+ v5 (5.35)

Where v5 can be written as an expression function of v4 and v4 as function of v3 and so on

and so forth. In the end, they are just functions of the initial condition v0 (other than λv,0 and

λs.) A similar expression can be written for s(λv,0, λs)|t=tf . The only unknowns remaining are

λv,0 and λs in Equation 5.34. Once written the expressions of each optimal speed and optimal

distance from the first phase to the last one, we explicitly obtain two equations function of λv,0

and λs. Then it is trivial to solve them as a system of two equations and two unknowns.

50

Naturally, depending on the initial and final condition chosen, some of the six phases can

disappear. We now provide the results for the control problem described at the beginning of

the chapter that we recall here:

vi = 0

vf = 0

s0 = 0

sf = 200m

tf = 24s

Tmax = 700Nm

Using these parameters all the six phases appears in the optimal solution. Graphical repre-

sentations of the optimal torque, speed and distance are shown in Figure 5 and Figure 6.

The analytical solution provided has a great potentiality due to its computational time.

Obtaining the optimal solution is much faster than with every other numerical approach.

The main disadvantage is that the optimal control problem proposed and solved contains

some approximations with respect to the original one. In particular, the air resistance force has

been neglected and this fact is acceptable just in low speed scenarios. It is not possible to include

the air resistance term in the analytical solution because it would not be possible to find an

explicit solution of the Hamiltonian function anymore. Furthermore, in the construction of this

51

Figure 5: Analytical optimal motor torque in its six different phases

simplified problem we did not keep into account state constraints such as the position of other

vehicles around the ego vehicle. Time-variant state constraints (the position of a target vehicle

would change at every time) cannot be added in the control problem and, furthermore, it would

be useless without the possibility to include the air drag reduction term in the formulation. Even

the constraint on the maximum speed cannot be included in this problem.

52

Figure 6: Analytical optimal speed (first plot) and position (second plot) profiles

In conclusion, the analytical solution permits to solve unconstrained finite time optimal

control problems which could be easily implemented in an on-line solution. Unfortunately, the

simplifications makes the solution not applicable to real-time problems.

5.3 Dynamic Programming

The aim of this section is to solve the same optimal control problem using a numerical

technique: the Dynamic Programming. The theory of this approach has been described in

53

Chapter 3. As a recall, it basically divides the problem into various subproblems and it evaluates

the optimal cost-to-go at each step proceeding backward from the final time until the initial

one.

Naturally, it solves a discretized version of the original (continuous) problem that,in this

case, can be written as:

minu∈U J =N∑k=0

gk(uk, xk) (5.36)

subj.to xk+1 = fk(uk, xk) (5.37)

v0 = vin vN = vf (5.38)

s0 = sin sN = sf (5.39)

The numerical computation has been obtained using a Dynamic Programming Matlab func-

tion [24] that can be downloaded at [25].

The results have been compared with the ones provided by the analytical solution for two

main reasons: to check the correctness of both solutions and to provide a comparison between

the two methods.

54

Figure 7: Torque comparison using analytical solution and dynamic programming approach

55

Figure 8: Optimal speed (first plot) and position (second plot)+ of the ego vehicle using

analytical solution and dynamic programming approach

The DP approach has the advantage that can theoretically solve any finite-time constrained

optimal control problem. Which means that we can include in the state dynamics also the

aerodynamics resistance and the time-variant state constraints. It converges to the optimal

solution even with non-convex problem.

56

The main problem is the computational time. It can be relatively fast when the number of

states is low but it increases exponentially with the increase of the number of states. Further-

more, the computational time strongly depends on the grid adopted in order to discretize the

problem.

The small differences in the optimal torque between the two methods derive from the dis-

cretization of the problem. In particular, the analytical solution represents the perfect solution

of this particular problem, while the Dynamic Programming obtained a sub-optimal solution. It

is also possible to verify this conclusion checking the total cost of the two solutions. It appears

that the total cost of the analytical solution is less then the other one.

For the optimization the following grid point has been used: 801 point for each of the states

and 101 point for the input. In particular we considered just one input, the motor torque (that

includes the regenerative brake) and we neglected the friction-based braking force since it is

almost useless except for the last seconds. Adding one input would consistently increase the

computational time in the DP. Due to the discretization chosen, the optimization already takes

a discrete amount of time to be completed. It is possible to note from Figure 7 that analytical

and DP solutions differ more in the last seconds of the problem, where the second input should

be different from zero.

In conclusion, this method presents serious issues, too. The computational time is already

very high and furthermore, the time-variant state constraints we would like to add in the

problem would considerably increase the computational time. Therefore, it is not possible to

57

utilize this solution as a real-time optimization. Its main use is to verify the correctness of other

solution adopted.

5.4 Numerical Optimization: YALMIP Toolbox

The third approach analyzed is a numerical optimization using a Matlab toolbox: YALMIP[26].

The toolbox is consistent with MATLAB interface and it relies on external solvers for the op-

timization. The optimization structure is much simpler to build with respect to the DP or

the creation of an analytical solution. Furthermore, numerical solvers usually provides very

accurate solution and they are very fast.

The solver adopted depends on the problem YALMIP needs to solve,( e.g. if the problem

is linear or non-linear, convex or non-convex) and it can be chosen by the user.

When the optimization problem is non-linear (as it happens including the aerodynamic

resistance in the vehicle dynamics) the solver chosen in this thesis is IPOPT [27]. The name is

an acronym for Interior Point OPTimizer; it utilizes interior point methods, a class of algorithms

that solves non-linear problems. However, when the problem is non-linear there is not any

guarantee that the solution obtained is the global minimum and not just a local one. Indeed, if

the problem is non-convex, it can presents more than one minimum and not always the solver

is able to converge to the global one. For this reason, when possible, it is preferable to use

other solutions, like an analytical solution or a Dynamic Programming approach, which always

converges to the global minimum.

For the class of problem solved during this work, the numerical approach turns out to be

the best one: it is very fast and it does not give convergence problems.

58

As mentioned before, the only aspect not included in the vehicle dynamics is ηsign(Tm(t))t but

its behavior has been included into the powertrain model, as described in the previous chapter.

CHAPTER 6

ONLINE OPTIMIZATION

In this chapter the optimal control problem used in the real-time Eco-CACC has been

analyzed.

6.1 Optimal Control Problem

The optimal control problem adopted for the online optimization assumes the following

form:

J∗(x(t)) = minu∈U(x(t))

N−1∑k=0

g(xt+k, ut+k) (6.1)

subj.to : dmin ≤ sref (t+ k)− s(t+ k) ≤ dmax (6.2)

δ ≤ vref (t+ k)− v(t+ k) ≤ δ (6.3)

x(t+ k + 1) = f(k + t, x(k + t), u(k + t)) (6.4)

Tm(t+ k) ∈ [Tmin, Tmax] (6.5)

v(t+ k) ∈ [vmin, vmax] (6.6)

v(t) = vin s(t) = sin (6.7)

where x(t) = [v(t), s(t)]T .

59

60

The last four constraints represent respectively the non-linear dynamics of the system, the

limitations on the torque and the speed and the initial conditions of the problem. These

constraints appear in the offline optimization as well.

The first two constraints are related with the target vehicle position (sref ) and speed (vref ).

The first one (Equation 6.2) states that the ego vehicle has to maintain a minimum and maxi-

mum distance from the target vehicle. It is a safety constraint that assures that the ego vehicle

is not going to bump with the preceding vehicle. The maximum distance assures that the spon-

taneous platoon formation between the two vehicles is maintained. Note that the controller

receives forecast about the position and the speed of the target vehicle at each time step.

Finally, Equation 6.3 limits the difference of speed between the two vehicles. Indeed, a huge

speed difference can easily lead to stability problems and it should be avoided. δ, dmin and

dmax are constant values.

The cost function g(xt+k, ut+k) is, as usual, the powertrain model of the vehicle and it is

represented by Equation 4.19.

6.2 Suboptimality of the Solution

The optimal control problem described would lead to an optimal solution just along the

horizon of the forecast. In other words, if the controller receives forecast 10 seconds long, the

result of the optimization would be the best trajectory for the ego vehicle in the next 10 seconds.

This solution is sub-optimal because it does not take into account what it will happen from the

tenth second on. Indeed, the controller does not take into account that the vehicle will continue

to travel from the tenth second on.

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In order to solve this issue, we modify the cost function of the control problem in such a

way it takes into account also what it will happen after the end of the horizon.

There are two main issues responsible for sub-optimality. First, the vehicle will tend to arrive

at the end of the horizon with the mimimum kinetic energy in order to use less fuel in those

10 seconds. But, the kinetic energy provided by the powertrain to the vehicle is conservative

and it can be used during the future trajectory. There is no need to minimize it. Therefore, it

should not be included in the cost function of the problem. Otherwise the vehicle would always

decelerate at the end of the horizon.

A similar concept applies for the distance traveled. The vehicle will tend to travel the

minimum possible distance along the horizon (which means that the ego and the target vehicle

will be at their maximum relative distance at the end of the horizon). However, the distance

that the vehicle does not travel along the horizon, will be traveled in the future seconds.

To address these issues, two terminal costs are introduced in the cost function of the problem

that now assumes the following form:

J∗(x(t)) = minu∈U(x(t))

N−1∑k=0

g(xt+k, ut+k) + p1(xt+N ) + p2(xt+n) (6.8)

where p1(xt+N ), p2(xt+n) represent two costs that take into account the behavior of the

vehicle after the end of the horizon and they are function only of the state of the vehicle at the

end of the horizon.

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6.3 Penalization of the Terminal Kinetic Energy

We consider a platoon formation in which the target vehicle is traveling at a constant speed

and the ego vehicle is already at the maximum relative distance. The optimal solution provided

by the Eco-CACC without any terminal cost (shown in Figure 9), consists in a first acceleration

of the ego vehicle and then a strong deceleration in order to conclude with the minimum kinetic

energy. This solution is optimal along the horizon but it does not take into account that in the

future the vehicle will have to accelerate again in order to follow the target vehicle. Globally,

this condition implies a huge dissipation of energy.

A penalization of the terminal kinetic energy is thus introduced:

p1(xt+N ) =1

2bkm(v2ref,fin − v2fin) (6.9)

where vref,fin and vfin are, respectively, the speed of the target vehicle and the speed of

the ego vehicle at the end of the horizon. The controller receives vref,fin as part of the forecast

and it computes the optimal vfin

The less the kinetic energy at the end of the horizon the more the penalty added by the

terminal cost.

bk is an empiric coefficient (> 1) that takes into account extra motor losses linked to speed

oscillations.

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Figure 9: Relative speed between the ego and the target vehicle with and without penalizationof the kinetic energy

Including this terminal cost in the formulation of the problem, the ego vehicle follows the

target vehicle at a constant speed without any acceleration or deceleration and the global energy

consumption is thus reduced.

64

6.4 Penalization of the Terminal Distance

The issue presented in the previous example has been addressed thanks to the penalization

of the terminal kinetic energy. However, in the previous scenario the two vehicles already were

at their maximum relative distance. If not so, the ego vehicle would have decelerated in order

to be as far as possible from the target vehicle at the end of the horizon. The trivial reason of

this behavior is that traveling more distance implies a higher cost. The point again is that this

optimal solution is not taking into account what it will happen after the end of the horizon:

the vehicle is going to continue its journey.

As a reminder, the distance traveled by the ego vehicle within the horizon is bounded by

the position of the target vehicle. The most conservative position for the ego vehicle at the end

of the horizon occurs when the two vehicles are at their maximum relative distance. Calling

sN the absolute position of the ego vehicle at the end of the horizon the condition appears

when sN = smin,N . Every more meter traveled will cause the ego vehicle to be closer to the

target vehicle. In terms of absolute position, the minimum relative distance between them is

represented by sN = smax,N .

The terminal cost on the distance should therefore compensate the quota of friction asso-

ciated to every extra meter the vehicle travels within the horizon. In this way, the terminal

position of the ego vehicle will not be affected anymore by this aspect.

The terminal cost should be a function of the relative distance between the two cars:

65

p2(xt+N ) = 4Efriction(smax,N )−4Efriction(sN ) (6.10)

where 4Efriction(smax,N ) is the energy dissipated during the journey over the horizon when

the ego vehicle has traveled the maximum distance. While 4Efriction(sN ) is function of the

final position of the ego vehicle.

Friction is a nonconservative force, so an average friction for all the trajectory has been

considered.

4Efriction = FM,friction 4s

where

FM,friction = cdρAf4s2

24t2+ crg m (6.11)

The first term represents the viscous force, the second one the rolling friction.

Therefore the terminal cost assumes the following form:

p2(xt+N ) = (As2max,N +B)smax,N − (As2N +B)sN (6.12)

66

with

A =cdρAf2N2

(6.13)

B = crg m (6.14)

However, a terminal cost with a cubic function of the states such as Equation 6.12 makes

the problem too complex to be solved in a reasonable time.

The best solution would be to obtain a linear functon of the states. In this way the terminal

cost would not add any complexity to the optimization procedure.

For these purposes, it is possible to write p2(xt+N ) as function of the relative distance d

between the two vehicles, since:

d = smax,N − sN (6.15)

Substituting, we obtain the following expression of the terminal cost:

p2(xt+N ) = A(d3 − 3 d2 smax,N + 3 d s2max,N ) +B d (6.16)

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Considering a prediction horizon of N = 10s and an average speed of the ego vehicle of

v = 20 ÷ 22m/s we obtain smax,N ≈ 200m while d can always vary from 2 to 20 meters.

Therefore, the first two terms in the expression are negligible since they appears to be two or

three orders of magnitude smaller than the third one, and the terminal cost can be written as:

p2(xt+N ) ≈ (3As2max,N +B)(smax,N − sN ) (6.17)

Equation 6.17 is now a linear function of the states, since the only variable is sN , the position

of the ego vehicle at the end of the horizon. A, B and smax,N are just constant values.

6.5 Results

Results on the impact of the terminal costs are now presented. The scenario considered

is again the one in which the target vehicle is traveling at a constant speed and no air drag

reduction is included in the simulation. The ego-vehicles starts at a intermediate relative

distance sin = 12m and the two vehicles starts at the same speed vin = 22m/s.

It is possible to note how in the baseline scenario (with no terminal costs) the ego vehicle

has the minimum possible speed at the end of the horizon and it has traveled the minimum

possible distance. Introducing the terminal cost on the kinetic energy, the vehicle would not

tend anymore to decrease its speed but still at the end of the horizon it is as far as permitted

from the target vehicle. Introducing both terminal costs, the ego vehicle follows the target

vehicle at a constant speed and at a constant distance.

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Figure 10: Relative speed (first plot) and distance (second plot) between the ego and the targetvehicle with and without the terminal costs

The main goal of introducing terminal costs is to approximate the behavior of the vehicle

after the end of the horizon. Therefore, they will be use in every simulation from now on. Their

validity is general since the theory presented has nothing to do specifically with the scenario

presented.

CHAPTER 7

CLOSED LOOP SIMULATIONS WITH EXPERIMENTAL FORECAST

In this chapter simulation results of the Eco-CACC developed are presented. The scenarios

have been chosen in order to show the behavior of the powertrain controller in different situ-

ations. A comparison with a traditional Adaptive Cruise Control in terms of energy savings

is also presented. Forecast of position and speed of the target vehicle are obtained using ex-

perimental data. In particular, data from a real vehicle have been collected while driving in

different roads and highways around the Silicon Valley . Furthermore, an external Simulink

toolbox provides a better estimate of the energy savings of the Eco-CACC.

7.1 QuasiStatic Simulation Toolbox

QuasiStatic Simulation Toolbox (QSS TB) is a Simulink toolbox that permits to estimate

fuel consumption for the majority of powertrain systems, including the electric ones. It is very

simple and fast to use. At the same time, it provides quite accurate results [20].

It reverses the relationship of cause-effect that exists in a dynamic system: it does not

uses forces to determine accelerations and speed profiles but the exact opposite. It calculates

acceleration and it determines the necessary forces starting from a given speed profile.

The QSS TB approach can not capture dynamic phenomena (represented by differential

equations) since it is a QuasiStatic approach. However, it takes into account various elements

which are not considered in the powertrain model adopted in the controller.

69

70

7.1.1 QSS Outline

Figure 11: QSS outline and elements not included in the model adopted in the Eco-CACC

As it possible to see from Figure 11, QSS receives as an input the speed profile and it

provides the fuel consumption of the electric battery of the vehicle considered. To do so, it

takes into account different elements not included in the powertrain model.

The equations of the vehicle dynamics do not change but a non-linear air drag dependence

function of the distance instead of a linear one has been considered. There are not real-time

issues in the energy savings calculation, so accuracy can be increased.

Regarding the transmission part of the model, the idle power loss is here included and it

is also possible to take into account the switching model caused by the transmission efficiency

71

property. In the model adopted by the controller we included the transmission efficiency in the

efficiency map of the motor, but in this case is possible to include the switching model directly

in the transmission model.

The electric motor model considered is far more complex than the one adopted by the

controller since it takes into account inertia, an efficiency map and even an overload check which

prevents the motor from working out of its speed and torque boundaries. Battery efficiency is

also considered, function of the current i and of the state of charge (SOC). These dependences

are usually non-linear and too complex to be included in the controller model.

7.1.2 QSS Goals

QSS is used in this work for two main puroposes:

• Model validation

• Fuel consumption estimation

QSS TB can be used as a model validator since it reverses the cause-effect relationship of

the dynamic systems. The powertrain controller finds the optimal torque for a certain system.

The torque is then applied to the model and the consequent acceleration is obtained. From the

acceleration profile a consequent speed profile is finally determined. This speed profile can be

adopted as input of QSS TB. Reversing the usual cause-effect relationship, the toolbox obtains

the consequent torque. If the QSS torque is similar to the Eco-CACC one, it means that the

simplified model adopted in the controller is acceptable.

The main function of the toolbox is to provide the fuel consumption for a powertrain sys-

tem. In the case of electric motor it gives the battery consumption for a given speed profile.

72

We can compare the battery consumption obtained using the Eco-CACC with respect to the

consumption of a traditional ACC.

7.2 Simulation Setup

7.2.1 Forecast Data

Figure 12: Experimental speed profile collected driving in the Highway 101 around the SiliconValley

73

The results presented in this chapter are simulations obtained in a Matlab environment.

However, the forecast data that the ego vehicle receives from the target vehicles are experimental

data which have been collected driving a real car in the highway and in urban roads around the

Silicon Valley. The trajectory of the ego vehicle is always bounded by the position of the target

vehicle. Using real data we assure that the optimal trajectory obtained for the ego vehicle can

be associated with realistic scenarios.

Note that even if all the trajectory of the target vehicle is known in advance, the controller

is set up to receive just the portion of data within its prediction horizon.

7.2.2 Energy Savings

The total energy consumption of the vehicle is an important parameter but is more conve-

nient to analyze the energy savings that the Eco-CACC can perform with respect to a baseline

controller. We consider as a baseline controller an Adaptive Cruise Control that simply follows

the target vehicle at a fixed distance of 12 meters. In both cases, the fuel consumptions are

calculated using the QuasiStatic Toolbox.

7.2.3 MPC Parameters

A table with the MPC parameters used in all the simulations is presented.

The sampling time choice is a trade-off between computational power and accuracy. In

literature different choices are described; generally they go from 0.05 seconds [28] to 0.5 seconds

[29]. The smaller the sampling time the better it is. For our purposes, a sampling time of 0.5

seconds would not make the solution feasible, taking into account that a car would travel several

meters between one sample time and the other. On the other hand, the smaller the sampling

74

TABLE II: MPC PARAMETERS

Name Symbol Value

Prediction Horizon [s] N 10Sampling time [s] Ts 0.1Minimum relative distance [m] dmin 2Maximum relative distance [m] dmax 20Initial relative distance [m] din 12

time, the more computational power is required with the risk that the controller can not work

in real time. A sample time of 0.1 seconds is considered a good trade-off for this work: it makes

the solution feasible in all the scenarios adopted and the controller turns out to be powerful

enough to be able to work in real time.

A prediction horizon of 10 seconds is an optimistic assumption. It means that the ego

vehicle knows in advance the trajectory of the target vehicle for the future 10 seconds. This

assumption is not always realistic in case of traffic or with a non-autonomous target vehicles.

In those conditions, the simulations can fully show the potentiality of the Eco-CACC as well

as an upper bound on the energy savings.

A minimum distance of two meters between vehicles is commonly adopted in the platoon

literature [30]. High speeds, acceleration or decelerations of the vehicles and communications

delays can easily lead to safety problems. However, in this work, we do not take care of safety

issues at a deep level. We analyze the full potentiality of the controller, taking into account

that an emergency braking system can be necessary in order to guarantee safety.

75

A maximum relative distance of 20 meters has been chosen accordingly to the prediction

horizon of 10 seconds. In order for the controller to work at its full potentiality, the ego vehicle

needs to be able to have full capacity of movement within its position boundaries. The ego

vehicle should be able to pass from dmax to dmin or vice versa in 10 seconds. Otherwise, the

optimal solution of the controller would become too much sensitive to the horizon chosen. Note

that this powertrain controller is meant to be part of a hierarchical system of two levels. The

high level should decide whether or not the ego vehicle should platoon with the target vehicle.

The assumption considered in this work is always that platooning is the best solution.

7.3 Scenarios

In this section three different simulation scenarios are proposed and analyzed. The first two

are highway scenarios, in which the vehicles travel in the highway 101. The last one presents

an urban scenario which explicitly includes traffic.

7.3.1 Highway Scenario 1

In this first case, the vehicles are traveling in the highway and the target vehicle is acceler-

ating. As it possible to note from Figure 17, initially the ego vehicle catches up with the target

vehicle. The reason is that being at a close inter-vehicular distance permits to reduce the fuel

consumption. At the same time the controller performs a speed smoothing : it tries to avoid

every speed oscillation, since this behavior would increase the amount of energy dissipated. As

a consequence, once the ego vehicle has caught up, it does not follow the target vehicle at a

fixed distance, but it continuously varies its relative distance in order to save more energy.

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Figure 13: Extract of speed profile of target and ego vehicles (first plot) and their relativedistance (second plot) of “Highway Scenario 1”

The entire simulation has a time frame of more than three minutes in which it is possible

to observe how the optimal solution is not just a simple catch up but it requires a continuous

variation of the distance between the target and the ego vehicle. As a result, the speed profile

of the ego vehicle is smoother.

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Figure 14: Speed profile of target and ego vehicles (first plot) and their relative distance (secondplot) of “Highway Scenario 1”

7.3.1.1 Model Validation

In this scenario a model validation of the controller is also proposed. The model adopted

by the Eco-CACC is compared with the more detailed one used in QSS TB and differences are

analyzed. As it possible to note from Figure 15 the torque obtained by the controller and the

one used for validation are very similar. The QSS torque is always a little bit higher. This result

78

Figure 15: Comparison between the controller optimal torque and the QSS validation using thesame speed profile

was expected: QSS Toolbox takes into account elements not included in the model adopted by

the controller and the majority of those elements are extra-losses. Therefore, in the QSS model

a higher torque is necessary in order to travel the same distance with the same speed.

7.3.1.2 Energy Savings

The main aim of this work is to develop a controller which provides energy savings with

respect to a more traditional form of Cruise Control. Therefore, we compare the energy con-

sumption of the vehicle using the Eco-CACC and a traditional ACC. The ACC-based ego

vehicle just follows the target vehicle at a fixed distance of din = 12m. There is neither a initial

79

Figure 16: Energy consumption comparison using the Eco-CACC and a traditional ACC re-sulting in energy saving of 15.6 %

catch up, nor a smoothing of the speed profile. Therefore, the energy savings of the Eco-CACC

can be divided into two contributes in this scenario. The first one is the fact that being at a

closer inter-vehicular distance reduce fuel consumption. The second one is that accelerations

and decelerations leads to more dissipations and the Ego-CACC tries to avoid them.

As a result, using the powertrain controller developed, energy savings of 15.6% have been

obtained with respect to an Adaptive Cruise Control.

Energy savings have been calculated using QSS TB in order to obtain a more precise esti-

mation.

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Figure 17: Speed profile of target and ego vehicles (first plot) and their relative distance (secondplot) in “Highway Scenario 2”

7.3.2 Highway Scenario 2

This scenario represents a slow-down scene in the highway. The target vehicle, after a first

acceleration, starts to decelerate. The optimal solution provided by the Eco-CACC has similar

characteristics of the one described in “Highway Scenario 1”. First, there is a catch up. After

that the ego vehicle does not follow the target one at a fixed distance but it continuously changes

81

the relative distance. Again, the ego vehicle speed profile is smoother than the one of the target

vehicle.

However, in this case, the first catch up would not be necessary. In the global optimal

solution, the ego vehicle would have waited more time before to catching up. This limitation is

given by the prediction horizon. The controller knows the behavior of the target vehicle for the

future 10 seconds, not for the whole trajectory and therefore, at time zero, it can not predict a

future deceleration of the target vehicle.

In this scenario, the energy savings obtained using the Ego-CACC with respect to an ACC

are 21.7 %. This value, as mentioned before, can be considered as an upper bound since it

is not always possible to know the behavior of the target vehicle in the future 10 seconds. At

the same time, the controller can work with any horizon. The shorter the horizon the less

the energy savings. In any case, the controller will compute the optimal trajectory using the

available data.

7.3.3 Urban Scenario

An urban scenario is now presented. In this case, the traffic conditions play an important

role in the simulation. The target vehicle speed profile presents wide oscillations due to the

traffic on the road. In this case, the optimal solution for the ego vehicle is not represented by an

initial catch up. The oscillations are too wide and the savings obtainable thanks to a reduced

aerodynamic force would be compensated by losses due to accelerations and decelerations. As

a consequence, the relative distance between the two vehicles varies a lot during the simulation.

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Figure 18: Speed profile of target ego vehicles (first plot) and their relative distance (secondplot) in an urban scenario

Very high energy savings of 73.4 % with respect to the baseline ACC have been obtained.

However in this scenario, due to the traffic conditions, availability of forecast is not always

possible. Also, the flexibility of the ego vehicle in deciding its speed and position may be limited

by other closeby vehicles. On the other hand, this scenario fully underlines the potentiality of

the Eco-CACC in terms of energy savings.

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It may be not possible to have a perfect forecast of 10 seconds due to the traffic but as an

alternative it is possible to consider an approximate forecast. For this case, the speed profile of

the target vehicle can be modeled as a sinusoidal trajectory. Even if the powertrain controller

does not receive a perfect forecast, it can still elaborate a future forecast using the frequency

and the amplitude of the modeled sinusoidal trajectory. Energy savings of 73.4 % demonstrate

that there is big room for acceptable results even without a perfect forecast. This topic will be

part of future work, though, and it is not discussed further in this thesis.

CHAPTER 8

EMBEDDED IMPLEMENTATION

All the simulations presented in the previous chapters have been developed using software

tools such as Matlab and Simulink, which are non-real-time environments. In this chapter,

the computational time issue is analyzed in more detail. A real software in a vehicle can not

run on Matlab environment, though. An embedded software has to be written for a particular

hardware. This fact usually implies time and memory constraints. The algorithm has been

rewritten in a suitable environment for embedded implementation and time issues have been

faced and solved. Moreover, hardware in the loop simulations have been tested, in order to

move a step forward for the implementation of the code in a real vehicle.

8.1 Time Constraint

One of the biggest issue in the development of a real-time algorithm is the time constraint.

In the previous simulations we adopted a prediction horizon of 10 seconds. Which means that

the controller needs to find an optimal solution for the next ten seconds. This procedure has

to be entirely repeated every sampling time, i.e. every 0.1 seconds. If the controller needs

more than 0.1 seconds to find the optimal solution, then it does not have time for the next

optimization. The controller can work in real-time only if every optimization requires less than

the sample time to be solved. If the condition is not satisfied, the only way is to increase the

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sampling time or to decrease the prediction horizon with repercussions on the quality of the

solution.

The algorithm has been written trying to avoid as many non-linearities as possible in the

model in order to decrease the process time during the optimization. Prediction horizon and

sampling time have been chosen in such a way that the algorithm always works in real-time

when it runs on Matlab. In Figure 19, a graphical representation is shown.

However, for a given scenario, computational time depends on the hardware used and on

the solver chosen for the algorithm. For this reason, there is no guarantee that the solver can

really work on a real vehicle yet.

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Figure 19: Computational time in Matlab environment of ”Highway Scenario 2” with N=10s,

Ts=0.1s

8.2 FORCES Pro Code Generator

FORCES Pro [31] is a tool for mathematical optimization. It converts a code written in

Matlab into a C-language code which can be implemented in an embedded system. In other

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words, this tool is a code generator. The optimal problem needs to be rewritten using FORCES

syntax but still in a Matlab environment. Then, the optimal control problem is not solved

anymore using IPOPT [27] but always with interior point methods.

Figure 20: Optimal torque, speed and relative distance of the ego vehicle in ”Highway Scenario3” are shown in the first second and last plot respectively

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Matlab environment is very convenient for pure software simulation but the code can not

be used for hardware simulation, in other words for real testing. The code needs to be written

using C language in order to be implemented on an embedded platform.

This tool has its own Matlab syntax and a new code has been developed. The main difference

with the previous one is the initial guess. The code generated by FORCES Pro requires an

initial guess for all the states of the system, not just for the initial ones. At any time different

from zero the initial guess is given by the optimal solution at the previous time step. At time

zero a input sequence of zero value is guessed and applied to the dynamic system.

The initial guess choice is important since it affects the computational time of the solution.

Once the optimal problem is written, the code is automatically generated on line using

FORCES Pro servers. It turns out that the solver, which uses C language is much faster that

the one created on Matlab environment. Furthermore, this code can be perfectly embeddable

on any implementation which has a C compiler.

In order to compare the two solvers and their computational time, we analyze a new scenario

(“Highway Scenario 3”). Similarly to the previous ones, a target vehicle is accelerating in the

highway and then suddenly decelerating. Every parameter is identical to the previous scenarios

except for the prediction horizon. It has been set to 12 s in order to stress more the time

constraints of the solvers.

As it is possible to note from Figure 20, both interior point solvers obtain an almost identical

solution for the same scenario. Since the problem is nonlinear, the same optimal solution is not

guaranteed and some small differences in the solution are expected. The most notable aspect

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Figure 21: Computational time comparison between IPOPT and FORCES solvers.

is highlighted in Figure 22, though. FORCES solver is about five time faster with respect to

IPOPT one. This is a positive result, since it is the C code generated by FORCES the one that

can be actually implemented in an embedded machine.

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8.3 Hardware in the Loop Simulation

Until now, a control software has been developed and tested in a non-real-time environment.

However, pure software based simulations can not capture sufficiently real-time conditions and

they can not provide enough confidence and reliability. Therefore, it is wise to consider an

intermediate phase in the middle between pure software and pure hardware testing on a real

machine. This phase is called Hardware in the Loop (HIL) Simulation.

This process consists in testing the control software on a target Electronic Control Module

(ECM) simulator in real-time. The hardware is then connected to another computer which

simulates the controlled process dynamics in real-time. In this way, it is possible to test the

control software on a real hardware without using a real plant (such as a real vehicle).

In this thesis we analyze the algorithm developed on the computational time point of view.

A dSPACE HIL simulator has been used in order to simulate an Electronic Control Module.

Until now, every simulation run on a generic computer processor and, therefore, it could not

provide a precise indication regarding whether or not the algorithm developed effectively works

in real-time.

dSpace MicroAutobox 1401, (IBM PowerPC processor capable of running at 800 MHz) has

been used for the HIL simulation.

The result obtained is shown inFigure 22 and it proves that the controller can be embeddable

in a real machine with a sample time of 0.1 and an horizon of 8 seconds.

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Figure 22: Computational time during a Hardware in the Loop simulation with Ts=0.1 s andN=8 s

CHAPTER 9

CONCLUSION

This chapter presents general conclusions of the work developed.

Optimal control theory has provided the basis for the creation of an Ecological Coopera-

tive Adaptive Cruise Controller that aims to minimize fuel consumption of electric vehicles.

The main idea consists in taking advantage of platooning in terms of fuel consumption while

maintaining a spontaneous approach.

Various steps have been necessary in order to develop a real time optimal controller.

First, we have discussed the longitudinal dynamic of the vehicle, capturing elements which

play an important role in terms of fuel consumption. Air drag coefficient has been modeled as a

function of the relative distance between vehicles. The powertrain model of the electric vehicle

has been also developed using experimental data from a real electric motor.

Second, we have analyzed different optimization techniques, both analytical and numerical.

We have compared them and outlined their potentialities and drawbacks, and we have finally

chosen the technique which best address our needs.

Third, we have designed a model predictive controller for real-time implementations of

the Eco-CACC. The terminal costs have especially been designed to give the controller the

capability to anticipate what is gonna happen to the vehicle after the horizon.

At this point, once the optimization technique have been chosen and the control problem

defined, we have extensively tested the Eco-CACC. We have evaluated different highway and

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urban scenarios using experimental data collected driving in the Highway 101 and in the urban

areas of the Silicon Valley around the city of San Francisco. Then, we have evaluated the

performances of the Eco-CACC in terms of fuel consumption, making a comparison with a

traditional ACC. Energy savings have been calculated using a high-accuracy model. The same

model has been used also to validate the simplified model adopted in the powertrain controller.

Finally, we have addressed the problem of computational time. We made sure the con-

troller is able to work in real-time, and not just during simulations in a Matlab environment.

Hardware in the loop simulations have been performed implementing a C-language code into

HIL simulator. The positive results make the controller ready to be tested on real autonomous

vehicles.

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VITA

NAME Lorenzo Bertoni

EDUCATION

Master of Science in “ Mechanical Engineering ”, University of Illinoisat Chicago, Dec 2016, USA

Specialization Degree in “ Energy and Nuclear Engineering ”, Oct 2016,Polytechnic of Turin, Italy

Bachelor’s Degree in Energy Engineering, Jul 2014, Polytechnic ofTurin, Italy

LANGUAGE SKILLS

Italian Native speaker

English Full working proficiency

2014 - IELTS examination (7.0)

A.Y. 2015/16 Two semesters of study abroad at University of Califor-nia, Berkeley

A.Y. 2015/16 One semester of study abroad in Chicago, Illinois

SCHOLARSHIPS

Fall 2015 Italian scholarship for final project (thesis) at UIC

Fall 2015 Italian scholarship for TOP-UIC students

TECHNICAL SKILLS

Advanced level Matlab and Simulink

Average level ERP SAP SD

WORK EXPERIENCE AND PROJECTS

2017/Present Consultant at Oliver Wyman

2014/2015 Research Assistantship (RA) position (10 hours/week) at Polytechnicof Turin, Italy

2014/2015 Co-Founder of a startup (Corner Pop)

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VITA (continued)

Ago 2014 Volunteer in Africa. Hospital La Croix, Cotonou (Benin)