an optimal control framework for connected and …

AN OPTIMAL CONTROL FRAMEWORK FOR CONNECTED AND

AUTOMATED VEHICLES IN A MIXED TRAFFIC ENVIRONMENT

by

A M Ishtiaque Mahbub

A dissertation proposal submitted to the Faculty of the University of Delawarein partial fulfillment of the requirements for the degree of Doctor of Philosophy inMechanical Engineering

Spring, 2020

c© 2020 A M Ishtiaque MahbubAll Rights Reserved

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vLIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Chapter

1 INTRODUCTION AND LITERATURE REVIEW . . . . . . . . . 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Optimal Control Framework for CAV Coordination . . . . . . 31.2.2 A Review of Decentralized Optimal Control Framework . . . . 41.2.3 A Review of the Partial CAV Penetration Problem . . . . . . 71.2.4 A Review of Platoon Formation and Stability . . . . . . . . . 8

1.3 Summary of Research Gaps in the Literature . . . . . . . . . . . . . . 11

2 PRELIMINARY RESULTS . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1 Conditions to Provable System-Wide Optimal Coordination ofConnected and Automated Vehicles [1, 2] . . . . . . . . . . . . . . . . 14

2.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 152.1.2 Constrained Optimization . . . . . . . . . . . . . . . . . . . . 19

2.1.2.1 Condition of Constraint Exclusion . . . . . . . . . . 202.1.2.2 Conditions of Constraint Activation . . . . . . . . . 22

2.1.3 Analytical Solution of Constrained Optimization . . . . . . . . 24

iii

2.1.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2 Decentralized Optimal Coordination of Connected and AutomatedVehicles for Multiple Traffic Scenarios [3] . . . . . . . . . . . . . . . . 40

2.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 402.2.2 Vehicle model, Constraints, and Assumptions . . . . . . . . . 422.2.3 Upper-level optimization problem . . . . . . . . . . . . . . . . 442.2.4 Low-level optimal control problem . . . . . . . . . . . . . . . . 47

2.2.4.1 State and control constraint is not active . . . . . . . 492.2.4.2 Rear-end safety constraint is active . . . . . . . . . . 50

2.2.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 52

2.3 Extension of the Preliminary Research . . . . . . . . . . . . . . . . . 56

3 PROPOSED DISSERTATION RESEARCH . . . . . . . . . . . . . . 59

3.1 Aim 1: Adaptive and Stochastic Coordination Approach . . . . . . . 613.2 Aim 2: Platoon-based Framework for Indirect Control of HVs . . . . 663.3 Aim 3: Optimal Control Framework with System Uncertainty . . . . 71

4 EXPECTED CONTRIBUTIONS . . . . . . . . . . . . . . . . . . . . 73

Appendix

A LIST OF PUBLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . 79B COURSEWORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81C CURRICULUM VITAE . . . . . . . . . . . . . . . . . . . . . . . . . . 82

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

iv

LIST OF TABLES

4.1 Provisional timeline to accomplish research aims [Spring, 2020 -Winter, 2020]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76



v

LIST OF FIGURES

2.1 Connected automated vehicles at a four-way intersection. . . . . . . 16

2.2 State and control unconstrained optimal profile for 1) ai < 0 (top)and 2) ai > 0 (bottom) cases. . . . . . . . . . . . . . . . . . . . . . 38

2.3 State constrained optimal speed (top-left) and acceleration(top-right) profile, and control constrained optimal speed(bottom-left) and acceleration (bottom-right) profile. . . . . . . . . 39

2.4 State and control constrained optimal speed (left) and acceleration(right) profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.5 Corridor with connected and automated vehicles. . . . . . . . . . . 41

2.6 Unconstrained (left) and rear-end safety constrained state trajectory(right) of CAV i with respect to its immediately preceding vehicle k. 53

2.7 The corridor in Mcity. . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.8 Vehicle trajectories inside the corridor for (a) baseline and (b)optimal controlled case. The control zone for each of the conflictzones are shown for comparison. . . . . . . . . . . . . . . . . . . . . 55

2.9 Accumulated fuel consumption over time for the baseline and optimalcontrolled vehicles. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.1 Corridor with connected and automated vehicles and human drivenvehicles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.2 Indirect control framework with connected and automated vehiclesand human driven vehicles. . . . . . . . . . . . . . . . . . . . . . . 66

vi

Chapter 1

INTRODUCTION AND LITERATURE REVIEW

1.1 Motivation

The growing demand for intelligent transportation systems and mobility solu-

tions worldwide are well-documented. About 55% of the world’s population currently

live in urban areas, which is expected to rise to 68% by 2050 [4]. Conventional mobility

technologies and traffic management systems cannot cope with this growing demand as

the capacity of the transportation networks are currently being overwhelmed. Based on

the 2014 Urban Mobility Report, Schrank et al. [5] reported that in 2014 congestion in

the US caused urban commuters to spend 6.9 billion additional hours on the road at an

additional cost of 3.1 billion gallons of fuel, resulting in a total cost estimated at $160

billion. A report by INRIX [6] projects the combined annual cost of traffic congestion

in US, UK, France and Germany to reach $293B by 2030, which is a 50% increase from

2013. According to a recent World Economic Forum report [7], traffic congestion has

cost the US economy nearly $87 billion in 2018. The transportation sector is one of the

major contributors to greenhouse gas emissions as well. According to the Inventory of

U.S. Greenhouse Gas Emissions and Sinks 1990–2017 [8], transportation accounted for

the largest portion (29%) of total U.S. greenhouse gas emissions in 2017. Congestion

at traffic scenarios such as ramp merging, roundabouts, intersections also consists of

a high degree of safety-risks leading to increasing traffic accidents. Apart from the

inherent complexity arising from a complicated driving maneuver in these scenarios,

additional safety-risks result from driver anxiety, fatigue, distraction and discomfort of

the manually driven vehicles [9]. The US National Highway Traffic Safety Administra-

tion (NHTSA) reports that in 2012 over 90% of accidents in the US were caused by

1

human mistakes. In the US, according to a recent NHTSA report [10], 36,560 people

lost their lives in road crashes in 2018, where 2,841 lives were claimed due to distracted

driving along with 9,378 speeding-related deaths.

Connectivity and automation in vehicles provide the most intriguing opportu-

nity for enabling users to better monitor transportation network conditions and make

better operating decisions. Connected and automated vehicles (CAVs) have attracted

considerable attention for the potential of improving mobility and safety along with

energy and emission reduction [11, 12]. With the current technological advancement,

the advent of CAVs adds a new dimension in terms of driver safety, passenger comfort,

throughput maximization and fuel efficiency. CAVs are able to subscribe to vehicle-

to-vehicle and vehicle-to-infrastructure communication, and derive and implement its

own optimal policy leading to possible emergence of automated highway system and

signal-free intersection eliminating the stop-and-go driving behavior. Tachet et al. [13]

indicated that transitioning from intersections with traffic lights to autonomous ones

has the potential of doubling capacity and reducing delays. CAVs can alleviate conges-

tion at the major transportation segments such as urban intersections, merging road-

ways, roundabouts, and speed reduction zones [14]. The impact of the CAV technology

has become evident by the ever increasing researches in developing feasible framework

and computational algorithm [15–19]. Investigating the impact of connected and auto-

mated vehicles on a transportation network and related implications on mobility and

safety has been of great interest in recent studies [20].

1.2 Literature Review

Several research efforts have been reported in the literature proposing opti-

mization and control approaches for coordinating CAVs at different traffic scenarios

that include merging at roadways and roundabouts, crossing intersections, cruising in

congested traffic, passing through speed reduction zones, and lane-merging or passing

2

maneuvers. Some earlier efforts have been reported in the literature proposing differ-

ent heuristic approaches for coordinating CAVs through urban bottlenecks including

fuzzy logic [21], genetic algorithms [22], and swarm optimization algorithms [23]. These

approaches are often computationally inefficient and pose difficulty for online imple-

mentation.

On the contrary, optimal control of CAVs to improve urban mobility and traffic

safety have gained traction in recent years. Both centralized and decentralized frame-

works are reported in the literature for optimal coordination of CAVs through traffic

congestion. In centralized approaches, there is at least one task in the system that is

globally decided for all vehicles by a single central controller, whereas in decentralized

approaches, the vehicles are treated as autonomous agents that collect traffic informa-

tion to optimize their specific performance criteria while satisfying physical constraints.

One of the main objectives of the optimal coordination of CAVs travelling through the

different traffic scenarios is to smooth the traffic flow by centralized or decentralized

vehicle control to reduce spatial and temporal speed variation and braking events, e.g.,

automated intersection crossing [17,24–27], cooperative merging [15,28,29], and speed

harmonization through optimal vehicle control [30].

1.2.1 Optimal Control Framework for CAV Coordination

One of the very early efforts in this direction was proposed by Athans [31]

for safe and efficient coordination of merging maneuvers with the intention to avoid

congestion. Since then, numerous approaches have been proposed on coordinating

CAVs to improve traffic flow [32–34], and to achieve safe and efficient control of traffic

through various traffic scenarios where potential vehicle collisions may happen [35–44].

In terms of energy impact, many studies have shown that significant fuel consumption

savings could be achieved through eco-driving and vehicle optimal control without

sacrificing driver safety [17, 28, 29, 45–47]. Considering near-future CAV deployment,

recent research work has also explored both traffic and energy implications of partial

3

CAV penetration under different transportation scenarios, e.g., [48–50].

In 2004, Dresner et al. [35] presented the use of the reservation scheme to control

a single intersection of two roads with vehicles traveling with similar speed on a single

direction on each road. Similar approaches have been reported in the literature for safe

and efficient coordination of CAVs at urban intersections; see [36–38, 51]. Colombo et

al. [51] constructed the invariant set for the control inputs that ensure lateral collision

avoidance. Some approaches have focused on coordinating vehicles at intersections

to improve the traffic flow [39, 40, 44]. The optimal control problem of coordinating

the CAV in intersection with the cost function involving travel time has also been

proposed [41–43, 52]. In [53], an energy optimal scheme has been proposed for CAVs

by probabilistic prediction of traffic lights in an intersection. Kim et al. [44] proposed a

model predictive control (MPC) based approach which allows each vehicle to optimize

its movement locally with a given cost function. Makarem et al. [54] formulated the

optimal control problem as a linear quadratic regulator (LQR), and solved it using

MPC. Qian et al. proposed a hierarchical control framework, where a high-level CAV

coordination, and a low-level multiobjective optimization scheme has been introduced.

Several papers have also focused on multi-objective optimization problems using a

receding horizon control solution either in centralized or decentralized approaches; see

[55–57].

1.2.2 A Review of Decentralized Optimal Control Framework

A decentralized optimal framework to minimize travel time has been proposed

in [58], where an optimization problem is solved to find the minimum time once the

merging sequence is determined. Kamal et al. [59] proposed Pontryagin minimum

principle (PMP) based numerical algorithms for CAV coordination in a signal-free

intersection. Dynamic programming (DP) has been used in [53, 60] to compute the

optimal control input, which is inherently not feasible for real-time application due to

high computational effort. Sciarretta et al. [61] developed an eco-driving controller for

4

CAVs for adaptive cruise control maneuver, where the optimal problem minimizes the

energy consumption with speed constraint. A speed advisory system has been pro-

posed in [62], where the longitudinal dynamics of each CAV is optimized to minimize

fuel consumption without considering the state and control constraints. In [60], the

authors provided a PMP-based speed profile optimization framework for minimizing

fuel consumption, where the CAV dynamics is not subject to any safety or accel-

eration/deceleration constraint. Detailed discussions of the research reported in the

literature to date on coordination of CAVs to improve vehicle-level operation can be

found in [20,63,64].

Recently, a decentralized optimal control framework was established for coor-

dinating online CAVs in different transportation segments. A closed-form, analytical

solution without considering state and control constraints was presented in [16], [65],

and [28] for coordinating online CAVs at highway on-ramps, in [66] at intersections,

and in [29] at roundabouts. The control framework of decentralized CAV coordination

has also been investigated for a traffic network with a corridor including interconnected

traffic scenarios [3, 67], where the safety constraints were guaranteed within the con-

trol zone along with interior boundary conditions. Similar analysis can also be found

in [68], where the impact of the control framework on isolated and interconnected

traffic scenarios were investigated. The solution of the unconstrained problem was

also validated experimentally at the University of Delaware scaled smart city using

robotic CAVs [69, 70] in a merging roadway scenario. The solution of the optimal

control problem considering state and control constraints was presented in [17] at an

urban intersection without considering rear-end collision avoidance constraint, and the

conditions under which the latter does not become active were presented in [30]. The

decentralized control framework for CAVs has also been rigorously tested and validated

in a real-world environment with an Audi A3 e-tron at the autonomous vehicle testing

facility Mcity as part of an ARPA-E NEXTCAR project [71,72]. These efforts consid-

ered a control zone inside of which the CAVs can communicate with each other and with

5

a coordinator who is not involved in any decision but it rather enables communication

of appropriate information among CAVs. Although the aforementioned frameworks

yield a closed-form analytic solution, they fail to account for any unwanted distur-

bances taking place within the control zone in a potential mixed traffic environment

of interacting HVs and CAVs. These frameworks do not consider model uncertainty,

noise and transmission latency as well.

Several decentralized optimal control framework can be found in recent litera-

ture [16, 48, 66], where the control problem to maximize throughput and energy effi-

ciency has been addressed. The solution to the state and control unconstrained opti-

mization problem discussed presented in these works shows acceleration spikes (jerk)

at the boundaries of the optimization horizon, possibly exceeding the vehicle’s phys-

ical limitation and giving rise to undesired passenger discomfort. Additionally, the

unconstrained optimal solution can only achieve the enforced speed at the predefined

terminal time, and fails to maintain the roadway speed limit within the control zone.

Therefore, the unconstrained optimal solution may not be admissible in real-world ap-

plication. To mitigate terminal jerk, Ntousakis et al. [28] reformulated the optimal

control problem with vehicle acceleration as an additional state and using jerk as the

control input. This approach provides safe state/control values at the terminal points,

but does not guarantee that the state and control constraints will not become active

within the optimization horizon. The optimal control problem considering state and

control constraints was addressed in [17,26,73] at an urban intersection, where result-

ing formulation relies on stitching the unconstrained and constrained arcs together,

and suggests recursive computation of the analytical solution until all of the constraint

activation are resolved. This procedure can lead to a complex structure of the solu-

tion process with computationally exhaustive numerical methods preventing possible

real-time implementation. Moreover, these works do not address the coupled nature

of different constraints. Several studies considered only partial constrained framework

for optimal coordination of CAVs. A PMP-based approach was employed in [60] for

6

optimizing the speed profile for fuel consumption minimization without considering the

acceleration or safety constraint. Han et al. [74] proposed a safety based eco-driving

control for CAVs without considering the control constraints in the formulation. Wang

et al. [75] formulated the multi-objective optimization problem for the CAVs approach-

ing intersection, and derived the state and control constrained solution with feasibility

approach rather than explicit inclusion into the Hamiltonian.

1.2.3 A Review of the Partial CAV Penetration Problem

In spite of the potential of all the added functionalities brought by the CAVs, the

realization of an automated merging scenario is restricted to a 100% CAV penetration

rate, where all the vehicles on the road are assumed to be CAVs. With the current

market and manufacturer trend, the existence of a 100% CAV penetration rate is not

estimated before 2060 [76]. As a result, the control frameworks developed for the CAVs

considering 100% CAV penetration are not feasible for real-world implementation, and

needs to be improved to take the mixed traffic environment into consideration, which

consists of interacting human-driven vehicles (HVs) and CAVs.

The impact of partial CAV penetration in terms of fuel consumption and traffic

throughput has been studied by Rios-Torres and Malikopoulos [77], where the authors

concluded that the fuel improvement is significant only in the case of 100% CAV pene-

tration rate near low traffic volume. Here, each CAV dynamics is optimized for energy

consumption which does not account for the uncertainties ensuing from the random

driving pattern of HVs modelled with a conventional car-following model [78]. The

longitudinal dynamics of each CAV is optimized for fuel consumption minimization

in [62] for mixed traffic environment in a signalized intersection, but the speed and ac-

celeration constraints were not considered while formulating the Hamiltonian. Jian [79]

investigated the effect on fuel consumption and traffic throughput in a signalized in-

tersection considering partial CAV penetration. The optimization problem was solved

using iterative method whereas the HV dynamics were modeled using intelligent driver

7

model car-following dynamics. Ntousakis et al. [28] formulated an LQR-based auto-

mated on-ramp merging framework to be solved by MPC, which takes into account

possible disturbances inside the control zone. They extended the problem to include

manually driven vehicles following a simple car-following dynamics. A distributed

MPC framework is proposed by Qian et. al. [80] for smooth transition of CAVs in an

automated intersection by selecting a quadratic running cost for penalizing the input

variable. The modeling of successive vehicle dynamics through MPC is computation-

ally exacting due to the limited computational capability of the vehicle on-board. A

common shortcoming of the reported frameworks [16, 18, 28, 79, 81] considering the

mixed traffic environment is that, the dynamics of the human driven vehicles are al-

ways modeled with simplified car-following models to numerically simulate the actual

human driven vehicles on the road. However, the stochastic driving pattern, coupled

car-following dynamics and other uncertainties related to the HVs has not been taken

into consideration in the mathematical formulation, leaving the frameworks vulnera-

ble to practical implementation. Complex vehicle dynamics for CAVs and HVs, and

trajectory uncertainties under different CAV penetration rate have not been address

in these papers as well. Therefore, although these approaches [28, 77, 79, 81, 82] deal

with the problem of CAV-HV interaction, a mathematically rigorous optimal control

framework is yet to be addressed in the literature to date for a stable traffic interaction

of CAVs and HVs in the context of an automated traffic coordination.

1.2.4 A Review of Platoon Formation and Stability

An important direction in the development and implementation of CAV tech-

nology is the platoon-based operation with the potential of additional energy and

travel-time benefits. Several research efforts have been reported in the literature in

terms of vehicle platoon formation and stability. The concept of forming platoons of

vehicles traveling at a high speed was a popular system-level approach to address traffic

congestion, that gained momentum in the 1980s and 1990s [83, 84]. Such automated

8

transportation system can alleviate congestion, reduce energy use and emissions, and

improve safety while increasing throughput significantly. The Energy ITS project in

Japan [85], the Safe Road Trains for the Environment program [86], and the California

Partner for Advanced Transportation Technology [87] are among the mostly-reported

efforts in this area. Maiti et al. [88] developed a formal model for vehicle platooning us-

ing ontology and maintained that the different derived operations like merge and split

can be expressed as an aggregate of primitive operations. Lioris et al. [89] analyzed

the potential mobility benefit from a platoon of CAVs under three different queuing

models. The authors concluded that at a signalized intersection, the formation of pla-

toon by the CAVs can increase the traffic throughput by double in contrast to the

conventional passing of the vehicles.

Both centralized and decentralized control framework have been addressed in

the literature regarding the string stability of the platoon. Kaku et al. [90] developed

a centralized controller for a platoon of three CAVs using LQR method for minimiz-

ing the overall fuel consumption of the platoon and compared its performance to the

decentralized one. Dunbar et al. [91] proposed a distributed receding horizon con-

trol for a platoon of CAVs having non-linear decoupled double integrator dynamics

and addressed the string stability of such platoon. The receding horizon control or

MPC formulation is applied to different vehicle connectivity topologies, namely leader-

follower and predecessor-follower topology. Details of this formulation can be found

in other literature [92, 93]. Morbidi et al. [94] synthesized an LQR control policy with

measurable disturbances for a two-vehicle platoon where the string stability is estab-

lished through controllers feedback and feed-forward gain. In addition, the H2,and H∞

performance criteria, denoting the group behavior and string stability of the platoon,

are simultaneously achieved using the proposed compensator-blending method. Wei

et al. [95] solved a distributed control policy using DP for maximizing vehicle through-

put in a signalized intersection with the vehicle reaction time as input and different

objective function. A simplified Newell’s car-following model has been considered to

9

model the inter-vehicle interaction and safety constraint, which addresses the coupled

dynamics of the CAV’s. Stankovic et al. [96] formulated a decentralized overlapping

control policy by decomposing the original system model and applied linear quadratic

optimization scheme to the locally extracted subsystems. Qiong et al. [97] formulated

the cooperative fuel-efficient vehicular platoon control problem using the distributed

receding horizon policy and solved it by Hamiltonian analysis.

In a string stable platoon, the following vehicles adjust their speed to return

the equilibrium position given an initial perturbation. In this area, Li et al. [82], Guo

et al. [98] proposed a hierarchical framework for cooperative braking of a CAV inside

a homogeneous vehicle platoon. The upper layer consists of a leading CAV with a

linear controller designed to ensure the asymptotic convergence of the vehicle’s Target

Set point after an braking event. On the other hand, the controllers for the follower

vehicles at the lower level is designed based on the Integrated sliding mode formulation

considering the constant spacing and constant time headway policy.

The literature addressing platoon stability and string stability mentioned so far

only considered distributed control of the each of the vehicles inside the platoon with

decoupled dynamics and an assumption of 100% penetration. As a result, the coupled

dynamics and string stability issue arising due to the interaction between CAVs and

HVs inside a platoon are yet to be fully investigated. On this note, the research

from Gong et al. [99] has been the most significant and pertinent in addressing the

coupled dynamics of a platoon of vehicles while considering the distributed control

based on constrained optimization. The error states are used to formulate a primal-

dual problem and solved using the dual based optimization scheme to ensure platoon

transient traffic smoothness with proven convergence. For an asymptotic dynamic

performance, a closed loop unconstrained linear system has also been established. Liu

et al. [81] designed a provably safe vehicle trajectory policy and coordination rules

for a longitudinal roadway under mixed traffic condition. The HVs are modeled with

car-following model while the CAVs follow the MPC protocol with the capability of

10

platoon formation. Li et al. [100] proposed a dedicated short range communication

based vehicle platoon control considering vehicle-to-everything communication. With

the leader-follower communication topology, stable platoon formation and merging was

achieved in field experiment validating the proposed framework. Ntousakis et al. [101]

reviewed the previous controller modeling efforts for the adaptive cruise control (ACC)

systems based on different car-following models, which gives an insight to the potential

coupled dynamics in a string of HVs.

1.3 Summary of Research Gaps in the Literature

Based on the literature review discussed above, we have outlined several gaps in

the state of the art of the optimal coordination of CAVs in different traffic scenarios.

We summarize the research gaps in the state of the art as follows.

1. Constrained optimization problem: The computational complexity to solve a

combined state, control and safety constrained decentralized optimal control framework

has not been addressed in the literature.

2. Traffic corridor problem: The decentralized coordination framework for

CAVs reported in the literature only considered single and isolated traffic scenario in the

formulation, e.g., on-ramp merging roadways, intersections, roundabouts etc. A com-

plete optimal control framework for a traffic corridor with multiple connected/unconnected

traffic scenarios has not been investigated in detail.

3. Mixed traffic scenario: The inherent problem of dealing with a mixed traffic

environment is the fact that the dynamics of HVs, with their embedded driving uncer-

tainties, are assumed to be controlled only by human drivers. This assumption inhibits

the formulation of a distributed control strategy during the automated CAV coordina-

tion at different traffic scenarios. Few papers in the literature only analyzed the impact

of mixed traffic environment without addressing the stochastic nature of the coupled

HV dynamics. To date, no rigorous mathematical framework has been presented in

the literature to address the problem of optimally coordinating CAVs through traffic

11

different scenarios in a mixed traffic environment with partial CAV penetration.

4. Formation and impact of mixed platoon: Although cooperative adaptive

cruise control and CAV platoon dynamics in a 100% CAV penetration setting have

been explored in the literature, formation and stability analysis of mixed platoons

consisting of CAVs and HVs has not been reported in the literature.

5. System uncertainty in a mixed traffic environment: The most relevant re-

search works regarding HV dynamics have reported in the literature in the context of

developing, calibrating and validating the car-following model to emulate the human

driving behavior. However, the interaction of CAVs and HVs in a mixed traffic envi-

ronment results in additional state estimation and HV dynamics prediction problem.

Predicting HV dynamics modeled with arbitrary car-following dynamics introduces

uncertainties in the system that has not been addressed in the literature.

6. Multi-lane operation: The bulk of the decentralized coordination framework

presented in the literature has been developed under the assumption of single lane

operation by considering only the longitudinal dynamics. The research efforts including

multi-lane scenario with lateral dynamics is sparse. Moreover, the HV car-following

dynamics has not been explored properly with the consideration of lateral lane change

dynamics.

7. Incomplete information and delay: Almost all of the optimal control frame-

works found in the literature with closed form analytic solution is formulated under

the assumption of instantaneous transmission of complete information among the con-

nected agents. A complete relaxation of such hard assumption has not been reported

in the literature in the context of coordination of CAVs and HVs.

In what follows, we address some of the research directions discussed above to

advance the state of the art. In the next chapter, we present some preliminary results

addressing the state and control constrained optimal control problem, and provide a

12

simplified framework that yields closed form analytic solution with higher computa-

tional efficiency (Section 2.1). Additionally, in Section 2.2, we address the problem of

coordinating CAV optimally through a corridor of multiple traffic scenarios with rear-

end and lateral collision avoidance constraints. Finally in Chapter 3, we address the

problem of optimal coordination of CAVs in a mixed traffic environment, and provide

a high-level exposition on research gaps 3, 4 and 5 as mentioned above.

13

Chapter 2

PRELIMINARY RESULTS

In this chapter, we present some of our preliminary research efforts on the de-

centralized time- and energy-optimal coordination of CAVs travelling through major

traffic congestion scenarios. In Section 2.1, we investigate the formulation presented

in [17] to provide useful insights about the state and control constraint activation to

increase computational efficiency. In this effort, we extend the formulation presented

in [17] to incorporate a simplified framework to derive the closed form analytical solu-

tion using the Hamiltonian analysis for real-time implementation without any recursive

procedure and numerical computation. In Section 2.2, we present a decentralized op-

timal control framework for CAVs coordination through multiple traffic scenarios such

as such as highway on-ramps, roundabouts, speed reduction zones, and urban intersec-

tions to increase the fuel efficiency. We formulate the upper-level CAV coordination

problem with dynamic re-sequencing queue, and explicitly consider rear-end collision

avoidance constraint in the low-level trajectory optimization problem.

2.1 Conditions to Provable System-Wide Optimal Coordination of Con-

nected and Automated Vehicles [1, 2]

In this section, we investigate the nature of the unconstrained problem and pro-

vide conditions under which the state and control constraints become active. We derive

a closed-form analytical solution of the constrained optimization problem and evaluate

the solution using numerical simulation. To this end, we state the main objectives of

this research as follows. In this research effort, we (1) provide conditions under which

the constraint activation space can be reduced, (2) provide conditions under which

14

specific combination of state and control constraint activation can be identified to

avoid redundant computation, (3) derive a closed-form analytical solution for the con-

strained optimization problem under different cases of constraint activation identified

in (2), and finally (4) validate the effectiveness of the analysis and the corresponding

optimal solution in simulation environment.

We advance the state-of-the-art with the following contributions. We (1) provide

an in-depth exposition of the characteristics of different cases of state and control

constraint activation, (2) eliminate the necessity for recursive solution structure for

the state and control constrained optimal control problems for CAV coordination by

considering the constraint activation conditions, and (3) increase the computational

efficiency for solving the state and control constrained optimal control problem by

presenting the closed-form analytical solution.

2.1.1 Problem Formulation

Although our theoretical framework can be applied to any traffic scenario, e.g.,

merging at roadways [28] and roundabouts, passing through speed reduction zones [30],

we use an intersection (Fig. 2.1) as a reference to present the fundamental ideas

and results of the developed framework, since an intersection provides unique features

making it technically more challenging compared to other traffic scenarios.

We define the area illustrated by the red square of dimension S in Fig. 2.1 as

the merging zone where possible lateral collision may occur. Upstream of the merging

zone, we define a control zone of length L, inside of which CAVs can coordinate with

each other before they pass the merging zone. The intersection has a coordinator that

only maintains the queue of the CAVs inside the control zone, and is not involved in

any decision-making process. When a CAV enters the control zone, the coordinator

receives its information and assigns a unique identity i ∈ N to it. The objective of each

CAV is to derive its optimal control input to cross the intersection without activating

any of the safety, state and control constraints.

15

Figure 2.1: Connected automated vehicles at a four-way intersection.

Vehicle Dynamics and Constraints: LetN (t) = {1, ..., N(t)}, where N(t) ∈N is the number of CAVs inside the control zone at time t ∈ R+, be the queue of CAVs

inside the control zone. We model each CAV i ∈ N (t) as a double integrator

pi = vi(t),

vi = ui(t), ∀t ∈ [t0i , tfi ], (2.1)

where pi(t) ∈ Pi, vi(t) ∈ Vi, and ui(t) ∈ Ui denote the position, speed and acceleration

(control input) of each CAV i in the corridor. The sets Pi, Vi, and Ui, i ∈ N (t),

are complete and totally bounded subsets of R. Let xi(t) = [pi(t) vi(t)]T denote the

state vector of each CAV i ∈ N (t), with initial value x0i = [p0

i v0i ]T

taking values in

Xi = Pi × Vi. The state space Xi for each CAV i ∈ N (t) is closed with respect to the

induced topology on Pi × Vi and thus, it is compact. Each CAV i enters the control

zone at t0i , enters the merging zone at tmi , and exits the merging zone at tfi .

To ensure that the control input and vehicle speed are within a given admissible

16

range, the following constraints are imposed

umin ≤ ui(t) ≤ umax, and

0 ≤ vmin ≤ vi(t) ≤ vmax, ∀t ∈ [t0i , tfi ], (2.2)

where umin, umax are the minimum and maximum acceleration for each CAV i ∈ N (t),

and vmin, vmax are the minimum and maximum speed limits respectively.

We impose the following rear-end safety constraint

si(t) = pk(t)− pi(t) ≥ δi(t), ∀t ∈ [t0i , tfi ], (2.3)

where si(t) denotes the distance of CAV i from CAV k which is physically immediately

ahead of i, and δi(t) denotes minimum safe distance which is a function of speed vi(t).

In the modeling framework described above, we impose the following assump-

tions:

Assumption 1. Each CAV is equipped with sensors to measure and share their

local information.

Assumption 2. Communication among CAVs occurs without any delays or

errors.

Assumption 1 restricts the problem to the case of 100% CAV penetration rate.

Assumption 2 may be strong, but it is relatively straightforward to relax as long as the

noise in the measurements and/or delays is bounded. For example, we can determine

upper bounds on the state uncertainties as a result of sensing or communication errors

and delays, and incorporate these into more conservative safety constraints.

Upper-Level Vehicle Coordination Problem: The queue of CAVs inside

the control zone N (t) can be determined as an outcome of an upper-level vehicle

coordination problem as described in [26, 102]. In what follows, we assume that the

time tmi that each CAV enters the merging zone is determined as the solution of some

upper-level coordination problem, and will focus only on a low-level energy optimization

17

problem that will yield the optimal control input ui(t) for each CAV i ∈ N (t) to achieve

the assigned merging time tmi .

Low-Level Energy Optimization Problem: For each CAV i ∈ N (t) travel-

ing inside the control zone, we formulate an optimal control problem to minimize the

L2-norm of the control input, i.e., 12u2i (t), so that the transient engine operation can

be minimized, and thus, fuel consumption [17],

minui∈Ui

Ji(u(t)) =

∫ tmi

t0i

1

2u2i (t) dt, (2.4)

subject to : (2.102), (2.2),

and given t0i , pi(t0i ) = 0, vi(t

0i ), t

mi , pi(t

mi ).

Note that, we do not provide a desired speed vi(tmi ) at tmi . Additionally, we do not

explicitly include the lateral and rear-end (2.3) safety constraints. We enforce the

lateral collision constraint by selecting the appropriate merging time tmi for each CAV

i in the upper-level vehicle coordination problem. The activation of rear-end safety

constraint can be avoided under proper initial conditions [t0i , vi(t0i )] as it was shown

in [30]. From (2.4) and the state equations (2.102), we formulate the state and control

constraint (2.2) adjoined Hamiltonian functionHi

(t,λ(t),µ(t),x(t), u(t)

)for each CAV

i ∈ N (t),

Hi

(t,λ(t),µ(t),x(t), u(t)

)=

1

2u2i + λpi · vi + λvi · ui

+µai · (ui − umax) + µbi · (umin − ui)

+µci · (vi − vmax) + µdi · (vmin − vi), ∀t ∈ [t0i , tmi ], (2.5)

where λpi , λvi are the co-state components corresponding to the state vector xi(t), and

µai , µbi , µ

ci , µ

di are the Lagrange multipliers satisfying the following KKT conditions,

µai =

> 0, ui(t)− umax = 0,

= 0, ui(t)− umax < 0,(2.6)

18

µbi =

> 0, umin − ui(t) = 0,

= 0, umin − ui(t) < 0,(2.7)

µci =

> 0, vi(t)− vmax = 0,

= 0, vi(t)− vmax < 0,(2.8)

µdi =

> 0, vmin − vi(t) = 0,

= 0, vmin − vi(t) < 0.(2.9)

The Euler-Lagrange equations are

λpi (t) = −∂Hi

∂pi= 0, (2.10)

λvi (t) = −∂Hi

∂vi= −λpi − µci + µdi , (2.11)

∂Hi

∂ui= ui + λvi + µai − µbi = 0. (2.12)

Unconstrained Optimization: If the inequality state and control constraints

(2.2) are not active, we have µai = µbi = µci = µdi = 0. From (2.10) and (2.12), for

each CAV i ∈ N (t) we derive the optimal control input u∗i (t), and the optimal states

p∗i (t), v∗i (t) as

u∗i (t) = ai · t+ bi, ∀t ∈ [t0i , tmi ], (2.13)

v∗i (t) =1

2ai · t2 + bi · t+ ci, ∀t ∈ [t0i , t

mi ], (2.14)

p∗i (t) =1

6ai · t3 +

1

2bi · t2 + ci · t+ di, ∀t ∈ [t0i , t

mi ], (2.15)

where ai, bi, ci, and di are constants of integration which can be computed by using

the boundary conditions in (2.4).

2.1.2 Constrained Optimization

To derive a closed-form analytical solution for (2.5), we (1) identify the con-

ditions for constraint(s) exclusion, (2) define the condition under which they become

active, and (3) derive the final constrained optimal solution without recursion.

19

2.1.2.1 Condition of Constraint Exclusion

Although we have two state and two control constraints from (2.2), there are 15

constraint combinations in total that can be activated. In this section, we show that

it is only possible for a subset of the constraints to be active within an unconstrained

solution. Therefore, it is not necessary to consider all the cases in the constrained

problem. In what follows, we delve deeper into the nature of the uncontrolled opti-

mal solution to derive useful information about the possible existence of constraint

activation within the control zone.

Lemma 1. If vi(tmi ) is not prescribed, and the terminal time tmi is fixed, then

tmi = − biai. (2.16)

Proof. For fixed terminal time tmi and unspecified state vi(tmi ), λvi (t

mi ) = 0. Using

λvi (tmi ) in (2.13), we have ui(t

mi ) = ait

mi + bi = 0, and the result follows.

Corollary 1. The constants ai and bi always have opposite signs.

Lemma 2. The unconstrained optimal control input ui(t) is linearly decreasing if

vi(t0i ) <

(pi(tmi )−pi(t0i ))

tmi. The unconstrained optimal control input ui(t) is linearly in-

creasing if vi(t0i ) >


tmi.

Proof. From (2.14) and (2.15), we can write vi(t0i ) = 1

2ai · (t0i )

2 + bi · t0i + ci and

pi(t0i ) = 1

6ai · (t0i )3 + 1

2bi · (t0i )2 + ci · t0i + di. With t0i = 0, we have

ci = vi(t0i ), di = pi(t

0i ). (2.17)

From (2.15), we have pi(tmi ) = 1

6ai · (tmi )3 + 1

2bi · (tmi )2 + ci · tmi + di. Using the results

from (2.17), and (2.16) in the above equation and solving for ai, we have

ai =3(vi(t

0i )t

mi − (pi(t

mi )− pi(t0i )))

(tmi )3. (2.18)

Since tmi > 0, ai is non positive if vi(t0i )t

mi − (pi(t

mi )− pi(t0i )) < 0, which in turn implies

negative slope of the linear control input u∗i (t), i.e., linearly decreasing acceleration

profile. The second part of Lemma 2 can be proved following similar steps.

20

Lemma 3. For a given unconstrained optimal profile, only a subset of state-control

constraints can remain active. Furthermore, if either vi(t)−vmax ≤ 0 or ui(t)−umax ≤ 0

is activated, none of the constraint pairs vmin − vi(t) ≤ 0 and umin − ui(t) ≤ 0 can be

activated. The reverse also holds.

Proof. The optimal solution derived from the unconstrained case yields a linear control

profile increasing or decreasing to the terminal value zero (Lemma 2). If the acceleration

profile is linearly decreasing, we have ui(t) = ait+bi ≥ 0 > umin, ∀t ∈ [t0i , tmi ], implying

that the constraint umin−ui(t) ≤ 0 will never be activated. Now, applying the necessary

condition of optimality in (2.14), we have ∂vi(t)∂t

= ait+bi = 0, which yields the inflection

point at time t = − biai

corresponding to the vertex of the concave parabola of (2.14),

and equal to tmi according to Lemma 1. As the inflection point of the concave parabola

is located at tmi and vmin < v0i < vmax, we have vi(t) > vmin, ∀t ∈ [t0i , t

mi ]. This

concludes the proof of the first part of Lemma 3. The second part of Lemma 3 can be

proved following similar steps.

Remark 1. The sign of the constant ai provides insight for the possible state and

control constraint activation.

The following result provides the condition under which certain state and control

constraints never become active.

Theorem 1. (i) The state constraint vmin−vi(t) ≤ 0 and the control constraint umin−ui(t) ≤ 0 are not activated if vi(t

0i ) <


tmi. (ii) The state constraint vi(t) −

vmax ≤ 0 and the control constraint ui(t) − umax ≤ 0 are not activated, if vi(t0i ) >


tmi.

Proof. If vi(t0i ) <


tmiholds, then from Lemma 2, ai is negative and the optimal

control input ui(t) is linearly decreasing. From Lemma 3, a decreasing optimal control

input ui(t) indicates that the state constraint vmin−vi(t) ≤ 0 and the control constraint

umin − ui(t) ≤ 0 will never be activated, which concludes the proof of the first part.

The second part of Theorem 1 can be proved following similar procedure.

21

2.1.2.2 Conditions of Constraint Activation

The following results provide the condition for which specific state and control

constraints are activated.

Lemma 4. Activation of the control constraint ui(t) − umax ≤ 0 or umin − ui(t) ≤ 0

occurs at time t = t0i .

Proof. For ai < 0, there is a possibility that either the state constraint vi(t)−vmax ≤ 0,

or the control constraint ui(t)− umax ≤ 0, or both to be activated (Lemma 3). In this

case, the control input linearly decreases to zero at time t = tmi . Therefore, any

activation of the control constraint ui(t) − umax ≤ 0 occurs at time t = t0i . Following

similar reasoning, it can be proved that the activation of umin − ui(t) ≤ 0 occurs at

time t = t0i

Theorem 2. (i) If ai < 0, the state constraint vi(t) − vmax ≤ 0 is activated if tmi <

3(pi(tmi )−pi(t0i ))

v0i +2vmax. (ii) If ai > 0, the state constraint vmin − vi(t) ≤ 0 is activated if

tmi >3(pi(t

mi )−pi(t0i ))

v0i +2vmin.

Proof. Suppose that for ai < 0, there exists a time ts ∈ (t0i , tmi ] at which the state

constraint vi(t) − vmax ≤ 0 is activated. Using (2.17), from (2.14) we have 12ait

2s +

bits + v0i = vmax. Solving for ts, we have ts =

−2bi±√

4b2i−8ai·(v0i−vmax)

2ai, yielding two

possible solutions, either ts,1 = tmi +√

4b2i−8ai·(v0i−vmax)

4a2ior ts,2 = tmi −

√4b2i−8ai·(v0i−vmax)

4a2i.

Since ts,1 is not feasible as√

4b2i − 8ai · (v0

i − vmax) cannot be negative or equal to zero

to satisfy ts < tmi , ts,2 is the only acceptable solution. To satisfy ts < tmi , we have√4b2i − 8ai · (v0

i − vmax) > 0 resulting in ai <2(v0i−vmax)

(tmi )2. Using (2.18), the first part of

Theorem 2 follows. The second part of Theorem 2 can be proved for ai > 0 following

similar arguments.

Theorem 3. (i) For ai < 0, the control constraint ui(t)−umax ≤ 0 is activated if tmi <−3v0i +

√9(v0i )2+12umax·(pi(tmi )−pi(t0i ))

2umax. (ii) For ai > 0, control constraint umin − ui(t) ≤ 0 is

activated if tmi <3v0i +√

9(v0i )2+12‖umin‖(pi(tmi )−pi(t0i ))

2‖umin‖ .

22

Proof. Based on Lemma 4, the control constraint activation occurs at t = t0i . For

ai < 0, we have ui(t0i ) = ait

0i + bi > umax. Without loss of generality, for t0i = 0, we

have bi = −aitmi > umax. Substituting ai from (2.18), we obtain umax · (tmi )2 + 3v0i tmi −

3(pi(tmi )− pi(t0i )) < 0. Solving the above quadratic term for tmi , we obtain

(tmi +3v0

i +√

9(v0i )

2 + 12umax · (pi(tmi )− pi(t0i ))2umax

)

·(tmi +3v0

i −√

9(v0i )

2 + 12umax · (pi(tmi )− pi(t0i ))2umax

) < 0. (2.19)

The first term of (2.19) is always positive since√

9(v0i )

2 + 12umax · (pi(tmi )− pi(t0i )) >0. Therefore, the second part has to be negative, and the result follows. Following

similar steps, the second part of Theorem 3 can be proved for ai > 0.

We have discussed so far the conditions under which the state and control

constraints are activated individually. With this information, we can solve the con-

strained optimization problem and derive corresponding analytical solutions, which

we will present in the following section. However, there is a possibility that the con-

strained optimal solution may result in additional activation of constraint arcs, i.e., a

state constrained optimal solution might cause a control constraint activation. Simi-

larly, a control constrained optimal solution might cause a state constraint activation.

In such cases, the optimization problem has to be resolved by piecing all constrained

arcs together to derive the corresponding solution. To overcome this recursive proce-

dure, we can identify a priori under which conditions whether other constraint arcs

might be activated.

Theorem 4. (i) The constrained optimal solution corresponding to the state constraint

vi(t) − vmax ≤ 0 with a junction point at t = τs may result in activating the control

constraint ui(t)−umax ≤ 0, if τs <−3v0i +

√9(v0i )2+12umax·(pi(τs)−pi(t0i ))

2umax. (ii) The constrained

optimal solution corresponding to state constraint vmin − vi(t) ≤ 0 with a junction

point at t = τs may result in activating the control constraint umin − ui(t) ≤ 0, if

τs <3v0i +√

9(v0i )2+12‖umin‖(pi(τs)−pi(t0i ))

2‖umin‖ .

23

Proof. Suppose that the state constrained (vi(t) − vmax ≤ 0) solution has a junction

point between the constrained and unconstrained arc at t = τs, where ui(τs) = 0.

Therefore, we focus on the unconstrained arc of the constrained solution with the

time horizon [t0i , τs] instead of [t0i , tmi ]. Following similar procedure as in the proof

of Theorem 3, we can construct the condition of the first part of Theorem 4 that

indicates whether the control constraint ui(t)− umax ≤ 0 is being activated within the

time horizon t ∈ [t0i , τs]. The second part of Theorem 4 can be proved following similar

arguments.

Theorem 5. (i) The constrained optimal solution corresponding to the control con-

straint ui(t) − umax ≤ 0 with a junction point at t = τc may result in activating the

state constraint vi(t) − vmax ≤ 0, if tmi < τc +3(pi(t

mi )−pi(τc))

v0i +umaxτc+2vmax. (ii) The constrained

optimal solution corresponding to the control constraint umin − ui(t) ≤ 0 with a junc-

tion point at t = τc may result in activating the state constraint vmin − vi(t) ≤ 0, if

tmi > τc +3(pi(t

mi )−pi(τc))

v0i +uminτc+2vmin.

Proof. Suppose that the control constrained (ui(t)−umax ≤ 0) solution has a junction

point between the constrained and unconstrained arc at t = τc > t0i . Therefore,

we focus on the unconstrained arc of the control constrained solution with the time

horizon [τc, tmi ] instead of [t0i , t

mi ]. Following similar procedure to the proof of Theorem

2, we can construct the condition stated in the first part of Theorem 5 which indicates

whether the state constraint vi(t)−vmax ≤ 0 is being activated within the time horizon

t ∈ [τc, tmi ]. Similar approach can be employed to prove the second part of Theorem

5.

2.1.3 Analytical Solution of Constrained Optimization

To derive the analytical solution of (2.4) using Hamiltonian analysis, we follow

the standard methodology used in optimal control problems with control and state

constraints [103] and [104]. We first start with Theorem 1 to reduce the set of possible

constraint activations. Then we evaluate the conditions in Theorem 2 and 3 to check

24

for possible constraint activation. If no activation is identified, we simply derive the

unconstrained arc. If any contraint activation is detected by checking the conditions

in Theorem 2 and 3, we then further check the conditions in Theorem 4 and 5 to

identify any additional constraint activation that might ensue from the previous case.

Once the nature of the current and additional constraint activation is specified, we

then piece together the relevant unconstrained and constrained arcs that yield a set of

algebraic equations which are solved simultaneously using the boundary conditions of

and interior conditions between the arcs.

Since we stitch together multiple constrained and unconstrained arcs together,

we denote the constant parameters corresponding to each arc as a group a(p)i , b

(p)i , c

(p)i , d

(p)i ,

where p represents the position of the arcs in terms of their appearance in the optimal

solution.

At time t = τ , we have a junction point between the unconstrained and con-

strained arcs due to state/control constraint activation. Note that, this junction point

will be an entry point from the unconstrained to constrained arc for the state constraint

activation, and an exit point from the constrained to unconstrained arc for the control

constraint activation. In our case, we do not have secondary junction point where the

state and control trajectory exits the constrained arc for state constraint activation.

According to Lemma 4, the control constraint activation takes place at t = t0i . Hence

there is not entry point to the constrained arc either in this case. Let τ− represent

the time instance just before τ , and τ+ signifies just after τ . The state trajectories are

always continuous at the junction points. Thus, we have

pi(τ−) = pi(τ

+) (2.20)

vi(τ−) = vi(τ

+). (2.21)

However, in optimal control problems involving inequality constraints on state and

control, we may encounter discontinuity in the control trajectory at the junction points.

25

The q-component vector function of tangency constraints N(xi(t), t) is

N(xi(t), t) =

Si(xi(t), t)

S(q−1)i (xi(t), t)

. (2.22)

The state trajectory entering onto the 1st-order speed constraint boundary must satisfy

the either of the following conditions on the tangency constraints based on the nature

of the activated state constraint,

Ni(xi(τ), τ) = vi(τ)− vmax = 0, or (2.23)

Ni(xi(τ), τ) = vmin − vi(τ) = 0. (2.24)

Additionally, the q derivative of Si(xi(τ), τ) will also vanish on the constraint arc such

that, S(q)i (xi(τ), τ) = 0. The tangency constraints in (2.23)-(2.24) also applies to the

state trajectory leaving the constraint boundary. The equations in (2.121) form a set

of interior boundary conditions where the co-states λpi (t) and λvi (t) in general have

discontinuity at the junction points, i.e., entry and exit points of the state trajectory

between the constrained and the unconstrained arcs. However, the control trajectory

may or may not have discontinuities at the junction points.

For the problem with state variable inequality constraints, we need to consider

the jump conditions to asses the discontinuities of the costates and Hamiltonian as,

λ(τ−)− λ(τ+) = πT∂N(t, x(t))

∂x

∣∣∣∣t=τ

, (2.25)

H(τ+)−H(τ−) = πT∂N(t, x(t))

∂t

∣∣∣∣t=τ

, (2.26)

where, πT is a vector of constant Lagrange multipliers. In the following, we address the

jump conditions at the junction points. For the problem with control variable inequality

constraints, we need to consider the jump conditions to asses the discontinuities of the

costates and Hamiltonian as,

λ(τ−)− λ(τ+) = 0, (2.27)

H(τ+)−H(τ−) = 0. (2.28)

26

Note that, from (2.27) and (2.28) both the costates and the lagrangian of the hamilto-

nian are continuous at the junction point for the case of control constraint activation.

This is due to the fact that, the tangency constraint is not an explicit function of

the control variable. The problem with both state and control variable inequality

constraint is subject to the following condition at the junction points,

Hu(τ+)−Hu(τ

−) = 0. (2.29)

We now consider the different cases of state and control constraint activations

to derive the optimal profile for the CAVs. Due to space limitation, we only present

derivation pertaining to the state constraint vi(t)− vmax(t) ≤ 0 and control constraint

ui(t)− umax(t) ≤ 0 activation.

Case 1: Only the state constraint vi(t)− vmax ≤ 0 is activated

In this case, we have µai = µbi = µdi = 0. The corresponding necessary condition

for optimality from (2.115), and the Euler-Lagrangian equations (2.10) and (2.11) for

the costates become

ui + λvi = 0, (2.30)

λpi = 0, (2.31)

λvi = −λpi − µci . (2.32)

Let’s consider that at time t = τs the state constraint vi(t) − vmax ≤ 0 is

activated, and let τ−s and τ+s be the time instance just before and after entering the

constraint arc. Since the q derivative of Si(xi(τs), τs) needs to vanish at the constraint

arc, we have S(1)i (xi(τs), τs) = ui(τs) = 0. Hence, from (2.23), we get the optimal speed

and control at the junction point as

v∗i (τs) = vmax, (2.33)

u∗i (τs) = 0. (2.34)

27

Using (2.25) and (2.26), we determine the jump conditions of the costates and the

Hamiltonian, the jump conditions at τs can be written as

λpi (τ−s ) = λpi (τ

+s ) + πT · ∂

∂pi(t)

[vi(t)− vmax(t)

] ∣∣∣∣t=τs

,

⇒ λpi (τ−s ) = λpi (τ

+s ) (2.35)

λvi (τ−s ) = λvi (τ

+s ) + πT · ∂

∂vi(t)

[vi(t)− vmax(t)

] ∣∣∣∣t=τs

,

⇒ λvi (τ−s ) = λvi (τ

+s ) + π. (2.36)

H(τ−s ) = H(τ+s )− πT · ∂

∂t

[vi(t)− vmax(t)

] ∣∣∣∣t=τs

. (2.37)

According to (2.29), the following condition must hold at the junction point t = τs,

∂H(t)

∂ui(t)

∣∣∣∣t=τ−s

=∂H(t)

∂ui(t)

∣∣∣∣t=τ+s

,

⇒ ui(τ−s ) + λvi (τ

−s ) = ui(τ

+s ) + λvi (τ

+s ). (2.38)

In (2.35)-(2.38), πT = [π] is a q-component vector of constant Langrange multipliers

to be determined so that the condition in (2.23) is satisfied. Note that, (2.35)-(2.37)

imply possible discontinuity of the costates and the Lagrangian of the Hamiltonian at

t = τs, whereas the state variables remain continuous from (2.20)-(2.123). From (2.35)

and (2.37), the position costate and the Lagrangian of the Hamiltonian is continuous

at t = τs. Hence, equation (2.37) yields,

1

2u2i (τ−s ) + λpi (τ

−s )vi(τ

−s ) + λvi (τ

−s )ui(τ

−s ) + µci(τ

−s )(vi(τ

−s )− vmax)

=1

2u2i (τ

+s ) + λpi (τ

+s )vi(τ

+s ) + λvi (τ

+s )ui(τ

+s ) + µci(τ

+s )(vi(τ

+s )− vmax) (2.39)

We use the following results to further simplify (2.39). From (2.33) and (2.67), we

have vi(τ+s ) = vmax and ui(τ

+s ) = 0 respectively on the constraint arc. With ad-

ditional information of the continuity of state (2.123) and position costate (2.35),

we have λpi (τ−s )vi(τ

−s ) = λpi (τ

+s )vi(τ

+s ). Finally, the KKT condition in (2.8) gives

28

µci(τ−s )(vi(τ

−s ) − vmax) = µci(τ

+s )(vi(τ

+s ) − vmax) = 0. With the above information at

hand, (2.39) simplifies to

1

2u2i (τ−s ) + λvi (τ

−s )ui(τ

−s ) = 0. (2.40)

From (2.40), we can have either ui(τ−s ) = 0 or 1

2ui(τ

−s ) + λvi (τ

−s ) = 0. Since the second

term contradicts (2.30), we have ui(τ−s ) = 0. Using ui(τ

−s ) = 0 in (2.38), we have

λvi (τ−s ) = λvi (τ

+s ) implying the continuity of λvi (τs) at t = τs, and we have π = 0

in (2.36). Using the terminal condition of λvi (t), we can write λvi (tmi ) = λvi (τs) = 0.

Finally, from (2.32), we have,

µci(t) =

0, if vi(t) < vmax,

−λpi , if vi(t) = vmax.(2.41)

Using (2.30), (2.31), (2.32), (2.41), the initial, final conditions, and the terminal

transversality condition at t = tmi , we now formulate a set of equations for stitching

the unconstrained and constrained arcs together at t = τs

1

2a

(1)i · (t0i )2 + b

(1)i · t0i + c

(1)i = vi(t

0i ), (2.42)

1

6a

(1)i · (t0i )3 +

1

2b

(1)i · (t0i )2 + c

(1)i · t0i + d

(1)i = pi(t

0i ), (2.43)

1

2a

(1)i · (τs)2 + b

(1)i · τs + c

(1)i = vmax (2.44)

a(1)i · τs + b

(1)i = 0, (2.45)

1

6a

(1)i · (τs)3 +

1

2b

(1)i · (τs)2 + c

(1)i · τs + d

(1)i + vmax · (tmi − τs) = pi(t

mi ), (2.46)

1

2a

(2)i · (τs)2 + b

(2)i · τs + c

(2)i = vmax, (2.47)

a(2)i · tmi + b

(1)i = 0, (2.48)

a(2)i · τs + b

(2)i = 0, (2.49)

1

6a

(2)i · (t0i )3 +

1

2b

(2)i · (t0i )2 + c

(2)i · t0i + d

(2)i = pi(t

mi ), (2.50)

where a(1)i , b

(1)i , c

(1)i , d

(1)i are the constants of integration for the unconstrained arc, and

a(2)i , b

(2)i , c

(2)i , d

(2)i are the constants of integration for the constrained arc. The constants

29

of integration and the optimal junction point τ ∗s can be determined numerically from

the above set of equations. However, if the optimal junction point τ ∗s can be derived

explicitly from the known initial parameters, then the above set of equations can be

expressed as a closed-form analytical solution, and solved by simple matrix inversion.

In the following Lemma, we state the nature of the optimal junction point τ ∗s and the

explicit expression.

Lemma 5. The Optimal junction point τ ∗s between the unconstrained arc and the con-

strained arc for vi(t)− vmax ≤ 0 activation is an explicit function of pi(tmi ), vmax, t

mi ,

and vi(t0i ), can be expressed as,

τ ∗s =3(pi(t

mi )− vmax · tmi )

(vi(t0i )− vmax)(2.51)

Proof. For the state constraint vi(t)− vmax ≤ 0 activation, we have an unconstrained

arc (with constant parameters a(1)i , b

(1)i , c

(1)i , d

(1)i ) followed by a constrained arc (with

constant parameters a(2)i , b

(2)i , c

(2)i , d

(2)i ) stitched together at the optimal junction point

t = τ ∗s . Let ai, bi, ci, di be the parameters of the unconstrained problem derived from

(2.13)-(2.15), where ai and bi are explicit function of pi(tmi ), vi(t

0i ) and t

mi , ci = vi(t

0i )

and di = 0 from Lemma 1 and 2. We resolve the constrained arc at the boundaries

t = τ ∗s and t = tmi ,

a(2)i · τ ∗s + b

(2)i = 0, (2.52)

a(2)i · tmi + b

(2)i = 0, (2.53)

1

2a

(2)i · (tmi )2 + b

(2)i · (tmi ) + c

(2)i = vmax (2.54)

1

6a

(2)i · (τ ∗s )3 +

1

2b

(2)i · (τ ∗s )2 + c

(2)i · (τ ∗s ) + d

(2)i + vmax · (tmi − τ ∗s ) = pi(t

mi ). (2.55)

From (2.52) and (2.53), we have a(2)i = 0 and b

(2)i = 0. Using this result in (2.54), we

get c(2)i = vmax. Finally, from (2.55) we get d

(2)i = (pi(t

mi )− vmax · tmi ). Now, we have

the unconstrained arc at the initial condition t = t0i as,

1

2a

(1)i · (t0i )2 + b

(1)i · (t0i ) + c

(2)i = vmax (2.56)

1

6a

(1)i · (t0i )3 +

1

2b

(1)i · (t0i )2 + c

(1)i · (t0i ) + d

(1)i = pi(t

0i ). (2.57)

30

Resolving (2.56) and (2.57) at t = t0i = 0, we have c(2)i = vi(t

0i ) and d

(2)i = 0. At τ ∗s , we

have the following set of equations for the unconstrained arc,

a(1)i · τ ∗s + b

(1)i = 0, (2.58)

1

2a

(1)i · (τ ∗s )2 + b

(1)i · τ ∗s + (vi(t

0i )− vmax) = 0, (2.59)

1

6a

(1)i · (τ ∗s )3 +

1

2b

(1)i · (τ ∗s )2 + (vi(t

0i )− vmax) · τ ∗s − (pi(t

mi )− vmax · tmi ) = 0. (2.60)

Inserting τ ∗s = − b(1)i

a(1)i

from (2.58) to (2.59) and simplifying, we have

(b(1)i )2

a(1)i

= 2(vi(t0i )− vmax). (2.61)

Inserting τ ∗s = − b(1)i

a(1)i

from (2.58) to (2.60) and simplifying, we have

1

3

(b(1)i )3

(a(1)i )2

+(b

(1)i )

a(1)i

· (vmax − vi(t0i ))− (pi(tmi )− vmax · tmi ) = 0. (2.62)

Finally, using the result from (2.61) into (2.62), we obtain

τ ∗s = −3(pi(tmi )− vmax · tmi

(vmax − vi(t0i ), (2.63)

where τ ∗s is an explicit function of the known parameters pi(tmi ), vmax, vi(t

0i ) and tmi .

Case 2: Only the control constraint ui(t)− umax ≤ 0 is activated

In this case, we have µbi = µci = µdi = 0. The corresponding necessary condition

for optimality, and the Euler-Lagrangian equations (2.10) and (2.11) for the costates

become

ui + λvi + µai = 0, (2.64)

λpi = 0, (2.65)

λvi = −λpi . (2.66)

31

According to Lemma 4, CAV i will enter the constrained arc at t = t0i . Let us assume

that at a time τc > t0i , the vehicle leaves the constrained arc. We denote τ−c and τ+c as

the immediate left and the right side of τc. On the constraint arc, we have the optimal

control input at the junction point as

u∗i (τc) = umax. (2.67)

At this entry point, using the jump conditions (2.25) and (2.26), we assess the

discontinuities of the costates and the Hamiltonian,

λpi (τ−c )− λpi (τ+

c ) = 0, (2.68)

λvi (τ−c )− λvi (τ+

c ) = 0, (2.69)

H(τ+c )−H(τ−c ) = 0. (2.70)

According to (2.68)-(2.70), the position and speed costates, and the Langrangian

to the Hamiltonian are continuous at the junction point t = τc. To determine the

continuity of the control input ui(t) at t = τc, we expand (2.70),

1

2u2i (τ−c ) + λpi (τ

−c )vi(τ

−c ) + λvi (τ

−c )ui(τ

−c ) + µai (τ

−c )(ui(τ

−c )− umax)

=1

2u2i (τ

+c ) + λpi (τ

+s )vi(τ

+c ) + λvi (τ

+c )ui(τ

+c ) + µai (τ

+c )(ui(τ

+c )− umax) (2.71)

We use the following results to simplify (2.71). According to the continuity of the

state in (2.123) and position costate in (2.68) at t = τc, we have λpi (τ−c )vi(τ

−c ) =

λpi (τ+c )vi(τ

+c ). Additionally, the KKT condition in (2.6) yields µai (τ

−c )(ui(τ

−c )−umax) =

µai (τ+c )(ui(τ

+c ) − umax) = 0. Simplifying (2.71), we have either ui(τ

+c ) = ui(τ

−c ) or

12(ui(τ

+c ) + ui(τ

−c )) + λvi (τ

+c ). Since the second term contradicts (2.64), we have conti-

nuity in control input at t = τc as,

ui(τ+c ) = ui(τ

−c ) = umax. (2.72)

Finally, from (2.64), we get the formulation for the lagrange multiplier µai

µai (t) =

0, if ui(t) < umax,

−λvi − umax, if ui(t) = umax.(2.73)

32

Using (2.64),(2.65), (2.66), (2.72), the initial, final conditions, and the terminal

transversality condition at t = tmi , we can formulate a constrained solution by stitching

the constrained and unconstrained arcs together at the junction point t = τc. In this

case, we have a constrained arc with constant parameters a(1)i , b

(1)i , c

(1)i , d

(1)i , followed

by an unconstrained arc with constant parameters a(2)i , b

(2)i , c

(2)i , d

(2)i . At time t = t0i

and t = τc, we have the following set of equations for the constrained arc,

a(1)i · t0i + b

(1)i = umax, (2.74)

a(1)i · τc + b

(1)i = umax, (2.75)

1

2a

(1)i (t0i )

2 + b(1)i · t0i + c

(1)i = vi(t

0i ), (2.76)

1

6a

(1)i (t0i )

3 +1

2b

(1)i · (t0i )2 + c

(1)i · t0i + d

(1)i = 0. (2.77)

From (2.74) and (2.75), we evaluate the constrained arc parameters a(1)i = 0 and

b(1)i = umax. Using this result in (2.76), we get c

(1)i = vi(t

0i ). Finally, resolving (2.77),

we get d(1)i = 0. We have the following set of equations to determine the parameters

of the exiting unconstrained arc and the corresponding junctions point:

a(2)i · τc − b(2)

i + umax = 0, (2.78)

a(2)i tmi + b

(2)i = 0, (2.79)

1

2a

(2)i τ 2

c + (b(2)i − umax) · τc + c

(2)i − v0

i = 0, (2.80)

1

6a

(2)i τ 3

c +1

2(b

(2)i − umax) · τ 2

c + (c(2)i − v0

i )τc + d(2)i = 0, (2.81)

1

6a

(2)i (tmi )3 +

1

2b

(2)i (tmi )2 + c

(2)i tmi + d

(2)i − L = 0. (2.82)

The constants of integration and the optimal junction point τ ∗c can be deter-

mined numerically from the above set of equations. Similar to the previous case, if the

optimal junction point τ ∗c can be derived explicitly from the known initial parameters,

then the above set of equations can be expressed as a closed-form analytical solution,

and solved by simple matrix inversion. In the following Lemma, we state the nature

of the optimal junction point τ ∗c to leverage the aforementioned concept.

33

Lemma 6. The Optimal junction point τ ∗c between the unconstrained arc and the con-

strained arc for ui(t)−umax ≤ 0 activation is an explicit function of the known param-

eters pi(tmi ), umax, t

mi , and vi(t

0i ).

Proof. For the control constraint ui(t)−umax ≤ 0 activation, we have a constrained arc

(with constant parameters a(1)i , b

(1)i , c

(1)i , d

(1)i ) followed by an unconstrained arc (with

constant parameters a(2)i , b

(2)i , c

(2)i , d

(2)i ) stitched together at the optimal junction point

t = τ ∗c . Solving (2.78) and (2.80)-(2.82), we have

a(2)i = umax ·

√− umaxφ(tmi , pi(t

mi ), vi(t0i ), umax)

, (2.83)

where, φ(tmi , pi(tmi ), vi(t

0i ), umax) = −3(tmi )2 · umax − 6tmi · vi(t0i ) + 6L. From (2.78) and

(2.79), we have

τ ∗c =umax

a(2)i

+ tmi . (2.84)

Finally, inserting (2.83) into (2.84), we get the expression of the junction point τ ∗c as

an explicit function of the known parameters tmi , pi(tmi ), vi(t

0i ), umax,

τ ∗c =umax

umax ·√− umax

φ(tmi ,pi(tmi ),vi(t0i ),umax)

+ tmi . (2.85)

Case 3. Both state constraint vi(t) − vmax ≤ 0 and the control constraint

ui(t)− umax ≤ 0 is activated

If both the control constraint ui(t)−umax ≤ 0 and state constraint vi(t)−vmax ≤0 are activated, we can derive the analytical solution following the procedure described

in the previous two cases combined.

34

We have the following set of equations for the control constrained arc with

parameters a(1)i , b

(1)i , c

(1)i , d

(1)i as,

a(1)i · t0i + b

(1)i = umax, (2.86)

a(1)i · τc + b

(1)i = umax, (2.87)

1

2a

(1)i (t0i )

2 + b(1)i · t0i + c

(1)i − vi(t0i ) = 0, (2.88)

1

6a

(1)i (t0i )

3 +1

2(b

(1)i ) · (t0i )2 + c

(1)i · t0i + d

(1)i = 0, . (2.89)

The control constrained arc parameters can be evaluated from (2.86) - (2.89), and are

given as a(1)i = 0, b

(1)i = umax, c

(1)i = v0

i and d(1)i = 0. The exiting unconstrained arc

with parameters a(1)i , b

(1)i , c

(1)i , d

(1)i can be expressed as the following set of equations,

a(2)i · τc − b(2)

i + umax = 0, (2.90)

1

2a

(2)i τ 2

c + (b(2)i − umax) · τc + c

(2)i − v0

i = 0, (2.91)

1

6a

(2)i τ 3

c +1

2(b

(2)i − umax) · τ 2

c + (c(2)i − v0

i )τc + d(2)i = 0, (2.92)

a(2)i · τs + b

(2)i = 0, (2.93)

1

2a

(2)i τ 2

s + b(2)i τs + c

(2)i − vmax = 0, (2.94)

1

6a

(2)i · (τs)3 +

1

2b

(2)i · (τs)2 + c

(2)i · τs + d

(2)i + vmax · (tmi − τs) = pi(t

mi ). (2.95)

Finally, the unconstrained arc enters the state constrained arc at t = τs. The state-

constrained arc with parameters a(3)i , b

(3)i , c

(3)i , d

(3)i can be expressed by the following set

of equations,

a(3)i · tmi + b

(3)i = 0, (2.96)

a(3)i τs + b

(3)i = 0, (2.97)

1

2a

(3)i τ 2

s − b(3)i τs − c(3)

i − vmax = 0, (2.98)

1

6a

(3)i (tmi )3 +

1

2b

(3)i (tmi )2 + c

(3)i tmi + d

(3)i − pi(tmi ) = 0, . (2.99)

35

The state constrained arc parameters can be evaluated from (2.96), (2.97), (2.98) and

(2.99), and are given as a(3)i = 0, b

(3)i = 0, c

(3)i = vmax and d

(3)i = pi(t

mi ) − vmax ·

tmi . The arc parameters a(1)i , b

(1)i , c

(1)i , d

(1)i , a

(2)i , b

(2)i , c

(2)i , d

(2)i , a

(3)i , b

(3)i , c

(3)i , d

(3)i , and the

optimal junction points τ ∗s and τ ∗c can be determined numerically from the above set of

equations. However, the optimal junction points can also be explicitly computed from

the known parameters using the following Lemma.

Lemma 7. The Optimal junction point τ ∗s between the unconstrained arc and the con-

strained arc for vi(t) − vmax ≤ 0 activation, and the junction point τ ∗c between the

unconstrained arc and the constrained arc for ui(t)−umax ≤ 0 activation is an explicit

function of pi(tmi ), vmax, umax, t

mi , and vi(t

0i ).

Proof. For the control constraint ui(t) − umax ≤ 0 activation, we have a constrained

arc (with constant parameters a(1)i , b

(1)i , c

(1)i , d

(1)i ) followed by an unconstrained arc

(with constant parameters a(2)i , b

(2)i , c

(2)i , d

(2)i ) stitched together at the optimal junction

point t = τ ∗c . Similarly, For the state constraint vi(t) − vmax ≤ 0 activation, we have

an unconstrained arc (with constant parameters a(1)i , b

(1)i , c

(1)i , d

(1)i ) followed by a

constrained arc (with constant parameters a(2)i , b

(2)i , c

(2)i , d

(2)i ) stitched together at the

optimal junction point t = τ ∗s . Solving (2.86)-(2.89) for the control constrained arc, we

have a(1)i = 0, b

(1)i = umax, c

(1)i = vi(t

0i ) and d

(1)i = pi(t

0i ). Again, solving (2.99)-(2.97)

and (2.98) for the state constrained arc, we get a(1)i = 0, b

(1)i = 0, c

(1)i = vmax and

d(1)i = pi(t

mi ) − vmax · tmi . Now, from (2.90) and (2.93), we have τ ∗c =

umax−b(2)i

a(2)i

and

τ ∗s = − b(2)i

a(2)i

respectively. Inserting these values into (2.91),(2.92), (2.94) and (2.95), and

solving the resulting system of equations, we get

a(2)i = −umax ·

√− 1

φ2

, (2.100)

b(2)i =

umax(−2vi(t0i )√− 1φ2

+ 2vmax√− 1φ2

+ 1)

2. (2.101)

where, φ2(tmi , pi(tmi ), vi(t

0i ), umax, vmax) = −24(−tmi ·umax · vmax + pi(t

mi ) ·umax− vi(t0i ) ·

vmax) + 12(v2i (t

0i ) + v2

max).

36

Finally, using the result from (2.100) and (2.101) into (2.90) and (2.93), we get

the expression of the junction point τ ∗c as an explicit function of the known parameters

tmi , pi(tmi ), vi(t

0i ), umax and vmax.

2.1.4 Simulation Results

We validate the analytical solution of the constrained optimization problem

through numerical analysis in MATLAB. We select the initial and final position as

pi(t0i ) = 0 m and pi(t

mi ) = 200 m, and the initial speed as vi(t

0i ) = 13.4 m/s. We

enforce maximum speed vmax = 21m/s and maximum acceleration umax = 1.4 m/s2.

We present only the cases with the state constraint vi(t) − vmax ≤ 0 and control

constraint ui(t)− umax ≤ 0 activation.

The unconstrained optimal trajectory for two different merging time tmi at 10 s

and 20 s is shown in Fig. 2.2. As stated in Lemma 2, we observe two different types of

optimal trajectories. This implies that based on the terminal conditions, a CAV may

speed up, or slow down optimally to satisfy the boundary conditions. We also observed

that both the predefined state and control constraints (for ai < 0) are activated in Fig.

2.2 (top-left and right).

We can save computational effort for producing the unconstrained results in

Fig. 2.2, if we know that any of the constraints might be activated. Once we employ

Theorem 1 to reduce possible constraint activation set, we use Theorems 2 and 3

to pinpoint the specific constraint activation case. We illustrate the optimal state

and control trajectories for only state constrained in Fig. 2.3 (top) and only control

constrained in Fig. 2.3 (bottom) cases. However, the results in Fig. 2.3 have to

be recalculated if additional constraint activation is encountered. We can avoid the

unnecessary computational effort to produce the intermediate result in Fig. 2.3 based

on the following observation.

37

Figure 2.2: State and control unconstrained optimal profile for 1) ai < 0 (top) and 2)ai > 0 (bottom) cases.

Note that, in the state constrained solution, the control constraint is activated

(Fig. 2.3, top-right) which was non-existent before, as discussed in theorem 4. Simi-

larly, we observe in Fig. 2.3 (bottom-right) that the control constrained optimal trajec-

tory creates a possibility of additional state constraint activation (Fig. 2.3, bottom-left)

due to the increased speed, as discussed in Theorem 5. Hence, we need to take the

additional constraint activation into account from Theorems 4 and 5, and enforce both

state and control constraints if needed. The unconstrained and fully constrained state

and control constrained trajectories are shown in Fig. 2.4. The four-point boundary

value problem is solved in this case, and two constrained and one unconstrained arcs

are pieced together to provide the optimal solution.

38

Figure 2.3: State constrained optimal speed (top-left) and acceleration (top-right)profile, and control constrained optimal speed (bottom-left) and acceleration (bottom-right) profile.

Figure 2.4: State and control constrained optimal speed (left) and acceleration (right)profile.

39

2.2 Decentralized Optimal Coordination of Connected and Automated Ve-

hicles for Multiple Traffic Scenarios [3]

In this section, we address the problem of optimally coordinating CAVs in a cor-

ridor consisting of multiple traffic scenarios to improve energy consumption and travel

time under the hard safety constraints of colision avoidance. We formulate a two-level

optimization problem in which we maximize traffic throughput in the upper-level prob-

lem, and derive a closed-form analytical solution that yields the optimal control input

for each CAV, in terms of fuel consumption, in the low-level problem. In earlier work,

we presented a preliminary analysis on coordinating CAVs in a corridor yet without

considering the rear-end safety constraint; see [29] and [26]. Thus, the key contribution

of this research is the formulation and analytical solution of an optimal control prob-

lem for coordinating CAVs in a corridor consisting of multiple traffic scenarios with

the explicit incorporation of the rear-end safety constraint.

2.2.1 Problem Formulation

We consider a corridor (Fig. 2.5) that consists of several conflict zones (e.g., a

merging area, an intersection, and a roundabout), where potential lateral collision of

vehicles may occur. Upstream of each conflict zone, we define a control zone, inside

of which, the vehicles can communicate with each other. The dimension of each con-

trol zone is restricted by the communication range of an associated coordinator, which

records the vehicle queue inside the control zone. Note that the coordinator is not

involved in any decision on the CAV operation and only serves to coordinate informa-

tion with the CAVs. The communication range of the coordinator can be adjustable

and its length could be extended as needed. For clarity, we illustrate the boundary of

the corridor as indicated by dashed lines and the limits of each control zone by shaded

rectangles (Fig. 2.5). Note that we only coordinate CAVs inside the control zone of

each conflict zone.

We consider a corridor with a set of conflict zones Z ⊂ N. Let Nz(t) =

40

Figure 2.5: Corridor with connected and automated vehicles.

{1, 2, . . . , Nz(t)} be a queue of CAVs inside the control zone of a conflict zone z, where

Nz(t) ∈ N is the number of CAVs in the control zone of z at time t ∈ R+. When a CAV

enters the control zone, it broadcasts its route information to the coordinator of this

conflict zone. The coordinator then assigns a unique integer i ∈ N that serves as the

identification of CAVs inside the corridor. Let t0,zi be the initial time that CAV i enters

the control zone of z ∈ Z, and tzi be the time for CAV i that enters z. For example,

for CAV #7 (Fig. 2.5), t0,17 is the time that it enters the control zone of conflict zone

#1, which is also the time that it enters the corridor, and t17 is the time that it enters

the conflict zone #1. Similarly, CAV #1 enters the control zone of conflict zone #3 at

t0,31 , and enters the conflict zone #3 at t31. The CAV index given by the coordinator is

removed from the queue Nz once the vehicle i exits the conflict zone z.

To avoid any possible lateral collision, there is a number of ways to compute tzi

for each CAV i. In what follows, we present a decentralized framework in which we

formulate an upper-level optimal control problem for determining the time tzi that each

CAV i will enter the conflict zone z ∈ Z, and then address a lower-level control problem

that will yield for each CAV the optimal control input (acceleration/deceleration) to

41

achieve the assigned time tzi (upon arrival of CAV i) without collision.

2.2.2 Vehicle model, Constraints, and Assumptions

Each CAV i ∈ Nz(t) is modeled by a second order dynamics

pi = vi(t),

vi = ui(t), (2.102)

where pi(t) ∈ Pi, vi(t) ∈ Vi, and ui(t) ∈ Ui denote the position, speed and control input

(acceleration/deceleration) of each CAV i in the control zone. Let xi(t) = [pi(t) vi(t)]T

denote the state of each CAV i, with initial value at the entry of the control zone of

conflict zone z ∈ Z given as x0,zi =

[p0,zi v0,z

i

]T, where p0,z

i = pi(t0,zi ) and v0,z

i = vi(t0,zi ).

To ensure that each vehicle control input and speed are within a given admissible range,

we impose the following constraints

ui,min ≤ ui(t) ≤ ui,max, and

0 ≤ vmin ≤ vi(t) ≤ vmax, ∀t ∈ [t0,zi , tzi ],(2.103)

where ui,min, ui,max are the minimum deceleration and maximum acceleration for each

CAV i ∈ Nz(t), and vmin, vmax are the minimum and maximum speed limits respec-

tively. For simplicity, we do not consider vehicle diversity, and thus we set ui,min = umin

and ui,max = umax.

For each CAV i ∈ Nz(t), the lateral collision is possible within the set Γi∆=

{t | t ∈ [tzi , tzi + ρ]}, where, ρ is the safety time headway to avoid lateral collision.

Lateral collision between any two CAVs i, j ∈ Nz(t) can be avoided if the following

constraint holds,

Γi ∩ Γj = ∅, ∀t ∈ [tzi , tzi + ρ], i, j ∈ Nz(t). (2.104)

To ensure the absence of rear-end collision of two consecutive CAVs traveling on the

same lane, we impose the following condition

si(t) = pk(t)− pi(t) ≥ δ(t), ∀t ∈ [t0,zi , tzi ]. (2.105)

42

Here, si(t) denotes the distance between CAV i and CAV k which is physically imme-

diately ahead of i. The minimum safe distance δ(t) is a function of speed vi(t). Since

we consider an urban traffic corridor, the average speed does not exhibit significant

variations. Therefore, we can consider that the safe distance δ(t) = δ is constant. In

the modeling framework described above, we impose the following assumptions:

Assumpation 1. All vehicles are connected and automated, i.e., 100% pene-

tration rate of CAVs.

Assumpation 2. For each CAV, none of the constraints (2.103) and (2.105) is

active at t0,zi .

Assumpation 3. Each CAV i has proximity sensors and can measure local

information without errors or delays.

Assumpation 4. The corridor only contains single-lane road segments. The

vehicles traveling in the corridor do not change lanes except to make necessary turns.

Assumpation 5. The speed of each CAV i inside the conflict zone is constant.

The first assumption limits the scope of this research to the control of CAVs

in an idealized environment where all vehicles are automated and connected to each

other. Addressing different penetration of CAVs is the subject of proposed dissertation

research. The second assumption ensures that the solution of the optimal control

problem starts from a feasible state and control input. The third assumption might

impose barriers in a potential deployment of the proposed framework. However, we

could extend our results in the case that this assumption is relaxed, if the noise in the

measurements and delays are bounded. The fourth assumption simplifies the upper-

level optimal control problem so as to avoid implications related to lane changing.

Finally, the last assumption is imposed to enhance safety awareness. However, it could

be modified appropriately, if necessary, as discussed by [17].

Definition 1. Let i − 1, i ∈ Nz(t) be two CAVs inside the control zone traveling

towards the corresponding conflict zone z. Depending on the physical location and

43

trajectory inside the control zone with respect to CAV i, CAV i− 1 belongs to one of

the following three subsets of Nz(t) with respect to CAV i ∈ Nz(t):

1. Rzi contains all CAVs that travel in the same lane with CAV i towards the conflict

zone z, having travel paths that can cause rear-end collision.

2. Czi contains all CAVs from different roads having travel paths that can causelateral collision with CAV i in conflict zone z.

3. Ozi contains all CAVs from different roads having travel paths that cannot causelateral or rear-end collision with CAV i in conflict zone z

Upon arrival at the entry of the control zone of conflict zone z ∈ Z at time t0,zi ,

CAV i ∈ Nz(t) needs to compute the time tzi . In general, a value of tzi that satisfies

the safety constraints (2.104) and (2.105) may depend on the preceding CAV in the

control zone. Next, we address the question of identifying the appropriate tzi for each

CAV through an upper-level optimization problem.

2.2.3 Upper-level optimization problem

To fully utilize the capacity of the traffic network of CAVs, we formulate a

throughput maximization problem, in terms of minimizing the gaps between CAVs

in the conflict zones, i.e., minimizing the total time to process CAVs in the network,

subject to the constraints (2.103), (2.104) and (2.105). Note that for i = 1, the

safety constraint is not active since there is no prior CAV in the control zone, which

implies that tz1 is not constrained and can be determined outside the optimization

framework. Thus, for each control zone of conflict zone z ∈ Z, we formulate the

following optimization problem:

mint(2:Nz(t))

Nz(t)∑i=2

(tzi − tzi−1) = mint(Nz(t))

(tzNz(t) − tz1), (2.106)

subject to : (2.102), (2.103), (2.104), (2.105),

44

where, t(2:Nz(t))= [tz2, . . . , t

zNz(t)]. The solution of (2.106) yields the optimal time tzi

∗, i ≥2, z ∈ Z, which designates the entry times of each CAV in each conflict zone so as to

maximize the throughput of the corresponding bottleneck.

In what follows, we discuss how the lateral collision safety constraint is addressed

in the solution of (2.106). We also show that the solution has an iterative structure and

depends only on the state and control constraint (2.103) as well as the safety constraint

(2.105). To obtain the optimal solution of tzi∗ for CAV i ∈ Nz(t) at the conflict zone

z ∈ Z, we first consider the case when Czi (t) is empty, thus the entry time for CAV i at

z depends only on some CAV k = (i−1) ∈ Rzi , where k is physically immediately ahead

of i on the roadway segment inside the control zone. In this case, the minimum time

tzi for CAV i to enter the conflict zone z is designated by the rear-end safety constraint

(2.105), and in particular, by the safe headway, ρ, that a CAV i should maintain while

following CAV k, i.e., tzi = tzk + ρ. In this context, we need to find the bound of tzi to

ensure feasibility of the solution. Consider the maximum and minimum speeds that

CAV i could achieve. The value of tzi is then given by

tzi = max{

min{tzk + ρ, tz,maxi }, tz,mini

}, (2.107)

where tz,maxi and tz,mini is the longest and shortest possible travel time of a CAV i be-

tween the entry and exit of the control zone of the conflict zone z ∈ Z corresponding

to the minimum, vmin, and maximum, vmax, speed limit respectively. Note that condi-

tion (2.107) ensures that the time tzi that CAV i will be entering the conflict zone z is

feasible and can be attained based on the imposed speed limits in the corridor. From

(2.107), the safety constraint between CAVs traveling in the same lane is guaranteed

at tzi . We now turn our attention to the case where possible lateral collision might

occur if Czi (t) is non-empty. In this case, the minimum time tz∗i for CAV i to enter the

conflict zone z is constrained by both the lateral collision (2.104) and rear-end collision

(2.105) constraints.

Definition 2. We define the set Azi ⊂ Czi , Azi := {j ∈ Czi | tzj ≥ tzi }, that includes any

CAV j ∈ Czi whose entry time at conflict zone z is later than tzi , and the set Lzi ⊂ Azi ,

45

Lzi := {j ∈ Azi | tzj + ρ ≤ tzj+1 − ρ}, that includes any CAV j ∈ Czi whose entry time

satisfy (2.105).

Considering possible lateral collisions at conflict zone z, for all z ∈ Z, we obtain

the following result.

Theorem 6. The solution tzi∗,∀i ≥ 2,∀z ∈ Z, of (2.106) is recursively determined

through

tzi∗ =

max{

max{tzc}+ ρi, tzi

}, if ∀c ∈ Czi and @ c ∈ Azi ,

tzi , if ∃ a ∈ Azi and tzi + ρ ≤ min{tza},

min{tzb}+ ρi, if ∃ b ∈ Lzi and tzi + ρi > min{tza},

max{tza}+ ρi, if ∃ a ∈ Azi and @ a ∈ Lzi .

(2.108)

Proof. Based on Definition 1, there are three cases to consider for tzi∗.

Case 1 : If Azi 6= ∅, all CAVs in Czi will be entering the conflict zone z earlier

than tzi , which, by Definition 1, implies tzi∗ =

{tzi ,

max{tzc}+ ρ,∀c ∈ Czi . Hence, we

have tzi∗ = max

{max{tzc}+ ρ, tzi

},∀c ∈ Czi .

Case 2 : If Azi 6= ∅ and Lzi 6= ∅, we consider two cases: (i) if the earliest entry

time of CAVs in the set Azi is later than tzi plus a safe headway ρ, then the minimum

entry time of CAV i is tzi , which satisfies the safety constraints to avoid both lateral

and rear-end collisions; (ii) the optimal value of tzi is the earliest possible time slot

between the entry times of two consecutive CAVs in the set Lzi . Hence, we have

tzi∗ =

tzi , if tzi + ρ ≤ min{tza},

min{tzb}+ ρ, if tzi + ρ > min{tza},

∀a ∈ Azi , ∀b ∈ Lzi . (2.109)

Case 3 : Finally, if Lzi 6= ∅, this implies that there is no available time slot

between the entry times of two CAVs in Azi . In this case, CAV i will be entering

46

the conflict zone after the last CAV in Azi to avoid lateral collision, which implies

tzi∗ = max{tza} + ρ,∀a ∈ Azi , if there not exist a ∈ Lzi . Combining the above results,

we obtain tzi∗ in (2.108), which completes the proof.

Theorem 6 yields the sequence that the CAVs will be traveling through each

control zone. Each CAV i follows the above policy to determine the time tzi∗ that it will

be entering the conflict zone z ∈ Z upon arrival at the entry of the control zone. Once

the entry time tzi∗ is computed, it is stored in the coordinator and it is not changed.

Thus, the next CAV i+ 1, upon its arrival at the entry of the control zone, will search

for feasible times to cross the conflict zone based on the available time slots. The

recursion is initialized when the first CAV enters the boundary of the corridor, i.e., it

is assigned i = 1. In this case, tz1,∀z ∈ Z, can be externally assigned as the desired

entry time of this CAV whose behavior is unconstrained.

2.2.4 Low-level optimal control problem

When a CAV i enters the corridor, it communicates with the other CAVs (As-

sumption 1) and the coordinator broadcasts information without any errors or delays

(Assumption 3). The coordinator assigns a unique identity to each CAV and receives

back some information at the time each CAV arrives at the entry of the corridor, as

defined next.

Definition 3. For each CAV i, we define the information set Yi(t) as

Yi(t) , {pi(t), vi(t), oi, tzi ∗}, z ∈ Z, t ∈ [t0i , tzi ], (2.110)

where pi(t), vi(t) are the position and speed of CAV i inside the corridor, oi is the

route that CAV i travels inside the corridor, and tzi∗ is the time for CAV i to enter the

conflict zone z given by (2.108).

As discussed in the previous section, the time tzi∗ for CAV i is determined in a

recursive manner based on the information received from the coordinator. Therefore,

47

once CAV i enters each of the control zones, immediately all information in Yi(t)

becomes available for i and is stored in the coordinator accessible for next arriving

CAV i+ 1.

In the low-level optimal control problem, the objective is to minimize the con-

trol input (acceleration/deceleration) for each CAV i ∈ Nz(t) from the time t0,zi that

i enters the control zone until the time tzi that it exits the control zone under the

hard safety constraint to avoid rear-end collision. By minimizing each CAV’s acceler-

ation/deceleration, we minimize transient engine operation. Thus, we can have direct

benefits in fuel consumption and emissions since internal combustion engines are opti-

mized over steady state operating points (constant torque and speed). Therefore, the

optimization problem for each CAV i ∈ Nz(t) is to minimize the L2-norm of the control

input in [t0,zi , tzi ], formulated as follows:

minui(t)∈Ui

Ji(u(t)) = minui(t)∈Ui

1

2

∫ tzi

t0,zi

u2i (t) dt, (2.111)

subject to : (2.102), (2.103), (2.105),

given t0,zi , pi(t0,zi ) = 0, v0,z

i , tzi∗, and pi(t

zi∗) = pz.

Note that we do not include the lateral collision constraint (2.104) in (2.111), since

it has been addressed in the upper-level optimization problem. On the contrary, we

explicitly include the rear-end safety constraint. The problem formulation with the

state and control constraints requires the constrained and unconstrained arcs of the

state and control input to be pieced together to satisfy the Euler-Lagrange equations

and necessary condition of optimality. Let S(t,xi(t), ui(t)) = [vi(t)− vmax vmin(t)−vi(t) pi(t) − pk(t) − δ]T be the vector of constraints that are not explicit functions

of the control input ui(t). We take successive total time derivatives of S(t,xi(t), ui(t))

until we obtain an expression that is explicitly dependent on ui(t). If q time derivatives

are required for a specific constraint of Si(t,xi(t), ui(t)), we refer to that constraint as

a qth-order state variable inequality constraint; see [104]. Note that we have 1st-order

speed constraint and 2nd-order rear-end safety constraint in Si(t,xi(t), ui(t)). The

48

2nd-order rear-end safety constraint plays the role of a control variable constraint on

the constrained arc,

S(2)i (xi(t), ui(t), t) = ui(t)− uk(t) = 0. (2.112)

From (2.111), the CAV dynamics (2.102), state and control constraints (2.103), and

the rear-end safety constraint (2.105) for each CAV i ∈ Nz(t), we formulate the Hamil-

tonian function

Hi

(t, pi(t), vi(t), ui(t)

)=

1

2u(t)2

i + λpi · vi(t) + λvi · ui(t)

+ηai · ui(t)− ηbi · ui(t) + ηci · (ui(t)− uk(t))

+µdi · (ui(t)− umax) + µei · (umin − ui(t)), (2.113)

where λpi , λvi are the co-state components, and ηai , η

bi , η

ci , µ

diµ

ei are the Lagrange multi-

pliers satisfying the complimentary slackness conditions based on the state and control

constraints in (2.103) and (2.105).

The Euler-Lagrange equations become

λpi (t) = −∂Hi

∂pi= 0, λvi (t) = −∂Hi

∂vi= −λpi . (2.114)

The necessary condition for optimality is

∂Hi

∂ui= ui(t) + λvi + ηai − ηbi + ηci + µdi − µei = 0. (2.115)

2.2.4.1 State and control constraint is not active

When the inequality state and control constraints are not active, ηai = ηbi =

ηci = µdi = µei = 0, applying the necessary condition (2.115), the optimal control is

ui(t) + λvi = 0, i ∈ Nz(t). (2.116)

From (2.114) we have λpi (t) = ai, and λvi (t) = −(ai · t + bi). The coefficients ai, bi

are constants of integration corresponding to each CAV i. From (2.116) the optimal

49

control input (acceleration/deceleration), speed and position as a function of time are

given by

u∗i (t) = (ai · t+ bi), ∀t ≥ t0,zi , (2.117)

v∗i (t) =1

2ai · t2 + bi · t+ ci, ∀t ≥ t0,zi , (2.118)

p∗i (t) =1

6ai · t3 +

1

2bi · t2 + ci · t+ di, ∀t ≥ t0,zi , (2.119)

where ci and di are constants of integration which can be computed at each time

t, t0i ≤ t ≤ tzi , using the values of the control input, speed, and position of each CAV i

at t, the position pi(tzi ), and λvi (t

zi ) = 0.

To derive the constrained solution of (2.111), we first start with the unconstrained

arc and derive the solution using (2.117). If the solution violates any of the con-

straints, then we re-solve the problem with the constrained and unconstrained arcs

pieced together. This process is repeated until the solution does not violate any other

constraints. The simple nature of the optimal control and states in (2.117) through

(2.119) makes the online solution of (2.111) computationally feasible, even with the

additional burden of checking for active constraints. In what follows, we address the

optimization problem (2.111) with the activation of the constrained case corresponding

to the rear-end collision only. The other constrained cases related to the state, i.e.,

speed, vi, and control, ui as in (2.103), are similar to the cases presented by [17] and [1],

and thus are omitted.

2.2.4.2 Rear-end safety constraint is active

Suppose a CAV starts from a feasible state and control at t = t0i and at some

time t = t1, the rear-end safety constraint (pi(t) − pk(t) + δ) ≤ 0 is activated. In this

case, ηai = ηbi = µdi = µei = 0. From (2.115), the optimal control is given by

ui(t) + λvi + µci(t) = 0, ∀t ≥ t1. (2.120)

50

The state trajectory entering onto the 2nd-order rear-end safety constraint boundary

must satisfy the following tangency conditions

N(xi(t), t) =

pi(t)− pk(t) + δ

vi(t)− vk(t)

= 0, (2.121)

where, N(xi(t), t) is the q-component vector function of the 2nd order safety tangency

constraints. The tangency constraints in (2.121) also apply to the state trajectory

at the exit of the constraint arc. Since the optimal solution of the preceding CAV

k ∈ Nz(t) is known a priori, from (2.112) and (2.121), the optimal solution for CAV

i ∈ Nz(t) in the constrained arc is derived from Si(t,xi(t), ui(t)) = 0 and is

u∗i (t) = u∗k(t), v∗i (t) = v∗k(t), and p∗i (t) = p∗k(t)− δ. (2.122)

The equations in (2.121) form a set of interior boundary conditions where the co-states

λpi (t) and λvi (t) in general have discontinuity at the junction points, i.e., entry and

exit points of the state trajectory between the constrained and the unconstrained arcs.

However, the control trajectory may or may not have discontinuities at the junction

points. Next, we address the jump conditions at the entry junction point t = t1. At

time t = t1, when the safety constraint is activated, we have a junction point between

the unconstrained and constrained arcs. Let t−1 represents the time instance just before

t1, and t+1 signifies just after t1. The state trajectories are continuous at the junction

points. Thus, we have

pi(t−1 ) = pi(t

+1 ), vi(t

−1 ) = vi(t

+1 ). (2.123)

The jump conditions at t1 can be written as

λpi (t−1 ) = λpi (t

+1 ) + πT · ∂N(xi(t), t)

∂pi(t)

∣∣∣∣t=t1

, (2.124)

λvi (t−1 ) = λvi (t

+1 ) + πT · ∂N(xi(t), t)

∂vi(t)

∣∣∣∣t=t1

, (2.125)

H(t−1 ) = H(t+1 )− πT · ∂N(xi(t), t)

∂t

∣∣∣∣t=t1

, (2.126)

∂H(t−1 )

ui(t)=∂H(t+1 )

ui(t). (2.127)

51

In (2.124)-(2.127), πT = [π1 π2] is a vector of constant Langrange multipliers to be

determined so that the condition in (2.121) is satisfied. From (2.126), using (2.122)-

(2.127), we obtain 12(ui(t

−1 )2− ui(t+1 )2) + λvi (t

+1 ) · (ui(t−1 )− ui(t+1 )) = 0. This yields two

cases: either (ui(t−1 )− ui(t+1 )) = 0 or 1

2(ui(t

−1 ) + ui(t

+1 )) + λvi (t

+1 ) = 0. Both conditions

lead to ui(t−1 ) = ui(t

+1 ), which indicates that the control trajectory is continuous at

the entry junction point at t = t1. Finally, using the continuity in control and (2.125)

in (2.127), we obtain ηci (t+1 ) = π2. With two junction points at time t = t1 and

t = t2, we have a constrained arc between two unconstrained arcs. Since we have

multiple arcs pieced together at the junction points, we differentiate the constants of

integration for the state and control trajectory by adding a superscript h representing

the order of appearance of the arcs. Therefore, we represent the constants of integration

as (ahi , bhi , c

hi , d

hi ), where h = 1, 2 corresponds to the first and the last unconstrained

arc respectively. The control trajectory of CAV i considering the constrained and

unconstrained arcs can be written as,

u∗i (t) =

a1i · t+ b1

i , t0,zi ≥ t ≤ t1,

u∗k(t), t1 < t < t2,

a3i · t+ b3

i , t2 ≥ t ≤ tzi .

(2.128)

The constants of integration, along with the junction points t1 and t2 can be computed

by solving (2.117)-(2.119) and (2.122)-(2.128) with appropriate initial, boundary and

transversality conditions.

2.2.5 Simulation Results

To validate the effectiveness of the rear-end safety constrained formulation, we

present two cases in Fig. 2.6, where a leading CAV k and a following CAV i are both

cruising with the optimal control input. In the left panel of Fig. 2.6, the following

CAV i derives its control input according to (2.117), and activates the rear-end safety

constraint with respect to the immediately preceding CAV k with two junction points.

52

0 5 10 15 20

Time [s]

0

20

40

60

80

100

120

140

160

180

200

Dis

tan

ce

[m

]

CAV i

CAV k

0 5 10 15 20

time [s]

0

20

40

60

80

100

120

140

160

180

200

Dis

tan

ce

[m

]

CAV i

CAV k

Figure 2.6: Unconstrained (left) and rear-end safety constrained state trajectory (right)of CAV i with respect to its immediately preceding vehicle k.

In the right panel of Fig. 2.6, CAV i derives its control input using (2.128) subject to

safety constrained optimization.

To validate the proposed approach for multiple traffic scenarios, we use a sim-

ulation network of Mcity created in PTV VISSIM environment. We define a corridor

consisting of four conflict zones: (1) a merging roadway, (2) a speed reduction zone, (3)

a roundabout, and (4) an intersection. Vehicles enter the network on the ramp, join the

traffic on the highway with desired speed of 22 m/s, and then enter the speed reduction

zone where the speed limit drops to 11 m/s. The vehicles exit the highway segment

and travel through the roundabout, where a desired speed of 13 m/s is imposed until

the exit of the roundabout, to the intersection (conflict zone #4).

To evaluate the network performance with the proposed control framework, we

define two scenarios as follows:

Scenario 1: baseline, i.e., 0% CAV penetration rate. All vehicles in the network

are non-connected and non-automated. In this case, the Wiedemann car following

model [105] built in VISSIM is applied. 1.2 s time headway is adopted to estimate the

53

Figure 2.7: The corridor in Mcity.

minimum allowable following distance.

Scenario 2: optimal control, i.e., 100% CAV penetration rate. The proposed

control framework is integrated to generate the optimal acceleration/deceleration pro-

file for each CAV in the network.

The CAV speed trajectories under 0% and 100% CAV penetration rate in the

corridor are illustrated in Fig. 2.8. In the baseline scenario with 0% CAV penetration

rate, CAVs traveling along the corridor need to yield to mainline traffic, and wait in

the signalized intersection. Thus, we observe high fluctuations in their speed profiles

under the baseline scenario at the proximity of the conflict zones (see upper panel of

54

Fig. 2.8). In the optimal control scenario under 100% CAV penetration rate, CAVs

travel through the corridor without stop-and-go driving (see lower panel of Fig. 2.8).

The latter enables CAVs to have smoother speed trajectory affecting the uncontrolled

upstream and downstream area of the control zone. We observe 9% improvement in

terms of travel time and an average of 47% savings in total fuel consumption in the

optimal control scenario compared to the baseline one.

Figure 2.8: Vehicle trajectories inside the corridor for (a) baseline and (b) optimal con-trolled case. The control zone for each of the conflict zones are shown for comparison.

We plot the accumulated fuel consumption for all the vehicles traveling through

the corridor considered here in one simulation replication in Fig. 2.9 to show energy

consumption under both scenarios. With smooth acceleration/deceleration profiles

throughout the entire corridor, vehicles’ stop-and-go driving behavior is eliminated

under Scenario 2 with 100% CAV penetration rate. Thus, transient engine operation

is minimized, leading to direct fuel consumption savings compared to the baseline

scenario as shown in Fig. 2.9.

With vehicle trajectory data collected every 1 s, fuel consumption is estimated

55

Figure 2.9: Accumulated fuel consumption over time for the baseline and optimalcontrolled vehicles.

by using the polynomial metamodel proposed in [106] that relates vehicle fuel consump-

tion as a function of speed v(t) and acceleration u(t). Overall, through the optimal

control algorithm, an average of 47% savings in total fuel consumption for vehicles trav-

eling along the corridor during is obtained. The reasons are mainly twofold: 1) while

CAVs are immediately preparing for the speed reduction zone/roundabout/intersection

with smooth maneuver, human-driven vehicles keep accelerating or cruising with a

much higher speed until they are aware of downstream conflict zones. We observe

that, in Fig. 2.8 deceleration is the major behavior around the conflict zones in the

baseline scenario; 2) CAVs are coordinated with each other to create enough gaps for

merging and crossing the intersection, whereas human-driven vehicles need to stop and

accelerate again to cross these conflict zones.

2.3 Extension of the Preliminary Research

In our preliminary research, we have addressed the problem regarding the decen-

tralized optimal coordination of CAVs through traffic scenarios with state and control

56

constraints. We specifically laid out the theorems and results that would enable us

to compute the state and control constrained optimal control policy for the CAVs

with real-time closed-form analytic solution. We have also extended the optimal co-

ordination approach in a traffic network of multiple scenarios, where the CAVs can

dynamically re-sequence their optimal coordination queue, and cross the traffic scenar-

ios without any rear-end and lateral safety collision. However, we have made several

explicit assumptions to simplify our modelling framework presented in the previous

section. We have considered 100% CAV penetration which enabled us to have full

control of all the agents within the traffic network. However, such idealized framework

is not implementable in real-world setting consisting of human-driven vehicles or HVs

due to the following aspects,

(a) State Estimation Problem: HVs do not transmit their state information to

the CAVs. Therefore, the CAVs need to estimate the neighboring HV state either using

their on-board sensors, or get it from a suitable infrastructure such as road side units,

coordinators, loop detectors etc.

(b) Dynamics Prediction Problem: Although several car-following models exist

in the literature [78], [107], [105] to model the HV dynamics, no car-following model can

capture fully the human driving behavior. This leads to the problem of predicting the

HV dynamics properly which is required for the CAVs to derive a closed-form analytic

solution. The uncertainty rising out of such unpredictable HV dynamics also needs to

be considered by the CAVs.

(c) Vehicle Control Problem: The HVs we consider in our modelling framework

do not have any driving automation (Level 0 automation), and are controlled only by

the human input, which prevents the implementation of a distributed control framework

in a centralized or decentralized way.

We, therefore, aim at developing a control framework for the CAVs capable of

handling the aforementioned uncertainties stemming from the HVs in a mixed traffic

environment. CAV-HV interaction in a mixed traffic environment can occur in a 1)

57

lane following scenario, and in the 2) traffic congestion scenarios. In a lane following

scenario, the CAV can employ its ACC module to follow the preceding HV. Several

ACC algorithms can be found in the literature [101], [108], where a following CAV

can maintain the desired rear-end safety collision gap from its preceding vehicle. Our

research plan is to focus on the optimal coordination problem of the CAVs in a mixed

traffic environment, where potential lateral and rear-end collision may take place at

the traffic scenarios.

58

Chapter 3

PROPOSED DISSERTATION RESEARCH

The overarching goal of the proposed dissertation research is to develop an

optimal control framework for coordinating connected automated vehicles through au-

tomated traffic scenarios in a mixed traffic environment consisting of conventional

human-driven vehicles and CAVs. The traffic scenarios considered here can be the

roadways with possibility of traffic congestion, e.g., highway on-ramp merging roads,

roundabouts, speed reduction zones, urban intersections etc.

To achieve the overarching goal stated above, we set the following aims.

Aim 1: Adaptive and Stochastic Coordination of CAVs.

We will develop a time- and energy-optimal control framework for each CAV

approaching a traffic scenario. Each CAVs will need to coordinate with other CAVs,

HVs and infrastructure within its communication range to traverse its route without

any rear-end or lateral collision. Since the HVs do not transmit their own state in-

formation, the CAVs need to solve the state estimation problem, and predict the HV

dynamics based on a suitable car-following model. Note that, both the state estima-

tion and HV dynamics prediction problem may have inherent approximation errors.

Hence, the CAVs need to update the state estimation and dynamics prediction at each

selected time step to minimize the approximation error. We will also provide both the

hierarchical and joint time- and energy-optimization framework with multi-objective

cost functions to explore the trade-offs among different performance metrics.

Aim 2: Platoon-based Framework for Indirect Control of HVs.

59

We will develop an optimal control framework for the CAVs to form a mixed

platoon of following human-driven vehicles, and indirectly control the HVs within the

traffic network by controlling platoon. Then the CAVs leading the platoons will guide

the platoons to an automated merging scenario in an on-ramp roadway, roundabout or

signal-free intersections without activating any state, control and safety constraints.

As the stochastic nature of the HVs in a mixed traffic environment presents

significant challenges in terms of practical implementation, here we propose a platoon-

based approach. Instead of considering the HVs as distributed agents subject to their

stochastic nature, their longitudinal movement can be controlled by the leading vehicle

leading to the formation of mixed platoon, giving rise to an indirect control of HVs.

Note that, a vehicle platoon can be described as a set of vehicles having trajectories

in a close and coordinated pack without any mechanical linkage. We will consider the

HV state estimation and dynamics prediction problem to formulate an optimal mixed

platoon formation problem. We will then formulate an optimal coordination policy for

the platoons for the automated merging scenarios. The operation platoon formation

and optimal control of the mixed platoon dynamics has to guarantee stability and

string stability with rigorous mathematical formulation.

Aim 3: Optimal Control Framework with System Uncertainty

We will address the effect of system uncertainties in the optimal control frame-

work of CAVs developed in Aim 1 and 2. The source of uncertainties can stem from

the unknown driving behavior of the HVs which can be different from the assumed

car-following dynamics in the modelling framework. In such case, we will develop a

robust optimization scheme to avert the worst case scenario, predominantly associated

with the safety constraints. We will also explore the possibilities of stochastic model

predictive control approach to solve such problems associated with uncertainties. Ad-

ditionally, by considering the constraints and stability conditions of the mixed platoon

framework discussed in Aim 1 and 2, we will derive conditions to put bounds on the fea-

sibility of states and control inputs. Finally, we will also develop and validate efficient

60

algorithms for real-time implementation of the developed frameworks.

In what follows, we briefly describe the a general modelling framework, under-

lying assumptions, and multi-level optimization schemes to accomplish the aims of our

research direction mentioned above.

3.1 Aim 1: Adaptive and Stochastic Coordination Approach

Figure 3.1: Corridor with connected and automated vehicles and human driven vehi-cles.

In Fig. 3.1, we illustrate a traffic network consisting of CAVs and HVs approach-

ing an on-ramp merging scenario. We define the merging zone of length S as shown by

the red rectangle in Fig. 3.1 where lateral collision between the vehicles can take place.

Upstream of the merging zone, we define a control zone (green roadway in Fig. 3.1),

the length of which is determined by the range of a coordinator which maintains the

queue of both CAVs and HVs, and does not take any control action. Upon arrival of

each vehicle, irrespective of CAV and HV, the coordinator assigns an unique identity

i ∈ N (t), where N (t) is the initial queue of the vehicles based on arrival at the control

61

zone. Based on some upper level coordination or scheduling approach, the initial vehi-

cle queue N (t) can be updated. Once a vehicle exits the merging zone, all the indices

of N (t) are reduced by one. However, as a slight modification from the coordinator

defined earlier in Section 2.1 and 2.2, we assume that this particular coordinator can

estimate and transmit the state information of the HVs within the control zone. Since

the HVs do not transmit their own state information, the coordinator can employ road

side units or computer-vision technology to estimate the state information of HVs. On

the other hand, the CAVs can communicate with each other and with the coordinator

to receive and transmit required information within the control zone. At a suitable

location of the ramp roadway, a type of ramp-metering system, namely an adaptive

speed advisory signal is located. Such traffic signal may resolve the traffic gridlock

resulting from a specific coordination case that we will discuss later. Note that, traffic

gridlock can be a situation within a traffic network where the automated coordination

among the vehicles cannot be realized, which leads to unavoidable stop-and-go driving

behavior. Each vehicle i ∈ N (t) enters the control zone and merging at time t0i and

tmi , and exits the merging zone at time tfi . The objective of each CAV i ∈ N (t) is to

derive a time- and energy-optimal trajectory and pass the merging scenrio without any

stop-and-go driving without any lateral or rear-end collision.

We will adopt the general assumptions presented in Section 2.2, except for the

fact that we do not consider 100% CAV penetration here, and allow the interaction

of CAVs and HVs. Furthermore, to address the problem discussed in this section, we

make the following additional assumptions. First, we assume that the HVs employ a

car-following model known to the CAVs. Second, the HVs abide by the speed advisory

within the on-ramp roadway. Finally, for simplicity we assume first-in-first-out (FIFO)

queue N (t), i.e., vehicle entering the control zone first will enter the merging zone first.

Let i − 1 ∈ N (t) be the index of the putative leading vehicle on the same

or different roadway as the vehicle i ∈ N (t), and has a possibility of collision at the

merging zone. The putative leader can be the same or different vehicle as the preceding

62

vehicle. For example, the putative leader of vehicle #4 is vehicle #3 whereas its

preceding vehicle is vehicle #2 in Fig. 3.1.

We will consider the CAVs to have a simple double integrator dynamics as in

(2.102) ∀t ∈ [t0i , tfi ]. On the other hand, we model the driving behavior each of HV

i ∈ N (t) with an appropriate car-following model.

pi(t) = vi(t),

vi(t) = f(si(t),∆vi(t), vi(t)), ∀t ∈ R+, (3.1)

where, si(t) and ∆vi(t) are respectively the distance headway and speed difference

between vehicle i, i−1 ∈ N (t), i.e., si(t) := pi−1(t)−pi(t) and ∆vi(t) := vi−1(t)−vi(t).The acceleration function f(si(t),∆vi(t), vi(t)) can take different form based on the

adopted car-following model.

As our first approach, to model the optimal coordination problem for each CAV

i ∈ N (t), we will first formulate an upper-level vehicle coordination problem and then

a low-level energy minimization problem. As an alternative approach, we can replace

the aforementioned hierarchical framework with a joint optimization framework.

Vehicle Coordination Problem

For a given merging queue, each CAV i ∈ N (t) needs to determine its own

optimal merging time tm∗

i for each vehicle i so that no rear-end and lateral collision

takes place. The HVs cannot compute such merging time since they are not connected

with other vehicles in the network, and are driven by the human driver without any

notion of optimality. Nevertheless, an estimated merging time tmi of each HV i ∈ N (t)

is needed for the CAVs to compute their own optimal merging time. We have the

following cases to consider:

Case 1. If vehicle i, (i−1) ∈ N (t) are CAVs, then the optimal merging time tm∗

i

can be determined with a recursive structure [17]. All the information needed for the

computation of tm∗

i for each CAV i can be obtained from the information transmitted

from CAV (i− 1).

63

Case 2. If vehicle i ∈ N (t) is a CAV but vehicle (i− 1) ∈ N (t) is an HV, then

CAV i needs to know the information tmi−1, i.e., the estimated merging time of the HV

i− 1 to compute the merging time tmi−1 of HV i− 1.

we assumed that all the required information can be obtained from the coor-

dinator at time t ∈ [t0i , tfi ]. This is the state estimation problem we discussed earlier.

The time step at which the HV state information is available within the horizon [t0i , tfi ]

depends on the technology employed by the coordinator. For example, a computer-

vision enabled device can detect HV compared to the typical road side units placed at

an interval on the roadway.

In a mixed traffic environment the merging time of each HV to enter the merging

zone are not deterministic. To avoid this complexity, in several approaches [109], [28],

the merging time of HVs are estimated under the assumption that the HVs maintain

constant speed within the control zone. If the entry state of each HV at the control zone

is known, the merging time of HVs computed under the constant-speed assumption is

deterministic, and fails to capture the intricate driving behavior of the car-following

dynamics within the control zone. However, we will not enforce such assumptions on

the HVs, and assume that the HV trajectory will be dictated only by its car-following

dynamics. Therefore, each CAV needs to predict the driving behavior of the HVs, and

approximate the merging time tmi−1 of HV i − 1 based on the car-following dynamics.

Hence, we propose several methods to estimate the HV merging time capturing the car-

following dynamics, such as (a) stochastic prediction approach, (b) robust estimation

approach and (c) machine learning approach. Once the merging time tmi−1 of HV i−1 is

computed, CAV i can derive its optimal merging time tm∗

i following the similar recursive

structure mentioned in Case 1. Note that, if the information of HV (i − 1) ∈ N (t) is

available at each time step of [t0i , tfi ], the optimal merging time tm

∗i can be recomputed

to account for any model uncertainty and deviation.

Case 3: If vehicle i, (i−1) ∈ N (t) are HVs, and the HV i ∈ N (t) located on the

ramp road must yield to the HV (i− 1) ∈ N (t) on the main road to avoid any possible

64

lateral collision at the merging zone. Since the HVs cannot coordinate themselves by

connectivity and automation, we will end up with a gridlock in this case, where all

the vehicles on the ramp road following the HV i ∈ N (t) will experience stop-and-go

driving. Hence, such gridlocks will defeat the purpose of automated merging scenario.

However, with an adaptive ramp metering approach, such gridlock can be resolved.

We consider a speed-advisory based ramp metering approach, where the on-ramp HV

i ∈ N (t) is advised to follow a recommended speed. The speed recommendation is

made based on the state and approximate merging information of HV i and i − 1 so

that HV i can enter the merging zone after the HV i − 1 without any rear-end and

lateral collision.

Energy Optimization Problem: With the optimal merging time tm∗

i at hand,

the optimization problem for each CAV i ∈ N (t) is to minimize the L2-norm of the

control input in [t0i , tm∗i ], formulated as in (2.4).

Outside the control zone, each CAV can employ an adaptive vehicle following

dynamics to follow the preceding CAV or HV with preset safety gap. When a CAV

enters the control zone, we impose our control framework to optimize the vehicle per-

formance. However, there can be cases where the optimal control framework cannot be

realized within the control zone. If (a) there does not exist any solution to the upper-

level coordination problem, such in the case of a traffic gridlock, or (b) the solution

to the upper-level coordination problem does not yield a feasible control input while

solving the low-level state, control and safety constrained problem, then the control

input of the CAV will need to revert back to the adaptive following dynamics. In this

case, we may have to settle for a sub-optimal solution to satisfy the collision safety

constraints.

In addition to the optimization framework discussed above, we will also consider

the following augmentations. Instead of adopting a hierarchical optimization frame-

work discussed above, we can pursue a joint optimization scheme as presented in [110]

enabling us to analyze the trade off between the throughput maximization problem

65

and energy minimization problem in a stochastic mixed traffic environment. Moreover,

The cost function in (2.4) can be reformulated as a multi-objective cost function to

account for additional performance optimization.

3.2 Aim 2: Platoon-based Framework for Indirect Control of HVs

We propose a decentralized optimal framework for the CAVs to form and control

the heterogeneous platoon of following human-driven vehicles, and guide the platoon

to an automated merging scenario in an on-ramp roadway, roundabout or signal-free

intersections.

Figure 3.2: Indirect control framework with connected and automated vehicles andhuman driven vehicles.

We illustrate in Fig. 3.2 an automated on-ramp merging roadway where the

traffic network consists of CAVs and HVs. We define a merging zone of dimension S

illustrated by the red rectangle in Fig. 3.2, where lateral collision of the vehicles may

occurs. Upstream of the merging zone, we define two zones, namely a) a platoon zone of

length Lp, where heterogeneous platoons are formed optimally by controlling the CAVs

directly, and b) the control zone, where the platoons of heterogeneous vehicles formed

66

in the platoon zone coordinate themselves to optimally cross the merging zone without

any lateral and rear-end collision. The merging zone has a coordinator that enables

communication between the CAVs within the control zone and the platoon zone. The

coordinator only provides essential information to the CAVs, and does not take any

control actions. When each vehicle enters the platoon zone, the coordinator assigns a

unique identity i ∈ N , where N (t) = {1, 2, ..., N(t)}. Here, N(t) ∈ N is the number of

vehicles at time t within the platoon and control zone. Each vehicle i ∈ N (t) enters

the platoon zone, control zone, and merging zone respectively at time t0i , tci , t

mi , and

exits the merging zone at time tfi . Note that, the as we move forward with additional

definitions, the subscript i may change based on vehicle-level or platoon-level operation

without any loss of generality.

For modelling the framework, we have the following assumptions, (a) each pla-

toon consists of only one CAV as the platoon leader, and (b) each platoon leader

maintains constant speed after exiting the merging zone. These assumptions enable

the formulation of the platoon formation framework with low CAV penetration rate.

The objective of each CAV i ∈ N (t) is to act as a platoon leader and form

a platoon of following HVs. Then each platoon enters the control zone, where the

platoon leader, i.e., the leading CAVs coordinate with other platoon leaders and opti-

mally direct its own platoon through the merging zone without any safety constraint

activation.

Consistent with our previously assumed vehicle model, we consider a simple dou-

ble integrator dynamics 2.102 for the CAVs, and a predefined car-following dynamics

3.1 for the HVs to emulate the human driving behavior. Again, the acceleration func-

tion f(si(t),∆vi(t), vi(t)) in (3.1) is a generic car-following dynamics, and can be found

in the literature [78], [105] etc. Based on different car-following models, the function

f may take additional arguments such as driver reaction time, vehicle’s heterogeneity

etc., which are out of the scope of our current framework, but can be considered in the

future

67

Control Architecture: We propose a control architecture that is divided into

the following four sequential stages.

Stage 1: Platoon Identification and State Estimation Problem

When a CAV i ∈ N (t) enters the platoon zone, it identifies the following HVs

j ∈ N (t), and determines the appropriate platoon size and vehicle composition. Since

a platoon is a collection of closely spaced vehicles, we denote a platoon led by CAV

i ∈ N (t) as Ri. Once the platoon size of CAV i is fixed, it communicates with the

coordinator to receive the initial states of the following HVs of the considered platoon.

Note that, the state information of the considered HVs are not be readily available,

since the HVs do not transmit their state information. Therefore, appropriate state

estimation method needs to be employed for correct HV state estimation.

Stage 2: Platoon Formation Problem

With initial state information of the following HVs, the leading CAV i ∈ N (t)

derives its own control input that would force the following HVs to fall into the platoon

formation with desired speed and headway. Note that, Stage 1 and 2 are have to be

completed within the platoon zone.

Let’s consider a single lane of traffic consisting of CAVs and HVs. Since we

assumed that the platoon identification and state estimation problem from Stage 1 is

given a priori, CAV i ∈ N (t) knows the composition of the platoon set Ri. Since we

have a vehicle-level operation, we define the platoon set Ri = {z, z = 1, 2, ..., n}, where

n is the cardinality of Ri. Note that, z = 1 denotes the leading CAV i ∈ N (t) of the

platoon Ri.

In this framework, we restrict the choice of the acceleration function f such that

a) it admits a unique equilibrium point, and b) at the equilibrium point the function

resolves to zero. Since at the equilibrium the vehicle’s acceleration is zero, we have

f(s∗z(t), 0, v∗z(t)) = 0, (3.2)

where, s∗z(t) and v∗z(t) are the equilibrium headway and speed.

68

The objective of the platoon formation strategy is to drive the platoon to reach

the equilibrium speed v∗z(t) and headway s∗z(t) in an energy optimal way by controlling

only the lead CAV z ∈ Ri, z = 1 for time t ∈ [t0z, tcz]. We define the following cost

functions,

J1 :=

∫ tcz

t0z

1

2u2z(t) dt, z ∈ Ri, z = 1 (3.3)

J2 :=

∫ tcz

t0z

z=n∑z=2

αz(sz(t)− s∗z(t))2 dt, (3.4)

J3 :=

∫ tcz

t0z

z=n∑z=2

βz(vz−1(t)− vz(t))2 dt, (3.5)

J4 :=

∫ tcz

t0z

(vz(t)− v∗z(t))2 dt, z ∈ Ri, z = 1, (3.6)

where, αz, βz, z = 2, ..., n are suitable weights to penalize comparatively higher head-

way and speed deviation respectively. Finally, we formulate a multiobjective optimiza-

tion problem for the CAV z, z = 1 leading the platoon Ri within the time horizon

[t0z, tcz] as following,

minuz(t)∈Uz

k=4∑k=1

ωkJk (3.7)

subject to : (2.102), (3.1), (2.103),

given t0z, pz(t0z) = 0, v0

z , pz(tcz) = Lp, vz(t

cz) = v∗z(t), and tci is free.

By the appropriate choice of the weight parameters ωk, k = 1, . . . , 4, we can

have an optimal platoon formation.

Stage 3: Optimal Coordination Problem

When each platoon Ri formed during Stage 2 enters the control zone at time tci ,

each platoon leading CAV i ∈ N (t) starts communicating with other platoon leaders

to coordinate the platoon trajectory through the mering zone. Therefore, we turn our

attention from vehicle-level operation to platoon-level operation. We define a set of

platoons L(t) within the control zone at time t ∈ R+.

69

With the state information of the platoon at hand, each CAV i ∈ N (t) then

determines the optimal merging time tm∗

i to enter the merging zone without any lat-

eral and rear-end collision. Note that, the derivation of the optimal merging time is

deterministic in nature due to the fact that the CAVs are sharing the platoons state

information with each other.

If the platoon Ri ∈ L(t) is the only platoon within the control zone at time

tci , it can maintain its current speed throughout the control zone. Hence, the optimal

merging time tm∗

i for the platoon led by CAV i ∈ N (t) will be, tm∗

i = tci + Lvi(tci )

.

For each platoon Ri ∈ L(t) led by CAV i ∈ Nc(t), we now formulate the

following optimization problem to maximize throughput with hard safety constraint

on rear-end and lateral collision.

mint(2:Np(t))

Np(t)∑i=2

(tmi − tmi−1) = mint(Np(t))

(tmNp(t) − tm1 ), (3.8)

subject to : (2.102), (3.1), (2.103), (2.104), (2.105),

where, Np(t) is the cardinality of set L(t), and t(2:Np(t)) = [tm2 , . . . , tmNp(t)]. The solution

of (3.8) yields the optimal time tmi∗, i ≥ 2, which designates the entry times of each

platoonRi led by CAV i ∈ N (t) so as to maximize the throughput of the corresponding

scenario.

Stage 4: Energy-Optimal Merging Problem

The derived optimal merging time derived from Stage 3 is then used by the

CAV i ∈ N (t) leading the platoon Ri ∈ L(t) to derive an energy-optimal path for its

own platoon. The derivation of such path must guarantee string stability to avoid any

rear-end collision within the control zone.

In our formulation, we assume that the solution to the platoon identification

and state estimation problem in Stage 1 is given a priori. This simplification allows us

70

to focus on the remaining stages. In what follows, we delve deeper into Stage 2-4 to

explore how we can formulate our problem, and propose a general framework.

With the optimal merging time tm∗

i at hand, the optimization problem for each

platoon Ri ∈ L(t) led by CAV i ∈ N (t) is to minimize the L2-norm of the control

input in [tci , tm∗i ], formulated as in (2.4).

3.3 Aim 3: Optimal Control Framework with System Uncertainty

For modelling the mixed traffic scenario, we have made several assumptions such

as (a) known HV car-following dynamics, (b) given estimated HV state information at

every time step within the control zone, (c) instantaneous and complete data transmis-

sion etc. In this particular aim, we will relax these assumptions. However, relaxation

of the aforementioned assumptions has certain implications in the context of CAV-HV

interaction in a mixed traffic scenario.

If the car-following model adopted by the HV is not known a priori, then the

CAV will not be able to estimate the coupled car-following dynamics of the HV, and

cannot approximate the merging time of the HVs passing a traffic scenario. In our

framework, we will also address the complexity resulting from the state estimation and

data transmission. Such complexity can be dealt with a rigorous stochastic optimal

control framework, which may have some inherent drawback in terms of real-time policy

derivation and control input implementation. Therefore, we may use such framework

to derive the benchmark for our proposed optimization schemes. In what follows, we

propose several potential ways to resolve the issues discussed above.

(a) Robust optimization scheme: We can employ a robust optimization frame-

work to predict probability distribution the merging time of HVs and avoid the worst

case scenario. Note that, such approach can be overly conservative leading to loss of

optimality.

(b) Receding horizon optimization framework: We can employ a receding horizon

71

optimization framework to minimize the estimation and prediction states at each time

step of the optimization horizon. In this case, the CAV needs to receive the state

information of the HVs at each time step, compute/update its optimal policy, and

apply the control input only for the first time step. At the next time step, the CAV

will solve the optimization problem with reduced horizon and update its optimal policy.

(c) Learning-based approach: We can also employ a data-driven approach to

approximate and predict the dynamics of HVs for a specific traffic scenario.

72

Chapter 4

EXPECTED CONTRIBUTIONS

The contribution and expected outcome of the proposed research can summa-

rized as below:

Contribution 1: Development of a rigorous mathematical model for a decentral-

ized time and energy optimization framework for coordinating CAVs through different

traffic scenarios in a mixed traffic environment considering stochastic human driving

behavior (original contribution: Aim 1).

Contribution 2: An optimal string-stable platoon-based coordination framework

for CAVs and HVs in a mixed traffic environment for coordination in automated traffic

scenarios. (original contribution: Aim 2).

Contribution 3: A stochastic optimal framework considering of system uncer-

tainties in a traffic network with interacting CAVs and HVs (original contribution:

Aim 3).

The first contribution addresses a major gap in the literature by providing a

mathematically rigorous optimization framework for CAVs operating within a traffic

network with interacting HVs. Several studies [109], [77], [48] can be found in the liter-

ature which only provides a general impact analysis without addressing the stochastic

dynamics of the HVs. The key aspect of our contribution is that we will try to cap-

ture the complete stochastic nature of the HVs in our formulation. Additionally, this

contribution will provide a mathematically rigorous optimal coordination scheme for

the CAVs and HVs to maximize traffic throughput and the energy efficiency. We will

73

also provide a joint time- and energy-optimization framework with multi-objective cost

functions to explore the trade-offs among different performance metrics.

The second contribution introduces a platoon-based vehicle coordination ap-

proach that has not been addressed in the literature to date. Generally, the HVs in

a mixed traffic network do not share their state information, and are not automated,

posing major difficulty in developing automated coordination frameworks at traffic dif-

ferent scenarios. This contribution will enable the CAVs present in the traffic network

to indirectly control the HVs through platoon formation, which can guarantee optimal

coordination of CAVs and HVs at traffic scenarios such as on-ramp merging roadway,

roundabout or signal-free intersections without any stop-and-go driving behavior. This

contribution will also address the optimal formation and string stability of a mixed pla-

toon with heterogeneous dynamics extensively, which has not been addressed before

in the literature. The improvement of vehicle performance metrics such as energy ef-

ficiency and travel time in the context of mixed platoon can also be explored through

this contribution.

Finally, the third contribution specifically addresses the problem of optimal

coordination of CAVs and HVs in a mixed traffic environment with different system

uncertainties. An MPC-based approach has been proposed in [28] considering the

system uncertainties coming from other CAVs. In this contribution, we will extend

this approach to address the system uncertainties stemming from incorrect HV state

estimation, arbitrary HV car-following dynamics and incomplete information flow. A

robust optimization framework will be developed in this contribution to mitigate the

worst-case scenario attributed to the system uncertainties.

In summary, the three contribution discussed above will provide a complete

rigorous mathematical solution to the problem of optimal control of CAVs in a mixed

traffic environment. The outcome of the proposed research direction not only provide

a vehicle-level performance improvement, but also realizes the automated coordination

and merging concept, which increases the traffic performance from a macroscopic view

74

point. Compared to the utopian scenario with 100% CAV penetration, the developed

control framework will be implementable online in real-world mixed traffic network

with variety of scenarios.

75

Table 4.1: Provisional timeline to accomplish research aims [Spring, 2020 - Winter,2020].

Time Period Research TasksSpring, 2020

• Literature review of car-following models andestimation of coupled dynamics

Summer, 2020

• Development of a framework for HV stateestimation and vehicle dynamics approximationbased on the stochastic car-following model

• Formulation of the decentralized control problemfor mixed CAV-HV traffic environment through aselected scenario (Aim 1)

• Solution of the traffic coordination problem withhierarchical and joint optimization strategy

Fall, 2020

• Numerical validation and performancequantification of the proposed control framework(Aim 1)

• Mathematical formulation of an optimal controlframework for CAVs to form mixed platoons (Aim2)

• Solution of the mixed platoon formation problemand development computationally efficientalgorithm for real-time implementation. (Aim 2)

Winter, 2020

• Validation of the developed control frameworkthrough simulation in commercial software (Aim 2)

• Literature review of platoon stability in the contextof heterogeneous vehicle composition (Aim 2)

76



• Analysis of the asymptotic and string stabilitycharacteristics of mixed platoons (Aim 2)

• Formulation of the optimal coordination frameworkof the mixed platoons travelling through differenttraffic scenarios with safety guarantees and stringstability (Aim 2)

• Numerical validation and performancequantification of the complete framework developedfor Aim 2 in the state-of-the-art commercialsoftware

Summer, 2021

• Reviewing the literature of modelling systemuncertainties in the context of HVs and vehicleplatoons.

• Rigorous mathematical formulation of the theCAV-HV interaction and coordination with systemand model uncertainties

Fall, 2021

• Solution of the optimal control problem withuncertainties with stochastic optimization, robustoptimization, or model-predictive control approach.

• Derivation of conditions to enable system responsewith less computational complexity and higherefficiency

Winter, 2021

• Validation of the developed control frameworkthrough simulation in commercial software

• Characterization of performance metrics

77



• Revising the research outcomes from Aim 1, 2 and3, and finalize a complete formulation satisfying theoverarching objective of the dissertation

Summer, 2022

• Defense of the Ph.D dissertation

78

Appendix A

LIST OF PUBLICATIONS

Journal Papers:

1. Mahbub, A M I.; Malikopoulos, A A; “Conditions to Provable System-WideOptimal Coordination of Connected and Automated Vehicles” (in review, arXiv:2005.14551)

2. Mahbub, A M I.; Malikopoulos, A A; Zhao, L; “Decentralized Optimal Coor-dination of Connected and Automated Vehicles for Multiple Traffic Scenarios”,Automatica, vol. 117, 2020.

3. Beaver, LE; Chalaki, B; Mahbub, A M I.; Zhao, L; Zayas, R; Malikopoulos,A A; “Demonstration of a Time-Efficient Mobility System Using a Scaled SmartCity”, Vehicle System Dynamics, pp. 1-18, 2020.

Conference Proceedings:

1. Mahbub, A M I.; Malikopoulos, A A; Zhao, Liuhui, “Impact of Connectedand Automated Vehicles in a Corridor”, Proceedings of 2020 American ControlConference, pp. 1185-1190, 2020.

2. Mahbub, A M I.; Malikopoulos, A A; “Concurrent Optimization of VehicleDynamics and Powertrain Operation Using Connectivity and Automation”, SAETechnical Paper, 2020-01-0580, 2020, doi:10.4271/2020-01-0580.

3. Mahbub, A M I.; Karri, Vasanthi; Parikh, Darshil; Jade, Shyam; Malikopou-los, A A; “A Decentralized Time- and Energy-Optimal Control Framework forConnected Automated Vehicles: From Simulation to Field Test”, SAE TechnicalPaper, 2020-01-0579, 2020, doi:10.4271/2020-01-0579.

4. Mahbub, A M I.; Malikopoulos, A A; “Conditions for State and Control Con-straint Activation in Coordination of Connected and Automated Vehicles”, Pro-ceedings of 2020 American Control Conference, pp. 436-441, 2020.

79

5. Mahbub, A M I.; Zhao, L; Assanis, D; Malikopoulos, A A; “Energy-OptimalCoordination of Connected and Automated Vehicles at Multiple Intersections”,Proceeding of 2019 IEEE American Control Conference, pp. 2664–2669, 2019.

6. Zhao, L; Mahbub, A M I.; Malikopoulos, A A; “Optimal Vehicle Dynamicsand Powertrain Control for Connected and Automated Vehicles”, Proceedings of2019 IEEE Conference on Control Technology and Applications, pp. 33-38, 2019.

80

Appendix B

COURSEWORK

Systems and Controls:

• CIEG 667 - Network Optimization

• MEEG 698 - Stochastic Optimal Control

• MEEG 867 - Nonlinear Programming

• MEEG 895 - Game Theory and Mechanism Design

• MEEG 621 - Linear Systems

• MEEG 829 - Applied Nonlinear Control

• CIEG 646 - Convex Optimization

• MEEG 866 - Independent Study on Powertrain Optimization

Mathematics:

• MEEG 690 - Intermediate Engineering Mathematics

• MATH 630 - Probability Theory and Application

Other:

• MEEG 690 - Intermediate Dynamics

81

Appendix C

CURRICULUM VITAE

82

A M ISHTIAQUE MAHBUB

+1-302-268-2089 � [email protected]

007 Spencer Laboratory (IDS Lab) � 130 Academy Street, DE 19716

linkedin.com/in/a-m-ishtiaque-mahbub � sites.udel.edu/mahbub

EDUCATION

University of Delaware, USA 2017 - PresentPhD in Mechanical EngineeringConcentration: Optimal Control, Connected Automated Vehicle, Intelligent Transportation Systems

University of Stuttgart, Germany 2013-2016M.Sc. in Computational Mechanics of Materials and Structures (COMMAS)Thesis: 3D Dynamic Simulation of Concrete Hammer Drilling for SDSmax Using FEM

Bangladesh University of Engineering and Technology, Bangladesh 2008-2013B.Sc. in Mechanical EngineeringThesis: Design Optimization of a Horizontal Axis Micro Wind Turbine Through Development of a CFDModel and Experimentation

RESEARCH INTERESTS

Optimal Control of Vehicle Dynamics, Powertrain optimization, Plugin Hybrid Electric Vehicles, Con-nected and Automated Vehicles, Intelligent Transportation Systems, V2X Communication Framework

RELEVANT COURSES

PhD Network Optimization, Optimal Control, Convex Optimization,Probability and random processes, Decentralized Control,Nonlinear Control,Nonlinear Programming,Advanced Engineering Mathematics Linear Systems

M.Sc. Numerical Programming, Optimization of Mechanical Systems,Software Development C++, Discretization Methods,Numerical Methods for Differential Equations,Advanced Finite Element Methods

SOFTWARE SKILLS

Programming Languages Python, C/C++, C# (Console and Webform),MATLAB/Simulink, MATLAB GUI, Python, MySQLC++ Libraries Eigen, Armadillo, BoostPython Packages Pandas, Matplotlib, Numpy, Scipy, Pygame,

Seaborn, Keras, TensorFlow, Flask, Jupyter NotebookEditor & Compiler Visual Studio, Eclipse, devcpp, Notepad++, Pycharm, AnacondaTraffic Simulation & Tools PTV VISSIM 7-11, PreScan 7.6,CAE Tools ABAQUS 6.13, ANSYS Workbench, DIGIMATCAD Software SolidWorks, CATIA, AutoCADMiscellaneous LaTeX, MS Office Packages, Maple

MQTT, UDP & TCP-IP Protocol

PUBLICATIONS

Journal Articles:

· Mahbub, A M I.; Malikopoulos, A A; “Conditions to Provable System-Wide Optimal Coordinationof Connected and Automated Vehicles” (in review, arXiv:2005.14551)

· Mahbub, A M I.; Malikopoulos, A A; Zhao, L; “Decentralized Optimal Coordination of Connectedand Automated Vehicles for Multiple Traffic Scenarios”, Automatica, vol. 117, 2020.

· Beaver, LE; Chalaki, B; Mahbub, A M I.; Zhao, L; Zayas, R; Malikopoulos, A A; “Demonstrationof a Time-Efficient Mobility System Using a Scaled Smart City”, Vehicle System Dynamics, pp. 1-18,2020.

· Abrar, M A; Mahbub, A M I; Mamun, M; “Design optimization of a horizontal axis micro windturbine through development of CFD model and experimentation”, Procedia Engineering, Vol. 90, pp.333-338, 2014.

Conference Papers:

· Mahbub, A M I.; Malikopoulos, A A; Zhao, Liuhui, “Impact of Connected and Automated Vehiclesin a Corridor”, Proceedings of 2020 American Control Conference, pp. 1185-1190, 2020.

· Mahbub, A M I.; Malikopoulos, A A; “Conditions for State and Control Constraint Activation in Co-ordination of Connected and Automated Vehicles”, Proceedings of 2020 American Control Conference,pp. 436-441, 2020.

· Mahbub, A M I.; Malikopoulos, A A; “Concurrent Optimization of Vehicle Dynamics and Pow-ertrain Operation Using Connectivity and Automation”, SAE Technical Paper, 2020-01-0580, 2020,doi:10.4271/2020-01-0580.

· Mahbub, A M I.; Karri, Vasanthi; Parikh, Darshil; Jade, Shyam; Malikopoulos, A A; “A Decen-tralized Time- and Energy-Optimal Control Framework for Connected Automated Vehicles: FromSimulation to Field Test”, SAE Technical Paper, 2020-01-0579, 2020, doi:10.4271/2020-01-0579.

· Mahbub, A M I.; Zhao, L; Assanis, D; Malikopoulos, A A; “Energy-Optimal Coordination of Con-nected and Automated Vehicles at Multiple Intersections”, Proceeding of 2019 IEEE American ControlConference, pp. 26642669, 2019.

· Zhao, L; Mahbub, A M I.; Malikopoulos, A A; “Optimal Vehicle Dynamics and Powertrain Controlfor Connected and Automated Vehicles”, Proceedings of 2019 IEEE Conference on Control Technologyand Applications, pp. 33-38, 2019.

· Mahbub, A M I.; Mawa, Z; “A numerical approach to drying process of hygroscopic polymeric gran-ulates with different drying configurations and parameter comparison”, AIP Conference Proceedings,Vol. 1851, Num. 1, pp. 020060, 2017.

HONORS AND AWARDS

UD Collection-Based Research Grant, University of Delaware September 2019

Professional Development Award, University of Delaware July 2019

Graduate Travel Grant,2019 Learning for Dynamic Control Conference, MIT, USA May 2019

Outstanding Presentation Award, 8th Annual Graduate Students’ Forum,University of Delaware, USA 10 May, 2019

Dean’s List for Academic Excellence, BUET 2012-2013

Award Winner (3rd Place) at Mechanical Engineering Project Show, BUET 2010

Government Merit Scholarship, Bangladesh 2005-2007

EXPERIENCE

Graduate Intern June 2020 - PresentNational Renewable Energy Lab (NREL), Colorado

· Advanced transportation analysis with machine learning

Graduate Research Assistant February 2018 - PresentARPA-e NEXTCAR Project (Award Number: DE-AR0000796)

· Powertrain optimization and simulation of plugin hybrid electric vehicles (PHEVs)

· Vehicle dynamics optimization and simulation of connected automated vehicles (CAVs)in a traffic corridor with highway on-ramp merging, roundabout, speed reduction zone etc.

· Vehicle dynamics and powertrain controller integration

· Vehicle dynamics under partial penetration

PhD Intern June 2019 - August, 2019Robert Bosch LLC, USA

· Vehicle dynamics (VD) controller development and testing for Audi A3-etron

· Establishing V2X communication between Audi A3-etron and the UMTRI framework

· Design, setup and conduct vehicle testing in Mcity with augmented reality

Graduate Teaching Assistant August 2017 - February 2018Dept. of Mechanical Engineering, University of Delaware

· Vibration and Controls Lab (MEEG 312)

Assistant Professor Jan 2017 - June, 2017Military Institute of Science and Technology (MIST), Bangladesh

· Theoretical courses taught and laboratory sessions conducted

Lecturer July 2016 - Dec 2016Military Institute of Science and Technology (MIST), Bangladesh

· Theoretical courses taught and laboratory sessions conducted

M.Sc. Thesis Internship June 2015 - November, 2015Robert Bosch GmbH, Power Tools Division, Germany

· Creation of Abaqus explicit CAE model and implementation of material routine

· Experimental data collection, post-processing and validation

· Reverse engineering of competitor products, investigation and comparison

· Material parameter identification and design optimization

Student Research Assistant June 2014 - March, 2015Institute of Polymer Technology (IKT), University of Stuttgart, Germany

· Effect of injection molding parameters on fiber-glass composite characteristics

· Modeling and experimental study of hygroscopic polymer drying kinetics using FDM Formulation

Mechanical Engineering Intern September 2014 - February, 2015Fraunhofer Institute of Manufacturing Engineering and Automation (IPA), Stuttgart, Germany

· Multi-scale simulation of electrical properties of CNT-based composites

· Analysis of percolation threshold and conductivity

· Microstructure modelling and numerical simulation

Intern June 2013 - August 2013Engineering Resources International, Dhaka

· Transformation of FCK brick manufacturing kilns into modern Zig- Zag kilns by developing and de-signing an efficient and environment friendly exhaust system and combustion chamber.

ACADEMIC & PROFESSIONAL ACTIVITIES

Conference Attended2020 Transportation Research Board Annual Meeting, Washington D.C., USA 11-16 January, 20202019 American Control Conference, Philadelphia, USA 10-12 July, 20192019 Learning for Dynamic Control Conference (L4DC), MIT, USA 30-31 May, 20197th BSME International Conference on Thermal Engineering 2016, BUET 22-24 December, 2016(Session Co-Chair)

Training CompletedPLC Programming with Logo and Siemens, IAT, Bangladesh Dhaka 2017Short Training on Computational Fluid Dynamics-ANSYS, MIST Dhaka 2017Industrial Training on HVAC, Novartis Limited, Dhaka Dhaka 2012

Seminar/Conference Presentation2020 Transportation Research Board Annual Meeting, Washington D.C., USAPoster: Energy Optimal Operation of CA-PHEVs 11-16 January, 20202019 American Control Conference, Philadelphia, USA 10-12 July, 20198th Annual Graduate Students’ Forum, University of Delaware, USA 10 May, 2019Fracture Mechanics and Crack Propagation in Woods, Uni. Stuttgart February 2016

Reviewer of Scholarly ArticlesDiscrete Event Dynamic Systems 2020Automatica 2019-PresentIEEE International Conference on Intelligent Transportation Systems 2019-Present58th IEEE Conference on Decision and Control 20192019 American Control Conference, Philadelphia, USA 2019IEEE Transaction on Intelligent Transportation System 2018-PresentIEEE Transaction on Intelligent Vehicles 2018-Present21st IEEE International Conference on Intelligent Transportation Systems 2018

Organizational AffiliationSecretary, Mechanical Engineering Graduate Association, UD 2019 - presentGeneral Secretary, Bangladesh Student Association, UD 2019 - presentStudent Member, Society of Automotive Engineers (SAE) 2019 - presentStudent Member, Institute of Electrical and Electronics Engineers (IEEE) 2018 - presentStudent Member, IEEE Young Professionals 2018 - presentStudent Member, Society for Industrial and Applied Mathematics (SIAM) 2018 - presentUD Student Chapter: Society for Industrial and Applied Mathematics (SIAM) 2018 - present

Theoretical Courses TaughtEngineering Mechanics Spring, 2017Instrumentation and Measurement Spring, 2017Engineering Thermodynamics Spring, 2017Computer Programming Language (C/C++) Fall, 2016Numerical Analysis Fall, 2016

Laboratory Courses TaughtVibration and Controls Lab Fall, 2017Mechanics of Machinery Sessional Spring, 2017Engineering Thermodynamics Sessional Spring, 2017Engineering Mechanics Sessional Spring, 2017Mechanical Engineering Drawing II (SolidWorks) Fall, 2016Numerical Analysis Sessional (MATLAB) Fall, 2016Computer Programming and Applications Sessional Fall, 2016Computer Programming Language Sessional Fall, 2016Workshop Technology Sessional II Fall, 2016Machine Shop Sessional Fall, 2016

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