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9 Figures# Joint-based control of a new Eulerian network model of air traffic flow

Abstract

An Eulerian network model for air traffic flow in the National Airspace System is developed and used to design flow control schemes which could be used by Air Traffic Controllers to optimize traffic flow. The model relies on a modified version of the Lighthill-Whitham-Richards (LWR) partial differential equation (PDE), which contains a velocity control term inside the divergence operator. This PDE can be related to aircraft count, which is a key metric in air traffic control. An analytical solution to the LWR PDE is constructed for a benchmark problem, to assess the gridsize required to compute a numerical solution at a prescribed accuracy. The Jameson-Schmidt-Turkel (JST) scheme is selected among other numerical schemes to perform simulations, and evidence of numerical convergence is assessed against this analytical solution. Linear numerical schemes are discarded because of their poor performance. The model is validated against actual air traffic data (ETMS data), by showing that the Eulerian description enables good aircraft count predictions, provided a good choice of numerical parameters is made. This model is then embedded as the key constraint in an optimization problem, that of maximizing the throughput at a destination airport while maintaining aircraft density below a legal threshold in a set of sectors of the airspace. The optimization problem is solved by constructing the adjoint problem of the linearized network control problem, which provides an explicit formula for the gradient. Constraints are enforced using a logarithmic barrier. Simulations of actual air traffic data and control scenarios involving several airports between Chicago and the U.S. East Coast demonstrate the feasibility of the method

804 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 14, NO. 5, SEPTEMBER 2006

Adjoint-Based Control of a New Eulerian Network

Model of Air Trafﬁc Flow

Alexandre M. Bayen, Member, IEEE, Robin L. Raffard, and Claire J. Tomlin, Member, IEEE

Abstract—An Eulerian network model for air trafﬁc ﬂow in the

National Airspace System is developed and used to design ﬂow

control schemes which could be used by Air Trafﬁc Controllers

to optimize trafﬁc ﬂow. The model relies on a modiﬁed version of

the Lighthill–Whitham–Richards (LWR) partial differential equa-

tion (PDE), which contains a velocity control term inside the diver-

gence operator. This PDE can be related to aircraft count, which

is a key metric in air trafﬁc control. An analytical solution to the

LWR PDE is constructed for a benchmark problem, to assess the

gridsize required to compute a numerical solution at a prescribed

accuracy. The Jameson–Schmidt–Turkel (JST) scheme is selected

among other numerical schemes to perform simulations, and evi-

dence of numerical convergence is assessed against this analytical

solution. Linear numerical schemes are discarded because of their

poor performance. The model is validated against actual air trafﬁc

data (ETMS data), by showing that the Eulerian description en-

ables good aircraft count predictions, provided a good choice of

numerical parameters is made. This model is then embedded as

the key constraint in an optimization problem, that of maximizing

the throughput at a destination airport while maintaining aircraft

density below a legal threshold in a set of sectors of the airspace.

The optimization problem is solved by constructing the adjoint

problem of the linearized network control problem, which provides

an explicit formula for the gradient. Constraints are enforced using

a logarithmic barrier. Simulations of actual air trafﬁc data and

control scenarios involving several airports between Chicago and

the U.S. East Coast demonstrate the feasibility of the method.

Index Terms—Adjoint-based optimization, control of partial dif-

ferential equations, LWR PDE.

I. INTRODUCTION

T

HE National Airspace System (NAS) consists of aircraft,

control facilities, procedures, navigation and surveillance

equipment, analysis equipment, decision support tools, and con-

troller pilots who operate the systems. In this article, the focus is

trafﬁc ﬂow management (TFM), which has the goal to maximize

throughput while maintaining safety. This entails the design of

efﬁcient methods to route aircraft, while preventing the density

of aircraft from becomingtoo large in regions of airspace, and op-

erating efﬁcient reroutes when the weather does not allow trafﬁc

to cross a given region of airspace. These tasks are not currently

Manuscript received May 11, 2005. Manuscript received in ﬁnal form March

27, 2006. Recommended by Associate Editor S. Devasia. This work was

supported in part by NASA under Grant NCC 2-5422, by the Ofﬁce of Naval

Research (ONR) under MURI Contract N00014-02-1-0720, by the Defense

Advanced Research Projects Agency (DARPA) under the Software Enabled

Control Program (AFRL Contract F33615-99-C-3014), and by a Graduate

Fellowship of the Délégation Générale pour l’Armement, France.

The authors are with the Department of Aeronautics and Astronautics, Stan-

ford University, Stanford, CA 94305-4035 USA and are also with the Depart-

ment of Electrical Engineering and Department of Civil and Environmental En-

gineering, University of California at Berkeley, Berkeley, CA 94720-1710 USA.

Digital Object Identiﬁer 10.1109/TCST.2006.876904

optimized with respect to throughput or maximal density. Rather,

they are prescribed by playbooks, which are procedures that have

been established over time, based on controller experience.

The key objective of this article is to design control strategies

in the form of “ﬂow patterns,” that is, to come up with ways to

route streams of aircraft by generating the corresponding aircraft

velocities, rather than optimizing local trajectories of aircraft.

Ideally, one would like to automatically generate procedures

implementable by air trafﬁc control (ATC), of the following

kind: “aircraft on airway 148 at 33 000 ft, ﬂy at 450 kn for the

next hour and then accelerate by 10 kn per half hour.” This sug-

gests following an Eulerian approach advocated by Menon

et al.

[24] and dividing the airspace into line elements corresponding

to portions of airways, on which the density of aircraft as a

function of time and of the coordinate along the line, is modeled.

A traditional way to describe the evolution of the density of

vehicles in a network is to use a partial differential equation

(PDE). This PDE appears naturally in highway trafﬁc and is

called the Lighthill–Whitham–Richards (LWR) PDE [22], [30].

In this work, we will derive a modiﬁed version of the LWR

PDE speciﬁcally applicable to the ATC problem of interest.

First, we show that despite the information loss inherent in

any Eulerian model, the aircraft count (which is a crucial ATC

metric deﬁned in this article) is predicted accurately. Second, we

show that fast numerical analysis tools can be applied efﬁciently

to this problem for simulation purposes. The main difference be-

tween ours and previous work using LWR models of air trafﬁc

[24] or highway trafﬁc [14], [26], [34], is that we generate an op-

timization problem (with throughput and maximal density as an

objective function) using the continuous PDE directly, instead

of its discretization. We show that the use of linear numerical

schemes to approximate the solution of the PDE perform very

poorly, which unfortunately precludes the use of standard linear

optimization programs to control the system.

Controlling transportation networks in general is extremely

challenging and numerically difﬁcult [15], [26]. In the present

case, the control consists of speed assignments and routing poli-

cies. We show that we may use ﬂow control techniques [8],

which are directly applicable to PDE-driven systems.

Namely, we pose the optimal control of the network as an

optimization program, whose variables are solutions to a set

of PDEs and satisfy additional inequality constraints. The op-

timization is performed by updating the control variables in the

opposite direction of the gradient of the objective function. The

gradient is derived using an adjoint method, specially adapted to

the case in which the system is described by a set of PDEs cou-

pled through the boundary conditions, in the presence of con-

straints. This algorithm does not provide proofs of convergence

to a global optimal. However, this method, as well as other ﬂow

1063-6536/$20.00 © 2006 IEEE

BAYEN et al.: ADJOINT-BASED CONTROL OF A NEW EULERIAN NETWORK MODEL OF AIR TRAFFIC FLOW 805

control approaches [19], [3], [1], [13], [21], [32], have been

shown to work extremely well in practice in ﬂuid mechanics.

In addition, though we consider networks of PDEs, the dimen-

sion of each PDE is one, enabling online implementations, as

solving a set of one dimensional PDEs may be done extremely

quickly. As such, we demonstrate the feasibility of generating

direct, open-loop control solutions to the air trafﬁc ﬂow control

problem using accurate numerical schemes.

There are a few beneﬁts of the above outlined approach over

Lagrangian methods, which incorporate all trajectories of all

aircraft.

• Most of the Lagrangian methods pose the control problem

as an integer optimization program, which is intractable

in real-time because it is NP complete. In addition, the

solution provided by these methods often takes advantage

of actuating single aircraft individually, which precludes

the derivation of global policies. The Eulerian framework

scales well with the number of aircraft (the larger the

number of aircraft is, the more accurate the model be-

comes, without further computational complexity).

• The method presented in this article is general and can

be very easily adapted to speciﬁc classes of controllers

(smooth, continuous, piecewise afﬁne, etc.): it is possible

to use this method to derive a control law in a required

format, which is compatible with aircraft capabilities.

• This method can be applied to highway trafﬁc with minor

modiﬁcations [6], and we believe can be extended to other

problems such as networks of irrigation channels [23].

This article is organized as follows. In Section II, we will ﬁrst

rederive the LWR PDE for the case of interest and generalize it

to a network. Then, we determine an analytical solution for the

case of time-invariant velocity control, which, in Section II-C,

will be used for numerical validation purposes. In Section III, we

explain how to identify the numerical values of the parameters

for the airspace of interest using enhanced trafﬁc management

system (ETMS) data. This model is validated against ETMS

data in Section IV. In Section V, we derive the adjoint system

to our problem, and show how to use it to determine the mean

velocity proﬁles along the links as well as the routing policy.

Finally, in Section VI, we show how to apply this to control a

very busy portion of airspace: the area enclosed by Chicago,

New York, Boston, and the eastern coast of Canada.

II. N

EW EULERIAN NETWORK MODEL OF

AIRSPACE

A. Modiﬁed LWR Model of Air Trafﬁc

In describing the air trafﬁc system, like the road system, one

has to ﬁrst look at aircraft (or cars) present in the system and esti-

mate a density of vehicles. Therefore, given a portion of airspace

(airway or sector), one needs to introduce the aircraft count [9]

deﬁned as the number of aircraft in that region. Let us consider a

portion airway of length

, described by a coordinate .

The number of aircraft in the segment

at time is called

. Thus, represents the aircraft count on the por-

tion of airway

. Assuming a static mean velocity proﬁle

deﬁned on represents the mean velocity of air-

craft at location

, and the motion of an aircraft is described by

the dynamical system

.

Introducing

, it is fairly easy to see that

if an aircraft were at location

at time , it would be at at

time

. Because of the sign of is

invertible, and, therefore,

is related to and by

.

Consider a point

and a distance such that

. The number of aircraft between and at

can be related to the number of aircraft at at locations

and (con-

servation of aircraft):

. In other

words, assuming that there is no inﬂow at 0

Some simple algebra (two successive applications of the chain

rule) shows that the space derivative and the time derivative of

are related by

We recognize this as a ﬁrst-order linear hyperbolic PDE, and

can now enunciate the following proposition.

Proposition 1: Let

be a func-

tion with a ﬁnite number of discontinuities at

on . Assume for all . Let

and . Then the fol-

lowing PDE:

in

in

in

(1)

admits a unique continuous (weak) solution, given by

if (2)

if

where , and is its inverse.

Proof: See the Appendix.

In (1),

represents the inﬂow at the entrance of the link

(i.e., at

). In highway trafﬁc ﬂow analysis, is sometimes

referred to as cumulative ﬂow. It can be related to the vehicle

density through the integral relation

(3)

806 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 14, NO. 5, SEPTEMBER 2006

where is the vehicle density. It can be checked that the

vehicle density satisﬁes the following PDE:

(4)

Equation (4) can be related to (1) by a simple integration of

along . Equation (4) is a mass conservation equation. Note

that it is very different from the original LWR PDE [22], [30],

[2], [15], [12], which is a ﬁrst order hyperbolic conservation law.

In particular, (4) does not have a fundamental diagram, i.e., there

is no functional relation between

and , or between and the

ﬂux. In the implementation studied in this paper, the function

will represent the control input. It is also possible to rewrite

the ﬁrst equation in (4) as

(5)

which provides the following corollary.

Corollary 2: The solution of (5) for

is given by

if

otherwise.

The interpretation of the corollary follows. The quantity

is

conserved along the characteristic curves

. At this stage, is deﬁned by and satisﬁes

(4). However, unlike for highway trafﬁc, the density

might

not be the best way to characterize the ﬂow situation at a given

time. If the number of aircraft in the system is small,

will be

a set of spikes, which is intractable numerically. Therefore, a

more tractable quantity to work with would be

, where

represents the number of aircraft contained in a ﬁnite interval of

length

. This quantity does not a priori satisfy the PDE (4). It

is meaningful to introduce an additional “density-like” quantity

called

, which satisﬁes the PDE and for which we can suggest

a physical interpretation

where is a reference time. represents the number

of aircraft included in a window of

time units around lo-

cation

and can be referred to as “time density.” This way of

accounting for density is meaningful for air trafﬁc control, since

it incorporates a time scale

into the density computation and

thus, provides access to the time separation between aircraft. It

is easy to show that

itself satisﬁes the same PDE as for any

value of

One can also show that when , and are the same

At this stage, we have three quantities: , and . The

meaning of

as we know it in ﬂuid mechanics assumes a large

number of particles (i.e., aircraft) per unit volume (the threshold

is deﬁned by the Knüdsen number). In the present case, the

number of aircraft we consider will almost certainly be below

this number, meaning that the ﬂuid approximation is question-

able. This means that instead of using

, we will use

in the PDE. We will justify this approximation with

appropriate validations.

B. Network Model

The model of the previous section describes trafﬁc on a single

portion of airway or line element. As was done earlier for high-

ways [15], this model can be generalized to airway networks, i.e.,

sets of interconnected airways, as shown in Fig. 1. We now derive

a framework to describe unidirectional air trafﬁc. We describe

the topology of the network by a unidirectional graph

,in

which

is the set of edges or links, and the set of vertices. We

index the links by

, rather than by the indices of the

two corresponding vertices. For all

, we call

the set of upstream links merging into link , and the set of

links for which the upstream links are only merging. The number

of links merging into a single link is not limited; it is possible to

have

. If there is a divergence at the end of a link ,

we assume for simplicity that there are only two emanating links

from the corresponding vertex. We index by

and the two em-

anating links (left and right), and call

the portion of the ﬂow

going from

to , and the proportion of the ﬂowgoing from

to . We call the set of links with a divergence at the end of it.

The

are not known a priori and have to be determined. These

coefﬁcients might depend on

as well and, therefore, a depen-

dence

is included in the model. We call the set of sources

in the network and

a sink of the network, at which we might

want to perform optimization. We index all variables of the pre-

vious section by

: the aircraft density on link is , the coordi-

nate is

, the main velocity proﬁle is , etc. Note that we are not

using Einstein’s notation; the notation is summarized in Table I.

The governing PDE system thus reads

(6)

BAYEN et al.: ADJOINT-BASED CONTROL OF A NEW EULERIAN NETWORK MODEL OF AIR TRAFFIC FLOW 807

Fig. 1. Top: Tracks of ﬂights incoming into Chicago (ORD). The upper stream comes from Canada, the lower from New York and Boston (BOS). Additional

streams merge into the network (Detroit and Hartford Bradley). Bottom: Network model for the tracks shown above, with waypoints labeled. The model includes

ﬁve links, merging into ORD. The corresponding inﬂow terms correspond to a single airport as in BOS or Detroit (DTW), or to a set of airports, as in New York

(EWR, JFK, LGA).

TABLE I

N

OTATION FOR THE NETWORK PROBLEM

In the previous system, the PDE (ﬁrst equation) describes the

evolution of

on each link. The notation represents the

LWR operator. The second equation is the initial condition (i.e.,

the initial density of aircraft on each link). The third equation

expresses the conservation of aircraft at the merging points. The

fourth and ﬁfth equations express the conservation of aircraft at

the divergence points. The last equation expresses the boundary

conditions (inﬂow at the sources of the network). The sinks of

the system are free boundary conditions and, therefore, do not

appear in the previous system. The solution of (6) enables the

computation of certain metrics useful for ATC. For example,

one quantity of interest is the aircraft count per sector.

C. Accuracy of Numerical Solutions

Even for a single link

it is, in general, not possible to solve

the system (6) analytically when

depends on time. The solu-

tions

of the LWR PDE in the system (6) have very undesir-

able properties for numerical integration: they are by construc-

tion discontinuous;

1

they can develop kinks if the velocity pro-

ﬁles are discontinuous. Several numerical schemes of the orig-

inal LWR PDE have been the focus of recent research [16] in

order to address similar difﬁculties encountered in the original

LWR PDE; they have proved extremely efﬁcient in the case of

highway trafﬁc. We have chosen to use three different schemes

to compare their respective beneﬁts.

1) The well-known Lax–Friedrichs scheme [17].

2) A left-centered scheme, inspired by the Daganzo scheme

[16] in light trafﬁc

3) The Jameson–Schmidt–Turkel (JST) scheme. This scheme

is nonlinear and has very desirable properties for this work:

it captures shocks (which are present in the solutions we

1

Unlike which is its primitive and is continuous.

808 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 14, NO. 5, SEPTEMBER 2006

Fig. 2. error due to the discretization method, as a function of the number of

grid points for both schemes. Lax-Friedrichs scheme (solid), Jameson-Schmidt-

Turkel scheme

, left-centered scheme (- -).

compute, as will be seen), and when the PDE has an en-

tropy solution, which is the case for highway trafﬁc in the

original LWR setting, it converges to the entropy solution

of the problem. Details of this scheme are available in [20].

Even if a numerical scheme is theoretically proven to con-

verge to the analytical solution of a PDE, one usually does not

know

a priori the required gridsize to guarantee that the nu-

merical solution is close to the analytical solution. This type of

validation is standard in numerical analysis [17], [16].

We use the method developed earlier to compute the analyt-

ical solution of three benchmark problems solvable by hand,

involving solutions with shocks and kinks (a detailed descrip-

tion of the benchmark examples is available in [4]). For each of

the numerical schemes used, we compute the

error due to

the discretization method, as a function of the number of grid

points. The result is shown in Fig. 2. This study leads to sev-

eral conclusions. The Lax–Friedrichs scheme is very diffusive.

Its behavior is representative of linear schemes to approximate a

hyperbolic PDE. Consequently, we do not think that it is a good

idea to use such linear numerical schemes, even if it would have

the advantage of making the constraints linear in the resulting

optimization program. The left centered scheme is less diffu-

sive, but fails to capture the kinks of the solution. However, it

still provides good

convergence. The JST scheme captures

shocks accurately because of its anti-diffusive term, and thus,

gives the best results overall. It will be used for the rest of this

study. Additionally, the JST scheme has the beneﬁt that we can

use it both for the direct problem, and for the adjoint. A detailed

study of the computational time required to solve this class of

problems is out of the scope of this study. For this, we refer the

reader to our ongoing work [33], in which we compare the fol-

lowing three models: the original Menon model [24], the present

model, and a new cell-based model [31].

III. S

ELECTION OF MODEL PARAMETERS

A. System Identiﬁcation: Main Velocity Proﬁles

In this section, we identify the mean velocity proﬁles

on

each link. We use enhanced trafﬁc management system (ETMS)

data, which we can obtain from NASA Ames (see [9] for a de-

scription of ETMS data). From ETMS data, we can obtain useful

Fig. 3. Example of velocity proﬁle used for the junction LGA-ORD. The hori-

zontal coordinate is the distance from ORD in nm. The corresponding links are

shown, as well as the location of the airspace ﬁxes between the links.

ﬂight information at a 3 min rate:

2

position of each aircraft in the

NAS, altitude, velocity, and ﬂight plan (i.e., set of airways and

waypoints). From this data, we are able to identify the routes in

which trafﬁc is concentrated. Note that in recent work, Menon et

al. [24] focused on creating an automated tool which performs

similar tasks automatically at a NAS-wide level, using FACET

[9], a tool developed by NASA Ames.

We analyzed 24 hours of ETMS data and selected all aircraft

using the links of the network shown in Fig. 1. We identiﬁed all

aircraft which used each of the links, and recorded all tracks and

corresponding speeds between takeoff and landing. For each of

the links shown in Fig. 1, we identiﬁed the mean velocity pro-

ﬁles as piecewise afﬁne functions, using a least squares ﬁt. The

total number of aircraft used is 220. The result for the ﬂight New

York–Chicago is displayed in Fig. 3. The curve is a piecewise

afﬁne ﬁt obtained using least squares. As can be seen, once the

En Route altitude is reached, the curve ﬁts are almost ﬂat, which

means that the aircraft are En Route at a high altitude cruise

speed. It can also be seen from Fig. 3 that the data is relatively

broadly spread (standard deviation 19.6 kn). This suggests a re-

ﬁnement using multilayer models: dividing the link in sublayers

corresponding to altitudes (with different speed proﬁles) has the

beneﬁt of being more precise. In this work, we consider a single

layer.

B. Initial and Boundary Conditions

Once the mean velocity proﬁles are computed, we identify the

initial density of aircraft and the inﬂow (boundary conditions) in

the network. The initial position of the aircraft is easy to extract

from the ETMS data: at the prescribed time, all airborne aircraft

which are on the relevant links are selected.

1) For any selected aircraft

, at location , on link , the

classical density

is taken to be a “box”

around

, of length , where is a reference length

2

Current ETMS data can now be obtained at a higher rate, which was not

available at the time this work was performed.

BAYEN et al.: ADJOINT-BASED CONTROL OF A NEW EULERIAN NETWORK MODEL OF AIR TRAFFIC FLOW 809

Fig. 4. Different predictions obtained by the use of and for aircraft density.

Above: density propagation through the PDE system (6); below: position update

from ETMS data and from the PDE.

relevant for the scale of the problem. Calling the char-

acteristic function of an interval

(equal to 0 outside of

and 1 inside), is .

Taking all aircraft initially airborne on link

, the density is

2) Similarly, the density-like function is computed using the

knowledge of the mean velocity proﬁle along link

, called

, and the parameter

These two equations thus, represent the initial conditions for the

density and the density-like functions, which we extract from

ETMS data.

The inﬂows (boundary conditions) can also be extracted from

ETMS data: each time an aircraft takes off, it will appear in the

ETMS data as soon as it is airborne. The ETMS data also shows

the ﬁled ﬂight plan, which we select when it intends to use the

links of interest to us.

is computed the following way. At

any instant when the data shows a new aircraft on one of the

source links

, the track is in general passed the entrance point

of that link (because of the sampling rate of 3 min). Calling

the position of this aircraft on link at the ﬁrst time it appears,

we compute the time

at which it crossed the location

(using the knowledge of the mean velocity proﬁle on the link).

We then use one of the two deﬁnitions above to compute

corresponding to either or .

C. Identifying the Numerical Parameters

As explained in Section III-C, we have two ways of de-

scribing the density of aircraft in the network, in terms of

a density function

and a “density-like” function , which,

respectively, account for spatial and temporal distribution of

aircraft. The function

depends on the numerical parameter

, which we need to adjust. The value of this parameter is

crucial for predicting aircraft count: Fig. 4 shows how errors

can occur in translating density functions into aircraft count.

We want to determine the choice of parameters leading to the

smallest error in aircraft count prediction.

We ﬁrst run the following set of experiments. For the link New

York–Chicago, we run a set of simulations involving

aircraft, where successively takes all values between 1

and 50. We vary

between 0 and 120 nm. For each value

of

and , we run 400 experiments. Each experi-

ment corresponds to a uniformly distributed random density of

aircraft along link 1 in (see Fig. 1). The simu-

lation starts at a time

, with the density computed as in the

previous section, and computes the solution of the LWR PDE

until the time

. For the experiments, was chosen

equal to 1 hr (note that the duration of the total ﬂight is on the

order of two and a half hours). This solution is compared with

the solution obtained by propagating the Lagrangian trajecto-

ries of each of the aircraft independently from

to

and computing the resulting density. In mathematical terms, we

compare the two following quantities:

•

computed by the LWR PDE [6];

•

,

where

is the position of aircraft at time

.

In order to characterize the best choice of numerical param-

eters, we compute the following two quantities (notations refer

to Fig. 1):

• relative density error, deﬁned by

This quantity represents the error in density prediction due

to the propagation of

by the PDE;

• absolute aircraft count error, deﬁned by

where means number. This quantity is the sum of count

error for all sublinks of links 1, 4, and 5. Typically, a

link is divided into sublinks which correspond to different

airspace sectors. For example, if link 1 goes through 8 sec-

tors, we divide it in 8 sublinks and are interested in the

aircraft counts on these sublinks. This error thus estimates

the difference between the number of aircraft predicted by

the PDE and the number obtained by a Lagrangian prop-

agation of aircraft, where the error is the sum of all errors

on the sublinks.

The computation of both quantities is illustrated in Fig. 5. In

this ﬁgure, for each of these sublinks, we compare the number of

aircraft predicted by the method (depicted by arrows, which are

computed from the density) with the number of aircraft obtained

by a Lagrangian propagation of the trajectories. The error is

the sum of errors for all sublinks, i.e., the sum of the errors in

sector counts. The relative density error and absolute aircraft

count error are averaged (over the 400 runs) and plotted for the

range of

and considered. The result is shown in Fig. 6.

The left plot shows the relative density error. As expected, the

error decreases when the number of aircraft increases and

increases. The right plot shows the absolute aircraft count error,

averaged over 400 simulations. For this plot, each of the links 1,

4, and 5 have been divided in sublinks (20 total), of about 50 nm

length. This is a worst case scneario, i.e., the number of relevant

sectors for a ﬂight of this length is never higher. One can see

810 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 14, NO. 5, SEPTEMBER 2006

Fig. 5. (a) Illustration of the computation of the relative density error depicted in Fig. 6. The difference between the two density curves (shaded are

a) is divided

by the area below the

curve. (b) Illustration of the computation of the error in aircraft count. The link is divided into sublinks (which can correspond to sectors).

Fig. 6. (a) Relative density error between the density predicted by the Eulerian PDE propagation of the density. (b) Absolute aircraft count error for the junction

New York-Chicago.

that for and , the average aircraft count

error is always extremely small.

The best choice for

is thus obtained at the intersection

of the lowest level sets of both plots of Fig. 6, i.e., for a range

of

and . Fig. 6 can also be in-

terpreted as follows. The region in the top-right corner and the

bottom-left corner are both regions in which the model might

not be applicable. As can be seen, the relative error or abso-

lute count error exceeds values that might be realistic for prac-

tical purposes (15% error and absolute aircraft count error of 7).

These regions are to be avoided.

IV. V

ALIDATION OF THE MODEL

In the previous section, we have shown that the use of the

modiﬁed LWR PDE either with

(with any )or

(with an appropriate choice of ) enables accurate aircraft

count predictions. In this section, we validate the model against

real data.

A. Static Validation

In the ﬁrst experiment, we use the static velocity proﬁles

determined in the previous sections for the validation of

the method. We use a 6-hr ETMS data set. From this data set,

we extract the position of the aircraft, at the initial time, con-

struct the corresponding initial aircraft density, and propagate

it through the PDE system. At any given time, we compare the

aircraft count predicted by our method and the aircraft count

provided by the ETMS data (which is exact, since it provides

the position of each aircraft). We compute the error in aircraft

count for a set of ten sublinks for the network shown in Fig. 1.

The result is shown in Fig. 7(a). The window width

was

taken equal to 15 nm. One can see on the left plot that the total

error (for all airborne aircraft in this airspace) is relatively low

(the maximal error is 7 aircraft). In fact, the results are much

better than they seem: most of the errors come from the fact

that the aircraft distribution is such that there is always at least

one or two aircraft close to a sublink boundary, which will

thus be counted in the wrong sublink. In fact, this is not really

a problem, as it is more an artifact of the computation rather

than a true error (Fig. 11 illustrates that the density unambigu-

ously shows where the aircraft is). Furthermore, some of the

errors in aircraft count are due to errors present in the ETMS

data (some have clearly erroneous data; this fact has also been

reported in [11]).

BAYEN et al.: ADJOINT-BASED CONTROL OF A NEW EULERIAN NETWORK MODEL OF AIR TRAFFIC FLOW 811

Fig. 7. (a) Error in aircraft count for the static validation over a ﬁve hour period. (b) Error in aircraft count for the dynamic validation over a 5-hr period.

B. Dynamic Validation

We extend the validation to a case in which the velocity pro-

ﬁles are time dependent, i.e.,

. The details of the identiﬁ-

cation of these proﬁles are technical and are not explained here.

The comparison is the same as in the static case. The results are

shown in Fig. 7 and are more accurate than the static results, as

expected. The same remarks apply, and the results are again af-

fected by the quality of ETMS data and the inclusion of the com-

putation artifact. The only weakness of this validation is that the

simulation is run using data from the same day as the data used

in identiﬁcation. A way to improve this would be to perform the

velocity identiﬁcation with data of a given day over a 24–hr pe-

riod, and validate it over the next 24–hr period, using the fact that

there is periodicity in the trafﬁc for normal days. This was not

done here due to lack of available data. An animation (.avi movie

ﬁle) corresponding to the snapshots of Fig. 7 is available.

3

In both cases, the validation is very encouraging and shows

strong predictive capability for our model. The model was also

tested successfully using data from the western states (Oakland

Center with trafﬁc incoming into Bay Area airports), though for

brevity these results are not included here.

V. N

ETWORK CONTROL VIA ADJOINT METHOD

Consider solving the following problem: maximize the

throughput (i.e., ﬂux of landing aircraft) at a destination airport,

while maintaining the density of aircraft everywhere lower than

a given threshold. Let us call

the maximal allowed den-

sity on link

and the maximal and minimal

achievable speeds onlink

(which can depend on location). Using

the notations of Section II-B, the optimization problem thus reads

(7)

The difﬁculty posed by the constraints can be avoided in prac-

tice by using a barrier function as commonly done in optimiza-

3

http://cherokee.stanford.edu/~bayen/TCST06.html.

tion [10], in which the cost is augmented by a logarithmic term,

which prohibits violation of the constraints.

(8)

We call

the augmented cost function. When ,

and

are used without indices, it means that they are vectors,

i.e.,

. Note that the two last constraints in the

optimization program (7) have disappeared into the cost func-

tion. This constrained optimization problem is easier to solve

in practice. It is asymptotically equivalent to the problem of

interest when

. We use an adjoint method to alge-

braically compute the gradient of the cost function. This method

was extensively used [8] in ﬂow control. We now adapt the

adjoint method to the case in which we have a set of PDEs

coupled through the boundary conditions, and subject to con-

straints. The adjoint method computes the gradient of the cost

function

when is an implicit function of and

via the dynamics (6). Let us denote the cost function of the

two variables

and ,

where

is the solution of the PDE system (6). We compute the

linearized (6), which we will use to compute the gradient of

the cost function in the optimization program (8). We denote by

the linearized quantities around a nominal value denoted by

. We call the linearized LWR operator

and

. In order to abbreviate the notation, we will write

and . We omit the time and space

dependence when they are obvious. The linearized (6) reads

(9)

812 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 14, NO. 5, SEPTEMBER 2006

The ﬁrst variation of is obtained from (8)

An integration by parts leads to the following identity for any

two functions

and

which can be rewritten using the standard inner product denoted

for the domain

(10)

where

We will denote by the standard inner product in .

is called the adjoint operator of . In order to express

the ﬁrst variation of

as a function of the and only, we

choose an adjoint density ﬁeld

that cancels all the terms con-

taining

in the cost function. First, in order to eliminate the

term

,

we choose

such that

(11)

This is a ﬁrst-order linear hyperbolic PDE, which is well posed

if

is known and both the boundary conditions at one location

and the initial conditions at one time are speciﬁed. This allows

Fig. 8. Network model.

us to enforce two other conditions for

in order to cancel all

the terms containing

. We can choose

(12)

These conditions have been chosen by necessity of the algebraic

derivation, in order to cancel appropriate terms in the perturba-

tion of the cost function. After some algebra, using (10)–(12),

we are able to express the ﬁrst variation

of as a function

of the ﬁrst variations control variables only (

and ), as well

as nominal and adjoint quantities, which we can evaluate. The

result reads

where again denotes the inner product for the domain

and for . The functions and

generated by this method might be ill-behaved and, thus

be inappropriate for practical air trafﬁc control applications. We

can alleviate this difﬁculty by projecting the descent direction

into

a vector space

of appropriate functions, for example the set

of continuous functions with bounded derivative, or the set of

continuous piecewise afﬁne functions.

VI. CONTROLLER DESIGN

In this section, we demonstrate the effectiveness of the adjoint

method by applying it to the air trafﬁc model. Fig. 8 shows the

area which we will control (enclosed by a box). The inﬂows into

the box are thus now

and as shown in Fig. 8. We want to

impose the following constraint: for all links, the density should

be below a threshold

which we impose. We allow the ﬂow

BAYEN et al.: ADJOINT-BASED CONTROL OF A NEW EULERIAN NETWORK MODEL OF AIR TRAFFIC FLOW 813

Fig. 9. Top left: Decrease of the cost as a function of the iterations for the three scenarios. The increases in are clearly visible (steps), while the gradient descent

is more subtle. Congested trafﬁc (solid); heavy trafﬁc (- -); normal trafﬁc

. Top middle: Decrease of the true cost as a function of the iterations. The true cost

is the cost

without the barrier terms. The method does not guarantee the monotonicity of the decrease but only the convergence. Top right: Evolution of the

parameter as a function of time. Bottom: Evolution of the velocity ﬁelds as a function of time for the different links. Each of the plots corresponds to a link, (see

top-left corner). The axis of each subplot are:

(arclength along the link), (time) and , the velocity distribution.

to be split into a new link (link 6), in order to aid satisfaction of

the maximal density constraints. We call

the corresponding

split factor:

is the fraction of the ﬂow which stays on link

1 (called 1 bis);

is the fraction which is routed through

link 6. This new link might use another arrival into the airport (it

enters the arrival airspace from another direction).

4

We simulate

the following three scenarios.

Scenario 1: Normal Trafﬁc. (Real data) We take ETMS data,

from which we extract initial conditions and inﬂows. We impose

a restriction on the density and control the ﬂow.

Scenario 2: Heavy Trafﬁc. (Modiﬁed real data) We take the

same data as for the previous case, and add additional aircraft in

order to more heavily overload the network.

4

Note that using is equivalent to using turning proportions in road trafﬁc

and might not be the best way to represent network trafﬁc. It could be better to

deﬁne an assignment proportion, i.e., a coefﬁcient indexed by destination. This

might be implemented in the future (as a part of the control strategy), using a

framework such as the one developed by Papageorgiou [25].

Scenario 3: Congested Network. We generate data with very

high densities of aircraft. This situation does not use ETMS data;

it is generated randomly.

Fig. 9 shows the decrease in cost for the three scenarios as

a function of the total number of iterations (i.e., iterations on

and gradient advances). As can be seen in this ﬁgure, the

more congested the situation is, the higher the cost. The evolu-

tion of the cost with iterations exhibits two distinct behaviors,

as often with barrier methods [10]: large jumps corresponding

to the increases in

, and shallower decreases corresponding

to the gradient advances. Convergence is clearly observed for

the three scenarios. We display some of the results for the third

case. An animation (in form of an .avi movie ﬁle) corresponding

to each of the three scenarios is available.

3

We now describe in

detail the scenario corresponding to Case 3. We run a one-hour

simulation. Fig. 9 shows the aircraft density on all links at var-

ious instants, in the absence of control: the velocity is the mean

velocity proﬁle determined for each link, and no aircraft is al-

814 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 14, NO. 5, SEPTEMBER 2006

Fig. 10. Top 6 subﬁgures: Evolution of the aircraft density on the different links in the absence of control. Each of the subplot shows the density distribution at a

given time on the corresponding link as in Fig. 9 (the horizontal coordinate represents location, the vertical represents density). The horizontal line represents the

density threshold (all quantities are nondimensionalized by

, so that the threshold density is 1). As can be seen, the density threshold is violated in link 5 at

and . Bottom 6 subﬁgures: Evolution of the aircraft density with control applied. Note that link 6 is now open and used. This prevents the

second violation of density threshold observed in Fig. 10

: some of the ﬂow is directly routed from link 1 to link 6. The ﬁrst violation seen in the top 6

subﬁgures is avoided by speed changes. This ﬁgure is also available in form of a .avi ﬁle.

3

lowed into link 6 (i.e., ). The initial density is shown

in the top-left corner. The inﬂow into links 1 and 3 is such that

at time

, the density threshold (represented by the hor-

izontal line on each subplot) is violated until time

.At

time

, it is violated again, until the end of the experiments.

Fig. 9 shows the same experiment when link 6 is now opened

to trafﬁc, and velocity control is enabled. As can be seen, about

half of the trafﬁc incoming into link 1 is rerouted into link 6,

and the other half into link 1 bis. Fig. 9 shows the variation of

with time. As can be seen, around min, there is a peak

of about 25% of aircraft routed into link 6, which settles to 50%

at

. The routing control enables avoidance of violation of

maximal density shown in Fig. 10. The ﬁrst violation is avoided

by velocity changes.

The velocity proﬁles

are shown in Fig. 9. Each of

the subplots corresponds to one of the links. For links 5 and 6,

one can clearly see the descent velocity proﬁles. Also, for link

6 (subﬁgure below), one can see a ridge. It corresponds to a set

of aircraft which have to ﬂy at high speed into the airport. One

can also see similar ridges on the other subplots, which have

the same interpretation. For any ridge, the Controller command

could be to the corresponding set of aircraft: “ﬂy direct at 420 kn

direct into [the next waypoint].” Note that in the absence of con-

trol, the ﬁrst violation of the aircraft density threshold occurs

33 min after the beginning of the experiment, almost at the end

of the network, which is not intuitive. This shows the efﬁciency

of the method, which is capable of generating the right routing

and speed assignments to prevent undesirable events from hap-

BAYEN et al.: ADJOINT-BASED CONTROL OF A NEW EULERIAN NETWORK MODEL OF AIR TRAFFIC FLOW 815

Fig. 11. Display of the trafﬁc situation for the static validation. The density of

the links is depicted by the color. The colored rectangles shown in this plot repre-

sent the density. The color scale is: white for zero density; black for highest den-

sity. The actual aircraft positions are superimposed (triangles). Trafﬁc is shown

at

(top),

, etc. As can be seen, the peaks of density

corresponds to the actual positions of the aircraft.

pening much later. Finally, the simulations are also depicted on

a U.S. map in Fig. 12. One can see that before

, all aircraft

choose the direct route through link 5 to Chicago (it is shorter).

After

, the excessive amount of ﬂow incoming into links

1 and 3 forces the ﬂow to be split through links 1 bis and 6.

The expression of the cost function

can be replaced by any

arbitrary user-deﬁned cost as long as the integrand is smooth.

The goal of this paper was to prove the feasibility of the tech-

nique (with a particular cost function), but extending this to any

cost function is a straightforward process (the only thing which

is needed is to recompute the expression of the gradient based on

the new cost, following the steps outlined here). In particular, in

the work of [27], the authors use an integral form with quadratic

penalty. This can be interpreted as penalizing the cumulative

delay minutes at each point in time, and penalizing more se-

verely large deviations from the scheduled ﬂow than small ones.

VII. C

ONCLUSION

We have derived an Eulerian model of the airspace based on a

modiﬁed LWR partial-differential equation. The network struc-

ture of the airspace was modeled as a set of coupled LWR PDEs.

Given initial positions of aircraft and airport inﬂows, this system

of PDEs enables the prediction of the aircraft density. An ana-

lytical solution was derived for a single link in the case, in which

the mean velocity proﬁles of aircraft along airways do not vary

with time (just with space). It can be used for multiple links as

well. ETMS data was used to identify the numerical parame-

ters associated with this model. The data is also used to validate

the model, i.e., to demonstrate good predictive capability of this

method.

We ﬁrst have shown how to use efﬁcient numerical schemes

to simulate the network. We have discarded linear numerical

schemes because of their poor performance. We have used

the Jameson–Schmidt–Turkel scheme as our main numerical

scheme to perform numerical simulations of the network.

We have posed the problem of maximizing throughput at

a destination airport while maintaining the aircraft density

below a certain threshold as an optimization program. The

inequality constraints of this program have been handled using

a log-barrier method. The adjoint problem was derived and

used to compute the gradient of the augmented cost function.

The resulting optimization and control schemes were applied

to a real air trafﬁc case. Simulations show that this method

enables automated control of realistic scenarios as well as

highly congested situations. The output of the code is a set of

time dependent velocity proﬁles to apply to the network, and a

policy telling how to split the ﬂow in areas of diverging trafﬁc.

These outputs could, thus, be used by the Trafﬁc Management

Unit in charge of managing the ﬂow: they provide high-level

policies to apply to the aircraft streams, which are directly

understandable by human controllers.

Finally, this formulation of the air trafﬁc ﬂow control problem

as an optimization program of PDEs allows for many reﬁne-

ments in the control procedure. For instance, gradient descent

may be replaced by more sophisticated optimization methods

such as approximate Newton method [29] in order to ensure

real time convergence of the algorithm. Furthermore, using this

model, a decentralized control policy can also be derived using

816 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 14, NO. 5, SEPTEMBER 2006

Fig. 12. Aircraft density in the network around Chicago in presence of control and velocity assignment. The density of the links is depicted by the color. The

colored rectangles shown in this plot represent the density. The color scale is: white for zero density; black for highest density. As can be seen and was shown in

Fig. 10, a good portion of the ﬂow is routed into link 6 starting at time

.

decomposition techniques in order to allow different airlines

to separately optimize the ﬂow of their aircraft, while main-

taining safety criteria [28]. This method has since been applied

to highway trafﬁc as well [5], [18] and looks promising for other

applications involving networks of partial differential equations.

VIII. P

ROOF OF PROPOSITION 1

Existence:

is well deﬁned because for all

. Its inverse exists because is (strictly) increasing. It

is easy to check that (2) satisﬁes (1) almost everywhere, and

that it is continuous. This solution has been constructed using a

technique analogous to the algorithm of Bayen and Tomlin [7]

based on the method of characteristics.

Uniqueness: Let us call

and two continuous weak solu-

tions of (1). Call

. satisﬁes:

a.e. in in and

in . Multiplying this PDE by and integrating from

to the ﬁrst discontinuity of gives

BAYEN et al.: ADJOINT-BASED CONTROL OF A NEW EULERIAN NETWORK MODEL OF AIR TRAFFIC FLOW 817

from which we deduce

Integrating by parts gives

since and . Using the fact that

for all . Then,

we use the fact that

so that we can rewrite the inequality as

Using Gronwall’s lemma, this last inequality implies

almost everywhere in . By continuity,

everywhere in , and, therefore, at . The same

proof applies to

since for all .

By induction on

and are equal everywhere in

and, therefore, in .

A

CKNOWLEDGMENT

The authors would like to thank Dr. P. K. Menon and Dr.

K. Bilimoria for conversations which inspired this work,

Dr. G. Chatterji for his ongoing support and suggestions which

went into modelling this work, and Dr. G. Meyer for his sup-

port in this project. They also thank Prof. T. Bewley for useful

conversations regarding the application of the adjoint method

to ﬂow control, and his help in the original formulation of the

control problem. Prof. T.-P. Liu helped deﬁne the PDE used

for this model.

R

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Alexandre M. Bayen (S’02–M’04) received the B.S.

degree in applied mathematics from the Ecole Poly-

technique, Palaiseau, France, in 1998, the M .S. and

Ph.D. degrees in aeronautics and astronautics from

Stanford University, Stanford, CA, in 1999 and 2004,

respectively.

He was a Visiting Researcher at NASA Ames,

Moffett Field, CA, from 2001 to 2003. From 2004

to 2005, he worked for the Department of Defense,

France, where he held the rank of Major. During that

time, he was the Research Director of the Labora-

toire de Navigation Autonome at the Laboratoire de Recherches Balistiques et

Aérodynamiques, Vernon, France. Since March 2005, he has been an Assistant

Professor in the Department of Civil and Environmental Engineering at the

University of California at Berkeley, Berkeley. His research interests include

control of distributed parameter systems, combinatorial optimization, hybrid

systems, and air trafﬁc automation.

Dr. Bayen is a recipient of the Graduate Fellowship of the Délégation

Générale pour l’Armement (1998–2002) from France, and the Ballhaus Prize

for best doctoral thesis from the Department of Aeronautics and Astronautics

at Stanford University (2004).

Robin L. Raffard received the M.S. degree in me-

chanical engineering from the Ecole Centrale Paris,

France, in 2002, and the M.S. degree in aeronautics

and astronautics from Stanford University, Stanford,

CA, in 2003. He is currently working towards the

Ph.D. degree in aeronautics and astronautics at the

same school.

His research interests include distributed opti-

mization and optimal control of systems governed

by differential equations, with applications in air

trafﬁc ﬂow, systems biology, and stochastic systems.

Claire J. Tomlin (S’93–M’99) received the Ph.D.

degree in electrical engineering from the University

of California at Berkeley, Berkeley, in 1998. She also

received the M.Sc. degree from Imperial College,

London, in 1993, and the B.A.Sc. degree from the

University of Waterloo, Canada, in 1992, both in

electrical engineering.

She is an Associate Professor in the Department of

Electrical Engineering and Computer Sciences at the

University of California at Berkeley, and is an As-

sociate Professor in the Department of Aeronautics

and Astronautics at Stanford University, Stanford, CA, where she also holds the

Vance D. and Arlene C. Coffman Faculty Scholarship in the School of Engi-

neering. She joined Stanford in September 1998, as a Terman Assistant Pro-

fessor, and received tenure at Stanford in November 2004. In July 2005, she

joined Berkeley as an Associate Professor. She has held visiting research po-

sitions at NASA Ames, Honeywell Labs, and the University of British Co-

lumbia. Her research interests include control systems, speciﬁcally hybrid con-

trol theory, and she works on air trafﬁc control automation, ﬂight management

system analysis and design, and modeling and analysis of biological cell net-

works.

Dr. Tomlin is a recipient of the Eckman Award of the American Automatic

Control Council (2003), MIT Technology Review’s Top 100 Young Innovators

Award (2003), the AIAA Outstanding Teacher Award (2001), an National Sci-

ence Foundation (NSF) Career Award (1999), and the Bernard Friedman Memo-

rial Prize in Applied Mathematics (1998).

- CitationsCitations78
- ReferencesReferences43

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