Joint-based control of a new Eulerian network model of air traffic flow
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.
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
 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 , .
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
 or highway trafﬁc , , , 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 , . In the present
case, the control consists of speed assignments and routing poli-
cies. We show that we may use ﬂow control techniques ,
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 , , , , , , 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
• 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 , and we believe can be extended to other
problems such as networks of irrigation channels .
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.
EW EULERIAN NETWORK MODEL OF
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 
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
, it is fairly easy to see that
if an aircraft were at location
at time , it would be at at
. 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
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-
admits a unique continuous (weak) solution, given by
where , and is its inverse.
Proof: See the Appendix.
represents the inﬂow at the entrance of the link
). In highway trafﬁc ﬂow analysis, is sometimes
referred to as cumulative ﬂow. It can be related to the vehicle
density through the integral relation
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:
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 , ,
, , , 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
which provides the following corollary.
Corollary 2: The solution of (5) for
is given by
The interpretation of the corollary follows. The quantity
conserved along the characteristic curves
. At this stage, is deﬁned by and satisﬁes
(4). However, unlike for highway trafﬁc, the density
not be the best way to characterize the ﬂow situation at a given
time. If the number of aircraft in the system is small,
a set of spikes, which is intractable numerically. Therefore, a
more tractable quantity to work with would be
represents the number of aircraft contained in a ﬁnite interval of
. This quantity does not a priori satisfy the PDE (4). It
is meaningful to introduce an additional “density-like” quantity
, 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-
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
One can also show that when , and are the same
At this stage, we have three quantities: , and . The
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
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 , 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
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
. 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
to , and the proportion of the ﬂowgoing from
to . We call the set of links with a divergence at the end of it.
are not known a priori and have to be determined. These
coefﬁcients might depend on
as well and, therefore, a depen-
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-
, 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
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).
OTATION FOR THE NETWORK PROBLEM
In the previous system, the PDE (ﬁrst equation) describes the
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-
of the LWR PDE in the system (6) have very undesir-
able properties for numerical integration: they are by construc-
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  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 .
2) A left-centered scheme, inspired by the Daganzo scheme
 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
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-
, 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 .
Even if a numerical scheme is theoretically proven to con-
verge to the analytical solution of a PDE, one usually does not
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 , .
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 ). 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 , in which we compare the fol-
lowing three models: the original Menon model , the present
model, and a new cell-based model .
ELECTION OF MODEL PARAMETERS
A. System Identiﬁcation: Main Velocity Proﬁles
In this section, we identify the mean velocity proﬁles
each link. We use enhanced trafﬁc management system (ETMS)
data, which we can obtain from NASA Ames (see  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:
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.  focused on creating an automated tool which performs
similar tasks automatically at a NAS-wide level, using FACET
, 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
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
is taken to be a “box”
, of length , where is a reference length
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
, and the parameter
These two equations thus, represent the initial conditions for the
density and the density-like functions, which we extract from
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
, 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
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
and computing the resulting density. In mathematical terms, we
compare the two following quantities:
computed by the LWR PDE ;
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
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
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
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.
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
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
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 ).
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.
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.
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
The difﬁculty posed by the constraints can be avoided in prac-
tice by using a barrier function as commonly done in optimiza-
tion , in which the cost is augmented by a logarithmic term,
which prohibits violation of the constraints.
the augmented cost function. When ,
are used without indices, it means that they are vectors,
. 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
. We use an adjoint method to alge-
braically compute the gradient of the cost function. This method
was extensively used  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
when is an implicit function of and
via the dynamics (6). Let us denote the cost function of the
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
. 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
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
which can be rewritten using the standard inner product denoted
for the domain
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-
in the cost function. First, in order to eliminate the
This is a ﬁrst-order linear hyperbolic PDE, which is well posed
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
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
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
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
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).
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.
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 .
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 : 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.
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.
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
, the density threshold (represented by the hor-
izontal line on each subplot) is violated until 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%
. 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
, 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).
, 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 , 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.
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
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  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 . This method has since been applied
to highway trafﬁc as well ,  and looks promising for other
applications involving networks of partial differential equations.
ROOF OF PROPOSITION 1
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 
based on the method of characteristics.
Uniqueness: Let us call
and two continuous weak solu-
tions of (1). Call
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 .
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
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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,
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
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
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-
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).