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

Article · October 2006with58 Reads
DOI: 10.1109/TCST.2006.876904 · Source: IEEE Xplore
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
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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 uid 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 trafc ow control
problem using accurate numerical schemes.
There are a few benets 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 specic classes of controllers
(smooth, continuous, piecewise afne, 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 trafc with minor
modications [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 rst
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 trafc 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 proles 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 trafc system, like the road system, one
has to rst 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]
dened 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 prole
dened 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 inow 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 rst-order linear hyperbolic PDE, and
can now enunciate the following proposition.
Proposition 1: Let
be a func-
tion with a nite 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 inow at the entrance of the link
(i.e., at
). In highway trafc ow analysis, is sometimes
referred to as cumulative ow. 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 satises 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 rst 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
ux. In the implementation studied in this paper, the function
will represent the control input. It is also possible to rewrite
the rst 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 dened by and satises
(4). However, unlike for highway trafc, the density
might
not be the best way to characterize the ow 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 nite 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 satises 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 trafc 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 satises 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 uid mechanics assumes a large
number of particles (i.e., aircraft) per unit volume (the threshold
is dened by the Knüdsen number). In the present case, the
number of aircraft we consider will almost certainly be below
this number, meaning that the uid 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 trafc 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 trafc. 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 ow
going from
to , and the proportion of the owgoing 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
coefcients 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 prole is , etc. Note that we are not
using Einsteins 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 ights 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
ve links, merging into ORD. The corresponding inow 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 (rst 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 fth equations express the conservation of aircraft at
the divergence points. The last equation expresses the boundary
conditions (inow 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-
les are discontinuous. Several numerical schemes of the orig-
inal LWR PDE have been the focus of recent research [16] in
order to address similar difculties encountered in the original
LWR PDE; they have proved extremely efcient in the case of
highway trafc. We have chosen to use three different schemes
to compare their respective benets.
1) The well-known LaxFriedrichs scheme [17].
2) A left-centered scheme, inspired by the Daganzo scheme
[16] in light trafc
3) The JamesonSchmidtTurkel (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 trafc 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 LaxFriedrichs 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 benet 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 Identication: Main Velocity Proles
In this section, we identify the mean velocity proles
on
each link. We use enhanced trafc 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 prole 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 xes between the links.
ight information at a 3 min rate:
2
position of each aircraft in the
NAS, altitude, velocity, and ight plan (i.e., set of airways and
waypoints). From this data, we are able to identify the routes in
which trafc 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 identied 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 identied the mean velocity pro-
les as piecewise afne functions, using a least squares t. The
total number of aircraft used is 220. The result for the ight New
YorkChicago is displayed in Fig. 3. The curve is a piecewise
afne t obtained using least squares. As can be seen, once the
En Route altitude is reached, the curve ts are almost at, 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-
nement using multilayer models: dividing the link in sublayers
corresponding to altitudes (with different speed proles) has the
benet of being more precise. In this work, we consider a single
layer.
B. Initial and Boundary Conditions
Once the mean velocity proles are computed, we identify the
initial density of aircraft and the inow (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 prole 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 inows (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 led ight 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 rst time it appears,
we compute the time
at which it crossed the location
(using the knowledge of the mean velocity prole on the link).
We then use one of the two denitions 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 rst run the following set of experiments. For the link New
YorkChicago, 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 ight 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, dened by
This quantity represents the error in density prediction due
to the propagation of
by the PDE;
absolute aircraft count error, dened 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 gure, 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 ight 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
modied 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 rst experiment, we use the static velocity proles
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 ve 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-
les are time dependent, i.e.,
. The details of the identi-
cation of these proles 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 identication. A way to improve this would be to perform the
velocity identication with data of a given day over a 24hr pe-
riod, and validate it over the next 24hr period, using the fact that
there is periodicity in the trafc for normal days. This was not
done here due to lack of available data. An animation (.avi movie
le) 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 trafc 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., ux 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 difculty 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 ow 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 rst 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 rst variation of
as a function of the and only, we
choose an adjoint density eld
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 rst-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 specied. 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 rst variation
of as a function
of the rst 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 trafc control applications. We
can alleviate this difculty 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 afne functions.
VI. CONTROLLER DESIGN
In this section, we demonstrate the effectiveness of the adjoint
method by applying it to the air trafc model. Fig. 8 shows the
area which we will control (enclosed by a box). The inows 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 ow
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 trafc (solid); heavy trafc (- -); normal trafc
. 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 elds 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 ow 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 Trafc. (Real data) We take ETMS data,
from which we extract initial conditions and inows. We impose
a restriction on the density and control the ow.
Scenario 2: Heavy Trafc. (Modied 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 trafc
and might not be the best way to represent network trafc. It could be better to
dene an assignment proportion, i.e., a coefcient 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 gure, 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 le) 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 prole 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 subgures: 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 subgures: 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 ow is directly routed from link 1 to link 6. The rst violation seen in the top 6
subgures is avoided by speed changes. This gure is also available in form of a .avi le.
3
lowed into link 6 (i.e., ). The initial density is shown
in the top-left corner. The inow 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 trafc, and velocity control is enabled. As can be seen, about
half of the trafc 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 rst violation is avoided
by velocity changes.
The velocity proles
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 proles. Also, for link
6 (subgure below), one can see a ridge. It corresponds to a set
of aircraft which have to y 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 rst 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 efciency
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 trafc 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). Trafc 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 ow incoming into links
1 and 3 forces the ow to be split through links 1 bis and 6.
The expression of the cost function
can be replaced by any
arbitrary user-dened 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 ow than small ones.
VII. C
ONCLUSION
We have derived an Eulerian model of the airspace based on a
modied 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 inows, 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 proles 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 rst have shown how to use efcient numerical schemes
to simulate the network. We have discarded linear numerical
schemes because of their poor performance. We have used
the JamesonSchmidtTurkel 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 trafc 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 proles to apply to the network, and a
policy telling how to split the ow in areas of diverging trafc.
These outputs could, thus, be used by the Trafc Management
Unit in charge of managing the ow: they provide high-level
policies to apply to the aircraft streams, which are directly
understandable by human controllers.
Finally, this formulation of the air trafc ow control problem
as an optimization program of PDEs allows for many rene-
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 ow is routed into link 6 starting at time
.
decomposition techniques in order to allow different airlines
to separately optimize the ow of their aircraft, while main-
taining safety criteria [28]. This method has since been applied
to highway trafc 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 dened because for all
. Its inverse exists because is (strictly) increasing. It
is easy to check that (2) satises (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
. satises:
a.e. in in and
in . Multiplying this PDE by and integrating from
to the rst 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 Gronwalls 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 ow control, and his help in the original formulation of the
control problem. Prof. T.-P. Liu helped dene the PDE used
for this model.
R
EFERENCES
[1] O. M. Aamo and M. Krstic, Flow Control by Feedback. New York:
Springer-Verlag, 2002.
[2] R. Ansorge, What does the entropy condition mean in trafc ow
theory?,Transp. Res., vol. 24B, no. 2, pp. 133143, Apr. 1990.
[3] B. Bamieh, F. Paganini, and M. A. Daleh, Distributed control of spa-
tially-invariant systems, IEEE Trans. Autom. Control, vol. 47, no. 7,
pp. 10911107, Jul. 2002.
[4] A. M. Bayen, Computational control of networks of dynamical sys-
tems: Application to the National Airspace System, Ph.D. disserta-
tion, Dept. Aeronautics and Astronautics, Stanford Univ., Stanford,
CA, 2004.
[5] A. M. Bayen, R. Raffard, and C. J. Tomlin, Eulerian network model of
air trafc ow in congested areas,in Proc. Amer. Contr. Conf., 2004,
pp. 55205526.
[6] A. M. Bayen, R. Raffard, and C. J. Tomlin, Network congestion
alleviation using adjoint hybrid control: Application to highways,
in Number 2993 in Lecture Notes in Computer Science. New York:
Springer-Verlag, 2004, pp. 95110.
[7] A. M. Bayen and C. J. Tomlin, A construction procedure using char-
acteristics for viscosity solutions of the Hamilton-Jacobi equation,in
Proc. 40th IEEE Conf. Dec. Contr., 2001, pp. 16571662.
[8] T. R. Bewley, Flow control: New challenges for a new renaissance,
Prog. Aerosp. Sci., vol. 37, no. 1, pp. 1119, Jan. 2001.
[9] K. Bilimoria, B. Sridhar, G. Chatterji, K. Seth, and S. Graabe, FACET:
Future ATM concepts evaluation tool, in Proc. 3rd USA/Eur. Air
Trafc Manage. R&D Seminar, 2001, on CDROM.
[10] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge,
U.K.: Cambridge Univ. Press, 2004.
[11] G. Chatterji, B. Sridhar, and D. Kim, Analysis of ETMS data qualiy for
trafc ow management decisions, in Proc. AIAA Conf. Guid., Nav.
Contr., 2003, AIAA-2003-5626.
[12] C. Chen, Z. Jia, and P. Varaiya, Causes and cures of highway
congestion, IEEE Control Syst. Mag., vol. 21, no. 4, pp. 2633,
Dec. 2001.
[13] P. D. Christodes, Nonlinear and Robust Control of Partial Differential
Equation Systems: Methods and Applications to Transport-Reaction
Processes. Cambridge, MA: Birkhäuser, 2001.
[14] C. Daganzo, The cell transmission model: A dynamic representation
of highway trafc consistent with the hydrodynamic theory, Trans-
port. Res., vol. 28B, no. 4, pp. 269287, Aug. 1994.
[15] C. Daganzo, The cell transmission model, Part II: Network trafc,
Transport. Res., vol. 29B, no. 2, pp. 7993, Apr. 1995.
[16] C. Daganzo, A nite difference approximation of the kinematic wave
model of trafc ow, Transport. Res., vol. 29B, no. 4, pp. 261276,
Aug. 1995.
[17] C. Hirsch, Numerical Computation of Internal and External Flows.
New York: Wiley, 1988.
[18] D. Jacquet, C. Canudas de Wit, and D. Koenig, Optimal ramp metering
strategy with an extended LWR model: Analysis and computational
methods,in Proc. 16th IFAC World Congr., 2005, to be published.
[19] A. Jameson, Aerodynamic design via control theory, J. Scientic
Comput., vol. 3, no. 3, pp. 233260, Sep. 1988.
[20] A. Jameson, Analysis and design of numerical schemes for gas dy-
namics 1: Articial diffusion, upwind biasing, limiters and their effect
on accuracy and multigrid convergence, Int. J. Computational Fluid
Dyn., vol. 4, pp. 171218, Sep. 1995.
[21] M. Krstic, On global stabilization of Burgersequation by boundary
control,Syst. Control Lett., vol. 37, pp. 123142, Jul. 1999.
[22] M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory
of trafc ow on long crowded roads, in Proc. Royal Soc. London,
1956, pp. 317345.
[23] X. Litrico, Robust IMC ow control of SIMO dam-river open-channel
systems, IEEE Trans. Control Syst. Technol., vol. 10, no. 5, pp.
432437, May 2002.
[24] P. K. Menon, G. D. Sweriduk, and K. Bilimoria, New approach for
modeling, analysis, and control of air trafc ow, AIAA J. Guid.,
Contr., Dyn., vol. 27, no. 5, pp. 7315090, Sep./Oct. 2004.
[25] A. Messmer and M. Papageorgiou, Route diversion control in mo-
torway networks via nonlinear optimization, IEEE Trans. Control
Syst. Technol., vol. 3, no. 1, pp. 144154, Mar. 1995.
[26] L. Munoz, X. Sun, R. Horowitz, and L. Alvarez, Trafc density esti-
mation with the cell transmission model, in Proc. Amer. Contr. Conf.,
2003, pp. 37503755.
[27] R. Raffard, S. L. Waslander, A. M. Bayen, and C. J. Tomlin, Toward
efcient and equitable distributed air trafc ow control,presented at
the Proc. Amer. Contr. Conf., 2006, Minneapolis, MN.
[28] R. Raffard, S. L. Waslander, A. M. Bayen, and C. J. Tomlin, A co-
operative, distributed approach to multi-agent Eulerian network con-
trol: Application to air trafc management,in Proc. AIAA Guid., Nav.
Contr. Conf. Exhibit, 2005, AIAA-2005-6050.
[29] R. L. Raffard and C. J. Tomlin, Second order optimization of ordinary
and partial differential equations with applications to air trafc ow,
in Proc. Amer. Contr. Conf., 2005, pp. 798803.
[30] P. I. Richards, Shock waves on the highway, Oper. Res., vol. 4, no.
1, pp. 4251, Feb. 1956.
[31] C. Robelin, D. Sun, G. Wu, and A. M. Bayen, En-route air trafc
modeling and strategic ow management using mixed integer linear
programming,in INFORMS Annu. Meeting, 2005.
[32] R. C. Smith and M. A. Demetriou, Research Directions in Distributed
Parameter Systems. Philadelphia, PA: SIAM, 2000.
818 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 14, NO. 5, SEPTEMBER 2006
[33] D. Sun, S. Yang, I. S. Strub, B. Sridhar, K. Sheth, and A. M. Bayen,
Assessment of the respective performance of three Eulerian air trafc
ow models, presented at the Proc. AIAA Conf. Guid., Nav. Contr.,
Keystone, CO, 2006.
[34] Y. Wang, M. Papageorgiou, and A. Messmer, Motorway trafc state
estimation based on extended Kalman lter, in
Proc. Euro. Contr.
Conf., 2003.
Alexandre M. Bayen (S02M04) 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 trafc automation.
Dr. Bayen is a recipient of the Graduate Fellowship of the Délégation
Générale pour lArmement (19982002) 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
trafc ow, systems biology, and stochastic systems.
Claire J. Tomlin (S93M99) 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, specically hybrid con-
trol theory, and she works on air trafc control automation, ight 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 Reviews 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).
  • ...flow on the network. In this model, traffic flow is modeled by a deterministic linear system with unit time delay. Another model is presented byMenon et al. (2004). The modeling technique is to aggregate the air traffic into control volumes, which are line elements. The model accounts ATC actions and handles merging and diverging air traffic flows.Bayen et al. (2006)use the partial differential equations derived from conservation of mass in a control volume and it relies on a modified version of the Lighthill-Whitham-Richards (LWR) partial differential equations (Lighthill andWhitham (1955), Richards (1956)). Controller design strategies are also applied to these models to regulate the aircraft coun...
  • ...Secondly, the same sets of Poisson distributions estimated from historical data are adopted to model the departure flow at each time step for different days, which ignores the fact that the departure traffic flow varies significantly during days, weeks and seasons under different traffic conditions. In addition, the split parameters used in these models bring up the diffusion and dispersion problem, which may result in the inaccurate predictions and optimizations ( Bayen et al., 2006). On the other hand, motivated by the widely used LWR theory (traffic flow theory proposed by Lighthill, Whitham and Richards) ( Prandtl and Tietjens, 1957;Lighthill and Whitham, 1955) and the cell transmission model ( Daganzo, 1993Daganzo, , 1995Zhang and Chang, 2014) in highway traffic modeling, Menon et al. derive an aggregate air traffic model by using a baseline Eulerian model, where aircraft are also aggregated into control centers ( Menon et al., 2004). ...
  • ...The model accounts ATC actions and handles merging and diverging air traffic flows. Bayen et al.[9]use the partial differential equations derived from conservation of mass in a control volume and it relies on a modified version of the Lighthill-Whitham-Richards (LWR) partial differential equations[5,6]. Controller design strategies are also applied to these models to regulate the aircraft count in differential sectors under a legal threshold[10][11][12]. ...
  • ...In this work, based on the objective minimizing the mismatch between the outlet temperature and the desired value, we are using adjoint method and two-level decomposition approach to simplify and solve the optimization problem. In particular, the proposed optimal algorithm is also easily applicable to other linear or nonlinear first order hyperbolic PDE systems and these system include several physical problems of interest, including traffic flows[22], shallow water flow dynamics[23], heat exchangers[24], chemical reactors[13], oil production systems[25], thermostatically controlled loads[26],[27], and as we shall see, input-delayed systems[8]and[28]. In this work, we provide an iterative algorithm to solve the optimal control problem, where a sequence of optimal control signals – oil flow rates at time t are generated such that the L 2 −norm of the mismatch between the outlet temperature at right boundary point and the desired temperature is minimized. ...
  • ...It adopted the divide-and-conquer strategy to divide the complex problem into several low-dimensional subproblems which becomes easier to deal with. So far these works took the minimization of the airspace congestion or the flight delay as the sole objective [14, 15]. However, it might be more appropriate to consider both airspace congestion and extra flight cost and try to seek a good trade-off between them. ...
  • ...For network flow problems, where subsystems correspond to partitions of a network into subnetworks, such an assumption does not hold. To see this, one can imagine a traffic light timing plan causing a traffic jam which spreads across the entire freeway network[16]or a bottleneck of planes in an airspace affecting flight times throughout the air network[1]. As a result, it is not possible to decompose the subsystems by only sharing control parameters without coupling each subsystem to all control variables and modeling the evolution of the entire network within each subsystem. ...