Distributed Particle Filter for Urban Trafﬁc Networks Using a …asarlet/Data/msb_tits2013.pdf · 2013. 12. 16. · Distributed Particle Filter for Urban Trafﬁc Networks Using

1918 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 14, NO. 4, DECEMBER 2013

Distributed Particle Filter for Urban TrafficNetworks Using a Platoon-Based Model

Nicolae E. Marinica, Alain Sarlette, and René K. Boel, Life Senior Member, IEEE

Abstract—Raw measurement data are too noisy to directly ob-tain queue and traffic flow estimates usable for feedback controlof urban traffic. In this paper, we propose a recursive filter to es-timate traffic state by combining the real-time measurements witha reduced model of expected traffic behavior. The latter is basedon platoons rather than individual vehicles in order to achievefaster implementations. This new model is used as a predictor forreal-time traffic estimation using the particle filtering framework.As it becomes infeasible to let a truly large traffic network bemanaged by one central computer, with which all the local unitswould have to communicate, we also propose a distributed versionof the particle filter (PF) where the local estimators exchangeinformation on flows at their common boundaries. We assess thequality of our platoon-based PFs, both centralized and distributed,by comparing their queue-size estimates with the true queue sizesin simulated data.

Index Terms—Bayesian estimation, hybrid systems, parallelparticle filters (PFs), stochastic systems, urban traffic networks.

I. INTRODUCTION

THE motivation for this paper is the design of feedbackcontrollers efficiently coordinating traffic lights in an ur-

ban area. This feedback control requires estimation of queuesizes and of other variables describing the traffic state at a giventime. Urban traffic exhibits complicated nonlinear dynamicbehavior. Moreover, unlike what is customary for freewaytraffic, data collection and control decisions are not usuallyconcentrated in a large traffic control center. Thus, we need todevelop distributed estimators of the traffic state.

At several locations in the network (including upstream anddownstream of each intersection), sensors detect the passagetimes of vehicles. However, these sensors are noisy, oftenfailing to detect vehicles, and sometimes generating outputwhen no vehicles are passing. Therefore, sensor outputs cannotbe used directly for queue and traffic flow estimation. Instead, arecursive filter must be used that combines available online datawith constraints imposed by a model of reasonably expected

Manuscript received September 24, 2012; revised April 5, 2013; acceptedJune 16, 2013. Date of publication July 22, 2013; date of current versionNovember 26, 2013. This work presents research results of the Belgian NetworkDynamical Systems, Control, and Optimization (DYSCO) supported by theInteruniversity Attraction Poles Programme and initiated by the Belgian FederalScience Policy Office. The scientific responsibility rests with its authors. Thiswork was also supported in part by EUFP7 under Project CON4COORD andProject DISC. The Associate Editor for this paper was D. Srinivasan.

The authors are with the SYSTeMS Research Group, Faculty of Engineer-ing, Ghent University, 9052 Zwijnaarde, Belgium (e-mail: [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TITS.2013.2271326

traffic behavior, e.g., consequences of the known red/greenswitching times of traffic lights, and relations between pas-sage times at successive locations along the same road. Wetherefore need an abstract dynamic model of urban traffic thatis sufficiently simple and computationally fast to allow real-time comparison of several possible alternative estimates (and,later, control actions). To meet this goal, we introduce a novelapproach that models urban traffic on the basis of platoons.This platoon-based model (PBM) groups into platoons vehiclesthat travel at approximately the same speed, closely followingeach other. The PBM describes the state of an urban trafficnetwork as the status of each traffic light; the queue sizes ateach approaching lane of each intersection; and the location ofthe head and the size of each platoon between intersections.

This model provides a suitable abstraction to be used in aparticle filter (PF) estimation of traffic. The PF replaces theinverse problem of estimation [4], i.e., retrieving parameterand state values (cause) from measurements (consequences)and a regularizing model, by a sampling of a priori possiblesystem trajectories (“particles”), which are then weighted bythe likelihood that they give the observed sensor outputs. It isa standard algorithm for recursive estimation problems where afull Bayesian update of the estimated quantity’s probability dis-tribution is computationally infeasible. The PF has been alreadyused for motorway traffic estimation [2], [8], [13]; we haverecently proposed to adapt it for urban traffic as well [10]. In[16], recursive traffic state estimation is achieved via a Kalmanfilter (KF) combining sensor data with conservation equationsof vehicles. Indeed, when working on a car-by-car basis, thenonlinearity of the dynamics remains low enough to allow goodresults with an extended KF (EKF). However, in large networks,the dimension of a car-by-car state becomes too large to use anEKF for online control. In this case, a PF can be a faster option.Furthermore, the real traffic sensors that we tested introducestrongly non-Gaussian perturbations (e.g., missed cars and fakedetections). In fact, their limited accuracy justifies the use ofa reduced model, i.e., our platoon-based proposal, that furtherincreases nonlinearity (in fact, discontinuity in a hybrid model)but drastically reduces the state dimension toward computationsfeasible in real time. The increased nonlinearity would makean EKF poorly robust but does not affect the PF, which fullybenefits from the reduced dimension.

Urban traffic estimation faces several difficulties. First, theinverse problem admits a large variety of plausible a priorisolutions, e.g., a low measured traffic flow can be equally dueto few vehicles (low traffic flow rate) or to a very low speed ofa dense queue. Moreover, the queue size at an intersection isthe integrated difference between observed inflow and outflow,

1524-9050 © 2013 IEEE

MARINICA et al.: DISTRIBUTED PARTICLE FILTER FOR URBAN TRAFFIC NETWORKS USING A PBM 1919

and integrating noise leads to a random walk that can quicklydiverge. It is therefore crucial to improve estimates by takinginto account the relationship between successive sensors andexpected dynamics, knowing the state of the traffic lights. Forinstance, the outflow immediately after a traffic light turns greencertainly reflects the queue size at its inflow at the end of the redphase; this dependence allows correcting previous queue-sizeestimations and alleviating, among others, the two issues justmentioned. Similarly, a large platoon estimated from flows at anupstream sensor should be confirmed by observing a high trafficflow a short time later at a downstream sensor. The PF estimatorefficiently combines all this information to build queue-sizeestimations that are coherent with the expected logic of trafficbehavior.

The PBM proposed in this paper can be efficiently im-plemented using a discrete-event system simulator, and thismakes the PF a viable solution for real-time traffic estimationin small networks (a few intersections). For larger networks,the computational complexity per particle grows linearly withthe number of platoons in the network, and the number ofparticles must be also increased to ensure good coverage ofthe increased number of possibilities (difficult to quantify). Weshow, however, that the PF can be efficiently parallelized bypartitioning the urban traffic network into subnetworks. Theresulting distributed PF (DPF) has the additional advantage towork with local information exchange only, unlike a centralizedPF (CPF) that would require communication of all the measure-ments to a central computer. We validate the PF and the DPFby letting them estimate the state of a network whose actualevolution is generated by a detailed urban traffic simulator [1].

This paper is organized as follows. In Section II, the PBM isintroduced. Section III is dedicated to the estimation problem,including theoretical and algorithmic aspects related to the CPFand DPF approaches. Section IV validates the implementationsof the PBM, the PF, and the DPF algorithms.

II. PBM FOR URBAN TRAFFIC

This section starts with a motivation for using platoons asa reduced model for urban traffic flows in Section II-A. Thedetailed description of the PBM is given in Section II-B, andSection II-C represents it as an input/output automaton using acomposition rule as in [9].

A. Defining Platoons

Grouping vehicles into platoons yields a smaller and thusfaster reduced model of traffic flow with respect to modelsbased on individual vehicles. To investigate how accurate sucha platoon model might be, we have analyzed measurementstaken over ∼40 days in the area of Dendermonde, Belgium,covering a traffic network with five signalized intersections,several unsignalized intersections, and two roundabouts (datacourtesy of the “Vlaamse overheid”, Belgium). Additional sen-sors between the intersections, embracing more than 100 sensorlocations in total, increase the quantity (and, thus, combined ac-curacy) of measurements significantly above the typical trafficestimation context. The sensors are tube traffic counters that

Fig. 1. Number of platoons (defined with maximal intervehicle time Δ = 5 s)during the working days of one week.

Fig. 2. Average size of the platoons (defined with maximal intervehicle timeΔ = 5 s) versus flow rate for one working day.

generate a pulse each time an axle of a vehicle crosses thesensor location. This does not allow distinguishing whether alarge time delay between two successive pulses is due to alarge distance between vehicles or to a very low speed of closevehicles. In this context, however, the consistent majority ofcars either have a speed just below the maximum legal limitor are stopped (e.g., at a red light). We can therefore defineplatoons from time units: Consecutive sensor pulses belong tothe same platoon if they are separated by less than Δ seconds.For Δ between 3 and 10 s, we found that most vehicles travelin platoons of more than one vehicle, thus indeed leading tomodel reduction (except in very light traffic, e.g., during thenight, when traffic control is not needed anyway). The numberof platoons does not significantly change when increasing Δbeyond 5 s; hence, we select Δ = 5 in further analysis. Asa conclusion, on the basis of available data, we intend todefine platoons in the context of urban traffic networks withintermediate traffic load as follows: Either vehicles move closeto nominal speed, and platoons are robustly defined on the basisof Δ = 5, or vehicles join/leave a queue at a traffic light, andplatoons are defined by how the cars merge into the queue andleave the intersection together at green.

Fig. 1 shows that the number of platoons present in thenetwork at any time does not vary by more than a factor 2over all daytime hours of a working week, although flow ratecan vary a lot. Indeed, further analysis (see Fig. 2) confirmsthat flow rate rather affects the average size of the platoons,which grows roughly proportionally with the number of axlesper second. Based on these statistics and the above argumentsfor using platoons, we formally define a platoon as follows.

Definition 1: Let T kr denote the time at which sensor r makes

its kth axle detection. Then, the nth platoon, denoted Pr,n, is


Fig. 3. Components of an urban traffic network.

defined as gathering the axle measurements kn to kn+1 − 1,where for each n we have

T kr − T k−1

r ≤ Δ for k = kn, . . . , kn+1 − 1

T knr − T kn−1

r > Δ.

To obtain a reduced model, we characterize each platoonPr,n by the summarized information as follows:

• Tr,n := T knr , the time when the first axle crosses the

location r;• Tr,n,tail := T

kn+1−1r , the time when its last axle crosses

the location r;• Nr,n, the measured number of vehicles1 passing location

r between Tr,n and Tr,n,tail.

The motion of a platoon between sensors is modeled byassuming that all its vehicles travel at (approximately) the samespeed. To reduce the model, we do not consider dispersionwithin platoons (see, e.g., [14]). This is a good approximationon typical urban roads where cars are observed to move atmaximal speed unless they are blocked in a queue; in fact, thismakes the platoon state to remain robust along a typical urbanlink (300 m). We will later treat a queue of vehicles stoppedbehind a red light (associated to sensor r) as a stationary“fictitious” platoon, where Nr,n is the size of the queue andTr,n is continuously incremented while the queue is stopped.

B. PBM

The platoons introduced in Section II-A are the basis ofa hybrid model for urban traffic, which we describe as adiscrete-event system (see also [12]). We define four networkcomponents, namely, sources, sections, intersections, and sinks.The queues themselves (one per section) are modeled as in-put/output automata that interact with these network compo-nents (see Section II-C). In our PBM, the signals that evolvethrough this network are expressed in terms of platoons. Fig. 3illustrates how the elements are connected to form an urbantraffic network.

Definition 2: A road section Si ∈ S consists of thefollowing:

• an inflow boundary, where platoon Pi,n enters at eventtime TSi,n;

1Axle detection, as opposed to vehicle detection, is a peculiarity of ourparticular sensors. To get a PF compatible with most other sensor types, we mapour measurements to vehicle detections by considering a new carlike vehicleevery two axles. This is not fundamental to the algorithm’s working.

• an outflow boundary, generating an output event whenPi,n leaves Si. The outflow event for Si corresponds toan inflow event for the component that follows Si in thenetwork (see below);

• a fixed capacity Ci, representing the maximum number ofvehicles that it can contain (i.e., when it is filled by a singlequeue);

• Vmax,i, the maximum speed allowed on Si: Each platoonPi,n will move through Si with an independently ran-domly chosen speed vn = p · Vmax,i, where p = 0.8, 0.9,or 1 with respective probabilities 0.05, 0.15, and 0.8(values determined from empirical data). We normalize thespeed to express it in vehicle lengths per second.

The outflow boundary of Si can be blocked so that platoonscannot exit it and pile up to form a queue, whose length at timet we note Qi(t) [veh]. The delay δi,n = (Ci −Qi(t))/vn [sec]expresses how long Pi,n will drive from the inflow boundaryof Si until it reaches the end of its queue (or of the section ifQi = 0). We denote the associated queue-reaching time

TQi,n = TSi,n + δi,n. (1)

The outflow boundary of Si can be connected to the inflowof a downstream section, an intersection, or a sink component,defined below. The input boundary can be connected to theoutflow of an upstream section, an intersection, or a sourcecomponent. We denote the component connected to the outflow(respectively, inflow) boundary of Si by S∗

i (respectively, ∗Si).The arrival time and size of a platoon Pi,n that enters Si

are inherited from ∗Si, and similarly, a leaving platoon ispassed with its characteristic size and leaving time to S∗

i . WhenQi(t) = Ci, the section is full and the outflow of its upstreamcomponent gets blocked (this is called a spill-back).

If a platoon Pi,n is moving faster than its predecessor Pi,n−1,then the head of Pi,n could catch up with the tail of Pi,n−1.To keep things reasonably simple, we consider that, in thiscase, the platoons will be merged at the outflow boundary ofSi. The modularity of the model allows adding more detailssuch as overtaking or splitting along a section if necessary.Furthermore, if there are parking areas in section Si orminor side streets without sensors, then we can modelNi,n = Ni−1,n +Nenter,i,n −Nleave,i,n, where Nenter,i,n andNleave,i,n are random numbers of vehicles that join and respec-tively leave the platoon along Si.

Definition 3: A traffic source Am ∈ M generates new pla-toons of random size and at random times. It is connected to theinflow of a section Si ∈ S , i.e., Am = ∗Si. The nth platoon isgenerated by Am at time

TAm,n = TAm,n−1,tail +Δ+GAm,n

where the time gaps GAm,n are independently exponentiallydistributed. The platoon’s size NAm,n is drawn from a time-dependent probability distribution

Proba(NAm,n = k) = pAm(k, t), k = 1, 2, . . . ,Kmax(t)

(built to match empirical data). The duration TAm,n,tail −TAm,n of the nth platoon is then drawn as the sum of NAm,n


random variables Dk, representing intervehicle time gap withinthe platoon, with mean value E[Dk] =: Th.

The source parameters determine the time-varying emer-gence rate of vehicles at the location Am

λAm(t) =

E [NAm(t)]

Δ + E[GAm] + E [NAm

(t)] · Th[veh/s].

Here, E[NAm] and E[GAm

] denote the expected values of thedistributions for NAm,n and GAm,n, respectively, adjusted tomatch historical measurements at the location of Am.

Definition 4: A sink Bm ∈ O is just a recipient for platoonsthat leave the modeled network at the outflow of some sectionSi, i.e., Bm ⊂ S∗

i .This component only serves to keep track of exit times

for performance evaluation: A platoon Pm,n crossing Bm isassigned a leaving time interval [TBm,n, TBm,n,tail], after whichit undergoes no further changes.

Definition 5: An intersection Ij ∈ I is defined by the 5-tupleIj = {Inj , Extj , ODj , TLPhasej , traf_lightj(t)}, where

• Inj is a list of all entrance points of intersection Ij (exitpoints of upstream sections);

• Extj is a list of all exit points of intersection Ij (entrancepoint of downstream sections);

• ODj ⊆ Inj × Extj is a list of all entrance/exit pairs of Ijbetween which a traffic flow is admissible;

• TLPhasej enumerates all the possible phases of thetraffic light at Ij , for each phase listing which traffic flows∈ ODj it enables;

• traf_lightj(t) ∈ TLPhasej is the current state (phase)of the traffic light.

Each entrance point of an intersection is subdivided in prese-lection lanes, indexed by q ∈ {left-turning, forward1, forward2,right-turning, . . .}. When a platoon Pi,n reaches the outflowboundary of a section Si connected to Ij , the Ni,n vehicles fromPi,n are randomly distributed among the preselection lanes ofthe corresponding entrance point according to a multinomialdistribution pq(t) determined from historical data at Ij . Ifthe priority and safety rules are satisfied (green traffic lightand right-of-way rules for left-turning traffic are satisfied anddownstream section is not blocked), then the platoon Pq,n nowassigned to lane q moves to the associated exit point in Extj ,determined by the ODj pair the platoon belongs to. This motiontakes a random time δIj , determined from historical data andpossibly depending on road conditions. If the green period istoo short for the complete platoon to pass, then Pq,n is split andthe size of the passing platoon is determined by the length ofthe green period.

In order to set up a hierarchical model structure for largenetworks, we introduce the following intermediate element.

Definition 6: A link Ll ∈ L is a sequence of sections Si ∈S glued together, such that proper renumbering yields Si =∗Si+1 for all Si,

∗Si+1 ∈ Ll.All the sections forming the link Ll use the same maximum

speed Vmax. The subdivision of links into sections has beenintroduced such that each section inflow boundary is associatedwith a sensor.

Combining the enumerated components leads to a reducedmodel of urban traffic, which allows reasonably simulatingnetworks of moderate size.

Definition 7: An urban traffic network U consists of a set oflinks (roads) Ll ∈ L, connecting intersections Ij ∈ I, sourcesAm ∈ M, and sinks Bo ∈ O. The topology T (U) of U ischaracterized by specifying, for each intersection Ij ∈ I

• the links ∈∗ Ij carrying traffic toward each of its entrancepoints ∈ Inj ;

• the links ∈ I∗j carrying traffic, leaving each of its exitpoints ∈ Extj .

Each “free” link inflow/outflow boundary is then connectedto a source/sink.

Definition 8: The state XSi(t) of section Si consists of

• a list, with their properties, of all the platoons inside Si

at time t, i.e., for all Pi,n such that TSi,n ≤ t ≤ TS∗i,n,tail,

the state remembers (Ti,n, Ni,n);• the size of the queue Qi(t) at the downstream boundary

of Si.The state of the link Lm is obtained by stacking the states

Xsi(t) of all Si ⊂ Ll.The state of the network U is obtained by stacking the state

of its components and can be equivalently written as the tuple

XU = {Pm, Ij ⊃ traf_lightj} (2)

where Pm(t) =⋃

Si∈S XSi(t) is the set of platoons (including

queues) present in the network at time t.Traffic lights and axle-detecting sensors are respectively the

inputs and outputs with which U will be controlled.

C. Hybrid Model for Queue Dynamics

The dynamics of a queue at the outflow boundary of a sectionor in the preselection lanes of an intersection are describedby the input/output automaton shown in Fig. 4. The state ofthe outflow boundary of section Si is represented by exiti ∈{blocked = 0, free = 1}. An exiti = 0 can occur if the down-stream component reaches its maximal queue length (Qi(t) =Ci) or if the downstream traffic light is red. Fig. 4 furtheruses generic notations TH and TT , respectively, for TQi,n andTQi,n,tail of a particular platoon. The guards, indicated alongthe arrows, specify the jump conditions that force an event, i.e.,a state transition, to take place.

• In state x1: Qi(t) = 0, no approaching platoon. When aplatoon arrives and the outflow is not blocked (t = TH ∧exiti = 1), there is a transition to state x2; if a platoonarrives and the outflow is blocked (t = TH ∧ exiti = 0),a transition to x4 occurs.

• In state x2 : Qi(t) = 0, exiti = 1 and a platoon is cur-rently crossing location i(∃ Pi,n : TH ≤ t < TT ). As soonas the platoon has passed (t = TT ∧Qi(t) = 0), there is atransition back to x1; if the exit becomes blocked whilethere are still vehicles to pass (exiti = 0 ∧ t < TT ), atransition to x4 takes place.

• In state x3 : 0 < Qi(t) ≤ Ci, a platoon is currently arriv-ing (∃ Pi,n : TH ≤ t < TT ), and exiti = 1. If the down-stream component (section S∗

i or traffic light) sends a


Fig. 4. Input/output automaton describing queues. Red states have exiti = 0(outflow blocked), and green states have exiti = 1 (outflow open), whereas forx1, the condition of exiti does not matter.

message that the exit becomes blocked (exiti = 0) andthere are still vehicles to pass t < TT , then a transition tox4 takes place; else if the tail of the arriving platoon Pi,n

reaches the queue (t = TT ), there is a transition to x5; elseif the queue becomes empty (Qi(t) = 0) but the arrivingplatoon has not completely passed (t < TT ), then there isa transition to x2.

• In state x4 : Qi(t) > 0, a platoon is currently arriving(∃ Pi,n : TH ≤ t < TT ), and exiti = 0. If the queuereaches the maximum capacity (Qi(t) = Ci), a transitionto x6 occurs; if Qi(t) < Ci and the arriving platoonmerges completely with the queue (t = TT ), transition tox7 takes place. If the exit becomes unblocked (exiti = 1),there is a transition to x3.

• In state x5: 0 < Qi(t) ≤ Ci, exiti = 1 and the queue isdepleting without new arrivals. If Qi(t) becomes empty,the state jumps to x1; if instead the exit becomes blocked(exiti = 0), then transition to x7 occurs; if a new platoonstarts to join the nonempty queue (t = TH), there is a statetransition to x3.

• In state x6 : exiti = 0 and the queue is full (Qi(t) = Ci),blocking the exit of the upstream section. Whenever atransition to/from x6 occurs, a message must be sent to theupstream component indicating that its exit has becomeblocked/unblocked. The only transition allowed from x6

is jumping to x5 when the exit of Si gets unblocked.• In state x7 : exiti = 0, 0 < Qi(t) ≤ Ci and there is no

arriving platoon. If a platoon starts to merge with the queue(t = TH for some Pi,n), then a transition to x4 occurs; ifexiti is unblocked, a transition to x5 occurs.

The input/output automaton associated to a queue in eachcomponent i exchanges messages with the automata of up-stream and downstream components ∗i and i∗. As a platoon

Fig. 5. Input/output automaton describing the evolution of a traffic light.

passes from i to i∗, its characteristic properties are transferredbetween the corresponding automata. Transitions involvingstate x6 of component i are particular, associating component∗i as follows. When Qi = Ci and exiti becomes unblocked,i switches to x5 and sends an “unblocked” message to ∗i.If Q∗i = 0, nothing more happens. However, often, a queueQ∗i > 0 would have built up behind the blocked outflow. Then,at reception of the “unblocked” message, the automaton of ∗ijumps to x3 or x5 and it sends a platoon to i, whose automatonjumps to x3. Then, the evolution of Q∗i will depend on thespeed at which vehicles leave (i.e., can enter) i; hence, i mustsend this speed information to ∗i, which itself will communicatethe properties of the transferred platoon to i

The evolution of a traffic light is modeled by the simpleautomaton shown in Fig. 5. Each state represents a greenperiod for one set of nonconflicting queues (≡ OD pairs),e.g., Q2k = {North−South, South−North traffic} and Q2k ={East−West, West−East traffic}. Switching can be inducedby an internal clock or by specific events generated by theother automata. More complicated traffic lights, optimizing theswitching process with orange and “buffer” phases, can berepresented similarly [11].

The model described above can be efficiently implementedin a discrete-event system simulation tool using an agendathat keeps track of all the future events, i.e., all transi-tions in queue automata, including platoon-enters-intersection,platoon-enters-link, platoon-leaves-intersection, and platoon-leaves-link; all traffic-light-switches; and platoon-generated-by-source. For each event, the agenda provides the following:

• t_event, the time at which it occurs;• action_type, the type of event (e.g., platoon arriving);• place, the list of locations (components of U) affected;• size, a parameter used for different purposes depending

on action_type (e.g., number of cars that have to move orduration of next green period).

For more details, see [12]. The model is currently imple-mented in MATLAB. A compiled implementation (e.g., C++)could significantly increase the computational speed.

III. ESTIMATION

First, we explain why state estimation is needed for feedbackcontrol of traffic lights, as proposed in [11]. Then, we showhow to perform this estimation with a PF based on the model inSection II. Section III-A details a standard PF algorithm,whereas Section III-B shows how its distributed implementa-tion enables application to large systems.


Fig. 6. Cumulative flow at two consecutive sensor locations on the same road.For a better comparison, the curve of the second sensor is shifted in timeaccording to the estimated [distance ≈1 km divided by mean vehicle speed≈70 km/h] between the two sensors.

To investigate the sensor accuracy, using the data alreadypresented in Section II-A, we have compared the cumulativeflow at two consecutive sensor locations (≈1 km apart) on along stretch of road without intersections, parking areas, or sideroads (see Fig. 6). Ideally, the flow should be conserved, i.e.,the number of vehicles that passes the first sensor should bepreserved 1 km downstream at the second sensor location. Inthe data, the flow is apparently not conserved. In this particularcase, we could identify the main cause to be a frequent failure ofthe second sensor to detect the vehicles. But however accuratelywe model the sensors, there will always be an uncorrelateduncertainty in their signals, such that the estimated cumulativeflow(s), analog to a queue-length, diverge from the real situationlike a classical random walk. In fact, the noise in current trafficsensors is very large that just applying low-pass filtering andconservation laws yields too large errors in queue-size estima-tions. To get viable traffic estimates, we must therefore use aBayesian recursive filter that confronts the data measured onlinewith predictions from a system and sensors’ model; this addsa causal relation in the estimation by taking into account howthe observed behavior must be a plausible consequence of thepreviously estimated state. Given the complexity of the model,a PF estimator is the method of choice because it only requiressimulating a large number of random evolution possibilities.For those simulations, we propose to use the PBM as a reducedmodel that enables fast computations without losing details thatare important for control (such as arrival times, which would beabsent from, e.g., a time-averaged model). For large networks, aCPF implementation would still need very large capabilities interms of computation and communication, as well as robustnessof the IT network. Fortunately, the modularity of the PBMand of its discrete-event system (DES) implementation allowsdeveloping a DPF algorithm that requires little communicationbetween local agents: Only arrival times and sizes of platoonsmust be exchanged at the boundaries of directly connectedcomponents.

Now, let us give our sensing model. We place a sensor at eachsection inflow boundary and at each selection lane of intersec-tion inflows. Denoting by Δs the duration of a communicationround in real-time traffic management, a sensor is modeled asgiving noisy estimates about the number of vehicles Mr that

crosses location r during each time interval [(k − 1)Δs, kΔs],with k integer. More precisely, the sensor output

yr,k = M ′r +mr(k) (3)

represents two effects. Missed detections are taken into accountby making M ′

r a binomial random variable on Mr trials, eachwith pr = 0.95 probability of success. In addition, false detec-tions are represented with a distribution: mr(k) = 0, 1, or 2with respective probabilities 0.95, 0.04, and 0.01. All theserandom events are considered independent in time and amongsensors, and their characteristics can be adjusted from specifi-cations or from a statistical data analysis.

A. Standard PF

Since new measurements are collected every Δs, we translatethe PBM into discrete time. A general, nonlinear, stochastic,and discrete-time state evolution is given by

xk = fk(xk−1, uk−1, vk−1) (4)

where xk−1 and xk are previous and current states, uk−1 isthe known input of the system at time kΔs, and vk−1 is thenoise (usually non-Gaussian) affecting the system. The onlinemeasurement at time kΔs is given by

yk = hk(xk, ek) (5)

where ek is measurement noise. Our goal is to estimate xk

from all the measurements up to time kΔs. Since (4) and(5) are stochastic, xk cannot be exactly determined from themeasurements. The best characterization of xk is p(xk|y1:k);its conditional probability distribution (CPD) given all the pastmeasurements y1:k = {y1, y2, . . . , yk}. A recursive filter canupdate this CPD for every new sensor information via Bayes’law as follows.

• Prediction step—Taking the estimation p(xk−1|y1:k−1) atstep k − 1 as a distribution of initial conditions, computetheir likely evolution under fk(·)

p(xk|y1:k−1) =

∫p(xk|xk−1)p(xk−1|y1:k−1)dxk−1. (6)

The state transition probability distribution function (PDF)p(xk|xk−1) is just a reformulation of fk(·).

• Update step—Correct the predicted estimation by favoringstate values that are more likely to cause the observedmeasurement value yk according to hk(·)

p(xk|y1:k) =p(yk|xk)p(xk|y1:k−1)∫p(yk|xk)p(xk|y1:k−1)dxk

. (7)

The measurement likelihood PDF p(yk|xk) is a reformu-lation of hk(·) from the model.

An exact implementation of this Bayesian recursive filter isoften intractable because efficient numerical representations ofthe full CPD only exist for particular a priori known distri-bution types. (The KF precisely takes advantage of the factthat linear systems, with Gaussian additive noise, preserve aGaussian distribution type for the CPD.) The PF (see, e.g., [6]


and [15] for tutorials) builds on a Monte Carlo representationof the CPD, i.e., it uses a fixed number NP of random samplesof the state xi

k, with associated weights wik(∑NP

i=1 wik = 1),

such that∑NP

i=1 wik g(x

ik) approximates

∫p(xk|y1:k)g(xk)dxk

as well as possible for any useful function g(·). Thus, each pair〈xi

k, wik〉, called a particle, represents a hypothesis of the real

state trajectory of xk, with an associated belief wik. The key

to computing the weights is the importance sampling principle.Consider a known proposal PDF q(x), from which NP samplesxi are drawn. Then, by taking

wi ∝ p(xi)

q(xi)(8)

the associated empirical histogram of 〈xi, wi〉 will approximatethe PDF p(x) with high probability for sufficiently large NP .From (8), it is clear that, for given 〈xi, wi〉 representing p(x), avalid representation of some other distribution p′(x) is obtainedby keeping xi and adapting the weights to

w′i ∝ p′(xi)

q(xi)=

p′(xi)

p(xi)wi. (9)

This principle directly fits the needs of recursive state esti-mation. Starting with a set P of particles supposed to representp(xk−1|y1:k−1), it suffices to execute a stochastic simulation ofthe model (4) over the time interval [(k − 1)Δs, kΔs) for eachxi and leave wi unchanged to get a Monte Carlo representationof p(xk|y1:k−1) (prediction step). The update step then takesadvantage of the special form of (7) to implement efficientimportance sampling: Keeping the same xi, the correctionfrom p(x) = p(xk|y1:k−1) to the wanted p′(x) ∝ p(x)p(yk|xi

k)simplifies to

wik ∝ wi

k−1p(yk|xi

k

). (10)

A well-known problem of PFs in practice is degeneracy [3].As xi evenly explores all a priori possible evolution cases,many particles, associated to options that are unlikely to havetaken place according to measurements, will have a weightwi close to 0 after a few iterations; those particles conveylittle relevant information about the situation, such that theefficient particle set for exploring future evolution is reducedto a number � NP . The solution is to resample the particlestates on a regular basis: Particles with small weights arereplaced by newly created particles with states xi close to morelikely situations, after which all weights wi are readjusted.There exist different resampling algorithms and methods todetermine when resampling is necessary. After exploring a fewpossibilities for our case, we decided to perform a deterministicresampling method at each iteration by duplicating the mostlikely half of the particles, with their weight, and droppingthe other half. This method keeps many likely particle op-tions for future evaluation, as traditional stochastic resamplingdoes. However, in contrast, it also keeps significant weightdifferences among particles that reflect their whole history;this facilitates interpretation of the results, allowing to evaluatepossibilities in terms of individual particle weights instead ofnecessitating heavy particle density computations.

PF for Urban Traffic: Consider an urban traffic network U,as defined in Section II-B, with Nsens sensors at the inflows ofsections and intersection lanes. The measurement vector, i.e.,yk = {y1k, y2k, . . . , y

Nsens

k }, contains the number of vehicles thatpasses by each sensor in the time interval Δs. The full state ofthe network XU is unknown. Based on the observations yk, wewant to determine the queue lengths in front of each intersectionand the location and size of all platoons inside the network U.Algorithm 1 describes how we apply the just presented PF forthis task.

Algorithm 1 CPF for urban traffic

Ensure: → Initialization of the particles at k = 01 for i = 1 to NP do2 generate the ith sample {xi

0}3 initialize ith weight with wi

0 = 1/NP4 end for5 for all k such that k ≥ 1

Ensure: → Prediction step6 for i = 1 to NP7 propagate ith sample {xi

k} with p(xk|xik−1)

8 end forEnsure: → Update step

9 for i = 1 to NP do10 update the weight of the ith sample according to

wik = wi

k−1 · p(yk|xik)

11 end forEnsure: → Normalization

12 compute new total weight Wk =∑NP

i=1 wik

13 for i = 1 to NP do14 replace wi

k by the normalized wik/Wk

15 end forEnsure: → Output

16 order particles by their weight.17 XU ⇐ the state of the particle with the highest weight.

Ensure: → Resampling18 for i = 1 to NP/2 do19 replace the state of the particle x

NP−i+1

k with the stateof the particle xi

k

20 winewk = (wi

k + wNP−i+1

k )/2

21 wik = winew

k and wNP−i+1

k = winewk

22 end forEnsure: → Time step update

23 k ⇐ k +Δs

24 end for

At time k = 0, we generate NP random particle states XiU

,each one including sizes and locations of platoons throughoutthe network and being assigned a weight wi

0 = 1/NP . Afterthis initialization, the following recursion is performed.

• The prediction step (line 7 in Algorithm 1) amounts torunning the DES simulator of the PBM NP times inparallel to generate for each particle the state evolution inthe interval [(k − 1)Δs, kΔs).

• When the information yk is received from the sensors, theupdate step (line 10 in Algorithm 1) multiplies the weight


of each particle by p(yk|xik), the likelihood that it would

give the observed measurement.• The weights are renormalized; this step can be also de-

layed until after resampling.• The state of the particle with the highest weight is sent to

the output as being the estimated state XU. This choicehas the advantage to work without assuming any structureon the state space and probability distribution.

• Resampling (lines 18–22 in Algorithm 1) duplicates theNP /2 most likely particles and throws away the NP /2least likely ones.

The PF replaces the very complicated explicit formula forthe transition probability distribution over high-dimensionalXi

Urequired by the Bayesian filter by NP randomly simulated

Monte Carlo samples. The larger NP , the better the empiricalhistogram generated by the particles will approximate the truecumulative density function p(XU(k)|yk). The speed of thesimulation step, which is still the most demanding one, is thuscrucial to ensure best possible performance by allowing largerNP . The PBM with DES implementation proposed in this paperis a first step toward a computationally tractable PF for urbantraffic (see the results in Section IV). To further speed upcomputations for large networks, we next propose a distributedPF implementation.

B. DPF

For very large networks, following all the platoons andqueues can become a heavy burden for a central agent, with anoverloaded DES agenda in the simulation step and an overflowof communication with all sensors. Due to the modularity of thePBM, a distributed implementation can solve this problem.

In a general description, we start by partitioning the stateand measurement vectors into NR subvectors x(r) and y(r),r = 1, 2, . . . ,NR, such that (4) and (5) write

x(r)k = f

(r)k

(x(r)k−1, u

(r)k−1, b

(r)k−1, v

(r)k−1

)(11)

y(r)k =h

(r)k

(x(r)k , e

(r)k

), for r = 1, . . . ,NR. (12)

The output of subsystem r thus only reflects its local state, butthe state can be updated as a function of the whole networkstate due to the vector b

(r)k−1 of variables whose values are

shared by communication with neighboring subnetworks. Thiscan of course be done in many ways. In an extreme case, eachvector b(r)k−1 would cover all xk, but for networks with a spatialstructure, it is reasonable to expect that natural partitioning willallow to keep the dimension of b(r)k−1 small and independent ofthe network size.

It is adequate to assume that the noise affects each partitionindependently, such that

p(xk|xk−1) =

NR∏r=1

p(x(r)k |x(r)

k−1, b(r)k−1

)(13)

p(yk|xk) =

NR∏r=1

p(y(r)k |x(r)

k

). (14)

Then, one can assign to each subnetwork r its own local PF

PF (r). Each PF (r) has its own NP particles⟨x(r),ik−1 , w

(r),ik−1

⟩and only indirectly shares, with each of its neighbors j, thepart of x

(r),ik−1 intervening in b

(j),ik−1 , which we denote b

(j),ik−1,r.

Message passing ensures that the group Pi = {⟨x(r),i, w(r),i

⟩:

r = 1, 2, . . . ,NR} of all particles with the same index i in thedifferent subnetworks forms a consistent description for thewhole traffic, with associated probability weight

wi =

NR∏r=1

w(r),i ,

NP∑i=1

wi = 1.

Before the prediction step k, a round of local message ex-changes ensures that each PF (r) collects the information of itsb(r),ik−1 . Then, PF (r) uses the latter for each i in conjunction with

x(r),ik−1 , u

(r),ik−1 and a random sample for the stochastic component

v(r),ik−1 , to generate a possible x

(r),ik with (11), much like for

the CPF. The update step adapts the weights to complete theimportance sampling Monte Carlo computation by implement-ing a distributed version of (10). The local output affectationassumptions (12) and (14) are fundamental here, allowing towrite

p(yk|xi

k

)wi

k =

(NR∏r=1

p(y(r)k

∣∣∣x(r),ik

))( NR∏m=1

w(m),ik−1

)

=

NR∏r=1

(p(y

(r)k

∣∣∣x(r),ik )w

(r),ik−1

)

where each factor on the last line can be computed locally byPF (r). Thus, we ensure that the product of local weights indeedsatisfies (10) by performing the local computation

w(r),ik ∝ w

(r),ik−1 , p

(y(r)k

∣∣∣x(r),ik

). (15)

A coordination of all PFr remains necessary for normaliza-tion because

∑i

∏r w

(r),i �= 1 if one uses local normalization∑i w

(r),i = 1 for all r. Resampling, for which we use thesame strategy as the CPF, i.e., copying the local states of themost plausible particle group Pi, also needs a central stationto consider the plausibility of a particle i over all r. However,these two operations are not too time consuming, requiringonly one network-wide exchange of real numbers at the endof each iteration. To save communication and computationalresources, one could even apply them only every few iterations,with virtually no loss in estimation performance.

DPF for Urban Traffic: The partitioning principle appliesvery well to the modular urban traffic network model: Neigh-boring components can share information about traffic flow attheir boundaries through the variables b(r)k−1, which describe theplatoons that cross the boundary in the interval Δs. Consideragain the network U defined in Section II-B, with Nsens sensorsdeployed along its links. We partition the network, as illustratedin Fig. 7. Typically, each component takes care of a few links,


sources, sinks, inner intersections, and the selection lanes ofintersections that serve as connecting components; the methodis flexible enough to accommodate other partitioning schemes,e.g., cutting between two sections of a link. The NR local PFsPFr associated to the partitions work in parallel by recursivelyapplying the steps described by Algorithm 2.

Algorithm 2 PF PFr of subnetwork r in a DPF for urban traffic

Ensure: → Initialization of the particles at k = 01 for i = 1 to NP2 generate the ith sample {x(r),i

0 }3 initialize ith weight with w

(r),i0 = 1/(NP)

NR

4 end for5 for all k such that k ≥ 1

Ensure: → Send b(r∗),ik−1,r to all neighbors r∗

Ensure: → Prediction step6 for i = 1 to NP do7 propagate ith sample {x(r),i

k } with p(x(r)k |x(r),i

k−1 , b(r),ik−1 )8 end for

Ensure: → Update step9 for i = 1 to NP10 update the weight of the ith sample according to

w(r),ik = w

(r),ik−1 · p(y(r)k |x(r),i

k )11 end for

Ensure: → Send {w(r),ik : i = 1, 2, . . . ,NP} to central station

Ensure: → Normalization12 get global weights {wi

k=∏NR

r=1 w(r),ik : i=1, 2, . . . ,

NP} from central station13 compute new total weight Wk =

∑NPi=1 w

ik

14 for i = 1 to NP do15 replace w

(r),ik by NR

√wi

k/Wk

16 end forEnsure: → Output

17 order particles by their weight.18 Xr

U⇐ the state of the particle with the highest weight.

Ensure: → Resampling19 for i = 1 to NP/220 replace the state of the particle x

(r),NP−i+1k with the

state of the particle x(r),ik

21 winewk = ((w

(r),ik )NR + (w

(r),NP−i+1k )NR)/2

22 w(r),ik = NR

√winew

k and w(r),NP−i+1k = NR

√winew

k

23 end forEnsure: → Time step update

24 k ⇐ k +Δs

25 end for

The steps are essentially the same as for the CPF, with a fewexceptions.

• Before each prediction step, PFr sends the shared in-formation b

(r∗),ik−1,r (traffic flow at boundaries) to all its

neighbors; in return, it receives from them all b(r),ik−1,∗r,

making up its own b(r),ik−1 needed for the prediction.

• A correct normalization requires that all local weights aresent to some central station that computes and broadcasts

the weight of each consistent particle group Pi that repre-sents a plausible overall network situation.

• After normalization, w(r),ik and wi

k contain the same in-formation so that one can work indifferently with either ofthem for ordering/output and resampling. The latter adaptsthe local weights, i.e., no further communication required,with a modified formula (lines 21 and 22 in Algorithm 2)that ensures correct weighting of the particle groups Pi.

The big advantage of the distributed approach is that thesize of the traffic network does not represent an issue forsimulation since each PFr runs on a local computer with localinformation only. Some global coordination is still necessaryfor resampling, but note that the heaviest variables, which arelocal measurements and states, never need to be broadcast.In addition, remember that these global operations need notnecessarily take place at every iteration but only when theoutput is needed. More flexibility can be added by adaptingthe local particle numbers to actual requirements, e.g., usingM · NR particles in a busy region to represent M > 1 possiblelocal situations compatible with a single particle group i of theother regions. The DPF has the disadvantage of introducing asmall approximation because platoons that cross the boundarybetween partitions are signaled only at communication rounds,i.e., every Δs, whereas the CPF takes them into account in theexact continuous-time agenda.

IV. VALIDATION RESULTS

Available data sets of real traffic give only parameters such asinflows, outflows, and turning ratios at different locations wheresensors are installed, but no accurate information on queuesizes nor a fortiori on instantaneous locations of platoons. Wetherefore validate the algorithms on synthetic data, produced bysimulation of a ground truth model (GTM). The latter generatesa trajectory for the whole state (including the size of queuesand locations of all vehicles), but only observations taken everyΔs = 3 s are transmitted to the PF. We first use the PBM itselfas the GTM to check correctness of the PF algorithm alone.Then, to validate the PBM, we generate synthetic traffic datawith the well-established microsimulator Simulation for UrbanMObility (SUMO; see [7] and [1]). We have manually addednoise to the SUMO perfect measurement outputs to match thedata quality that our sensors seemed to give. For the simula-tions, the parameters of the model for the PF prediction stepare taken over from the GTM; in a real-world situation, theywould be estimated from historical data. An advanced versionof the algorithm could consider adaptive parameter estimation.The results are presented below by showing the queue sizesgenerated by the GTM, and those estimated by the particle withthe highest weight at each given instant for entrance points ofintersections; we show the sum of the queues on all preselectionlanes. Due to our particular resampling strategy, it indeed makesreasonable sense to approximate the most likely estimated stateby the most weighted particle, although this is not strictlycorrect. Like every filter, the PF in fact gives as estimation a“belief histogram,” i.e., a full probability distribution of queuesizes associated to the model and measurements at each time.


Fig. 7. Small urban traffic network and its partitions for the DPF.

Fig. 8. Road topology and the sensor’s locations marked by − · ·−.

The reason for representing only the most likely particle is thatcomputing and analyzing this histogram would not be feasiblefor real-time control purposes. The drawback is that, for flatbelief distributions (high uncertainty in the PF), selecting onemost likely value leads to high variability from one time instantto another time instant, explaining the jumpy estimation resultsin the figures. For better robustness, controllers could alsoconsider a simple uncertainty estimator, e.g., the weight valueof the selected particle, or the similarity between the five orten most likely particles. Such indicators require evaluation incombination with the controller; thus, they are left for futurework. For easier interpretation, green blocks at the bottomindicate when the traffic light is red for the section for whichthe queue size is shown.

The NP values are chosen to illustrate realistic possibilities;to be able to run the PF in real time, the computation time forrunning NP particles over an interval of length Δs must besmaller than Δs.

1) CPF With the PBM as GTM: Fig. 8 shows the networkused for this validation using sensors located at Sin, Smid, andSout. The first example shows how the PF can cope with anaccident that suddenly (at t = 360 s in Fig. 9) decreases the

Fig. 9. (Pink dashed line) Estimated queue versus (full red line) real queuegenerated by the PBM in the presence of a sudden accident.

Fig. 10. (Pink dashed line) Queue estimated by the PBM-based central PFversus (full red line) real queue generated by SUMO.

speed Vmax of platoons that enter a link (from 60 to 25 km/hin Fig. 9). The PF allows for an “accident” possibility byletting a random number of particles embed a random dropin speed at each Δs step. The algorithm adapts the weightsof those “accident” particles according to their likelihood,given the probabilistic model and measurements (this requiresimportance sampling to deal with the rare event of an accident).Fig. 9 shows the evolution of the estimated queue size (pinkdashed) and of the real queue (full red). The traffic lights switch


Fig. 11. CPF and DPF (PF1 and PF2) queue estimates versus ground truth for two labeled links in the network in Fig. 7.

every 30 s. The PF follows the real queue most of the time until≈270 s, where the estimator sees many vehicles arrive up to50 s later than they really do; this might be due to occasionalabnormally bad sensor measurements. Just after the accident,the PF is underestimating the real queue, until it recovers after≈100 s. The computation time for this example was 2.2 min toanalyze 12.1 min of real time using 202 particles, with a naiveMATLAB implementation. The accident is not investigatedanymore in the following examples.

2) CPF With SUMO as GTM: We consider the same net-work in Fig. 8. Here, we let SUMO randomly generate theparameters for traffic lights (switching every 45 s) and routecharacteristics; model parameters (e.g., 5-m vehicle length,vmax = 50 km/h, . . .) are taken over in the PBM implementedby the PF, assuming that, in practice, they can be reliablyestimated. Fig. 10 shows how the PF estimate follows the realqueue. The different ground truth model leads to a slower con-vergence time and mostly an overestimation of the real queue.This result is obtained with 300 particles, with a computationtime of 2.7 min for 20.05 min of traffic time.

3) DPF With SUMO as GTM: Since the proposed DPF al-gorithm is fully modular, our validation only needs to considerinteraction between two subsystems. We therefore considerthe network with three intersections shown in Fig. 7. Thestate vector of the GTM is, in this case, partitioned in twosubvectors. The middle intersection connects PF1 and PF2.The same SUMO simulator is used to generate synthetic datawith the network in Fig. 7 and traffic lights switching every40 s. The prediction errors of the queue sizes are presented inFig. 11 using 500 particles in the CPF and 300 particles foreach subsystem in the DPF. The running time for the DPF issimilar to the previous cases. As expected, the distributed andcentralized versions appear to estimate comparably well. Thejumping behavior is typical of representing a whole probabil-ity distribution by just one most likely particle: In case thedistribution is flat, very small changes in weights can lead toselecting very different particles. Convergence is similar to theprevious case, often giving an overestimation of the real queue.A meaningful metric to exactly quantify how well the respectiveestimators serve their purpose would be how well each oneallows controlling the traffic system; the controller developmentis subject of ongoing work.

V. CONCLUSION

This paper has presented a hybrid model for urban trafficthat is efficiently implementable as a discrete-event system.Our key step is to aggregate the traffic flow in entities calledplatoons, instead of describing it at the individual vehicle level,to get a reduced abstraction of the real system, which allowsfaster computations. We have shown that PBMs make sense byanalyzing field data of urban traffic. We then have developeda full modular description that is suitable for modeling highlycomplex traffic networks on the basis of platoons. The obtainedmodel can be directly used to estimate true traffic flow fromunreliable measurements by using a PF algorithm. The latteris an exceedingly general dynamic and measurement-basedestimation method, which easily accommodates sensor failures,data coming from different sources, and any type of nonlinear/hybrid system model. After developing a CPF algorithm, wehave taken advantage of the modular structure of our model topropose a DPF that is scalable to very large networks. We havevalidated our estimation algorithms on synthetic data producedby a more accurate model called SUMO [1]. This illustratesthat the method is robust against some model uncertainties,including the many small differences between SUMO and ourreduced model, as well as rare events such as an accident.

The proposed PF is a first step toward urban traffic con-trol. In addition, investigating potential improvements of thisestimation algorithm, e.g., benchmarking on larger networks, acomparison of performance versus communication demands, ora validation with more real data to add/remove more (ir)relevantaspects in the modular description, the main goal of our futurework is to propose distributed feedback control strategies thatlet the traffic lights react to the network’s state as estimatedby the PF. The control algorithms, for instance, could be re-quired to guarantee good performance of all the particles whoseweights are above some threshold.

REFERENCES

[1] M. Behrisch, L. Bieker, J. Erdmann, and D. Krajzewicz, “SUMO—Simulation of urban mobility: An overview,” in Proc. 3rd Int. Conf. Adv.Syst. Simul., 2011, pp. 55–60.

[2] R. Boel and L. Mihaylova, “A compositional stochastic model forreal-time freeway traffic simulation,” Transp. Res. B, vol. 40, no. 4,pp. 319–334, May 2006.


[3] M. Bolic, P. M. Djuric, and S. Hong, “Resampling algorithms for parti-cle filters: A computational complexity perspective,” EURASIP J. Appl.Signal Process., vol. 2004, pp. 2267–2277, Jan. 2004.

[4] C. Claudel, “Convex formulations of inverse modeling problems on sys-tems modeled by Hamilton–Jacobi Equations: Applications to traffic flowengineering,” Ph.D. dissertation, Univ. California Berkeley, Berkeley, CA,USA, 2011.

[5] R. Douc and O. Cappe, “Comparison of resampling schemes for particlefiltering,” in Proc. 4th Int. Symp. Image Signal Process. Anal., 2005,pp. 64–69.

[6] M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, “A tutorialon particle filters for online nonlinear/non-Gaussian Bayesian tracking,”IEEE Trans. Signal Process., vol. 50, no. 2, pp. 174–188, Feb. 2002.

[7] S. Krauss, “Microscopic modeling of traffic flow: Investigation ofcollision free vehicle dynamics,” Ph.D. dissertation, DLR Köln,Hauptabteilung Mobilität und Systemtechnik, Cologne, Germany, 1998.

[8] R. Boel, L. Mihaylova, and A. Hegyi, “Freeway traffic estimation withinparticle filtering framework,” Automatica, vol. 43, no. 2, pp. 290–300,Feb. 2007.

[9] N. A. Lynch and M. R. Tuttle, “An introduction to input/output automata,”CWI Quarterly, vol. 2, no. 3, pp. 219–246, Sep. 1989.

[10] N. Marinica and R. Boel, “Particle filters state estimator for large urbannetworks,” in Proc. Aust. Control Conf., 2011, pp. 374–380.

[11] N. Marinica and R. Boel, “A leader/follower approach for distributedcoordination of interacting components,” in Proc. 20th Int. Symp. Math.Theory Netw. Syst., 2012, pp. 1–3.

[12] N. Marinica and R. Boel, “Platoon based model for urban traffic control,”in Proc. Amer. Control Conf., 2012, pp. 6563–6568.

[13] L. Mihaylova and R. Boel, “A particle filter for freeway traffic estimation,”in Proc. 43rd IEEE Conf. Decision Control, 2004, pp. 2106–2111.

[14] D. I. Robertson, “Transyt: A Traffic Network Study Tool,” Road Res. Lab.,Crowthorne, Berkshire, U.K., Rep. RL-253, 1969.

[15] R. Y. Rubinstein, “Monte Carlo integration and variance reduction tech-niques,” in Simulation and the Monte Carlo Method. Hoboken, NJ,USA: Wiley Series in Probability and Statistics, 1981, pp. 114–157.

[16] G. Vigos and M. Papageorgiou, “Real-time estimation of vehicle-countwithin signalized links,” Transp. Res. Part C, Emerg. Technol., vol. 16,no. 1, pp. 18–35, Feb. 2008.

Nicolae E. Marinica received the B.Sc. degree inautomatic control and applied informatics and theM.Sc. degree in electronics from the University Po-litehnica of Bucharest, Bucharest, Romania, in 2005and 2007, respectively. He is currently working to-ward the Ph.D. degree with the SYSTeMS ResearchGroup, Faculty of Engineering, Ghent University,Zwijnaarde, Belgium.

His research interests include hybrid systems, em-bedded systems, distributed control of networks, anddiscrete-event systems.

Alain Sarlette received the Master’s (appliedphysics) and Ph.D. degrees (systems modeling andcontrol) in engineering from the University of Liège,Liege, Belgium, in 2005 and 2009, respectively.

He is currently an Assistant Professor at GhentUniversity, Zwijnaarde, Belgium. His research in-terests cover control design and behavior analysisfor interconnected systems, distributed systems, andcoordination algorithms, particularly in a nonlinearcontext, geometric tools for dynamical systems ingeneral, and control of quantum systems.

René K. Boel (S’71–M’75–SM’85–LSM’11) re-ceived the Master’s degree in electromechanicalengineering and the Master’s degree in nuclearengineering from Ghent University, Ghent, Belgium,in 1969 and 1970, respectively, and the M.Sc. andPh.D. degrees in electrical engineering and computersciences from the University of California, Berkeley,CA, USA, in 1972 and 1974, respectively.

He is currently a Professor in the SYSTeMS Re-search Group, Faculty of Engineering, Ghent Uni-versity, Zwijnaarde, Belgium. His current research

interests are in the fields of distributed control and estimation for stochastic,discrete-event, and hybrid systems, with applications to traffic networks andpower systems, as well as in bioinformatics.

Prof. Boel has been an Associate Editor of the IEEE TRANSACTIONS ON

AUTOMATIC CONTROL, Automatica, and the Journal of Applied StochasticModels in Business and Industry, and he is currently the Department Editorof the Journal on Discrete Event Dynamic Systems, an Associate Editor of theIET Control Theory and Applications, and a member of the editorial board ofSystems and Control Letters. He was the Chairman of the Steering Committeefor the Workshop on Discrete Event Systems (2000–2002 and 2006–2008), andhe has been on the organizing committee for a large number of conferences incontrol, statistics, and communication networks.

Documents

Distributed Particle Filter for Urban Trafﬁc Networks Using a …asarlet/Data/msb_tits2013.pdf · 2013. 12. 16. · Distributed Particle Filter for Urban Trafﬁc Networks Using