10.1.1.7.5232

7/30/2019 10.1.1.7.5232

http://slidepdf.com/reader/full/101175232 1/7

Aircraft and Weather Models for Probabilistic

Collision Avoidance in Air Traffic Control

John Lygeros ∗and Maria Prandini †

Copyright c 2002 IEEE.

Reprinted from IEEE 2002 Conference on Decision and Control.This material is posted here with permission of the IEEE. Such per-mission of the IEEE does not in any way imply IEEE endorsementof any HYBRIDGE products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish

this material for advertising or promotional purposes or for creat-ing new collective works for resale or redistribution must be ob-tained from the IEEE by sending a blank email message to [email protected].

By choosing to view this document, you agree to all provisionsof the copyright laws protecting it.

∗J. Lygeros is with the Department of Engineering, University of Cambridge, CambridgeCB2 1PZ, U.K., [email protected]

† M. Prandini is with Department of Electronics for Automation, University of Brescia, viaBranze, 38, 25123 Brescia, ITALY, [email protected]

7/30/2019 10.1.1.7.5232


Aircraft and Weather Models for Probabilistic Collision

Avoidance in Air Traffic Control

John Lygeros 1 and Maria Prandini 2

Abstract

We propose a discrete time probabilistic model for pre-dicting the future position of airliners, based on infor-mation about their current positions and ight plans.The model is used to derive an algorithm for detectingpossible conicts between aircraft, situations where the

aircraft may come closer than a certain distance to oneanother with high probability.

Keywords : Air Traffic Management; Air TrafficControl; probabilistic collision avoidance.

1 Introduction

Despite technological advances in navigation, commu-nication, computation and control, the Air Traffic Man-agement (ATM) system is still, to a large extent, builtaround a rigidly structured airspace and a centralised,mostly human-operated system architecture. The in-creasing demand for air travel is stressing current ATMpractices to their limit. Moreover, projections indicatethat air traffic could double over the next ten years.This is likely to cause both safety and performancedegradation in the near future, and place an additionalburden on the already overloaded human operators. Itis believed that by increasing the level of automation,the efficiency of ATM can be improved and the tasksof human operators can be simplied. This will allowthem to handle the increased demand in air traffic in

a more reliable way, enhancing the level of safety overthe current system.

The primary concern of all advanced ATM systems isto guarantee safety . Safety is typically quantied interms of critical situations, for example, conict sit-uations where two aircraft come closer than a certaindistance to one another. In this context, the main tasksinvolved in ATM safety analysis are conict detection (estimating the criticality of a given situation) and Con- ict resolution (designing algorithms for preventing orresolving safety critical situations).

1 J. Lygeros is with the Department of Engineering, Universityof Cambridge, Cambridge CB2 1PZ, U.K., [email protected]

2 M. Prandini is with Department of Electronics for Automa-tion, University of Brescia, via Branze, 38, 25123 Brescia, ITALY,[email protected]

In this paper we propose a new method for conictdetection at the Air Traffic Control (ATC) level of the ATM system. The main features of the proposedmethod are:

Model Based . A model of the aircraft motion fromthe point of view of ATC will be used to predict thepositions of aircraft in the future.

Probabilistic . The model will produce a probabilitydistribution for aircraft positions. This will allow us toincorporate uncertainty (due primarily to wind) in ourpredictions. Conict detection will involve estimatingthe probability of conict.

Mid-range . Conict detection will be carried out overhorizons of the order of tens of minutes.

Level Flight . The development is carried out in twodimensions, assuming level ight. We believe that theextension of the algorithms to three dimensions will berelatively easy conceptually, but may be signicantlyharder computationally.

We start by putting our work in context with respectto the state of the art in ATM research (Section 2). Wethen present the model we propose for modelling theight of airliners from the ATC point of view (Section 3)and discuss how it can be incorporated in a probabilisticconict detection algorithm (Section 4). We concludewith a brief discussion of our current research on thisproblem (Section 5).

2 Background

Conict detection and resolution for ATM relies onmethods for predicting the position of aircraft in thefuture, based on measurements about their current po-sition and information about their intents (e.g. theiright plans).

One of the key elements in the prediction is modellingthe uncertainty inherent in aircraft motion, due to thewind, the responses of human operators, and the er-

rors in tracking, navigation, and control. Predictionschemes can be classied in three categories, accordingto the method they use for taking this uncertainty intoaccount. The nominal approach assumes that the air-

7/30/2019 10.1.1.7.5232


craft will continue to move along their current path oright plan. This prediction method is fairly simple andwidely used in practice; for example, it is the predictionmethod used by the Traffic alert and Collision Avoid-ance System (TCAS) [1]. However, it is not robust,since uncertainty is neglected, and may lead to opti-mistic predictions. The worst case approach, on theother hand, assumes that the aircraft will follow theworst possible path within the allowable uncertaintyset. This approach (which is adopted in [2] for exam-ple) is essential for establishing absolute performanceguarantees. It tends, however, to be too conservative inpractice. Finally, in a probabilistic approach the uncer-tainty is taken into account by considering the ensembleof sample paths generated by a stochastic model of themotion of the aircraft, and assessing the criticality of the situation in terms of the “probability of conict”.

A thorough overview of the state of the art for all of these methods can be found in [3]. The present papertakes a probabilistic approach to the problem. The rea-son for this choice is that it avoids the conservativenessof the worst-case approach, but is more robust than thenominal approach.

A number of probabilistic models have already been de-veloped to capture different aspects of ATM. One of themost popular ones is the model developed by NASA [4],as part of the Centre TRACON Automation System(CTAS). This model is designed to operate at the ATClevel and takes a centralised view of the ATM process.The motion of the aircraft is characterised by a prob-abilistic deviation of the aircraft position with respectto the deterministic ight plan. The probabilistic devi-ation takes the form of a Gaussian distribution, whosecovariance matrix grows in time, to reect the fact thatthe validity of our predictions decreases the further wetry to project the aircraft position into the future.

This model has a number of drawbacks that limit theaccuracy of predictions based on it. Rather than tryingto capture the evolution of individual trajectories, the

model attempts to match the ensemble behaviour of allthe trajectories of the aircraft. This leads to predictionsthat overlook fundamental facts such as the statisticalcorrelation between positions of the same aircraft atdifferent points in time, the correlation among the po-sitions of nearby aircraft, etc. Here we propose a newprobabilistic model that succeeds in alleviating most of these drawbacks (Section 3).

A number of different methods have also been proposedfor computing the probability of conict for two aircraftencounters. A closed-form expression for the probabil-ity of conict for level ight is derived in [4], under a

number of simplifying assumptions. In spite of its sim-plicity, which makes it very attractive for on-line imple-mentation, this method does not allow one to formally

analyse the performance of the algorithm. Moreover,the exact interpretation of the results obtained by ap-plying the closed-form formula is unclear when, as isoften the case, the simplifying assumptions are not sat-ised. In [5], Monte Carlo simulation is used to com-pute the probability of conict. This approach doesnot require particular assumptions and can be appliedto many different scenarios. However, it is also notamenable to formal analysis. Moreover, it is computa-tionally intensive and therefore may not be suitable foron-line implementation.

In earlier work [6] the authors demonstrated how ran-domised algorithms can be used to alleviate the short-comings of both these approaches. In addition to be-ing efficient computationally, the proposed algorithmsprovide explicit (albeit probabilistic) performance guar-

antees, and allow one to optimise the tradeoff betweenaccuracy and computation time. In Section 4 we extendthis approach to the probabilistic model developed inSection 3.

3 Modelling

Our model attempts to predict the position of aircraftover a horizon of T in the future (typically, T = 20minutes). It deals with level ight and comprises thefollowing components:

Flight Plan . A sequence of way points and speedsthat determine the nominal path of the aircraft.

Aircraft Kinematics . A discrete time system tomodel the horizontal movement of aircraft.

Flight Management System . A controller for thekinematic model, whose gains determine how the FMStracks straight line paths.

Nominal Wind Model . A function (possibly givenas a periodically updated look-up table) providing thenominal wind at each point in the airspace.

Stochastic Wind Perturbation . A two dimensionalrandom eld to capture the stochastic deviation be-tween the nominal and actual wind.

The rest of this section is devoted to lling in the detailsfor all these components.

3.1 Flight PlanWe assume that during the time horizon, T , of interestthe aircraft is ying at a constant altitude, following a

ight plan dened by a sequence of way points and airspeeds {(O i , vi )}N i =0 with O i ∈ R 2 and vi ∈ R + . The

coordinates of the way points are typically measured innautical miles and are assumed to be given in a global

7/30/2019 10.1.1.7.5232


P ( t )

V (t )ψ ( t )

θ( t )

d( t )

l( t )

v i

O i

φ i

O i +1

O i +2Flight Path

Global Coordinate Frame

Figure 1: A segment of the ight plan and the relatednotation.

coordinate frame. The speed is typically measured innautical miles per hour.

Two consecutive way points, O i and O i +1 , dene astraight line segment O i +1 − O i with slope φi in theglobal coordinate frame (Figure 1). This straight linesegment constitutes the nominal path that the aircraftis supposed to follow when travelling from way pointO i to way point O i +1 . vi is the nominal speed of theaircraft when moving between these way points. Wewill use the term ight plan to refer to the sequence of way points and speeds {(O i , v i )}N

i =0 , and ight path torefer to the sequence of straight line segments denedby the ight plan in R 2 .

Because of the uncertainty inherent in the path trackingprocess, it is likely that at any point in time the aircraftwill not be exactly on the ight path. The role of theight management system is to get the aircraft to trackthe ight path as closely as possible. Let P (t) ∈ R 2

denote the position and ψ(t) the heading of the aircraft(dened here as the direction of the velocity) at timet∈[0

, T ] in the global coordinate frame. The deviationof the aircraft from the ight path can be computed

using a change of coordinates. Let us consider the ightpath segment i . Denote by R (φi ) the rotation matrix

R (φi ) =cos(φi ) sin(φi )

− sin(φi ) cos(φi ),

and dene

X (t) = R (φi )(P (t) − O i )(1)

θ(t) = ψ(t) − φi .

If we denote by l(t ) the projection of P (t) along theight path and by d(t) the deviation from the ight path(see Figure 1), one can verify that X (t) = [ l(t ) d(t)]T .

3.2 Aircraft KinematicsFix a sampling time interval ∆ > 0. The aircraft mo-tion can be described through the following discretetime kinematic model

P (t + ∆) = P (t ) + cos(ψ(t))sin(ψ(t)) vi ∆ + W (t, P (t ))∆

+ N (t, P (t)) , (2)ψ(t + ∆) = ψ(t) + ω(t)∆ ,

where (P (t), ψ (t)) ∈ R 3 is the aircraft position andheading (system state), ω(t) ∈R is the rate of changeof the heading (control input), W (t, P ) ∈ R 2 is thenominal wind velocity (a measured, deterministic dis-turbance), and {N (t, P ) ∈R 2 } is the wind random eld(an unmeasured, stochastic disturbance).

The model rests on a number of underlying assump-tions. For example, the air speed of the aircraft is as-sumed to be constant and equal to V (t ) = vi . Thisassumes that the aircraft adjusts its heading but notits speed in response to deviations from the ight plan.This assumption is reasonably realistic for aircraft with3D FMS, where the FMS is only concerned with gettingto a way point ignoring timing constraints. We also as-sume that the air speed of the aircraft is bounded ina certain range, dictated by ATC practice, limitationsof the air frame and engines, and the need to generateenough lift. Finally, to ensure that it is possible to de-

sign a stable FMS to track the ight path, we assumethat the wind speed does not exceed the aircraft airspeed.

3.3 Flight Management SystemWe assume a 3D FMS whose main function is to adjustthe aircraft heading in order to track the ight path,without trying to meet timing requirements. Morespecically, the FMS measures the path tracking error,d(t ), and the heading, ψ(t), and uses their values to se-lect ω(t) in an attempt to keep d(t) small. We proposeto model the FMS by the following linear, dynamic,

feedback controller:

s(t + ∆) = s(t ) + ∆ d(t),(3)

ω(t) = − k1 d(t) − k2 (ψ(t ) − φi ) − k3 s (t).

Since the conict detection computations in Section 4are based on a linearisation of the model about a nomi-nal trajectory they are easier to carry out in coordinatesaligned with the ight path. If we let

W (t, X ) = R (φi )W t, R (φi )− 1 X + O i

N (t, X ) = R (φi )N t, R (φi )− 1 X + O i

the closed loop system modelling the aircraft motion

7/30/2019 10.1.1.7.5232


between way point O i and way point O i +1 becomes

X (t + ∆) = X (t) + sin(θ(t))cos(θ(t)) vi ∆ + W (t, X (t))∆

+ N (t, X (t))θ(t + ∆) = θ(t) − k2 ∆ θ(t) − k3 ∆ s(t) (4)

− k1 ∆ 0 1 X (t)s(t + ∆) = s(t ) + ∆ 0 1 X (t),

where θ(t) is dened in (1).

3.4 Modelling TurnsTo facilitate the analysis we assume that the turn fromone segment to the next happens when the projection,l(t), of the aircraft position along the direction φi of the current segment of the ight plan rst satises

l(t) ≥ O i +1 − O i .

We further assume that the turns between one segmentand the next are instantaneous. This type of turningmakes the analysis of the model somewhat simpler andis easy to implement in simulation, by switching thevalues of the parameters from O i , φi , vi to O i +1 , φi +1 ,vi +1 when the aircraft turns. We conjecture that thisis also reasonably realistic from the point of view of theair traffic controller.

The resulting model is a nonlinear, stochastic, hybridsystem. The discrete states of the system are the seg-ments of the ight plan and are identied by the triple(O i , φ i , vi ). The continuous state corresponds to theaircraft position, heading, and the state of the FMS,i.e., l(t), d(t ), θ(t), s (t). The dynamics within eachdiscrete state ( O i , φ i , vi ) is governed by equation (4),which depends on the discrete state through the nomi-nal speed vi . Evolution can go on in each discrete state(O i , φ i , vi ) as long as l(t) < O i +1 − O i . Transitionbetween discrete states are “guarded” by conditions of the form l(t) ≥ O i +1 − O i . The continuous state isthen reset according to equation (1). Notice that in theX − θ coordinates, the state of the system will undergoa jump at the turn points, since a change in φi and O i

will induce an instantaneous change in X and θ.

3.5 Nominal Wind Model, W (t, P )Strictly speaking the model proposed here could oper-ate without any information about the nominal wind.In this case the term W (t, P (t)) can be set to zero, andthe wind can be assumed to be entirely an unknownrandom disturbance that enters the model throughN (t, P (t )). This assumption is quite common in the lit-erature; for example, the model of [4] does not includeany information about the nominal wind. Of course the

more a-priori information about the nominal wind weincorporate in the model, the more accurate the predic-tions based on the model are going to be. Our currentassumption is that the nominal wind data will consist

of a look up table that returns a vector W (t, P ) for dif-ferent times t and different points P in the airspace.Data like this is made available both to the air trafficcontrollers and is updated periodically.

3.6 Stochastic Wind Perturbation, N (t, P )We model N (t, P ) using a two dimensional randomeld, i.e. a family of random variables taking valuesin R 2 , dened on a common probability space and in-dexed by the 3-dimensional parameter vector ( t, P ).Motivated by realistic considerations (and to keep theanalysis tractable) we assume that the wind randomeld is Gaussian, stationary, zero mean and isotropic.These assumptions imply that all nite dimensional dis-tributions of N (t, P ) are Gaussian, and for all t 1 , t 2 ,P 1 , P 2 , E [N (t 1 , P 1 )] = 0 and E [N (t 1 , P 1 )N (t 2 , P 2 )T ] =R (|t 1 − t 2 |, P 1 − P 2 ).

Considerable effort was made to form an idea of whatthe covariance function R (·, ·) looks like and obtain typ-ical values for the variances and other parameters. Thisinformation proved to be surprisingly elusive. Expertsin ATC, in dispersion of pollutants and in turbulencewere consulted, but apparently none of these disciplinesdirectly addresses the problem in question. Some infor-mation is available in the meteorology literature [7], butnot at the level of detail that one would hope for. Weare currently focusing on the Dryden-Kolmogorov at-mospheric turbulence model [8] as a possible source of

information about these parameters.

4 Conict Detection

Conict detection using the model proposed here iscomplicated because the model is nonlinear and thenominal wind and correlation structure of the randomeld depend on the state of the system. The conictdetection algorithm proposed in this section attemptsto address these difficulties by systematic approxima-tions. In particular, we propose to simulate the non-linear model to generate a nominal trajectory for eachaircraft. We then linearise the nonlinear model aroundthe nominal trajectory and use the nominal trajectoryinstead of the state in the nominal wind look-up tableand the correlation function of the wind random eld.

The algorithm consists of a number of steps. The en-tire sequence of steps will be repeated periodically, totake into account new information (new radar measure-ments, changes in the ight plan, changes in the windconditions, etc.) An obvious choice for when to repeatthe conict detection computation is whenever a new

radar measurement comes in (typically every 12 sec-onds). Because the algorithm may be computationallydemanding, however, we may want to repeat the com-putation more infrequently (e.g. every minute).

7/30/2019 10.1.1.7.5232


Step 0: Initial Data . For each iteration of theconict detection algorithm we assume that we aregiven a prediction horizon , T ∈ R + , the current po-sition , P (0) ∈ R 2 , of each aircraft, the ight plan {(O i , vi )}N

i=0 for each aircraft, and the nominal wind

prole W : R + × R 2 → R 2 . P (0) is assumed to be a nor-mally distributed random variable, with mean P 0 ∈R 2

and covariance matrix Q 0 ∈R 2 × 2 . The interpretationis that P 0 ∈ R 2 is the radar measurement, given ina global coordinate frame, and Q 0 reects our uncer-tainty about this measurement. We assume that at thecurrent time the aircraft is on its way from way pointO0 to O1 ; way points further in the past are discarded.

Step 1: Nominal Trajectory . Simulate the non-linear hybrid system starting from the discrete state(O0 , φ0 , v0 ) and with the stochastic perturbation term

N (t, P (t )) set equal to 0. The outcome of the simula-tion a sequence of nominal positions, {P t }, P t ∈ R 2

for t = 0 , ∆ , 2∆ , . . . . The simulation also generates afunction WP : {P k ∆ }m

k =0 → {(O i , vi , φ i )}N i =0 . The in-

terpretation is that if WP (P k ∆ ) = ( O i , vi , φ i ), then attime k∆ in the future the aircraft will be on its wayfrom way point O i to way point O i +1 .

Step 2: Time Varying Coordinate Change . Sup-pose that at time t = k∆, WP (P t ) = ( O i , vi , φ i ). Wedene

R t = cos φi sin φi

− sin φi cos φi,

wφ i (t)

wφ ⊥i (t)

= R t W (t, P t ),

θt = − sin− 1 wφ ⊥

i(t)

vi, s t = −

k2

k3θt .

The coordinate change to linearising coordinates cannow be written as

X δθδs

=R t 0 00 1 00 0 1

P ψs

−R t O i

φi + θt

s t

, (5)

or, in other words,

Y = T tP ψs

+ S t , (6)

where we set Y = [X,δθ,δs ]T , and T t and S t are thematrices that appear in equation (5).

Step 3: Linear Time Varying Model . A somewhattedious computation reveals that the evolution of thesystem from time t to t + ∆ in linearising coordinatesY can be approximated by

Y (t + ∆) = A t Y (t) + B t + C t N (t, P t ), (7)for appropriate choices of A t , B t and C t that dependon the time varying coordinate change of Step 2, the

FMS gains and the sampling interval ∆. Y is a dis-crete time stochastic process which evolves from time taccording to the linear time-varying equation (7), un-til the switching time t i +1 is reached. At time t i +1 , Y is reset, and it then evolves according to equation (7)with appropriately redened matrices A t , B t , C t .

Step 4: Pairwise Detection . Select a pair of aircraftmoving in the same region of the airspace. The remain-ing steps will be repeated until all pairs of aircraft havebeen tested.

Let Y (t ) and Y (t) denote the state of the two aircraftat time t in the respective linearised coordinates. Theevolution of the two aircraft system can be approxi-mated by a 8 × 8, discrete time, linear, time varying,stochastic system

Y Y (t + ∆) = A t 0

0 A t

Y Y (t ) + B t

B t

+ C t 00 C t

N (t, P t )N (t, P t )

, (8)

until one of the aircraft turns to a different ight plansegment. For ease of explanation we refer to the casewhen no aircraft turns in the considered time interval[0, t ]. Taking into account the occurrence of a turningevent will only make the expressions derived below morecomplex.

We treat ( Y (t), Y (t)) as a Gaussian random variablewith mean ( m (t), m (t)) ∈ R 8 and covariance matrixQ (t) ∈ R 8 × 8 . This is the case if all the random vari-ables N (k, P k ), N (k, P k ), k = 0 , . . . , t − ∆, and theinitial state ( Y (0) , Y (0)) are jointly Gaussian. We as-sume that N (k, P ) and ( Y (0) , Y (0)) are uncorrelatedfor all k ≥ 0 and P ∈R 2 , and that Y (0) is normally dis-tributed with mean m (0) and covariance matrix Q (0).The mean of ( Y (t ), Y (t)) can then be computed recur-sively by

mm (t + ∆) =

A t 00 A

t

mm (t) +

B t

Bt

,

where we use the fact that N (k, P ) is assumed to bezero mean. The covariance matrix can also be com-puted recursively (using the fact that N (k, P ) is sta-tionary, uncorrelated with ( Y (0) , Y (0) ), and isotropic),but the equations are rather tedious and are omitted.

Step 5: Separation and Overlap Probability. Theseparation of the two aircraft at time t is D (t) =P (t ) − P (t). The overlap probability at time t is theprobability that the random variable D (t) falls in a discof radius h centred at the origin, where h is the mini-

mum allowed horizontal separation. Using equation (6),it is easy to show that D (t) is an affine function of Y (t)and Y (t). This implies that the aircraft separationD (t) at time t is a Gaussian random variable taking

7/30/2019 10.1.1.7.5232


values in R 2 , whose mean and covariance matrix canbe computed using the formulas in Step 4. Thereforethe overlap probability can be easily approximated ei-ther numerically, or using randomised extractions.

5 Concluding Remarks

One problem that needs to be addressed to ensure thatthe proposed model is realistic is stability analysis. Themodel of (4) is too complicated to analyse its stability“at one go”, since it involves the interaction of nonlinearcontinuous dynamics, discrete dynamics, and stochasticterms. None of the methods currently in the literatureare capable of dealing with this type of system. We pro-pose to develop a method for studying the stability of such stochastic hybrid systems. A good starting pointare methods developed for switched systems [9]. Mostsuch methods deal with deterministic continuous timesystems and will have to be adapted to be applied toour stochastic discrete time model.

The proposed model contains a number of “free” pa-rameters (the FMS gains, the variances of the wind,etc.) whose values need to be chosen to improve the pre-dictions of the model. To make the system realistic, theexact parameter values should be chosen to match theobserved behaviour of real aircraft. This can be donethrough a process of system identication . When we

try to pose the problem of selecting parameter values ina system identication framework we immediately runagainst the problem of time scales . FMS control takesplace at fairly high frequency (of the order of one sam-ple every second, i.e. 1Hz). Ideally the data used foridentication should be sampled at the same frequency.Unfortunately, because the available data is obtainedthrough radar, it is likely to be much more infrequent:less than 0 .1Hz if we assume a standard radar samplingtime of 12 seconds. In real life data collection exper-iments, the data may be even more sparse: roughly 1sample every minute, or 0 .017Hz. System identica-tion methods designed specically to deal with missingdata [10] are unlikely to work in this case, since theyare designed for situations where the missing samplesare sparse (in our case possibly 59 out of 60 samplesare missing).

In the long run, one would hope to be able to use mod-els like the one developed here not only to warn airtraffic controllers about potential problems, but also toprovide suggestions on how to resolve them. As a rststep in this direction we are currently investigating howthe results of our algorithms can be coupled with theframework of “no-go” zones used by HIPS [11].

Acknowledgements : The authors are grateful to DrVu Duong for supporting their work and for helpful dis-cussions providing insight into different aspects of air

traffic control. The work was supported by EurocontrolExperimental Center, by the European Commission un-der project HYBRIDGE, IST-2001-32460, and by theEPSRC under GR-R62663-01.

References

[1] The MITRE Corporation, “TCAS II colli-sion avoidance subsystem requirements specication,”September 1996.[2] C. Tomlin, G. J. Pappas, and S. Sastry, “Con-ict resolution for air traffic management: A study inmultiagent hybrid systems,” IEEE Transactions on Au-tomatic Control , vol. 43, no. 4, pp. 509–521, 1998.[3] J. Kuchar and L. Yang, “A review of conictdetection and resolution modeling methods,” IEEE Transactions on Intelligent Transportation Systems ,vol. 1, pp. 179–189, December 2000.[4] R. A. Paielli and H. Erzberger, “Conict proba-bility estimation for free ight,” Journal of Guidance,Control and Dynamics , vol. 20, no. 3, pp. 588–596,1997.[5] L. Yang and J. Kuchar, “Prototype conict alert-ing logic for free ight,” Journal of Guidance Control and Dynamics , vol. 20, no. 4, pp. 768–773, 1997.[6] M. Prandini, J. Hu, J. Lygeros, and S. Sastry,“A probabilistic framework for aircraft conict detec-

tion,” IEEE Transactions on Intelligent Transportation Systems , vol. 1, pp. 199–220, December 2000.[7] C. Bysouth, “Providing meteorologiacal data tothe nal approach spacing tool,” Tech. Rep. ForecastingResearch 354, Met Office, U.K., May 2001.[8] J. McMinn, “Extension of a Kolmogorov atmo-spheric turbulance model for time-based simulation im-plementations,” tech. rep., NASA Langley ResearchCenter, 1997.[9] D. Liberzon and A. Morse, “Basic problems instability and design of switched systems,” IEEE Con-trol Systems Magazine , vol. 19, pp. 59–70, October1999.[10] R. Sanchis, A. Sala, and P. Albertos, “Scarce dataoperating conditions: Process model identication,” inIFAC System Identication , pp. 453–458, Kitakyushy,Japan 1997.[11] C. Meckiff and P. Gibbs, “PHARE highly inter-active problem solver,” Tech. Rep. EEC 273/94, Euro-control Experimental Centre, November 1994.

10.1.1.7.5232

Documents