NONLINEAR MODEL PREDICTIVE CONTROL FOR OPTIMAL …

NONLINEAR MODEL PREDICTIVE CONTROL FOR OPTIMALAIRCRAFT SEQUENCING

Benedikt Grüter∗ , Johannes Diepolder∗ , Patrick Piprek∗ , Tugba Akman∗ , Florian Holzapfel∗∗Institute of Flight System Dynamics, Technical University of Munich

Keywords: Nonlinear Model Predictive Control, Optimal Control, Trajectory Optimization, ArrivalManagement, Aircraft Sequencing

Abstract

A real-time, bi-level feedback control algorithmfor assigning arrival times is presented, which areoptimal with regard to the cumulated fuel con-sumption of inbound aircraft. The bi-level struc-ture implements the solution of optimal controlproblems for each individual aircraft to determinethe minimum fuel consumption for the assignedarrival time. Additional constraints are includedin the upper level to assure safe time separationsat the merge point. After obtaining the solution ofthe original problem at the initial time point, thebi-level problem is approximated by a blendedquadratic program. This program provides a pre-diction and correction of the optimal arrival timesfor the next time point and the respective changein initial states of all aircraft. Furthermore, exactsolutions for the lower level optimal control prob-lems are calculated for the updated arrival times.The algorithm is successfully validated for a ex-ample of nine aircraft and a frequency of six up-dates per minute for the arrival time assignment.

1 Introduction

Global air traffic has grown steadily over thelast decades and is expected to continue grow-ing at an exponential rate [1]. Consequently,challenges arise concerning the performance ofthe air traffic system, which has already reachedits limits in some areas, such as runway capac-ity at highly frequented airports. Several re-search programs, e.g. the Single European Sky

ATM Research Joint Undertaking (SESAR JU)[2] and Next Generation Air Transportation Sys-tem (NextGen) in the USA, have been initiated tomodernize the air traffic system, thus increasingefficiency and air space capacity.

A special focus has been drawn to the ter-minal maneuvering areas (TMA) of large aero-dromes, which are a bottleneck of the air traf-fic system [3]. SESAR’s User Preferred Routingand i4D trajectories as well as Extended ArrivalManagers (E-AMAN) facilitate challenges by in-creasing flexibility and predictability. Similarly,programs like System Wide Information Man-agement (SWIM) increase the amount of dataavailable to all stakeholders of air traffic. [2]

Regarding challenges of the future air trafficsystem, it is imperative to focus research on smartalgorithms, which can make use of the combinedpotential of modern technologies. This studyis concerned with the solution of the sequenc-ing problem under the conditions that safety con-straints, such as temporal and spatial separationsof aircraft, are fulfilled and each aircraft is oper-ated optimally.

The problem of calculating an optimal se-quence for arriving aircraft along with opti-mized trajectories has been formulated in differ-ent ways: BITTNER et al. [4, 5] combine severalaircraft in a large scale optimal control problem(OCP), which is solved using a direct colloca-tion method. TORATANI et al. [6] proposed asimultaneous optimization of the trajectory andsequence by calculating a criterion function us-ing indirect optimal control methods and varying

1

B. GRÜTER , J. DIEPOLDER , P. PIPREK , T. AKMAN , F. HOLZAPFEL

the final time. Subsequently, a mixed integer lin-ear program (MILP) including constraints for thetime separation at the runway is formulated andsolved. A similar approach is utilized in [7]. Abi-level algorithm is proposed in [8], where theupper level sequencing problem is formulated asa combinatorial program and solved using a ge-netic algorithm. The embedded lower level OCPsare solved sequentially by applying a direct collo-cation method. Other formulations can be foundin [9, 10].

A similar problem is addressed by HULT et al.[11] and ZANON et al. [12]: The problem of co-ordinating traffic at an intersection is solved us-ing a primal decomposition and a subsequent so-lution of the distributed problem. These are em-bedded within an upper level problem that pro-vides occupancy time slots for each car.

In this study, the bi-level formulation of theoptimal arrival time assignment is solved in a firststep to generate nominal arrival times and corre-sponding optimal trajectories. Furthermore, thealgorithm is extended by a quadratic approxima-tion of the optimal solution over all variations offixed parameters. Specifically, these are the cur-rent position of each aircraft, which acts as theinitial boundary condition for the OCPs. In orderto achieve a good approximation of the optimalsolution, the optimization vector is updated usingthe blended solution of two problems [13]:

• a linear prediction of the optimal solutionfor a parameter variation ("predictor")

• a Newton step of the problem at the newparameter ("corrector").

The remaining paper is structured as follows:The original parametric optimization problem forthe optimal arrival time assignment is presentedin section 2. As outlined above, the problem isconstrained by optimal control problems, whichare described in section 3. Furthermore, thequadratic approximation and respective calcula-tion of the updated problem is detailed in sec-tion 4. Finally, a case study is presented in sec-tion 5 before concluding the paper in section 6.

2 Arrival Time Assignment

The optimization problem associated to the se-quencing problem considered here is defined asfollows: Find the optimal final times t∗f ,i for eachaircraft i, which minimizes the overall fuel con-sumption by maximizing the final mass of the air-craft:

t∗f (x0, t0) = argmin(tf,mf)

n

∑i=1−mf,i (1)

where n represents the number of aircraft. Theoptimization variables are the arrival times tf,iand the optimal final mass mf,i of each aircraft,which implicitly depend on the initial states x0and times t0 of each trajectory.

To respect operational constraints imposed bythe safety regulations within the air traffic sys-tem,

nc =n2(n−1) (2)

algebraic inequalities as follows are added to theoptimization problem:

ct,ij =(tf,i− tf,j−∆tij

)2−(∆tij)2 ≥ 0, i 6= j

∆tij =∆tWTC,ji−∆tWTC,ij

2(3)

∆tij =∆tWTC,ji +∆tWTC,ij

2

These inequalities ensure a pairwise time sep-aration ∆tWTC,ij or ∆tWTC,ji between aircraft i andj, depending on which of the aircraft arrives atthe final approach fix first, as depicted in Fig. 1.The required time separation as defined in ICAODoc. 4444 [14] in dependence on the aircrafts’wake turbulence category is displayed in Table1a.

To facilitate the solution of the bi-level prob-lem and the calculation of real-time updates, thedistances separation constraints specified in Ta-ble 1b are approximated by a time separation.The maximum value of this estimation and therequired time separation stated in Table 1a is con-sidered within the respective constraints given inequation (3).

2

Nonlinear Model Predictive Control for Optimal Aircraft Sequencing

−4 −3 −2 −1 0 1 2 3

−4

−2

0

2

4

6

8

tfi− tfj / min

c tij

Fig. 1 : Time separation constraint function be-tween aircraft i (H) and j (M).

Table 1: Separation based on the wake turbu-lence categories L(ight), M(edium), H(eavy), andS(uper Heavy) of preceding (row) and followingaircraft (column). [14]

(a) Time separation in min

L M H S

L 0 0 0 0M 3 2 0 0H 3 2 0 0S 3 3 2 0

(b) Horizontal distance separation in nm

L M H S

L 3 3 3 3M 5 3 3 3H 6 5 4 4S 8 5 4 4

Finally, the lower level OCPs are added as ad-ditional equality constraints for each aircraft:

cm,i = m∗f,i (x0i, t0i)− mf,i = 0 (4)

where mf,i is a slack variable for the optimal finalmass introduced in equation (1). Furthermore,m∗f,i represents the value function of the respec-tive optimal control problem, which is a functionof the initial state x0,i and time t0,i of the aircraft.

3 Optimal Control Problems

The lower level problems receive the assigned ar-rival times from the upper level problem in eachiteration. Furthermore, additional quantities arefixed, such as the current states of the aircraft andthe current time. This determines the durationuntil reaching the fixed final boundary condition,i.e. the final approach fix.

However, the problem of finding a trajectoryrespecting the above mentioned constraints hasinfinite dimensions. In order to obtain an optimalsolution it is assumed, that each pilot tries to findthe optimal solution regarding the operating cost.In this paper, these are assumed to be only depen-dent on the fuel consumption. The calculation ofthe described trajectories leads to the OCP:

m∗f,i (x0,i, t0,i) = minu∈U−mf,i (5)

s.t.xi = f(xi,ui) (6)

ΨΨΨ(x0,i,xf,i, t0,i) ≤ 0 (7)ci (xi,ui) ≤ 0 (8)

tf,i− tf,i = 0 (9)

Besides the objective function given in equa-tion (5), several constraints have to be respected,such as the aircraft’s dynamics (6), the initial andfinal boundary conditions (7), as well as path in-equality constraints (8). Finally, the arrival timetf,i of each aircraft i is determined by the upperlevel problem as denoted in equation (9).

Each aircraft is modeled as a point mass con-sidering a quasi static decrease in the aircraftmass due to the fuel consumption according to

3


[15]. The resulting seven states are listed in Ta-ble 2. The propagation in WGS84 coordinatesand consideration of a round and rotating earthmodel in the translation equations of motion [16]provide the possibility to extend the problem to alarger AMAN horizon:

~VK +(~ωωω

OK)

K×(~VK

)E

K

+(~ωωω

EO)

K×(~VK

)E

K︸︷︷︸round earth

+ 2 ·(~ωωω

IE)

K×(~VK

)E

K−(~ωωω

IE)

K×[(~ωωω

IE)

K×(~rG)

K

]︸︷︷︸

rotating earth

= 1m ∑

(~F)

K(10)

The forces acting on the aircraft as well asthe fuel consumption are modeled according tothe Base of Aircraft Data family 3 (BADA 3)published by EUROCONTROL [17]. Hence, thecontrol variables are chosen as the kinematic an-gle of attack, bank angle, and the thrust lever po-sition as displayed in Table 2.

While the final boundary conditions in equa-tion (7) are fixed to the final approach fix (FAF),we introduce slack variables for the initial bound-ary conditions to respect the current state of eachaircraft:

ΨΨΨIBC (x0,i, t0,i) =

[xi (t0)−x0,itcurrent− t0,i

]= 0 (11)

where the current state of the aircraft x0,i as wellas the current time tcurrent are fixed parametersto the optimization problem. The value functionm∗f,i (x0i, t0i) of the OCP, i.e. the optimal objec-tive, is a function of these parameters.

The OCPs are solved using a direct colloca-tion transcription method implemented by the op-timal control toolbox FALCON.m [18]. The re-sulting nonlinear program (NLP) is solved by uti-lizing the interior point solver IPOPT [19]. Fur-thermore, the gradients

∇(x0i,t0i)m∗f,i (x0i, t0i)

and hessian matrices of the value functions

∇2(x0i,t0i)

m∗f,i (x0i, t0i)

with respect to the current state of the aircrafthave to be calculated and provided to the upperlevel problem. To this end, post-optimal sensi-tivities [20, 21] are employed. The derivatives ofthe upper level problem are constructed accord-ing to [15].

4 Near-Optimal Feedback Control

Solving the bi-level optimization problem includ-ing all lower level OCPs may require a significantamount of time, depending predominantly on theinitial guess and the solver settings. Hence, anexact solution is not feasible for a real-time ap-plication. Instead, information contained in thenominal solution is utilized to approximate theoptimization problem for parameter variations inthe neighborhood of the nominal solution. Ina first step this can be achieved by obtaining apiecewise linear prediction of the solution fora change in the parameters. Furthermore, thequadratic approximation can be utilized to per-form a correction.

4.1 Prediction

For the prediction of the optimal solution fora variation of parameters, a quadratic program(QP) is formed using the Hessian matrix of theoptimization problem at the current point with re-spect to the optimization variables and the fixedparameters:

∇2(z,p)T L =

[H ∇2

pzLT

∇2pzL ∇2

ppL

](12)

The solution of the resulting QP

min∆z12∆zT H∆z+∆pT ∇pzL∆z

s.t. c+∇zcT ∆z+∇pcT ∆p≤ 0(13)

approximates the optimal deviation of the opti-mization variables with respect to a change inthe parameters. Hereby, the constraints of theoriginal problem are respected in a neighbor-hood of the current parameters values, specifi-cally changes in the active set are detected [13].

4


Table 2: States and controls for the point mass aircraft model along with their min and max values.

Description Min Max

ϕ geodetic latitude −90◦ 90◦

λ geodetic longitude −180◦ 180◦

h altitude above reference ellipsoid 5000 f t ∞

VK kinematic speed Vk,min Vk,maxχK flight path azimuth angle −180◦ 180◦

γK flight path climb angle −45◦ 0m aircraft mass ZFWk m0

αK kinematic angle of attack −8◦ 12◦

µK kinematic bank angle −30◦ 30◦

δT thrust lever position 0 1

4.2 Correction

Despite the fact that the QP approximation is ableto detect active set changes and thus "corners" inthe optimal solution, the linear prediction of thesolution of the perturbed optimization problemmay include errors depending on the perturbationmagnitude of the disturbed parameters p. Hence,the obtained solution may also be seen as an ini-tial guess to the perturbed problem, which can besolved to optimality in a next step. As this maybe computationally expensive, a correction canbe achieved by performing a Newton step instead[13], which can be calculated by the followingQP:

min∆z12∆zT H∆z+∇JT ∆z

s.t. c+∇zcT ∆z≤ 0(14)

The correction step provides accurate results asthe prediction increases the accuracy of the lin-earization at the current point [13].

4.3 Real-time Iteration

Reducing the computational effort further, theproblem of finding prediction and correctionsteps can be blended into a single QP as fol-lows [13]:

min∆z12∆zT H∆z+∇JT ∆z+∆pT ∇pzL∆z

s.t. c+∇zcT ∆z+∇pcT ∆p≤ 0(15)

It is important to notice that the evaluationof the Hessian matrix and gradients requires in-formation of the lower level problems. There-for, the solution of all lower level problems un-der the updated constraints is required, whichcan be obtained in different ways, e.g. by per-forming similar updates for the lower levels. Forthe present application it has proven to be suffi-cient to resolve the lower level problems utilizingthe warm start functionality of the interior pointsolver IPOPT [19]. Please note that the proposedproblem formulation allows for a fully indepen-dent solution of the lower level problems. Specif-ically, the problems can be solved in parallel oron distributed hardware.

5 Case Studies

A validation of the performed algorithm is con-ducted by a two case studies: The first study con-cerns an ideal case, where the aircraft are able tofollow the optimal trajectory perfectly. Secondly,a more realistic case is considered by introduc-ing uncertainties. For this case, a normally dis-tributed noise term is added to the current posi-tion of the aircraft, which is provided to the feed-back control. The standard deviation for the noiseterms are set to ten meters in the north, east, anddown direction, with a mean value of zero.

For both case studies, nine aircraft are consid-ered approaching the final approach fix of run-

5


0 100 200 300 400 500 600−20

0

20

time / s

∆x N

/m

0 100 200 300 400 500 600

0

2

4

6

time / s

∆m∗ f,i

/kg

Fig. 2 : Variation of the optimal final mass of anaircraft due to a disturbance in the current posi-tion. The red, dashed curves represent the idealcase, the blue, solid lines show the case subjectto deviations from the optimal trajectories.

way 08L of Munich Airport. The aircraft enterthe TMA at a uniformly distributed time differ-ence with a mean value of 2 minutes and a stan-dard deviation of 30 seconds, i.e. aircraft haveto be delayed/accelerated to meet the separationconstraints at the final approach fix. This can beobserved for the nominal solution displayed inFig. 3a, where the yellow aircraft takes a slightdetour.

The algorithm is run on a Core [email protected] GHz, 16 GB RAM desktop computerrunning Windows 10. The nominal solution isobtained by an exact bi-level optimization in230.13 seconds. Subsequently, the solution is up-dated at a frequency of 0.1 Hz using the algorithmproposed in section 4 for the current states of allaircraft. The solution for the case of disturbedtrajectories is shown in Fig. 3b.

Figure 2 shows the deviation of the positionof aircraft 1 with respect to the optimal trajec-tory, which is zero for the ideal case. Below,the current estimation of the optimal final mass isshown. Again, the dashed graph depicts the idealcase and represents the accuracy of the algorithm.

The solid line illustrates the updated solution forthe current state of the aircraft.

Finally, Fig. 4 depicts the CPU times requiredto solve the lower level OCPs for each aircraft,which feature a perturbed initial boundary condi-tion according to the current position of each air-craft. The maximum CPU time for the solutionof an optimal control problem is 5.23 seconds(mean: 1.06 s, std: 0.94 s), which is well belowthe required 10 seconds to achieve an update rateof 0.1 Hz.

6 Conclusion

A bi-level algorithm for finding optimal arrivaltimes and trajectories for multiple aircraft ap-proaching an airport was presented. The algo-rithm is extended by an approximation of the per-turbed solution by a quadratic program, account-ing for the current position of each aircraft. Thevalidation of the algorithm shows the feasibilityof a real-time application for an approach sce-nario, if the solution of the lower level optimalcontrol problems is performed in parallel.

Future work will include the implementationof a quadratic approximation of the perturbed so-lutions of the lower level optimal control prob-lems, which may be implemented at a higher fre-quency to act as an optimal guidance algorithmfor the individual aircraft. Furthermore, the de-velopment of the implementation on distributedhardware will enable the application in inboundaircraft. Finally, the algorithm may be combinedwith global optimization techniques to provide anoptimal initial guess and help to overcome thedrawbacks of gradient-based optimization algo-rithms, such as convergence to a local minimum.

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(a) Nominal solution before entry of aircraft into theTMA.

(b) Updated solution at 830 seconds, shortly beforethe first aircraft reaches the FAF.

Fig. 3 : Optimal trajectories at different time points.

0 100 200 300 400 500 600 700 800 900 1,000 1,100 1,200 1,300 1,400 1,500 1,600 1,7000

2

4

6

time / s

solu

tion

time

/s

Fig. 4 : CPU times required for the solution of the perturbed optimal control problem.

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7 References

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Holzapfel. Optimal Sequencing in ATMCombining Genetic Algorithms and Gradi-ent Based Methods to a Bilevel Approach.In ICAS 30th International Congress of theInternational Council of the AeronauticalSciences, 2016.

[9] Florian Fisch, Matthias Bittner, and FlorianHolzapfel. Optimal Scheduling of Fuel-Minimal Approach Trajectories. Journal ofAerospace Operations, 2014.

[10] Hang Guo. Application of Genetic Al-gorithms in the New Air Traffic Manage-ment Simulation System. Physics Procedia,33:604–611, 2012.

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[15] Benedikt Grüter, Matthias Bittner, MatthiasRieck, Johannes Diepolder, and FlorianHolzapfel. Bi-Level Homotopic AircraftSequencing Using Gradient-Based Arrival

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[16] Rudolf Brockhaus, Wolfgang Alles, andRobert Luckner. Flugregelung. Springer,Dordrecht, 2011.

[17] A. Nuic. User Manual for the Base of Air-craft Data (BADA) Revision 3.11. Atmo-sphere, 2010:001, 2010.

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8 Contact Author Email Address

Benedikt Grüter: benedikt.grueter tum.de

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NONLINEAR MODEL PREDICTIVE CONTROL FOR OPTIMAL …