NASA/TM–2015–218734

Minimum-Cost Aircraft Descent

Trajectories with a Constrained Altitude

Profile

Minghong G. Wu

Research Scientist, University of California, Santa Cruz, Moffett Field, California, 94035-1000

Alexander V. Sadovsky

Aerospace Engineer, NASA Ames Research Center, Moffett Field, California, 94035-1000

April 2015
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NASA/TM–2015–218734

Minimum-Cost Aircraft Descent

Trajectories with a Constrained Altitude

Profile

Minghong G. Wu

Research Scientist, University of California, Santa Cruz, Moffett Field, California, 94035-1000

Alexander V. Sadovsky

Aerospace Engineer, NASA Ames Research Center, Moffett Field, California, 94035-1000

National Aeronautics and

Space Administration

Ames Research Center

Moffett Field, California, 94035-1000

April 2015
The use of trademarks or names of manufacturers in this report is for accurate reporting and does not constitute

an offical endorsement, either expressed or implied, of such products or manufacturers by the National Aeronautics

and Space Administration.

Available from:

NASA Center for AeroSpace Information

7115 Standard Drive

Hanover, MD 21076-1320

443-757-5802
Abstract

An analytical formula for solving the speed profile that accrues minimum cost during an

aircraft descent with a constrained altitude profile is derived. The optimal speed profile

first reaches a certain speed, called the minimum-cost speed, as quickly as possible using

an appropriate extreme value of thrust. The speed profile then stays on the minimum-cost

speed as long as possible, before switching to an extreme value of thrust for the rest of the

descent. The formula is applied to an actual arrival route and its sensitivity to winds and

airlines’ business objectives is analyzed.

Nomenclature

A Area on the xs-plane enclosed by a closed contour

CD Drag coefficient

CL Lift coefficient

Cd0 Zero-lift drag coefficient

D Drag

D Effective drag per unit mass

F cfD

H Hamiltonian of the Problem

I Integrand in the area integral on xs-space

 ̃I (V + w)

2

I

J Direct Operating Cost

K Lift-induced drag coefficient

L Aircraft’s Lift

M, P Functions of x and V

R Local turn radius

S Aircraft’s wing Area

T Thrust

T Thrust per unit mass

T max Maximum-thrust per unit mass

T min Minimum-thrust per unit mass

T mc Thrust per unit mass on the minimum-cost speed

V Airspeed

V ̇ Airspeed change rate

V

i

Initial airspeed

V

f Final airspeed

Vmc Minimum-cost speed

W Aircraft’s weight

X

fe Flat-Earth Coordinate along a constant latitude

Y

fe Flat-Earth Coordinate along a constant longitude

cf Thrust-specific fuel consumption

g Gravity of earth

h Aircraft’s altitude

h ̇ Aircraft’s altitude rate

1
pf Fuel Price

r Cost Index

r Cost Index per unit mass per unit fuel price

t Time

w Wind component along the route

x Ground Path distance

x ̇ Path distance rate

x

i

Initial ground path distance

x

f Final ground path distance

Γ Trajectory on the xs-plane

α Best-range speed modulating factor

γi Inertial flight-path angle

γa Aerodynamic flight-path angle

λs Costate of s

λx Costate of x

ρ Air density

1 Introduction

Efficient arrival operations in the terminal airspace have been the focus of air traffic management research for years. In recent years, significant efforts have been made to integrate

existing and new technologies as well as new procedures to manage arrival flights in a

trajectory-based, precision-scheduling fashion [1, 2]. Arrival flights are scheduled based on

specific arrival routes defined all the way from the top-of-descent to the runway threshold [3, 4]. With this trajectory-based, precision-scheduling approach, air traffic controllers

use speed clearances to guide flights to meet the schedule and maintain separation while

reserving vectoring clearances for exceptions only. This speed-centric approach is in contrast to the conventional approach, in which aircraft routinely follow air traffic controllers’

vectoring clearances as soon as they enter the terminal airspace.

An arrival procedure that supports trajectory-based, precision-scheduling requires a flight

to traverse a specific route consisting of a ordered sequence of waypoints1. Such a procedure

also imposes altitude restrictions at certain waypoints. The altitude restrictions provide vertical separation for different traffic flows, keep specific arrival flows in the same controller’s

sector, and account for terrain constraints [6]. An altitude restriction can have an upper

bound, a lower bound, or both. While the upper and lower bounds can be made different to

accommodate aircraft types’ varying performance envelopes in the en route airspace, ATC

can rarely afford such flexibility in a congested terminal airspace with complex traffic flows.

Therefore, altitude restrictions in the terminal airspace as well as some transition airspace

always have identical upper and lower bounds. When such altitude restrictions are closely

spaced, arrival flights are observed to fly a sequence of constant flight-path angle (FPA)

segments.

While an arrival flight’s altitude profile is much constrained by considerations related to

1A waypoint is a geographical position defined by its latitude and longitude coordinates [5].

2
traffic flow, terrain, and safety, its speed profile leaves room for optimization. Cost-effective

speed strategies are of common interest to airlines and ATC, and insight of such can benefit

both parties. Specifically, the insight can be used to improve scheduling efficiencies in

the ground automation tools for ATC. For example, the Terminal Sequencing and Spacing

(TSS) system in the United States requires route-specific nominal speed profiles as input

parameters for computing the arrival flights’ schedule. Each aircraft is scheduled by TSS’s

scheduler at route merge points such as the meter fix2 and the runway threshold [1, 3].

Route-specific nominal speed profiles are used for ordering aircraft on the schedule. TSS

then computes for each aircraft a speed profile that would meet the schedule, perturbing

the speed profile from the nominal one to meet separation restrictions at the merge points if

necessary [8]. While the nominal speed profiles are critical to the efficiency of the schedule,

little attention has been paid to their selection. They have been selected only to ensure

flyability without any check for systemic fuel or cost efficiency. This lack of consideration

motivated the study in this work.

Trajectories that minimize fuel or cost have been studied extensively. A fairly comprehensive review of work in this area can be found in [9]. Early work considered optimization

of trajectories in climb, cruise, and descent phases combined [10–13]. Such analysis can be

formulated as a multi-phase optimization problem [14]. Some recent work analyzed descent

only or cruise/descent trajectories [15–18] and investigated trade-off between fuel and flight

time [19]. The optimal descent trajectory obtained from these work typically utilizes idle

thrust engine control. However, none of these work considered altitude restrictions during

the descent. Therefore, the optimal trajectory may violate the altitude restrictions in a

trajectory-based arrival procedure and cannot be executed in operations without modification.

This paper derives the minimum-cost speed profile for an arrival flight following a specific

route with a constrained altitude profile. The direct operating cost [20] serves as the objective functional to be minimized. The total engine thrust, referred to as thrust, is the control

variable. The central result consists of an analytical formula for the speed profile. It reveals

that an optimal speed profile generally gets to a minimum-cost speed as quickly as possible.

It then stays on the minimum-cost speed as long as possible, until it has to leave to reach

the final speed. The minimum-cost speed profile is computed for Boeing 737-800 along an

actual route, and its sensitivity to wind and airlines’ business objectives is investigated.

The rest of the paper is organized as follows: Section 2 gives additional background information and discusses the modeling assumptions; Section 3 formulates the optimal control

model; Section 4 derives the analytical formula for the speed profile; Section 5 applies the

formula to an actual arrival route and compares the minimum-cost trajectory to that of a

simulated flight; Section 6 summarizes the findings and discusses the potential applications

of the minimum-cost speed profile.

2A meter fix is a waypoint at the boundary of the terminal area that can have scheduled times of arrival

for flights during high-density traffic [7].

3
2 Background

2.1 Minimum-Cost Descent

An arrival flight transitions from cruise to descent at a distance of 100 to 150 nmi from

the destination airport. In the absence of wind and altitude restrictions, the most fuelefficient operation for an arrival flight is to maintain at the best-range speed in cruise until

the top-of-descent point for an idle-thrust descent at the aircraft’s minimum-drag speed [20].

However, airlines can save time-related costs by flying an aircraft faster to the destination

airport. If time saving is the sole objective, the ideal cruise and descent speeds would be

the maximum speeds allowable by the aircraft performance envelope. In actual operations,

airlines consider both fuel and time costs and the minimum-cost speeds lie between these

two extremes. The relative importance of expediency to fuel is modeled by the Cost Index,

a parameter selected by airlines during flight planning. The Cost Index varies from route

to route and from aircraft type to aircraft type [21]. A zero Cost Index represents the case

in which flight time does not contribute to the cost. A large Cost Index represents that

expediency is more important than fuel.

2.2 Arrival Procedure

In the United States, arrival flights to major airports follow published arrival procedures

called the Standard Terminal Arrival Route (STAR). While current STARs do not connect

the route all the way to the runway threshold, they are expected to be extended to the

runway in the near future to support trajectory-based operations [1]. For the rest of the

paper, it is assumed that a specific route is defined for an arrival flight all the way to the

runway threshold.

The aircraft descent trajectory considered in this work begins from the first altitude restriction along the arrival route and ends at the initial approach fix that is usually 5 to 10 nmi

from the airport. The total path distance of the trajectory is between 50 and 100 nmi.

3 An Optimal Control Model for Aircraft Descent

3.1 Modeling Assumptions

The following assumptions are made throughout this section, grouped here by the different

aspects of the operation.

1. Route geometry: The descent trajectory is modeled as a continuous curve in 3-

dimensional airspace, parameterized by the ground path distance.

2. Aircraft kinematics and kinetics:

(a) The aircraft is modeled as a point mass.

(b) The weight of the aircraft is treated as a constant.

4
(c) The thrust acts along the direction of the flight path.

(d) Fuel flow is linear with thrust.

3. Flight conditions:

(a) The wind velocity is constant in time. Crosswind and vertical wind are small

compared to the airspeed and, therefore, is neglected.

(b) Both the inertial and aerodynamic flight path angles are small enough that their

cosines are assumed to equal 1.

3.2 Model Formulation

Let x denote the ground path distance of the aircraft’s position along the coordinate defined

by the horizontal path. By assumption (1), the altitude h of the aircraft, the air density ρ,

and inertial flight path angle, γi, of the aircraft can each be represented as a function of x

alone,

h = h(x), (1)

ρ = ρ(x),

tan γi =

dh

dx.

The last equation is based on the sign convention that γiis negative for descent.

Let V = V (t) denote the speed of the aircraft with respect to the air, i.e., the true airspeed.

In the following sections, speed refers to the true airspeed unless noted otherwise. The wind

along the route is denoted by

w = w(x) (2)

and, by assumption (3a), is independent of time. By assumption (3b), the difference between

the horizontal component of the true airspeed and the true airspeed itself is ignored:

Vhoriz ∼ V

whoriz ∼ w

In the following state equations, T denotes thrust, D drag, γa the aerodynamic flight path

angle, W the aircraft’s weight, g the gravity of Earth, and the dot over a symbol denotes

differentiation with respect to time:

x ̇ = V + w, (3a)

V ̇ = g



T − D

W

− sin γa



− (V + w)

dw

dx , (3b)

where D = D(x, V ) and x = x(t). The right-hand side of Eq. (3b) consists of forces

from the thrust, drag, gravity, and the inertial force as a result of wind. Note the use of

assumptions (2b) and (2c) in Eq. (3b). The FPAs γa and γi are related by

h ̇ = V sin γa = (V + w) sin γi. (4)

5
The use of γa in the state equations has been preferred to that of γi for brevity.

The aircraft’s total engine thrust is bounded,

Tmin ≤ T ≤ Tmax,

with the bounds determined by the engine’s performance envelope.

The objective to be minimized is the direct operating cost, which is a sum of two terms

corresponding to fuel and flight duration, respectively:

J =

Z t

f

0







Fuel

z }| {

pf × cf × [T(t) − Tmin] + f0 +

Time

z }| {

pf × r







dt. (5)

Here, t

f

stands for the arrival time at the end of the route, pf the fuel price, cf the

thrust-specific fuel consumption, f0 the minimum-thrust fuel rate, and r a constant Cost

Index [20].

In what follows, it will be convenient to combine the drag, gravity, and wind gradient into

the “effective drag per unit mass”,

D =

g

W

(D + W sin γa) + (V + w)

dw

dx , (6)

and to normalize thrust to a “thrust per unit mass”,

T =

g

W

T. (7)

These definitions simplify the state equations to

x ̇ = V + w, (8a)

V ̇ = T − D (8b)

The optimal control problem central to this paper is the problem with state variables x and

V , control variable T, state equations (8), initial time t = 0, final time t

f

free, boundary

conditions

x(0) = x

i

,

V (0) = V

i

,

x(t

f

) = x

f

,

V (t

f

) = V

f

,







(9)

control constraints

T min ≤ T ≤ T max, (10)

and the objective to minimize being the normalized cost per unit aircraft mass:

J =

Z t

f

0



cfT(t) + r



dt, where r =

g

W

(−cfTmin + f0 + r). (11)

6
4 An Analytical Optimal Solution in Feedback Form

4.1 Comparison of Two Feasible State Trajectories

The optimal control problem (8), (9), (10), (11) has two state variables and one control

variable which, furthermore, enters the state equations and the cost linearly. This special

form of the problem allows one to find optimal control strategies using a technique based

on Green’s Theorem; see, e.g., [22]. This technique consists of comparing two feasible (i.e.,

compliant with (8), (9), (10), but possibly suboptimal) state trajectories for performance.

Let

Γ1 = (x1(t), V1(t)), 0 ≤ t ≤ t1

f

,

and

Γ2 = (x2(t), V2(t)), 0 ≤ t ≤ t2

f

,

be two feasible state trajectories corresponding to control strategies T1(t) and T2(t) with

generally different final times. The respective operating costs (11) of the two trajectories

can be briefly written

Jk =

Z t

f

k

0



cfTk(t) + r



dt, k = 1, 2.

The direct operating cost can be written as a path integral, along the state trajectory, of a

differential form P(x, V ) dx + M(x, V ) dV (derived explicitly in Appendix A), i.e.,

Jk =

Z

Γk

[P(x, V ) dx + M(x, V ) dV ] , k = 1, 2. (12)

At constant airspeed, i.e. dV = 0, the quantity P has the interpretation of steady-state

cost per unit ground path distance.

The difference

J1 − J2 (13)

is the difference in performance between the two state trajectories. Henceforth, assume that

the two trajectories share no other states besides (x

i

, V i) and (x

f

, V f). This assumption,

which entails no loss of generality but simplifies analysis, implies that Γ1 and reverseoriented Γ2 together constitute a closed contour [23], denoted here by Γ1 ∪ −Γ2 and, for

definiteness, assumed oriented counter-clockwise. See Figure 1 for a notional illustration.

Expression (13) can be written as a path integral along the aforementioned closed contour:

I

Γ1∪−Γ2

(P dx + M dV ).

By Green’s Theorem, the latter integral can be written as an integral over the area A

enclosed and oriented by the contour:

Z Z

A



∂M

∂x −

∂P

∂V 

dx dV. (14)

The two state trajectories can now be compared for performance by using the knowledge

7
x

V

P

f

P

i

2

1

I = I(x, V)

A

Γ

Γ

Figure 1. Green’s Theorem predicts J1 < J2 if I < 0 everywhere in the enclosed area; and

J1 > J2 if I > 0 everywhere in the enclosed area.

of the integrand

I ≡

∂M

∂x −

∂P

∂V

to examine the sign of the difference (13). In particular, if I is nonnegative in and on the

boundary of A, then Γ2 performs as well as does Γ1 or better. Consequently, a comparison of

two feasible state trajectories reduces to an examination of the sign of I in the appropriate

regions of the state space. This examination will now be carried out.

By the derivation in Appendix A,

P =

cfD + r

V + w

, M = cf , (15)

and, consequently,

I =

∂cf

∂x −

∂

∂V 

cfD + r

V + w



.

For typical commercial aircraft and descent FPAs, cf does not vary substantially with the

ground path distance. Therefore, the first term on the right hand side can be ignored

without noticeable effects on the results, i.e.,

I ' −

∂

∂V 

cfD + r

V + w



. (16)

For aircraft with typical drag formulas (described in detail below) and, within the range

of airspeeds considered, one finds that the zero-level set [I = 0] is the graph of a function

Vmc(x) called the minimum-cost speed. The subscript mc stands for minimum cost because

∂P

∂V









V =Vmc

= 0

8
and

∂

2P

∂V 2

> 0, (17)

which will be shown at the end of this section. It follows that

I(x, V )







< 0 if V > Vmc(x),

= 0 if V = Vmc(x),

> 0 if V < Vmc(x).

(18)

To conclude the section, we justify stipulation (17). A typical drag polar [24] for modeling

drag is

D =

1

2

ρV 2SCD =

1

2

ρV 2S



Cd0 + KCL

2



,

where CL denotes the lift coefficient, CD the drag coefficient, Cd0 the zero-lift drag coefficient, K the lift drag constant, S the aircraft’s wing area, and ρ = ρ(x) the air density.

Replacing CL with its definition [24],

L = W

s

1 +

V 4

g

2R2

=

1

2

ρV 2SCL,

where R is the instantaneous turn radius, the drag can be written

D =

1

2

ρV 2SCd0 +

KW2

1

2

ρV 2S



1 +

V

4

g

2R2



. (19)

Equations (6), (16), (19) and the dependency of D on V together imply that

I ∼







−

g

2W

ρSCd0 −

2KW

ρSgR2

< 0, for a large V,

6KW

gρSV 4

> 0, for a small V.

(20)

It can also be shown that

∂

2P

∂V 2

= −

∂I

∂V =

24KW

gρV 5S

> 0, if w = r = 0. (21)

Results (20) and (21) together imply (18).

Numerical evidence obtained for the case when w and r have realistic nonzero values suggests

that the inequality in Eq. (21) always holds for the range of V considered, hence confirming

(17) and (18).

4.2 Optimal State Trajectories and Control Strategies

The following observation will be instrumental throughout this section: the curve in the

state space traversed by a feasible state trajectory (x(t), V (t)) can be parameterized by x

9
instead of by t; i.e., can be viewed as the graph of a function V = V (x). This follows from

the fact that, by dividing Eq. (8b) by Eq. (8a), one obtains

dV

dx =

T − D

V + w

is finite.

Therefore, in what follows, whenever convenient, a state trajectory (x(t), V (t)) will be

treated as a function V = V (x) without further explanation.

The following lemma is a direct consequence of the results of Section 4.1. The five parts of

the lemma are illustrated, respectively, by the five panels of Figure 2. It will be convenient

to use the control-theoretic terms of admissibility [25] and attainability: A control value

T satisfying (10) is said to be admissible. This term will be applied as well to a control

strategy with the value T(t) admissible at every t. A state trajectory corresponding to an

admissible control strategy will be referred to as attainable. The last part of the following

lemma addresses the situation when not all segments of [I = 0] are attainable trajectories.

Lemma 1 Consider an initial state a = (xa, Va) and a final state b = (xb, Vb), with

xa < xb. For (a) through (d), consider two attainable state trajectories (x1(t), V1(t)) and

(x2(t), V2(t)) that have the same initial state a and the same final state b:

(a) If a is above [I = 0], b is on [I = 0], and

Vmc(x) ≤ V1(x) ≤ V2(x) for xa ≤ x ≤ xb,

then V1(x) outperforms V2(x). (This is because the integrand I in the area A enclosed

by Γ1 ∪ −Γ2 is negative.)

(b) If a is below [I = 0], b is on [I = 0], and

V2(x) ≤ V1(x) ≤ Vmc(x) for xa ≤ x ≤ xb,

then V1(x) outperforms V2(x).

(c) If a is on [I = 0], b is above [I = 0], and

Vmc(x) ≤ V1(x) ≤ V2(x) for xa ≤ x ≤ xb,

then V1(x) outperforms V2(x).

(d) If a is on [I = 0], b is below [I = 0], and

V2(x) ≤ V1(x) ≤ Vmc(x) for xa ≤ x ≤ xb,

then V1(x) outperforms V2(x).

(e) Suppose a and b are on the curve [I = 0], and let ab denote the segment of [I = 0]

connecting these two states. If a segment a

0b0 of [I = 0] is contained in ab and is not

an attainable state trajectory (i.e., state b

0

cannot be reached from a

0 using admissible

controls), then the optimal state trajectory from a to b consists of three segments: the

segment of [I = 0] from a to a state a1 on aa0, one (attained by a suitable extreme value

of control) from a1 to a state b1 on b

0b, and the segment of [I = 0] from b1 to b.

10
The optimal trajectory described in the last part, (e), of the Lemma is completely determined by the point of intersection (the gray dot in Fig. 2(e)) between a

0b0 and trajectory

a1b1; i.e., by the choice of that state on a

0b0 which minimizes the cost of getting from a to

b. Since segment a

0b0

is closed and bounded, such a state necessarily exists.

In summary, the optimal trajectory goes towards [I = 0] as rapidly as possible, as in

Figure 2(a) and 2(b), and leaves [I = 0] as late as possible, as in Figure 2(c) and 2(d). This

observation along with Eq. (8b) suggests that extreme values of T must be used in getting

the optimal trajectory to or from [I = 0]. That the optimal trajectory must, in fact, use

T min or T max when it is not on the [I = 0] curve, can be proved using the Hamiltonian [26]

of the problem,

H = cfT + ci + λx(V + w) + λs



T − D



, (22)

where λx and λs are the respective costate variables of x and V . The optimal control

minimizes (per the sign convention in [26]) the Hamiltonian at every point of the trajectory.

The Hamiltonian H is linear in T and is minimized by:

T =

(

T min if ∂H

∂T

> 0

T max if ∂H

∂T

< 0.

(23)

If ∂H/∂T = 0, then the optimal control is on a singular arc [26]. The curve [I = 0] turns out

to be the singular arc of the Hamiltonian. In particular, for an optimal trajectory passing

through a state in I < 0 and later intersecting [I = 0], as in Lemma 1(a), the optimal

control at that state is T min. Similarly, for a state on an optimal trajectory passing through

a state in I > 0 and later intersecting [I = 0], as in Lemma 1(b), the optimal control at

that state is T max.

This classification of optimal control strategies can also be carried out by more elementary

means, without involving the Hamiltonian formalism, but expending more verbiage. This

simpler approach to optimal control classification is presented in Appendix B.

4.3 Control on the Minimum-Cost Speed Curve

The control that keeps the aircraft on the minimum-cost speed curve, T mc, is computed

here by setting the right-hand side of Eq. (16) to zero and using the state equations (8).

For convenience, define

 ̃I ≡ (V + w)

2

I. (24)

Since (V + w) is never zero in the range of airspeed and wind considered, the curves [I = 0]

and [ ̃I = 0] coincide.

The following equality holds on [  ̃I = 0]:

d V

d x









I ̃=0

= −

 

∂  ̃I

∂V !−1

∂  ̃I

∂x. (25)

Defining

F ≡ cfD,

11
(a)

Vmc(x)

I < 0

I > 0

x

V

a

b

V1(x)

(b)

Vmc(x)

I < 0

I > 0

x

V

a

V1(x) b

(c)

Vmc(x)

I < 0

I > 0

x

V

a

b

V1(x)

(d)

Vmc(x)

I < 0

I > 0

x

V

a

b

V1(x)

(e)

Vmc(x)

I < 0

I > 0

x

V

a

b

a

0

b

0

a1

b1

Figure 2. A notional illustration of Lemma 1. In panels (a) through (d), the trajectories

from state a to state b shown in solid outperform those shown dashed.

12
and substituting Eqs. (24) and (16) into the right-hand side of Eq. (25), one obtains

d V

d x









I ̃=0

=



(V + w)

∂

2F

∂V 2

−1 

−

∂F

∂V

dw

dx +

∂F

∂x − (V + w)

∂

2F

∂x ∂V 

, (26)

From Eqs. (8) and (26), one obtains

T mc = D +



∂

2F

∂V 2

−1 

−

∂F

∂V

dw

dx +

∂F

∂x − (V + w)

∂

2F

∂x ∂V 

. (27)

With analytic expressions for the drag and the wind, the right-hand side of Eq. (27) can be

evaluated for specific values of x and V . The thrust Tmc on [I = 0] is given by

Tmc =

W

g

T mc.

The thrust T mc computed using Eq. (27) can turn out to violate the admissibility condition (10) on an interval of the x-axis, leading to the situation of Lemma 1(e). In this

case, the optimal trajectory leaves the minimum-cost speed curve with an extremal value

of thrust until it either rejoins the minimum-cost speed curve or reaches the final state.

4.4 Vmc with Constant cf and without Wind

The content of this section is a number of consequences derived from the above results

for the special case when cf is assumed constant, and wind speed assumed is zero at all

altitudes. Under the assumption of constant cf , Eq. (24) simplifies to

 ̃I = −(V + w)

2

cf

∂

∂V 

D + r/cf

V + w



= 0. (28)

Furthermore, if the wind effect is negligible, then

 ̃I = −V

2

cf

∂

∂V 

D + r/cf

V



= 0. (29)

By using Eqs. (6) and (19) to model the drag, Eq. (29) is solved analytically for V . The

minimum-cost speed is found to be

Vmc(x)

2 =

(W/S)

Cd0 ρ(x)

h

α +

p

α2 + 12KCd0

i

. (30)

Here

α = sin γi +

r

cf g

captures the combined effect of the FPA and of the Cost Index on the minimum-cost speed,

In particular, α increases with r and with γi

in a neighborhood of γi = 0.

Equality (30) has the following interpretations and implications

13
* The minimum-cost speed Vmc can be regarded as a modified best-range speed modulated

by the factor α. For a flight segment which is level (i.e., with FPA zero) and has a zero

Cost Index, the resulting special case α = 0 corresponds to an Vmc equal to the best-range

speed at an altitude specified by h(x) [24].

* The minimum-cost speed Vmc(x) decreases with altitude due to air density change.

* The minimum-cost speed Vmc(x) increases with α, which means Vmc increases with γi

and with the Cost Index.

5 Case Study: GEELA Arrival Route to the Phoenix Airport

5.1 Route Definition and Test Conditions

This section investigates the sensitivity of the optimal speed profile to the wind condition

and Cost Index, explores the effects of certain modeling assumptions on the fuel computation, and compares the computed optimal trajectory to a simulated flight trajectory. The

specific example of an arrival procedure used as a basis for this section is a portion of

the GEELA route from waypoint HYDRR to waypoint PHX08 (runway 8) of the Phoenix

Airport.3

Figure 3 shows the waypoints and altitude restrictions specified in this procedure. The

coordinates Xfe and Yfe in Fig. 3 increase Eastward and Northward, respectively. Note

that the origin of the coordinates is arbitrary, since this figure is mainly for demonstrating

the scale of an arrival route. This procedure is closely based on current operations but

has additional waypoints to precisely guide the arrival flights all the way to the runway.

This procedure has been used in simulations conducted to test arrival management technologies [3]. The ground path distance of the entire route was constructed by connecting

consecutive waypoints’ positions and span roughly 54 nmi. The coordinate x, representing

the ground path distance, starts negative and increases to 0 at the runway threshold. Turns

were modeled as having a five nautical mile turn radius each. This approximate treatment

of the turns decoupled the horizontal path from the speed. Lift was treated approximately

as being equal to the aircraft weight all the time. These approximations have been used by

trajectory generators of the ground automation systems [27].

Figure 4 shows the modeled altitude h(x) of the GEELA route. The FPA is modeled as

a constant on each segment between two consecutive waypoints, except varying near the

waypoint so as to remain continuous. This change of FPA takes place after the aircraft has

crossed the waypoint. A fixed change rate of ±1

◦ per nmi was used to model these transient

segments.

The air density was a function of the altitude computed from the standard atmosphere

model. It was 0.778 kg3/m at the beginning of the descent and 1.187 kg3/m at the end.

The initial airspeed at the waypoint HYDRR, V

i

, for the aircraft was chosen to be the

calibrated airspeed (CAS) [24] of 260 knots (174.7 m/sec), required at HYDRR by the

3The GEELA route has two upstream branches that merge at HYDRR. Only the portion from HYDRR

to the airport is analyzed in this work.

14
180 190 200 210 220 230

300

305

310

315

320

X

fe (nmi)

Yfe (nmi)

HYDRR 

15000 ft

GEELA 

 

PUNNT 

10000 ft

TEICH 

8000 ft 

TESLE 

7000 ft 

HIKID 

6000 ft 

ILIKE 

5000 ft 

JAMIL 

4000 ft 

WAZUP 

3000 ft 

PHX08 

1111 ft 

Figure 3. The GEELA arrival route to the Phoenix Airport.

−50 −40 −30 −20 −10 0

0

5

10

15

20

Ground Path Distance (nmi)

h (10

3 ft)

Figure 4. The altitude profile along the GEELA arrival route.

arrival procedure. The final speed at the runway PHX08, V

f

, was set to a typical landing

speed of 135 knots CAS (69.4 m/sec).

Boeing 737-800 aircraft type was modeled using the Base of Aircraft Database (BADA)

3.8 [28]. Flap schedule was not considered and only the clean configuration was used. The

minimum-cost speed curve, Vmc(x) (see Fig. 2), was computed using Eq. (28) for the entire

range of x. Substituting Eq. (19) into Eq. (28), a 5th-order polynomial in V was obtained

whose coefficients depended on x. The polynomial reduces to 4th order when w = 0, in

which case Vmc can be computed from Eq. (30). In all test conditions, one and only one

real root between V

i and Vf was found for each x.

The optimal control strategy were applied to four benchmark conditions, labeled as MinFuel, Headwind, Tailwind, and Fuel-and-Time. These conditions differ only in the choice

of wind function and Cost Index, listed in Table 1 and justified in the following sections.

Table 1. Parameters that define the four benchmark conditions.

Condition

Wind Gradient Cost Index

(see Eq. (2)) (see Eq. (5))

w(x) (knots; x in nmi) r ($/hr / (cents / lb))

Min-Fuel 0 0

Headwind −3.79 + 0.226x 0

Tailwind 1.03 − 0.301x 0

Fuel-and-Time 0 30

15
Table 2. Flight times and fuel burn for the four conditions.

Condition Flight Time (sec) Fuel Burn (lb)

Min-Fuel 846 127

Headwind 830 166

Tailwind 854 82

Fuel-and-Time 801 150

140

220

300

V (knots)

Q1

Q2

Pi

Pf

(a)

−50 −40 −30 −20 −10

0

10

20

x (nmi)

T (10

3 lbf)

(b)

Figure 5. The minimum-cost speed profile and control strategy for the Min-Fuel condition.

In all conditions, the initial airspeed is above the minimum-cost speed and the final airspeed

is below the minimum-cost speed, corresponding to Lemma 1(a) and 1(c), respectively. The

discussion in Section 4.2 indicates that an extreme value of thrust, T min, must be used in

the initial and final phases of the state trajectory. The speeds for those phases were found

by integrating the state equations with that value of thrust forward from the first two

boundary conditions of (9) and backward from the last two boundary conditions of (9).

The integration was carried out numerically using MATLAB’s [29] ode45 function. The

thrust T mc was computed as a function of x using Eq. (27).

Table 2 summarizes the flight times and fuel burn obtained for the four conditions. Details

of the control strategy and speed profile will be discussed for each condition in the following

sections.

5.2 Min-Fuel Condition: No Wind, No Time Cost

Figure 5(a) shows the optimal speed profile. The thin line represents the minimum-cost

speed, Vmc. Q1 and Q2 stand for points that join a Tmin speed profile with a Tmc speed

profile. The value Vmc was computed essentially from Eq. (30). The rapid changes of Vmc at

some values of x were due to change of the FPA before and after a waypoint. The general

trend was that Vmc increased with increasing (shallower) γi. For example, the increase of

Vmc near x = −34 nmi corresponded to the waypoint of PUNNT. Between PUNNT and

TEICH, the FPA was shallower than before PUNNT (see Fig. 4), resulting in higher values

of Vmc.

16
-20

-15

-10

-5

0

5

10

15

20

PUNNT, 10000 ft

PHX08, 1111 ft

Days of Year 2011

w (m/s

e

c)

Figure 6. Daily averages of the along-the-route wind components at PUNNT and PHX08.

Figure 5(b) shows T as a function of x, which has non-zero thrust during the period of time

when the trajectory follows the minimum-cost speed curve. The thrust for this period of

time was equal to Tmc. Shallower FPAs result in higher values of Tmc (e.g., on the segment

from PUNNT to TEICH; ). This corresponds to the physical intuition that a shallower

descent does not convert as much potential energy to kinetic energy to compensate for the

drag and therefore must use more thrust instead.

The speed profile shown in Fig. 5 turns out to be unattainable near x = −24 nmi and near

x = −20 nmi. This situation corresponds to Lemma 1(e). In physical flight control, the

effect of negative thrust could have been attained by using speed brakes. This use, however,

would not result in the reduction of cost (11) as would negative thrust. In this model, the

trajectory segment is computed, following the notation in Lemma 1(e), by iterating through

possible segments of a1b1 until the minimum cost between ab is obtained. As expected,

the actual optimal speed has a flatter |dV /dx| than that of |dVmc/dx| on these segments.

5.3 Headwind and Tailwind Conditions

The main purpose of studying these two wind conditions is to understand the effect of

wind variation on the optimal control strategy and the resulting speed profile. To select

representative winds conditions, winds along the GEELA route were computed using the

two-hour look-ahead time, 40-km Rapid Update Cycle weather forecast [30] for year 2011.

Twenty-four forecast winds were generated for each day on an hourly basis. Daily averages

were computed for every day of year 2011 at the waypoints PUNNT and PHX08, two of

the waypoints on the GEELA route (see Fig. 3).

Figure 6 shows the daily averages of the along-the-route components of wind at PUNNT

and PHX08. The wind components for PUNNT and PHX08 were computed at 10,000 ft

and 1,111 ft, respectively. The data indicated predominantly tailwinds for PUNNT all year

long. Wind components at PHX08 were smaller in magnitude and varied somewhat by the

season.

17
140

220

300

V (knots)

Q1

Q2

Pi

Pf

(a)

−50 −40 −30 −20 −10

0

10

20

x (nmi)

T (10

3 lbf)

(b)

Figure 7. The speed profile and control strategy for the Headwind condition.

The Headwind and Tailwind conditions were selected from the two days of year 2011 that

had the most negative and positive wind gradients, dw/dx, respectively. The Headwind

condition was defined based on February 3’s average wind, which had the most negative wind

gradient. Both PUNNT and PHX08 experienced headwinds along the route on February 3.

The Tailwind condition was defined based on February 19’s average wind, which had the

most positive wind gradient. Both PUNNT and PHX08 experienced tailwinds along the

route on February 19. The wind gradient was assumed to be constant throughout the

path, and was computed by dividing the difference of the wind components at PHX08 and

PUNNT by the ground path distance between these two waypoints. The resulting wind

functions are listed in Table 1.

The optimal speed profile and control strategy for the Headwind condition are shown in

Fig. 7. The wind perturbed the minimum-cost speed curve noticeably, increasing Vmc by

16 knots at the beginning (xi) of the trajectory and by only 4 knots at the end of the

trajectory. The initial and final arcs descended faster in speed (more negative dVmc/dx),

causing the aircraft to stay longer on the minimum-cost speed curve than in the Min-Fuel

condition. The thrust on the minimum-cost speed curve increased slightly when compared

to the Min-Fuel condition. For transient segments on which the FPA increases rapidly

around x = −34 nmi, the thrust takes relatively high values.

The optimal speed profile for the Tailwind condition is shown in Fig. 8(a). The wind

perturbed the minimum-cost speed curve in the opposition direction, decreasing Vmc by

15 knots of Vmc at the beginning of the trajectory and by only 1 knot at the end of the

trajectory. Fig. 8(b) shows the Tmin arcs descended slower in speed, causing the aircraft to

stay only a short distance on the minimum-cost speed curve. The thrust on the minimumcost speed curve decreased slightly compared to the Min-Fuel condition.

The effect of wind on the minimum-cost speed is consistent with the dependency of the

best-range speed on winds [24]: higher for headwind and lower for tailwind. Comparing the

fuel burn to the Min-Fuel condition, the trajectory for the Headwind condition burns 36 lb

more fuel while the trajectory for the Tailwind condition burns 31 lb less fuel (see Table 2).

18
140

220

300

V (knots)

Q1

Q2

Pi

Pf

(a)

−50 −40 −30 −20 −10

0

10

20

x (nmi)

T (10

3 lbf)

(b)

Figure 8. The speed profile and control strategy for the Tailwind condition.

140

220

300

V (knots)

Q1

Q Pi 2

Pf

(a)

−50 −40 −30 −20 −10

0

10

20

x (nmi)

T (10

3 lbf)

(b)

Figure 9. The minimum-cost speed curve and control strategy for the Fuel-and-Time Condition.

5.4 Fuel-and-Time Condition

The Fuel-and-Time condition considered both fuel and time costs. The Cost Index selected

by airlines varies greatly from route to route and among aircraft types [20]. For B737, a

typical value [20, 31] of

r = 30 

$/hr

cents $/lb

was used for the analysis. The fuel price, pf , was chosen to be 0.43 ($/lb).

The optimal speed profile for the Fuel-and-Time condition is shown in Fig. 9(a). Compared

to the Min-Fuel condition, the minimum-cost speed curve moves toward higher speeds

significantly. This is because the aircraft must fly faster in order to reduce the time cost

at the expense of the fuel cost. The minimum-cost speed curve was reached here much

earlier than in the Min-Fuel condition. The speed profile for the Fuel-and-Time condition

saved 45 seconds compared to that for the Min-Fuel condition (see Table 2), at the cost of

23 lb of fuel. Fig. 9(b) shows the required thrust T as a function of x. A large value of

19
thrust was required to accelerate the aircraft around x = −34 nmi due to the increase of

the minimum-cost speed.

6 Conclusions

The key contribution of this paper is an exact feedback-control formula for the speed strategy

that accrue minimum cost during a descent under altitude restrictions. The sensitivity of the

optimal speed strategy to winds and airlines’ business objectives was analyzed by applying

the formula to an actual arrival route.

This analytical formula has the following potential applications:

Evaluation of arrival procedures: While most altitude restrictions are set by terrain, noise,

and separation of traffic flows, speed restrictions can be compared to the minimum-cost

speed profile to see whether the procedure can be refined to reduce aircrafts’ direct operational costs. Since weather, airlines’ business objectives, and aircraft types all affect

the minimum-cost speed profile, a static arrival procedure should at best achieve systemic

fuel/cost savings on an aggregated basis.

Scheduling efficiency of ground automation tools: Ground automation tools that assist ATC

in scheduling arrival aircraft can take advantage of the minimum-cost speed profile and

use this information to improve its scheduling efficiency. Specifically, the TSS system in

the United States can use this information to improve the nominal speed profiles used as

reference points for scheduling aircraft.

Guidance of small jets: While large commercial jets equipped with performance-based Flight

Management Systems (FMS) have similar capabilities of computing company-preferred

speed profile based on the airlines’ business objective, aircraft performance model, and

limited wind information, small jets such as regional and business jets are equipped with

simpler FMS that cannot compute minimum-cost speed profiles. These jets accounts for

25% to 30% of the operations in the United States [32]. The methodology developed in this

paper can be used to aid the airlines in the selection of speed profiles for these jets. The

selection can presumably be made during the flight-planning phase.

A reference point for multi-aircraft operations: During periods of medium to heavy traffic,

separation constraints must be taken into account and minimum-cost speed profiles may

not be always feasible for individual aircraft. Nonetheless, the minimum-cost speed profiles

can still be used as a reference point for evaluating the level of optimality of the system and

for computing systemic cost-effective strategies.

Among the directions for future research, the following seem reasonable:

* Compare the minimum-cost speed profile to actual trajectories and determine how far-off

is the current-day operations from optimality.

* Predict a systemic minimum-cost speed profile for a group of aircraft and a period of

future time, using airlines’ flight schedules and weather forecast.

20
* Accommodate hard-coded speed restrictions from ATC in the model formulation such as

the 250 knots-or-below rule.

* Develop ways of applying the results herein to minimize the operating cost of multiple

arrival flights in the terminal area. One key difficulty is the imposition of the separation

constraint: every pair of aircraft must be separated by a minimum distance imposed by

the FAA.

* Implement a generic solver for the minimum-cost speed in an extensible way such that it

can work with various aircraft performance models.

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23
Appendix A

Direct Operating Cost as a Path Integral

The direct operating cost in Eq. (11) can be written as a path integral along the state

trajectory [22]. First, rewrite the equations of motion in Eqs. (8) in a matrix form:

"

x ̇

V ̇

#

=

"

(V + w) 0

−D 1

# " 1

T

#

. (A1)

Invert the two-by-two matrix on the right hand side to write the control T as a linear

combination of  ̇x and V ̇[22],

T =



D

V + w



x ̇ + V .  ̇ (A2)

Substituting Eq. (A2) into Eq. (11),

J =

Z t

f

0



cf

 D

V + w



x ̇ + V ̇



+ r



dt.

Multiplying the last term by a factor of unity,  ̇x/(V + w),

J =

Z t

f

0



cf

 D

V + w



x ̇ + V ̇



+

r

V + w

x ̇



dt,

and changing the variables of integration,

J =

Z

Γ

cf

D

V + w



+

r

V + w



dx + cf dV . (A3)

Here Γ stands for the trajectory, and dx and dV are related by the equations of motion and

the control T. Eq. (A3) expresses the cost function as a line integral of a differential form

P(x, V )dx + M(x, V )dV , where

P(x, V ) = cfD + r

V + w

, and M(x, V ) = cf .

24
Appendix B

Optimal Control Strategy Connecting A State to the [I = 0]

Curve

The following lemma proves that the optimal control strategy for the problem considered

in this work must use either T min or T max when connecting a state to the [I = 0] curve.

Lemma 2 A state trajectory that has no part in I > 0, has the final state on [I = 0], and

passes through a state (x(t), V (t)) in I < 0 at some instant t∗, with the control strategy T(t)

strictly greater than T min in some neighborhood of t∗, is not optimal.

Proof. Construct a new control strategy, denoted here by T∗(t), as follows. Replace T(t)

by T min on the interval [t∗, t1], where t1 is the earliest instant at which the newly obtained

state trajectory, (x∗(t), V∗(t)), intersects either (x(t), V (t)) (Case (a)) or the curve [I = 0]

(Case (b)), whichever happens first (if the two cases coincide, the situation is treated as as

Case (a)).

In Case (a), let T∗(t) = T(t) for t ≥ t1, to obtain

T∗(t) =







T(t) for t < t∗,

T min for t∗ ≤ t ≤ t1,

T(t) for t1 < t.

In Case (b), let T∗(t) = T mc(t) from t = t1 until such earliest instant t2 when (x∗(t), V∗(t))

intersects the (x(t), V (t)) (possibly at the final state). Assume this portion of arc on [I = 0]

is attainable. From t = t2 on, let T∗(t) = T(t). This yields

T∗(t) =







T(t) for t ≤ t∗,

T min for t∗ ≤ t ≤ t1,

T mc(t) for t1 ≤ t ≤ t2.

T(t) for t2 ≤ t.

In both cases, V∗(x) by construction satisfies both of the following conditions: V∗(x) ≤ V (x)

for every x, and V∗(x) < V (x) in some right-sided neighborhood of x(t∗). By Lemma 1a at

Section 4.2, (x∗(t), V∗(t)) outperforms (x(t), V (t)). This completes the proof.

Immediate consequences of Lemma 2, and of results analogous to it, are as follows:

(i) An optimal trajectory passing through a state (x, V ) in I < 0 and later reaching

[I = 0] has in state (x, V ) control T min.

(ii) An optimal trajectory passing through a state (x, V ) in I > 0 and later reaching

[I = 0] has in state (x, V ) control T max.

25
(iii) If an optimal trajectory passes through two states on [I = 0], then the portion of

the trajectory between these two states lies entirely on [I = 0]. (In other words, an

optimal trajectory cannot leave [I = 0] and then come back to [I = 0].)

26

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1. REPORT DATE (DD-MM-YYYY)

01-04-2015

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Technical Memorandum

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4. TITLE AND SUBTITLE

Minimum-Cost Aircraft Descent Trajectories with a Constrained Altitude

Profile

5a. CONTRACT NUMBER

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5c. PROGRAM ELEMENT NUMBER

5d. PROJECT NUMBER

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6. AUTHOR(S)

Minghong G. Wu and Alexander V. Sadovsky

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NASA Ames Research Center

Moffett Field, California, 94035-1000

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L–00000

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National Aeronautics and Space Administration

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NASA

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Unclassified-Unlimited

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Availability: NASA CASI (443) 757-5802

13. SUPPLEMENTARY NOTES

An electronic version can be found at http://ntrs.nasa.gov.

14. ABSTRACT

An analytical formula for solving the speed profile that accrues minimum cost during an aircraft descent with a constrained

altitude profile is derived. The optimal speed profile first reaches a certain speed, called the minimum-cost speed, as quickly

as possible using an appropriate extreme value of thrust. The speed profile then stays on the minimum-cost speed as long as

possible, before switching to an extreme value of thrust for the rest of the descent. The formula is applied to an actual arrival

route and its sensitivity to winds and airlines’ business objectives is analyzed.

15. SUBJECT TERMS

EDA, fixed flight path angle descent, fuel burn

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