Aircraft Fossen 2011

MATHEMATICAL MODELS FOR CONTROL OFAIRCRAFT AND SATELLITES

THOR I. FOSSENProfessor of Guidance, Navigation and Control

Department of Engineering CyberneticsNorwegian University of Science and Technology

January 20112nd edition

Copyright c©1998-2011 Department of Engineering Cybernetics, NTNU.1st edition published February 1998.

2nd edition published January 2011.

Contents

Figures iii

1 Introduction 1

2 Aircraft Modeling 32.1 Definition of Aircraft State-Space Vectors . . . . . . . . . . . . . . . . . . . . 32.2 Body-Fixed Coordinate Systems for Aircraft . . . . . . . . . . . . . . . . . . 4

2.2.1 Rotation matrices for wind and stability axes . . . . . . . . . . . . . 5

2.3 Aircraft Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.1 Kinematic equations for translation . . . . . . . . . . . . . . . . . . . 62.3.2 Kinematic equations for attitude . . . . . . . . . . . . . . . . . . . . 7

2.3.3 Rigid-body kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.4 Sensors and measurement systems . . . . . . . . . . . . . . . . . . . . 8

2.4 Perturbation Theory (Linear Theory) . . . . . . . . . . . . . . . . . . . . . . 82.4.1 Definition of nominal and perturbation values . . . . . . . . . . . . . 92.4.2 Linearization of the rigid-body kinetics . . . . . . . . . . . . . . . . . 92.4.3 Linear state-space model based using wind and stability axes . . . . . 11

2.5 Decoupling in Longitudinal and Lateral Modes . . . . . . . . . . . . . . . . . 122.5.1 Longitudinal equations . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5.2 Lateral equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6 Aerodynamic Forces and Moments . . . . . . . . . . . . . . . . . . . . . . . 142.6.1 Longitudinal aerodynamic forces and moments . . . . . . . . . . . . . 162.6.2 Lateral aerodynamic forces and moments . . . . . . . . . . . . . . . . 16

2.7 Standard Aircraft Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . 172.7.1 Dynamic equation for coordinated turn (bank-to-turn) . . . . . . . . 17

2.7.2 Dynamic equation for altitude control . . . . . . . . . . . . . . . . . . 182.8 Aircraft Stability Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.8.1 Longitudinal stability analysis . . . . . . . . . . . . . . . . . . . . . . 202.8.2 Lateral stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.9 Design of flight control systems . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Satellite Modeling 233.1 Attitude Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1 Euler’s 2nd Axiom Applied to Satellites . . . . . . . . . . . . . . . . . 23

i

ii CONTENTS

3.1.2 Skew-symmetric representation of the satellite model . . . . . . . . . 233.2 Satellite Model Stability Properties . . . . . . . . . . . . . . . . . . . . . . . 243.3 Design of Satellite Attitude Control Systems . . . . . . . . . . . . . . . . . . 24

4 Matlab Simulation Models 254.1 Boeing-767 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.1 Longitudinal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.1.2 Lateral model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 F-16 Fighter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2.1 Longitudinal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.3 F2B Bristol Fighter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.3.1 Lateral model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

List of Figures

1.1 Sketch showing a modern fighter aircraft (Stevens and Lewis 1992). . . . . . 1

2.1 Definition of aircraft body axes, velocities, forces, moments and Euler angles(McLean 1990). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Definition of stability and wind axes for an aircraft (Stevens and Lewis 1992). 52.3 Control inputs for conventional aircraft. Notice that the two ailerons can be

controlled by using one control input: δA = 1/2(δAL + δAR). . . . . . . . . . 142.4 Aircraft longitudinal eigenvalue configuration plotted in the complex plane. . 21

4.1 Schematic drawing of the Bristol F.2B Fighter (McRuer et al 1973). . . . . . 28

iii

iv LIST OF FIGURES

Chapter 1

Introduction

This note uses a vectorial notation to describe aircraft and satellites. The notation is similarto the one used for marine craft (ships, high-speed craft and underwater vehicles). Theequations of motion are based on:

Fossen, T. I. (1994). Guidance and Control of Ocean Vehicles(John Wiley & Sons Ltd), Chapter 2.

Fossen, T. I. (2011). Handbook of Marine Craft Hydrodynamics and Motion Control(John Wiley & Sons Ltd.), Chapters 2 and 3.

The kinematic and kinetic equations of a marine craft can be modified to describe aircraftand satellites by minor adjustments of notation and assumptions.

Figure 1.1: Sketch showing a modern fighter aircraft (Stevens and Lewis 1992).

The note is organized according to:

• Chapter 2: Aircraft Modeling

• Chapter 3: Satellite Modeling

• Chapter 4: Matlab Simulation Models

1

2 CHAPTER 1. INTRODUCTION

This note is in addition to the textbook Handbook of Marine Craft Hydrodynamics andMotion Control in the course TTK4109 Guidance and Control that is given at the Depart-ment of Engineering Cybernetics, NTNU.

Other useful references on flight control are:

Blakelock, J. H. (1991). Aircraft and Missiles (John Wiley & Sons Ltd.)

Etkin, B. and L. D. Reid (1996). Dynamics of Flight: Stability and Control (JohnWiley & Sons Ltd.)

McLean, D. (1990). Automatic Flight Control Systems (Prentice Hall Inc.)

McRuer, D., D. Ashkenas and A. I. Graham (1973). Aircraft Dynamics and Auto-matic Control (Princeton University Press)

Nelson, R. C. (1998). Flight Stability and Automatic Control (McGraw-Hill int.)

Roskam, J. (1999). Airplane Flight Dynamics and Automatic Flight Controls (Darcor-poration)

Stevens, B. L. and F. L. Lewis (1992). Aircraft Control and Simulation (John Wiley& Sons Ltd.)

Information about the author as well as the graduate coursesTTK4109 Guidance and Controland TK8109 Advanced Guidance and Control are found on the web-pages:

Thor I. Fossen: http://www.itk.ntnu.no/ansatte/Fossen_Thor

TTK4109 Guidance and Control: http://www.itk.ntnu.no/emner/ttk4190

TK8109 Advanced Guidance and Control: http://www.itk.ntnu.no/emner/tk8109

Thor I. FossenTrondheim —3 January 2011

Chapter 2

Aircraft Modeling

This chapter gives an introduction to aircraft modeling. The equations of motion are lin-earized using perturbation theory and the final results are state-space models for the longi-tudinal and lateral motions. The models can be used for aircraft simulation and design offlight control systems.

2.1 Definition of Aircraft State-Space Vectors

The aircraft velocity vector is defined according to (see Figure 2.1):

ν :=

UVWPQR

=

longitudinal (forward) velocitylateral (transverse) velocityvertical velocityroll ratepitch rateyaw rate

(2.1)

η :=

XE

YEZE, hΦΘΨ

=

Earth-fixed x-positionEarth-fixed y-positionEarth-fixed z-position (axis downwards), altituderoll anglepitch angleyaw angle

(2.2)

Forces and moments are defined in a similar manner:XYZLMN

:=

longitudinal forcetransverse forcevertical forceroll momentpitch momentyaw moment

(2.3)

Comment 1: Notice that the capital letters L,M,N for the moments are different from

3

4 CHAPTER 2. AIRCRAFT MODELING

Figure 2.1: Definition of aircraft body axes, velocities, forces, moments and Euler angles(McLean 1990).

those used for marine craft—that is, K,M,N. The reason for this is that L is reserved aslength parameter for ships and underwater vehicles.

Comment 2: For aircraft it is common to use capital letters for the states U, V,W, etc.while it is common to use small letters for marine craft.

2.2 Body-Fixed Coordinate Systems for Aircraft

For aircraft it is common to use the following body-fixed coordinate systems:

• Body axes

• Stability axes

• Wind axes

The axis systems are shown in Figure 2.2 where the angle of attack α and sideslip angle βare defined as:

tan(α) :=W

U(2.4)

sin(β) :=V

VT(2.5)

whereVT =

√U2 + V 2 +W 2 (2.6)

2.2. BODY-FIXED COORDINATE SYSTEMS FOR AIRCRAFT 5

Figure 2.2: Definition of stability and wind axes for an aircraft (Stevens and Lewis 1992).

is the total speed of the aircraft. Aerodynamic effects are classified according to the Machnumber :

M :=VTa

(2.7)

where a = 340 m/s = 1224 km/h is the speed of sound in air at a temperature of 20o C onthe ocean surface. The following terminology is speed:

Subsonic speed M < 1.0

Transonic speed 0.8 ≤M ≤ 1.2

Supersonic speed 1.0 ≤M ≤ 5.0

Hypersonic speed 5.0 ≤M

An aircraft will break the sound barrier at M = 1.0 and this is clearly heard as a sharpcrack. If you fly at low altitude and break the sound barrier, windows in building will breakdue to pressure-induced waves.

2.2.1 Rotation matrices for wind and stability axes

The relationship between vectors expressed in different coordinate systems can be derivedusing rotation matrices. The body-fixed coordinate system is first rotated a negative sideslipangle −β about the z-axis. The new coordinate system is then rotated a positive angle ofattack α about the new y-axis such that the resulting x-axis points in the direction of thetotal speed VT . The first rotation defines the wind axes while the second rotation defines


the stability axes. This can be mathematically expressed as:

pwind = Rz,−βpstab =

cos(β) sin(β) 0− sin(β) cos(β) 0

0 0 1

pstab (2.8)

pstab = Ry,αpbody =

cos(α) 0 sin(α)0 1 0

− sin(α) 0 cos(α)

pbody (2.9)

The rotation matrix becomes:Rwindbody = Rz,−βRy,α (2.10)

Hence,

pwind = Rwindbodyp

body (2.11)

m

pwind =

cos(β) sin(β) 0− sin(β) cos(β) 0

0 0 1

cos(α) 0 sin(α)0 1 0

− sin(α) 0 cos(α)

pbody (2.12)

m

pwind =

cos(α) cos(β) sin(β) sin(α) cos(β)− cos(α) sin(β) cos(β) − sin(α) sin(β)− sin(α) 0 cos(α)

pbody (2.13)

This gives the following relationship between the velocities in body and wind axes:

vbody =

UVW

= (Rwindbody)

>vwind = R>y,αR>z,−β

VT00

=

VT cos(α) cos(β)VT sin(β)VT sin(α) cos(β)

(2.14)

Consequently,U = VT cos(α) cos(β)V = VT sin(β)W = VT sin(α) cos(β)

(2.15)

2.3 Aircraft Equations of Motion

2.3.1 Kinematic equations for translation

The kinematic equations for translation and rotation of a body-fixed coordinate systemABC with respect to a local geographic coordinate system NED (North-East-Down) can beexpressed in terms or the Euler angles: XE

YEZE

= Rnedabc

UVW

= Rz,ΨRy,ΘRx,Φ

UVW

(2.16)

2.3. AIRCRAFT EQUATIONS OF MOTION 7

Expanding this expression gives: XE

YEZE

=

cΨ −sΨ 0sΨ cΨ 00 0 1

cΘ 0 sΘ0 1 0−sΘ 0 cΘ

1 0 00 cΦ −sΦ0 sΦ cΦ

UVW

m (2.17) XE

YEZE

=

cΨcΘ −sΨcΦ + cΨsΘsΦ sΨsΦ + cΨcΦsΘsΨcΘ cΨcΦ + sΦsΘsΨ −cΨsΦ + sΘsΨcΦ−sΘ cΘsΦ cΘcΦ

UVW

2.3.2 Kinematic equations for attitude

The attitude is given by: PQR

=

Φ00

+R>x,Φ

0

Θ0

+R>x,ΦR>y,Θ

00

Ψ

(2.18)

which gives: Φ

Θ

Ψ

=

1 sΦtΘ cΦtΘ0 cΦ −sΦ0 sΦ/cΘ cΦ/cΘ

PQR

, cΘ 6= 0 (2.19)

2.3.3 Rigid-body kinetics

The aircraft rigid-body kinetics can be expressed as (Fossen 1994, 2011):

m(ν1 + ν2 × ν1) = τ 1 (2.20)

ICGν2 + ν2 × (ICGν2) = τ 2 (2.21)

where ν1 := [U, V,W ]T ,ν2 := [P,Q,R]T , τ 1 := [X, Y, Z]T and τ 2 := [L,M,N ]T . It is as-sumed that the coordinate system is located in the aircraft center of gravity (CG). Theresulting model is written:

MRBν +CRB(ν)ν = τRB (2.22)

where

MRB =

[mI3×3 O3×3

O3×3 ICG

], CRB(ν) =

[mS(ν2) O3×3

O3×3 −S(ICGν2)

](2.23)

The inertia tensor is defined as (assume that Ixy = Iyz = 0 which corresponds to xz−planesymmetry):

ICG :=

Ix 0 −Ixz0 Iy 0−Ixz 0 Iz

(2.24)

The forces and moments acting on the aircraft can be expressed as:

τRB = −g(η) + τ (2.25)


where τ is a generalized vector that includes aerodynamic and control forces. The gravita-

tional force fG = [0 0 mg]T acts in the CG (origin of the body-fixed coordinate system) andthis gives the following vector expressed in NED:

g(η) = −(Rnedabc)

>[

fGO3×1

]=

mg sin(Θ)

−mg cos(Θ) sin(Φ)−mg cos(Θ) cos(Φ)

000

(2.26)

Hence, the aircraft model can be written in matrix form as:

MRBν +CRB(ν)ν + g(η) = τ (2.27)

or in component form:

m(U +QW −RV + g sin(Θ)) = X

m(V + UR−WP − g cos(Θ) sin(Φ)) = Y

m(W + V P −QU − g cos(Θ) cos(Φ)) = Z

IxP − Ixz(R + PQ) + (Iz − Iy)QR = L (2.28)

IyQ+ Ixz(P2 −R2) + (Ix − Iz)PR = M

IzR− IxzP + (Iy − Ix)PQ+ IxzQR = N

2.3.4 Sensors and measurement systems

It is common that aircraft sensor systems are equipped with three accelerometers. If theaccelerometers are located in the CG, the measurement equations take the following form:

axCG =X

m= U +QW −RV + g sin(Θ) (2.29)

ayCG =Y

m= V + UR−WP − g cos(Θ) sin(Φ) (2.30)

azCG =Z

m= W + V P −QU − g cos(Θ) cos(Φ) (2.31)

In addition to these sensors, an aircraft is equipped with gyros, magnetometers and a sensorfor altitude h and wind speed VT . These sensors are used in inertial navigation systems (INS)which again use a Kalman filter to compute estimates of U, V,W, P,Q and R as well as theEuler angles Φ,Θ and Ψ. Other measurement systems that are used onboard aircraft areglobal navigation satellite systems (GNSS), radar and sensors for angle of attack.

2.4 Perturbation Theory (Linear Theory)

The nonlinear equations of motion can be linearized by using perturbation theory. This isillustrated below.

2.4. PERTURBATION THEORY (LINEAR THEORY) 9

2.4.1 Definition of nominal and perturbation values

According to linear theory it is possible to write the states as the sum of a nominal value(usually constant) and a perturbation (deviation from the nominal value). Moreover,

Total state = Nominal value + Perturbation

The following definitions are made:

τ := τ 0 + δτ =

X0

Y0

Z0

L0

M0

N0

+

δXδYδZδLδMδN

, ν := ν0 + δν =

U0

V0

W0

P0

Q0

R0

+

uvwpqr

(2.32)

Similar, the angles are defined according to: ΘΦΨ

:=

Θ0

Φ0

Ψ0

+

θφψ

(2.33)

Consequently, a linearized state-space model will consist of the following states u, v, w, p, q, r, θ, φand ψ.

2.4.2 Linearization of the rigid-body kinetics

The rigid-body kinetics can be linearized by using perturbation theory.

Equilibrium condition

If the aerodynamic forces and moments, velocities, angles and control inputs are expressedas nominal values and perturbations τ = τ 0 + δτ , ν = ν0 + δν and η = η0 + δη, the aircraftequilibrium point will satisfy (it is assumed that ν0 = 0):

CRB(ν0)ν0 + g(η0) = τ 0 (2.34)

This can be expanded according to:

m(Q0W0 −R0V0 + g sin(Θ0)) = X0

m(U0R0 − P0W0 − g cos(Θ0) sin(Φ0)) = Y0

m(P0V0 −Q0U0 − g cos(Θ0) cos(Φ0)) = Z0

(Iz − Iy)Q0R0 − P0Q0Ixz = L0 (2.35)

(P 20 −R2

0)Ixz + (Ix − Iz)P0R0 = M0

(Iy − Ix)P0Q0 +Q0R0Ixz = N0


Perturbed equations

The perturbed equations—that is, the linearized equations of motion are usually derived bya 1st-order Taylor series expansion about the nominal values. Alternatively, it is possibleto substitute (2.32) and (2.35) into (2.27) and neglect higher-order terms of the perturbedstates. This is illustrated for the first degree of freedom (DOF):

Example 1 (Linearization of surge using perturbation theory)

m[U +QW −RV + g sin(Θ)] = X

⇓ (2.36)

m[U0 + u+ (Q0 + q)(W0 + w)− (R0 + r)(V0 + v) + g sin(Θ0 + θ)] = X0 + δX

This can be written:

sin(Θ0 + θ) = sin(Θ0) cos(θ) + cos(Θ0) sin(θ)θ small≈ sin(Θ0) + cos(Θ0)θ (2.37)

Since U0 = 0 and

m(Q0W0 −R0V0 + g sin(Θ0)) = X0 (2.38)

Equation (2.36) is reduced to:

m[u+Q0w +W0q + wq −R0v − V0r − vr + g cos(Θ0)θ] = δX (2.39)

If it is assumed that the 2nd-order terms wq and vr are negligible, the linearized modelbecomes:

m[u+Q0w +W0q −R0v − V0r + g cos(Θ0)θ] = δX (2.40)

�

Linear state-space model for aircraftIf all DOFs are linearized, the following state-space model is obtained:

m[u+Q0w +W0q −R0v − V0r + g cos(Θ0)θ] = δX

m[v + U0r +R0u−W0p− P0w − g cos(Θ0) cos(Φ0)φ+ g sin(Θ0) sin(Φ0)θ] = δY

m[w + V0p+ P0v − U0q −Q0u+ g cos(Θ0) sin(Φ0)φ+ g sin(Θ0) cos(Φ0)θ] = δZ

Ixp− Ixz r + (Iz − Iy)(Q0r +R0q)− Ixz(P0q +Q0p) = δL (2.41)

Iy q + (Ix − Iz)(P0r +R0p)− 2Ixz(R0r + P0p) = δM

Iz r − Ixzp+ (Iy − Ix)(P0q +Q0p) + Ixz(Q0r +R0q) = δN

This can be expressed in matrix form as:

MRBν +NRBν +Gη = τ (2.42)

2.4. PERTURBATION THEORY (LINEAR THEORY) 11

where

MRB =

m 0 0

m 0 03×3

0 0 mIx 0 −Ixz

03×3 0 Iy 0−Ixz 0 Iz

NRB =

0 −mR0 mQ0 0 mW0 −mV0

mR0 0 −P0 −W0 0 U0

−mQ0 mP0 0 mV0 −mU0 0−IxzQ0 (Iz − Iy)R0 − IxzP0 (Iz − Iy)Q0

03×3 (Ix − Iz)R0 −2IxzP0 (Ix − Iz)P0 − 2IxzR0

(Iy − Ix)Q0 (Iy − Ix)P0 + IxzR0 IxzQ0

G =

0 mg cos(Θ0) 0

03×3 −mg cos(Θ0) cos(Φ0) mg sin(Θ0) sin(Φ0) 0mg cos(Θ0) sin(Φ0) mg sin(Θ0) cos(Φ0) 0

03×3 03×3

In addition to this, the kinematic equations must be linearized.

2.4.3 Linear state-space model based using wind and stability axes

An alternative state-space model is obtained by using α and β as states. If it is assumedthat α and β are small such that cos(α) ≈ 1 and sin(β) ≈ β, Equation (2.15) can be writtenas:

U = VT

V = VTβ

W = VTα

⇒

U = VT

β = VVT

α = WVT

(2.43)

Furthermore, the state-space vector:

x =

uβαpqr

=

surge velocitysideslip angleangle of attackroll ratepitch rateyaw rate

(2.44)

is chosen to describe motions in 6 DOF. The relationship between the body-fixed velocityvector:

ν = [u, v, w, p, q, r]T (2.45)

and the new state-space vector x can be written as:

ν = Tx = diag{1, VT , VT , 1, 1, 1, 1}x (2.46)


where VT > 0. If the total speed is VT = U0 = constant (linear theory), it is seen that:

α =1

VTw (2.47)

β =1

VTv (2.48)

VT = 0 (2.49)

If nonlinear theory is applied, the following differential equations are obtained:

α =UW −WU

U2 +W 2(2.50)

β =V VT − V VTV 2T cos β

(2.51)

VT =UU + V V +WW

VT(2.52)

In the linear case it is possible to transform the body-fixed state-space model:

ν = Fν +Gu (2.53)

tox = Ax+Bu (2.54)

whereA = T−1FT, B = T−1G (2.55)

For VT 6= 0 this transformation is much more complicated. The linear state-space transfor-mation is commonly used by aircraft manufactures. An example is the Boeing B-767 model(see Chapter 4).

2.5 Decoupling in Longitudinal and Lateral Modes

For an aircraft it is common to assume that the longitudinal modes (DOFs 1, 3 and 5)are decoupled from the lateral modes (DOFs 2, 4 and 6). The key assumption is that thefuselage is slender—that is, the length is much larger than the width and the height of theaircraft. It is also assumed that the the longitudinal velocity is much larger than the verticaland transversal velocities.In order to decouple the rigid-body kinetics (2.41) in longitudinal and lateral modes it

will be assumed that the states v, p, r and φ are negligible in the longitudinal channel whileu,w, q and θ are negligible when considering the lateral channel. This gives two sub-systems:

2.5. DECOUPLING IN LONGITUDINAL AND LATERAL MODES 13

2.5.1 Longitudinal equations

Kinetics:

m[u+Q0w +W0q + g cos(Θ0)θ] = δX

m[w − U0q −Q0u+ g sin(Θ0) cos(Φ0)θ] = δZ (2.56)

Iy q = δM m 0 00 m 00 0 Iy

uwq

+

0 mQ0 mW0

−mQ0 0 −mU0

0 0 0

uwq

+

mg cos(Θ0)mg sin(Θ0) cos(Φ0)

0

θ =

δXδZδM

(2.57)

Kinematics:θ = q (2.58)

2.5.2 Lateral equations

Kinetics:

m[v + U0r −W0p− g cos(Θ0) cos(Φ0)φ] = δY

Ixp− Ixz r + (Iz − Iy)Q0r − IxzQ0p = δL (2.59)

Iz r − Ixzp+ (Iy − Ix)Q0p+ IxzQ0r = δN m 0 00 Ix −Ixz0 −Ixz Iz

vpr

+

0 −mW0 mU0

0 −IxzQ0 (Iz − Iy)Q0

0 (Iy − Ix)Q0 IxzQ0

vpr

+

−mg cos(Θ0) cos(Φ0)00

φ =

δYδLδN

(2.60)

Kinematics: [φ

ψ

]=

[1 tan(Θ0)0 1/ cos(Θ0)

] [pr

](2.61)


Figure 2.3: Control inputs for conventional aircraft. Notice that the two ailerons can becontrolled by using one control input: δA = 1/2(δAL + δAR).

2.6 Aerodynamic Forces and Moments

In the forthcoming sections, the following abbreviations and notation will be used to describethe aerodynamic coeffi cients:

Xindex = ∂X∂ index Lindex = ∂L

∂ index

Yindex = ∂Y∂ index Mindex = ∂M

∂ index

Zindex = ∂Z∂ index Nindex = ∂N

∂ index

In order to illustrate how control surfaces influence the aircraft, an aircraft equipped withthe following control inputs will be considered (se Figure 2.3):

2.6. AERODYNAMIC FORCES AND MOMENTS 15

δT Thrust Jet/propeller

δE ElevatorControl surfaces on the rear of the aircraft used for pitch andaltitude control

δA AileronHinged control surfaces attached to the trailing edge of the wing usedfor roll/bank control

δF FlapsHinged surfaces on the trailing edge of the wings used for brakingand bank-to-turn

δR Rudder Vertical control surface at the rear of the aircraft used for turning

Linear theory will be assumed in order to reduce the number of aerodynamic coeffi cients.Control inputs and aerodynamic forces and moments are written as:

τ = −MF ν −NFν +Bu (2.62)

where −MF is aerodynamic added mass, −NF ia aerodynamic damping and B is a matrixdescribing the actuator configuration including the force coeffi cients. The actuator dynamicsis modeled by a 1st-order system:

u = T−1(uc − u) (2.63)

where uc is commanded input, u is the actual control input produced by the actuators andT = diag{T1, T2, ..., Tr} is a diagonal matrix of positive time constants. Substitution of (2.62)into the model (2.42) gives:

(MRB +MF )ν + (NRB +NF

)ν +Gη = Bu

m (2.64)

Mν +Nν +Gη = Bu

The matricesM and N are defined asM = MRB +MF and N = NRB +NF. The linearized

kinematics takes the following form:

η = Jνν + Jηη (2.65)

The resulting state-space models are:

Linear state-space model with actuator dynamics ηνu

=

Jη Jν 0−M−1G −M−1N M−1B

0 0 −T−1

ηνu

+

00T−1

uc (2.66)

Linear state-space model neglecting the actuator dynamics[ην

]=

[Jη Jν

−M−1G −M−1N

] [ην

]+

[0

M−1B

]u (2.67)


2.6.1 Longitudinal aerodynamic forces and moments

McLean [7] expresses the longitudinal forces and moments as:

dXdZdM

=

Xu Xw Xq

Zu Zw ZqMu Mw Mq

uwq

+

Xu Xw Xq

Zu Zw ZqMq Mw Mq

uwq

+

XδT XδE XδF

ZδT

ZδE ZδFMδT MδE MδF

δTδEδF

(2.68)

which corresponds to the matrices −MF ,−NF and B in (2.62). If the aircraft cruise speedU0 = constant, then δT = 0. Altitude can be controlled by using the elevators δE. Flaps δFcan be used to reduce the speed during landing. The flaps can also be used to turn harderfor instance by moving one flap while the other is kept at the zero position. This is commonin bank-to-turn maneuvers. For conventional aircraft the following aerodynamic coeffi cientscan be neglected:

Xu, Xq, Xw, XδE , Zu, Zw,Mu (2.69)

Hence, the model for altitude control reduces to: dXdZdM

=

0 0 Xq

0 0 Zq0 Mw Mq

uwq

+

Xu Xw 0Zu Zw ZqMq Mw Mq

uwq

+

XδE

ZδEMδE

δE (2.70)

If the actuator dynamics is important, aerodynamic coeffi cients such as XδT, XδE

, ... mustbe included in the model.

2.6.2 Lateral aerodynamic forces and moments

The lateral model takes the form [7]:

dYdLdN

=

Yv Yp YrLv Lp LrNv Np Nr

vpr

+

Yv Yp YrLv Lp LrNv Np Nr

vpr

+

YδA YδRLδA LδRNδA NδR

[ δAδR

](2.71)

which corresponds to the matrices −MF ,−NF and B in (2.62). For conventional aircraftthe following aerodynamic coeffi cients can be neglected:

Yv, Yp, Yp, Yr, Yr, YδA,Lv, Lr, Nv, Nr (2.72)

This gives:

dYdLdN

=

0 0 00 Lp 00 Np 0

vpr

+

Yv 0 0Lv Lp LrNv Np Nr

vpr

+

0 YδRLδA

LδRNδA NδR

[ δAδR

](2.73)

2.7. STANDARD AIRCRAFT MANEUVERS 17

2.7 Standard Aircraft Maneuvers

The nominal values depends on the aircraft maneuver. For instance:

1. Straight flight: Θ0 = Q0 = R0 = 0

2. Symmetric flight: Ψ0 = V0 = 0

3. Flying with wings level: Φ0 = P0 = 0

Constant angular rate maneuvers can be classified according to:

1. Steady turn: R0 = constant

2. Steady pitching flight: Q0 = constant

3. Steady rolling/spinning flight: P0 = constant

2.7.1 Dynamic equation for coordinated turn (bank-to-turn)

A frequently used maneuver is coordinated turn where the acceleration in the y-direction iszero (β = 0), sideslip β = 0 and zero steady-state pitch and roll angles—that is,

β = β = 0 (2.74)

Θ0 = Φ0 = 0 (2.75)

Furthermore it is assumed that VT = U0 = constant. Since β = β = 0 and β = V/U0 itfollows that V = V = 0. This implies that the external forces δY = 0. From (2.28) it is seenthat:

m[V + UR−WP − g cos(Θ) sin(Φ)] = Y (2.76)

⇓m[(U0 + u)(R0 + r)− (W0 + w)(P0 + p)− g cos(θ) sin(φ)] = Y0 (2.77)

Assume that the longitudinal and lateral motions are decoupled—that is, u = w = q = θ = 0.If perturbation theory is applied under the assumption that the 2nd-order terms ur = pw =0,we get:

m(U0R0 + U0r −W0P0 −W0p− g sin(φ)) = Y0 (2.78)

The equilibrium equation (2.35) gives the steady-state condition:

m(U0R0 −W0P0) = Y0 (2.79)

Substitution of (2.79) into (2.78) gives:

m(U0r −W0p− g sin(φ)) = 0 (2.80)

or

r =W0

U0

p+g

U0

sin(φ) (2.81)


The aircraft is often trimmed such that the angle of attack α0 = W0/U0 = 0. This impliesthat the yaw rate can be expressed as:

r =g

U0

sin(φ)φ small≈ g

U0

φ (2.82)

which is a very important result since it states that a roll angle angle φ different from zerowill induce a yaw rate r which again turns the aircraft (bank-to-turn). With other words, wecan use a moment in roll, for instance generated by the ailerons, to turn the aircraft. Theyaw angle is given by:

ψ = r (2.83)

An alternative method is of course to turn the aircraft by using the rear rudder to generatea yaw moment. The bank-to-turn principle is used in many missile control systems since itimproves maneuverability, in particular in combination with a rudder controlled system.

Example 2 (Augmented turning model using rear rudders)Turning autopilots using the rudder δR as control input is based on the lateral state-spacemodel. The differential equation for ψ is augmented on the lateral model as shown below:

vpr

φ

ψ

=

a11 a12 a13 a14 0a21 a22 a23 0 0a31 a32 a33 0 00 1 0 0 00 0 1 0 0

vprφψ

+

b11

b21

b31

00

δR (2.84)

�

Example 3 (Augmented bank-to-turn model using ailerons)Bank-to-turn autopilots are designed using the lateral state-space model with ailerons ascontrol inputs, for instance δA = 1/2(δAL + δAR). This is done by augmenting the bank-to-turn equation (2.82) to the state-space model according to:

vpr

φ

ψ

=

a11 a12 a13 a14 0a21 a22 a23 0 0a31 a32 a33 0 00 1 0 0 00 0 0 g

U00

vprφψ

+

b11

b21

b31

00

δA (2.85)

�

2.7.2 Dynamic equation for altitude control

Aircraft altitude control systems are designed by considering the equation for the the verticalacceleration in the center of gravity expressed in NED coordinates; see (2.28). This gives:

azCG = W + V P −QU − g cos(Θ) cos(Φ) (2.86)

2.7. STANDARD AIRCRAFT MANEUVERS 19

If the acceleration is perturbed according to azCG = az0 + δaz and we assume that W0 = 0,the following equilibrium condition is obtained:

az0 = V0P0 −Q0U0 − g cos(Θ0) cos(Φ0) (2.87)

Equation (2.86) can be perturbed as:

az0 + δaz = w + (V0 + v)(P0 + p)− (Q0 + q)(U0 + u)− g cos(Θ0 + θ) cos(Φ0 + φ) (2.88)

Furthermore, assume that the altitude is changed by symmetric straight-line flight withhorizontal wings such that V0 = Φ0 = Θ0 = P0 = Q0 = 0. This gives:

az0 + δaz = w + vp− q(U0 + u)− g cos(θ) cos(φ) (2.89)

Assume that the 2nd-order terms vp and uq can be neglected and subtract the equilibriumcondition (2.87) from (2.89) such that:

δaz = w − U0q (2.90)

Differentiating the altitude twice with respect to time gives the relationship:

h = −δaz = U0q − w (2.91)

If we integrate this expression under the assumption that h(0) = U0θ(0)−w(0) = 0, we get:

h = U0θ − w (2.92)

The flight path is defined as:γ := θ − α (2.93)

where α = w/U0. This gives the resulting differential equation for altitude control:

h = U0γ (2.94)

Example 4 (Augmented model for altitude control using ailerons)An autopilot model for altitude control based on the longitudinal state-space model withstates u,w(alt. α), q and θ is obtained by augmenting the differential equation for h to thestate-space model according to:

uwq

θ

h

+

a11 a12 a13 a14 0a21 a22 a23 a24 0a31 a32 a33 0 00 0 1 0 00 −1 0 U0 0

uwqθh

+

b11 b12

b21 b22

b31 b32

0 00 0

[δTδE

](2.95)

�


2.8 Aircraft Stability Properties

The aircraft stability properties can be investigated by computing the eigenvalues of thesystem matrix A using:

det(λI−A) = 0 (2.96)

For a matrix A of dimension n× n there is n solutions of λ.

2.8.1 Longitudinal stability analysis

For conventional aircraft the characteristic equation is of 4th order. Moreover,

(λ2 + 2ζphωphλ+ ω2ph)(λ

2 + 2ζspωspλ+ ω2sp) = 0 (2.97)

where the subscripts ph and sp denote the following modi:

• Phugoid mode

• Short-period mode

The phugoid mode is observed as a long period oscillation with little damping. In somecases the Phugoid mode can be unstable such that the oscillations increase with time. ThePhugoid mode is characterized by the natural frequency ωph and relative damping ratio ζph.The short-period mode is a fast mode given by the natural frequency ωsp and relative

damping factor ζsp. The short-period mode is usually well damped.

Classification of eigenvalues

• Conventional aircraft: Conventional aircraft have usually two complex conjugatedpairs of Phugoid and short-period types. For the B-767 model in Chapter 4, theeigenvalues can be computed using damp.m in Matlab:

a = [-0.0168 0.1121 0.0003 -0.5608-0.0164 -0.7771 0.9945 0.0015-0.0417 -3.6595 -0.9544 0

0 0 1.0000 0]

damp(a)Eigenvalue Damping Freq. (rad/sec)-0.0064 + 0.0593i 0.1070 0.0596-0.0064 - 0.0593i 0.1070 0.0596-0.8678 + 1.9061i 0.4143 2.0943-0.8678 - 1.9061i 0.4143 2.0943

From this is is seen that ωph = 0.0596, ζph = 0.1070, ωsp = 2.0943 and ζsp = 0.4143.

2.8. AIRCRAFT STABILITY PROPERTIES 21

Figure 2.4: Aircraft longitudinal eigenvalue configuration plotted in the complex plane.

• Tuck Mode: Supersonic aircraft may have a very large aerodynamic coeffi cient Mu.This implies that the oscillatory Phugoid equation gives two real solutions where oneis positive (unstable) and one is negative (stable). This is referred to as the tuck modesince the phenomenon is observed as a downwards pointing nose (tucking under) forincreasing speed.

• A third oscillatory mode: For fighter aircraft the center of gravity is often locatedbehind the neutral point or the aerodynamic center—that is, the point where the thetrim moment Mww is zero. When this happens, the aerodynamic coeffi cient Mw takesa value such that the roots of the characteristic equation has four real solutions. Whenthe center of gravity is moved backwards one of the roots of the Phugoid and short-period modes become imaginary and they form a new complex conjugated pair. Thisis usually referred to as the 3rd oscillatory mode. The locations of the eigenvalues areillustrated in Figure 2.4.

2.8.2 Lateral stability analysis

The lateral characteristic equation is usually of 5th order:

λ5 + α4λ4 + α3λ

3 + α2λ2 + α1λ+ α0 = 0 (2.98)

⇓λ(λ+ e)(λ+ f)(λ2 + 2ζDωDλ+ ω2

D) = 0 (2.99)

where the term λ in the last equation corresponds to a pure integrator in roll—that is, ψ = r.The term (λ+e) is the aircraft spiral/divergence mode. This is usually a very slow mode.

Spiral/divergence corresponds to horizontally leveled wings followed by roll and a divergingspiral maneuver.


The term (λ+f) describes the subsidiary roll mode while the 2nd-order system is referredto as Dutch roll. This is an oscillatory system with a small relative damping factor ζD. Thenatural frequency in Dutch roll is denoted ωD.If the lateral B-767 Matlab model in Chapter 4 is considered, the Matlab command

damp.m gives:

a = [-0.1245 0.0350 0.0414 -0.9962-15.2138 -2.0587 0.0032 0.6450

0 1.0000 0 0.03571.6447 -0.0447 -0.0022 -0.1416

damp(a)Eigenvalue Damping Freq. (rad/sec)-0.0143 1.0000 0.0143-0.1121 + 1.4996i 0.0745 1.5038-0.1121 - 1.4996i 0.0745 1.5038-2.0863 1.0000 2.0863

In this example we only get four eigenvalues since the pure integrator in yaw is notinclude in the system matrix. It is seen that the spiral mode is given by e = −0.0143 whilethe subsidiary roll mode is given by f = −2.0863. Dutch roll is recognized by ωD = 0.0745and ζD = 1.5038.

2.9 Design of flight control systems

A detailed introduction to design of flight control systems are given by [7], [11], [8] and [12].The control system are based on linear design techniques using linearized models similar tothose discussed in the sections above.

Chapter 3

Satellite Modeling

When stabilizing satellites in geostationary orbits only the attitude of the satellite is ofinterest since the position is given by the Earth’s rotation.

3.1 Attitude Model

3.1.1 Euler’s 2nd Axiom Applied to Satellites

The rigid-body kinetics (2.21) gives:

ICGω + ω × (ICGω) = τ (3.1)

were ω = [p, q, r]> and ICG = I>CG > 0 is the inertia tensor about the center of gravity givenby:

ICG =

Ix −Ixy −Ixz−Ixy Iy −Iyz−Ixz −Iyz Iz

(3.2)

If the Euler angles are used to represent attitude, the kinematics become:

θ = J(θ)ω (3.3)

where θ = [Φ,Θ,Ψ]> and

J(θ) =

1 sin(Φ) tan(Θ) cos(Φ) tan(Θ)0 cos(Φ) − sin(Φ)0 sin(Φ)/ cos(Θ) cos(Φ)/ cos(Θ)

, cos(Θ) 6= 0 (3.4)

The satellite has three controllable moments τ = [τ 1, τ 2, τ 3]> which can be generated usingdifferent actuators.

3.1.2 Skew-symmetric representation of the satellite model

The dynamic model (3.1) can also be written:

ICGω − (ICGω)× ω = τ (3.5)

23

24 CHAPTER 3. SATELLITE MODELING

where we have used the fact that a × b = −b × a. Furthermore, it is possible to find amatrix S(ω) such that S(ω) = −S>(ω) is skew-symmetric. With other words:

ICGω + S(ω)ω = τ (3.6)

The matrix S(ω) must satisfy the condition:

S(ω)ω ≡ −(ICGω)× ω (3.7)

A matrix satisfying this condition is:

S(ω) =

0 −Iyzq − Ixzp+ Izr Iyzr + Ixyp− IyqIyzq + Ixzp− Izr 0 −Ixzr − Ixyq + Ixp−Iyzr − Ixyp+ Iyq Ixzr + Ixyq − Ixp 0

(3.8)

3.2 Satellite Model Stability Properties

Consider the Lyapunov function candidate:

V =1

2ω>ICGω + h(θ) (3.9)

where h(θ) is a positive definite function depending of the attitude. Differentiation of Vwith respect to time gives:

V = ωT ICGω + θ> ∂h

∂θ(3.10)

= ω>(−S(ω)ω + τ ) + ω>J>(θ)∂h

∂θ(3.11)

Since S(ω) = −S>(ω), it follows that:

ω>S(ω)ω = 0 ∀ω (3.12)

This suggests that the control input τ should be chosen as:

V = ω>(τ + J>(θ)

∂h

∂θ

)≤ 0 (3.13)

One control law satisfying this is:

τ = −Kdω − J>(θ)∂h

∂θ(3.14)

where Kd > 0. This finally gives:

V = −ω>Kdω ≤ 0 (3.15)

and stability and convergence follow from standard Lyapunov techniques.

3.3 Design of Satellite Attitude Control Systems

Design of nonlinear control systems based on the model (3.6) is quite common in the litera-ture. One example is the nonlinear and passive adaptive attitude control system of Slotineand Di Benedetto [10]. An alternative representation is proposed by Fossen [3].

Chapter 4

Matlab Simulation Models

4.1 Boeing-767

The longitudinal and lateral B-767 state-space models are given below. The state vectorsare:

xlang =

u (ft/s)α (deg)q (deg/s)θ (deg)

,xlat =

β (deg)p (deg/s)φ (deg/s)r (deg)

(4.1)

ulang =

[δE (deg)δT (%)

],ulat =

[δA (deg)δR (deg)

](4.2)

Equilibrium point:

Speed VT = 890 ft/s = 980 km/hAltitude h = 35 000 ftMass m = 184 000 lbsMach-number M = 0.8

4.1.1 Longitudinal model

a = [ -0.0168 0.1121 0.0003 -0.5608-0.0164 -0.7771 0.9945 0.0015-0.0417 -3.6595 -0.9544 0

0 0 1.0000 0];

b = [ -0.0243 0.0519-0.0634 -0.0005-3.6942 0.0243

0 0 ];

25

26 CHAPTER 4. MATLAB SIMULATION MODELS

Eigenvalues:

lam = [-0.8678 + 1.9061i-0.8678 - 1.9061i-0.0064 + 0.0593i-0.0064 - 0.0593i];

4.1.2 Lateral model

a = [-0.1245 0.0350 0.0414 -0.9962-15.2138 -2.0587 0.0032 0.6458

0 1.0000 0 0.03571.6447 -0.0447 -0.0022 -0.1416];

b = [-0.0049 0.0237-4.0379 0.9613

0 0-0.0568 -1.2168];

Eigenvalues:

lam = [-0.1121 + 1.4996i-0.1121 - 1.4996i-2.0863-0.0143];

4.2 F-16 Fighter

The lateral model of the F-16 fighter aircraft is based on [11], pages 370—371. The statevectors are:

xlat =

β (ft/s)φ (ft/s)p (rad/s)r (rad)δA (rad)δR (rad)rw (rad)

,ulat =

[uA (rad)uR (rad)

],ylat =

rw (deg)p (deg/s)β (deg)φ (deg)

(4.3)

Equilibrium point:

Speed VT = 502 ft/s = 552 km/hMach-number M = 0.45

4.2. F-16 FIGHTER 27

4.2.1 Longitudinal model

a = [-0.3220 0.0640 0.0364 -0.9917 0.0003 0.0008 00 0 1 -0.0037 0 0 0

-30.6492 0 -3.6784 0.6646 -0.7333 0.1315 08.5396 0 -0.0254 -0.4764 -0.0319 -0.0620 0

0 0 0 0 -20.2 0 00 0 0 0 0 -20.2 00 0 0 57.2958 0 0 -1 ];

b = [0 00 00 00 0

20.2 00 20.20 0 ];

c= [0 0 0 57.2958 0 0 -10 0 57.2958 0 0 0 0

57.2958 0 0 0 0 0 00 57.2958 0 0 0 0 0 ];

Eigenvalues:

lam = [-1.0000-0.4224+ 3.0633i-0.4224- 3.0633i-0.0167-3.6152-20.2000-20.2000 ];

Notice that the last two eigenvalues correspond to the actuator states.


Figure 4.1: Schematic drawing of the Bristol F.2B Fighter (McRuer et al 1973).

4.3 F2B Bristol Fighter

The lateral model of the F2B Bristol fighter aircraft is given below [8]. This is a Britishaircraft from World War I (see Figure 4.1). The aircraft model is for β = 0 (coordinatedturn). The state-space vector is:

xlat =

p (deg/s)r (deg/s)φ (deg)ψ (deg)

,ulat = [δA (deg)] (4.4)

where the dynamics for ψ satisfies the bank-to-turn equation:

ψ =g

U0

φ =9.81

138 · 0.3048φ = 0. 233φ (4.5)

Equilibrium point:

Speed VT = 138 ft/s = 151.4 km/hAltitude h = 6 000 ftMach-number M = 0.126

4.3. F2B BRISTOL FIGHTER 29

4.3.1 Lateral model

a = [-7.1700 2.0600 0 0-0.4360 -0.3410 0 01.0000 0 0 0

0 0 0.2330 0];b = [

26.1000-1.6600

00];

Eigenvalues:

lam = [00

-0.4752-7.0358];

Bibliography

[1] Blakelock, J. H. (1991). Aircraft and Missiles (John Wiley & Sons Ltd.)

[2] Etkin, B. and L. D. Reid (1996). Dynamics of Flight: Stability and Control (JohnWiley & Sons Ltd.)

[3] Fossen, T. I. (1993). Comments on ”Hamiltonian Adaptive Control of Spacecraft”,IEEE Transactions on Automatic Control, TAC-38(5):671—672.

[4] Fossen, T. I. (1994). Guidance and Control of Ocean Vehicles (John Wiley & SonsLtd.)

[5] Fossen, T. I. (2011). Handbook of Marine Craft Hydrodynamics and Motion Control.(John Wiley & Sons Ltd.)

[6] Hughes, P. C. (1986). Spacecraft Attitude Dynamics (John Wiley & Sons Ltd.)

[7] McLean, D. (1990). Automatic Flight Control Systems (Prentice Hall Inc.)

[8] McRuer, D., D. Ashkenas og A. I. Graham (1973). Aircraft Dynamics andAutomatic Control. (Princeton University Press, New Jersey 1973).

[9] Nelson R. C. (1998). Flight Stability and Automatic Control (McGraw-Hill Int.)

[10] Slotine, J. J. E. og M. D. Di Benedetto (1990). Hamiltonian Adaptive Controlof Spacecraft, IEEE Transactions on Automatic Control, TAC-35(7):848—852.

[11] Stevens, B. L. og F. L. Lewis (1992). Aircraft Control and Simulation (John Wiley& Sons Ltd.)

[12] Roskam, J. (1999). Airplane Flight Dynamics and Automatic Flight Controls (Dar-corporation)

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