Aircraft Dynamics and Control
Aircraft motion is more complicated
than spacecraft motion due to the
dependence of forces and moments on
attitude and velocity
Typical first investigations of
aircraft dynamics and control focuses
on the longitudinal and lateral
motions of symmetric (left-to-right)
aircraft near straight and level
flight
Motion Equations for a Rigid Aircraft
Fb : v̇
=
Fb : ω̇
=
Fi : ṙ
θ̇
¤
1 £
−ω v +
faero + fthrust + Rbi agrav
m
¤
£
−1
×
I
−ω Iω + gaero + gthrust
×
=
Rib v
=
S−1 (θ)ω
agrav = [0 0 g]T
where
Note that v and ω are expressed in Fb , and r is expressed
in Fi
Usually, aircraft d & c folks write
v
=
[u v w]T ,
ω = [p q r]T
Assuming left-right symmetry (but not up-down symmetry) implies that Ixy = Iyz = 0, but in general Ixz 6= 0
Reference Frames for Aircraft D&C
Inertial Frame, Fi . Typically assume flat Earth, but rotating
Earth is used for some problems. The inertial frame has 3-axis
down, and origin at aircraft mass center
Velocity Frame, Fv . A 3-2 rotation from Fi through the
horizontal flight path angle ξ and the vertical flight path angle
γ: Rvi = R2 (γ)R3 (ξ)
Wind Frame, Fw . A 1 rotation from Fv through bank angle
μ: Rwv = R1 (μ)
Body Frame, Fb . From Fw , a -3 rotation through sideslip angle β, followed by a 2 rotation through α: Rbw = R2 (α)R3 (−β)
Body Frame, Fb . A 3-2-1 rotation from Fi through yaw (ψ),
pitch (θ), and roll (φ) angles: Rbi = R1 (φ)R2 (θ)R3 (ψ)
Reference Frames for Aircraft D&C (2)
Illustration (from R. Stengel’s
Flight Dynamics) shows relationship between pitch, yaw,
angle of attack, sideslip, and
flight path angles
Note that the illustration is
based on a zero-roll and zerobank angles
Roll and bank are similar
to each other: roll is about
the body 1-axis, and bank is
about the velocity vector
Motion Equations for a Rigid Aircraft (2)
Typical kinematics variables are yaw (ψ), pitch (θ), and
roll (φ) angles, a 3-2-1 sequence (could use quaternions or
other Euler angle sequence)
The aerodynamic forces and moments depend on airspeed
(Va ), angle of attack (α), and sideslip angle (β)
Relationship between velocity vector components (u, v, s)
and airflow angles (α, β):
⎤
⎡
⎤
⎡
⎤
⎡
⎤ ⎡ √
2
2
2
u
cαcβ
V
u +v +w
⎣ v ⎦ = V ⎣ sβ ⎦
⎣ α ⎦ = ⎣ tan−1 (w/u) ⎦
w
sαcβ
β
sin−1 (v/V )
Straight and Level Flight
The state vector is
h
iT
x = v T ω T rT θ T
=
[u v w p q r x y z φ θ ψ]
T
For straight and level flight, the aircraft must have a non-zero
angle of attack (to generate lift=weight), and hence non-zero
pitch angle
The straight and level state vector is
x∗
=
T
[Vo cos αo 0 Vo sin αo 0 0 0 xo yo zo 0 θo 0]
The control to achieve this steady motion is
u∗
=
T
[δEo δAo δRo δTo ]
(Elevator angle, Aileron angle, Rudder angle, Thrust)
Linearized Motion
Linearizing about the straight and level flight leads to a linear
system with n = 12, p = 4 (12 states, 4 inputs)
The equations are block-decoupled into two systems: a longitudinal motion system and a lateral motion system, each with six
states and two inputs
Longitudinal Motion:
x
=
[∆u ∆w ∆q ∆x ∆z ∆θ]
u
=
[∆δE ∆δT ]
T
T
Lateral Motion:
T
x
=
[∆v ∆p ∆r ∆y ∆φ ∆ψ]
u
=
[∆δA ∆δR]
T
Longitudinal Motion
Longitudinal Motion: Elevator and Thrust control position, velocity in vertical plane, pitch angle and pitch rate
⎡
A
=
x
=
[∆u ∆w ∆q ∆x ∆z ∆θ]
u
=
[∆δE ∆δT ]
Xu
⎢ Zu
⎢ 1−Zẇ
⎢ M + aZ
u
u
⎢
⎢ cos θ
o
⎢
⎣ − sin θo
0
Xw
Zw
1−Zẇ
Mw + aZw
sin θo
cos θo
0
T
T
Xq − wo
Zq +uo
1−Zẇ
Mq + a(Zq + uo )
0
0
1
0
0
0
0
0
0
Xz
Zz
1−Zẇ
0
0
0
0
−go cos θo
⎤
⎥
⎥
Mθ ago sin θo ⎥
⎥
⎥
b
⎥
⎦
c
−go sin θo
1−Zẇ
0
The Xu , Mu , Mθ etc. terms represent the partial derivatives of the forces and
moments with respect to velocity and angular velocity; a, b, c also depend on
aerodynamic properties
All terms are evaluated at the reference flight condition, typically using extensive
tabulated aerodynamic data for a particular aircraft
Longitudinal Motion (2)
Longitudinal Motion: Elevator and Thrust control position, velocity in vertical plane, pitch angle and pitch rate
x
=
[∆u ∆w ∆q ∆x ∆z ∆θ]
u
=
[∆δE ∆δT ]
⎡
B
=
XδE
T
T
⎢ ZδE
⎢ 1−Zẇ
⎢ MδE + aZδE
⎢
⎢ 0
⎢
⎣ 0
0
XδT
ZδT
1−Zẇ
MδT + aZδT
0
0
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
All terms are evaluated at the reference flight condition, typically using extensive
tabulated aerodynamic data for a particular aircraft
Because the coupling of the position components with the velocity and angular
components is light, a fourth-order model is frequently used by omitting x and
z
Longitudinal Motion (3)
Longitudinal Motion (4th order model): Elevator and Thrust control velocity in vertical plane, pitch angle and pitch rate
A
=
⎡
x
=
[∆u ∆w ∆q ∆θ]
u
=
[∆δE ∆δT ]
Xu
⎢ Zu
⎢ 1−Zẇ
⎣ Mu + aZu
0
Xw
Zw
1−Zẇ
Mw + aZw
0
T
T
Xq − wo
Zq +uo
1−Zẇ
Mq + a(Zq + uo )
1
The B matrix simply loses the 4th and 5th rows
−go cos θo
⎤
⎥
⎥
Mθ ago sin θo ⎦
0
−go sin θo
1−Zẇ
Longitudinal Motion (4)
Business Jet Example of Longitudinal Motion (4th order model): Elevator and Thrust control velocity in vertical plane, pitch angle and pitch rate
⎡
⎤
−0.0121
0.0960 −6.4500 −9.7870
⎢ −0.1160 −1.2770 100.8000 −0.6202 ⎥
⎥
A = ⎢
⎣ 0.0050 −0.0781 −1.2790
0.0000 ⎦
0.0000
0.0000
1.0000
0.0000
⎡
⎤
0.0065 4.6740
⎢ −13.1700 0.0000 ⎥
⎥
B = ⎢
⎣ −9.0690 0.0000 ⎦
0.0000 0.0000
Stability check:
eig(A) ⇒ λ ∈ (−1.2767 ± 2.8101i, −0.0074 ± 0.1256i)
Stable, and block-diagonalizable
Longitudinal Motion (5)
Business Jet Example of Longitudinal Motion (4th order model) continued....
Controllability
Mc
rank Mc
⎡
0.0000 0.0047
0.0572
⎢ −0.0132
0 −0.8973
= ⎢
⎣ −0.0091
0
0.0126
0
0 −0.0091
= 4 ⇒ Controllable
−0.0001
−0.0005
0.0000
0
−0.0795
2.4178
0.0542
0.0126
⎤
−0.0002
0.0031 ⎥
⎥ × 103
0.0000 ⎦
0.0000
Observability
If any state is observable (C is a row of the identity matrix), then the system is observable.
Modal Analysis
Eb
Λb
⎡
−0.0444
0.0145
⎢ 0.9985
0
= ⎢
⎣ 0.0000
0.0278
0.0082 −0.0037
⎡
−1.2767
2.8101
⎢ −2.8101 −1.2767
= ⎢
⎣ 0.0000 −0.0000
−0.0000
0.0000
⎤
−0.9992
0
−0.0375 0.0017 ⎥
⎥
−0.0016 0.0001 ⎦
0.0012 0.0128
−0.0000
−0.0000
−0.0074
−0.1256
⎤
0.0000
0.0000 ⎥
⎥
0.1256 ⎦
−0.0074
Longitudinal Motion (6)
Business Jet Example of Longitudinal Motion (4th order model) continued....
Res ponse to Initial Conditions
To: Out(1 )
0.05
0
-0.05
Amplitud e
To: Out( 2)
1
0.5
0
-0.5
To: Out(3 )
0.01
0
-0.01
-0.02
To: Ou t(4)
10
x 10
-3
Note
Final
Time
5
0
-5
0
0.5
1
1.5
2
2.5
3
3. 5
4
4.5
Time (sec )
Short-period, heavily damped motion resulting from initial conditions in first pair of modes, with
λ = −1.2767 ± 2.8101i
Longitudinal Motion (7)
Business Jet Example of Longitudinal Motion (4th order model) continued....
Res ponse to Initial Conditions
To: Out( 1)
1
0.5
0
-0.5
-1
Amplitud e
To: Out(2 )
0.05
0
-0.05
To: Ou t(3)
2
x 10
-3
1
0
-1
-2
To: Out(4 )
0.01
Note
Final
Time
0
-0.01
-0.02
0
100
200
300
400
500
600
700
800
Time (sec )
Long-period, lightly damped motion resulting from initial conditions in second pair of modes,
with λ = −0.0074 ± 0.1256i. This mode is called the phugoid mode
Lateral Motion
Lateral Motion: Aileron and Rudder control position, velocity perpendicular
to vertical plane, roll and yaw angles and rates
x
=
[∆v ∆p ∆r ∆y ∆φ ∆ψ]
u
=
[∆δA ∆δR]
T
T
As with longitudinal motion, A and B depend on numerous aerodynamic force
and moment terms that depend on the reference state
Typically, there are two zero eigenvalues, one real positive eigenvalue, one real
negative eigenvalue, and a complex conjugate pair
The two zero eigenvalues correspond to crossrange and yaw modes; the unstable
eigenvalue corresponds to a spiral divergence mode, the stable real eigenvalue
corresponds to a roll mode, and the complex pair corresponds to the so-called
Dutch roll mode