Aircraft Flight Dynamics Robert F. Stengel Lecture15 Transfer Functions and Frequency Response

Transfer Functions and
Frequency Response

Robert Stengel, Aircraft Flight Dynamics

MAE 331, 2012"
•  Frequency domain view of initial
condition response"
•  Response of dynamic systems
to sinusoidal inputs"
•  Transfer functions"
•  Bode plots"
Copyright 2012 by Robert Stengel. All rights reserved. For educational use only.!
/>!
/>!
Laplace Transform of Initial
Condition Response
Laplace Transform of
a Dynamic System "

Δ

x(t ) = F Δx(t ) + G Δu(t) + LΔw(t )
•  System equation!
•  Laplace transform of system equation!
sΔx(s) − Δx(0) = F Δx(s ) + GΔ u(s) + LΔw(s)
dim(Δx) = (n × 1)
dim(Δu ) = (m × 1)
dim(Δw) = (s ×1)
Laplace Transform of
a Dynamic System "
•  Rearrange Laplace transform of dynamic equation!
sΔx(s) − FΔ x(s) = Δx(0) + GΔ u(s) + LΔw(s)

sI − F
[ ]
Δx(s) = Δx(0) + GΔ u(s) + LΔw(s)
Δx(s) = sI − F
[ ]
−1
Δx(0) + G Δu(s) + LΔw(s)
[ ]
Initial !
Condition!
Control/Disturbance !
Input!
4
th
-Order Initial
Condition Response "
•  Longitudinal dynamic model (time domain)!
Δx(s) = sI − F
[ ]
−1
Δx(0)

Δ

V (t)
Δ

γ
(t)
Δ


q(t)
Δ

α
(t)
$
%
&
&
&
&
&
'
(
)
)
)
)
)
=
−D
V
−g −D
q
−D
α
L
V
V

N
0
L
q
V
N
L
α
V
N
M
V
0 M
q
M
α
−
L
V
V
N
0 1 −
L
α
V
N
$
%
&
&

&
&
&
&
&
'
(
)
)
)
)
)
)
)
ΔV(t)
Δ
γ
(t)
Δq(t)
Δ
α
(t)
$
%
&
&
&
&
&
'

(
)
)
)
)
)
,
ΔV(0)
Δ
γ
(0)
Δq(0)
Δ
α
(0)
$
%
&
&
&
&
&
'
(
)
)
)
)
)
given

•  Longitudinal model (frequency domain)!
ΔV(s)
Δ
γ
(s)
Δq(s)
Δ
α
(s)
$
%
&
&
&
&
&
'
(
)
)
)
)
)
= sI − F
Lon
[ ]
−1
ΔV(0)
Δ
γ

(0)
Δq(0)
Δ
α
(0)
$
%
&
&
&
&
&
'
(
)
)
)
)
)
Elements of the Characteristic
Matrix Inverse"
sI − F
Lon
≡ Δ
Lon
(s)
= s
4
+ a
3

s
3
+ a
2
s
2
+ a
1
s + a
0
Adj sI − F
Lon
( )
=
n
V
V
(s) n
γ
V
(s) n
q
V
(s) n
α
V
(s)
n
V
γ

(s) n
γ
γ
(s) n
q
γ
(s) n
α
γ
(s)
n
V
q
(s) n
γ
q
(s) n
q
q
(s) n
α
q
(s)
n
V
α
(s) n
V
α
(s) n

V
α
(s) n
V
α
(s)
$
%
&
&
&
&
&
&
'
(
)
)
)
)
)
)
sI − F
Lon
[ ]
−1
=
Adj sI − F
Lon
( )

sI − F
Lon
=
C
T
s
( )
Δ
Lon
(s)
(4 × 4)
1×1
( )
•  Denominator
is scalar!
•  Numerator is an (n x n) matrix of polynomials!
(sI – F)
–1
Distributes and Shapes the
Effects of Initial Conditions"

sI −F
Lon
[ ]
−1
=
n
V
V
(s) n

γ
V
(s) n
q
V
(s) n
α
V
(s)
n
V
γ
(s) n
γ
γ
(s) n
q
γ
(s) n
α
γ
(s)
n
V
q
(s) n
γ
q
(s) n
q

q
(s) n
α
q
(s)
n
V
α
(s) n
V
α
(s) n
V
α
(s) n
V
α
(s)
$
%
&
&
&
&
&
&
'
(
)
)

)
)
)
)
s
4
+ a
3
s
3
+ a
2
s
2
+ a
1
s + a
0
 A
Lon
s
( )
(4 × 4)
1×1
( )
•  Denominator determines the modes of motion"
•  Numerator distributes each element of the initial
condition to each element of the state!
Δx(s) =
Adj sI −F

Lon
( )
sI − F
Lon
Δx(0) = A
Lon
s
( )
Δx(0) 4 ×1
( )
Relationship of (sI – F)
–1
to
State Transition Matrix, (t,0)"
•  Initial condition response!
Δx(s) = sI − F
[ ]
−1
Δx(0) = A s
( )
Δx(0)
Δx(t ) = Φ t, 0
( )
Δx(0)
Time !
Domain!
Frequency !
Domain!
•  Δx(s) is the Laplace transform of Δx(t)!
Δx(s) = A s

( )
Δx(0) = L Δx(t )
[ ]
= L Φ t,0
( )
Δx(0)
#
$
%
&
= L Φ t,0
( )
#
$
%
&
Δx(0)
Relationship of (sI – F)
–1
to
State Transition Matrix, (t,0)"
sI − F
[ ]
−1
= A s
( )
= L Φ t, 0
( )
#
$

%
&
= Laplace transform of the state transition matrix
•  Therefore,!
Initial Condition Response of a
Single State Element "

A s
( )
 sI − F
[ ]
−1
•  Typical (ij
th
) element of A(s)!
a
ij
(s) =
k
ij
n
ij
(s)
Δ(s)
=
k
ij
s
q
+ b

q−1
s
q−1
+ +b
1
s + b
0
( )
s
n
+ a
n−1
s
n−1
+ + a
1
s + a
0
( )
=
k
ij
s − z
1
( )
ij
s − z
2
( )
ij

s − z
q
( )
ij
s −
λ
1
( )
s −
λ
2
( )
s −
λ
n
( )
Initial Condition Response of a
Single State Element "

p
i
(s) = k
i1
s
q
1
+ +b
0
( )
1

Δx
1
(0)++ k
in
s
q
n
+ +b
0
( )
n
Δx
n
(0)
 k
p
i
s
q
max
+ + b
0
( )
•  All terms have the same denominator polynomial"
•  Terms sum to produce a single numerator polynomial!

Δx
i
(s) = a
i1

s
( )
Δx
1
(0)+ a
i2
s
( )
Δx
1
(0)++ a
in
s
( )
Δx
n
(0) 
p
i
s
( )
Δ s
( )
•  Initial condition response of Δx
i
(s)!
Real, scalar"
Partial Fraction Expansion of the
Initial Condition Response"
•  Scalar response can be expressed with n parts,

each containing a single mode!

Δx
i
(s) =
p
i
s
( )
Δ s
( )
=
d
1
s −
λ
1
( )
+
d
2
s −
λ
2
( )
+
d
n
s −
λ

n
( )
$
%
&
&
'
(
)
)
i
, i =1,n
where, for each i, the (possibly complex) coefficients are
d
j
= s −
λ
j
( )
p
i
s
( )
Δ s
( )
s=
λ
j
, j = 1,n
Partial Fraction Expansion of the

Initial Condition Response"
•  Time response is the inverse Laplace transform!

Δx
i
(t) = L
−1
Δx
i
(s)
[ ]
= L
−1
d
1
s −
λ
1
( )
+
d
2
s −
λ
2
( )
+ 
d
n
s −

λ
n
( )
$
%
&
'
(
)
i
= d
1
e
λ
1
t
+ d
2
e
λ
2
t
+  + d
n
e
λ
n
t
( )
i

, i = 1, n
Each element’s time response contains
every mode of the system (although
some coefficients may be zero)"
Scalar and Matrix
Transfer Functions
Response to a Control Input"
•  Neglect initial condition"
•  State response to control"

sΔx(s) = FΔx(s)+ GΔu(s)+ Δx(0), Δx(0)  0
Δx(s) = sI − F
[ ]
−1
G Δu(s)
•  Output response to control"
Δy(s) = H
x
Δx(s)+H
u
Δu(s)
= H
x
sI − F
[ ]
−1
GΔu(s)+ H
u
Δu(s)
= H

x
sI − F
[ ]
−1
G + H
u
{ }
Δu(s)
Transfer Function Matrix"
•  Frequency-domain effect of all inputs
on all outputs"
•  Assume control effects do not appear
directly in the output: H
u
= 0"
•  Transfer function matrix!

H (s) = H
x
sI − F
[ ]
−1
G  H
x
A s
( )
G
r × n
( )
n × n

( )
n × m
( )
= r × m
( )
First-Order Transfer Function "
y s
( )
u s
( )
= H (s) = h s − f
[ ]
−1
g =
hg
s − f
( )
(n = m = r = 1)
•  Scalar transfer function (= first-order lag)!


x t
( )
= fx t
( )
+ gu t
( )
y t
( )
= hx t

( )
•  Scalar dynamic system!
Second-Order Transfer Function "
H(s) = H
x
A s
( )
G =
h
11
h
12
h
21
h
22
!
"
#
#
$
%
&
&
adj
s − f
11
( )
− f
12

− f
21
s − f
22
!
"
#
#
$
%
&
&
det
s − f
11
( )
− f
12
− f
21
s − f
22
( )
(
)
*
*
+
,
-

-
g
11
g
12
g
21
f
22
!
"
#
#
$
%
&
&
(n = m = r = 2)
•  Second-order transfer function matrix!
r × n
( )
n × n
( )
n × m
( )
= r × m
( )
= 2 × 2
( )


x t
( )
=

x
1
t
( )

x
2
t
( )
!
"
#
#
$
%
&
&
=
f
11
f
12
f
21
f

22
!
"
#
#
$
%
&
&
x
1
t
( )
x
2
t
( )
!
"
#
#
$
%
&
&
+
g
11
g
12

g
21
f
22
!
"
#
#
$
%
&
&
u
1
t
( )
u
2
t
( )
!
"
#
#
$
%
&
&
y t
( )

=
y
1
t
( )
y
2
t
( )
!
"
#
#
$
%
&
&
=
h
11
h
12
h
21
h
22
!
"
#
#

$
%
&
&
x
1
t
( )
x
2
t
( )
!
"
#
#
$
%
&
&
•  Second-order dynamic system!
Longitudinal Transfer
Function Matrix "
•  With H
x
= I, and assuming"
–  Elevator produces only a pitching moment"
–  Throttle
affects only the rate of change of velocity"
–  Flaps

produce only lift!
H
Lon
(s) = H
x
Lon
sI − F
Lon
[ ]
−1
G
Lon
= H
x
Lon
A
Lon
s
( )
G
Lon
=
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
"
#
$
$

$
$
%
&
'
'
'
'
n
V
V
(s) n
γ
V
(s) n
q
V
(s) n
α
V
(s)
n
V
γ
(s) n
γ
γ
(s) n
q
γ

(s) n
α
γ
(s)
n
V
q
(s) n
γ
q
(s) n
q
q
(s) n
α
q
(s)
n
V
α
(s) n
γ
α
(s) n
q
α
(s) n
α
α
(s)

"
#
$
$
$
$
$
$
%
&
'
'
'
'
'
'
0 T
δ
T
0
0 0 L
δ
F
/ V
N
M
δ
E
0 0
0 0 −L

δ
F
/ V
N
"
#
$
$
$
$
$
%
&
'
'
'
'
'
Δ
Lon
s
( )
Longitudinal Transfer
Function Matrix "
H
Lon
(s) =
n
δ
E

V
(s) n
δ
T
V
(s) n
δ
F
V
(s)
n
δ
E
γ
(s) n
δ
T
γ
(s) n
δ
F
γ
(s)
n
δ
E
q
(s) n
δ
T

q
(s) n
δ
F
q
(s)
n
δ
E
α
(s) n
δ
T
α
(s) n
δ
F
α
(s)
$
%
&
&
&
&
&
&
'
(
)

)
)
)
)
)
s
2
+ 2
ζ
P
ω
nP
s +
ω
n
P
2
( )
s
2
+ 2
ζ
SP
ω
n
SP
s +
ω
n
SP

2
( )
•  There are 4 outputs and 3 inputs!
Douglas AD-1 Skyraider!
Longitudinal Transfer
Function Matrix "
ΔV(s)
Δ
γ
(s)
Δq(s)
Δ
α
(s)
$
%
&
&
&
&
&
'
(
)
)
)
)
)
= H
Lon

(s)
Δ
δ
E(s)
Δ
δ
T (s)
Δ
δ
F(s)
$
%
&
&
&
'
(
)
)
)
•  Input-output relationship!
Scalar Transfer Function
from Δu
j
to Δy
i
"
H
ij
(s) =

k
ij
n
ij
(s)
Δ(s)
=
k
ij
s
q
+ b
q−1
s
q−1
+ + b
1
s + b
0
( )
s
n
+ a
n−1
s
n−1
+ + a
1
s + a
0

( )
# zeros = q!
# poles = n"
•  Just one element of the matrix, H(s)"
•  Denominator polynomial contains n roots"
•  Each numerator term is a polynomial with q zeros,
where q varies from term to term and ≤ n – 1
!
=
k
ij
s − z
1
( )
ij
s − z
2
( )
ij
s − z
q
( )
ij
s −
λ
1
( )
s −
λ
2

( )
s −
λ
n
( )
Scalar Frequency Response
Function"
H
ij
(j
ω
) =
k
ij
j
ω
− z
1
( )
ij
j
ω
− z
2
( )
ij
j
ω
− z
q

( )
ij
j
ω
−
λ
1
( )
j
ω
−
λ
2
( )
j
ω
−
λ
n
( )
• Substitute: s = j
ω
!
• Frequency response is a complex function of
input frequency,
ω
"
–  Real and imaginary parts, or"
–  ** Amplitude ratio and phase angle **
!

= a(
ω
)+ jb(
ω
) → AR(
ω
) e
j
φ
(
ω
)
Transfer Function Matrix for
Short-Period Approximation "
•  Transfer Function Matrix (with H
x
= I, H
u
= 0)"

H
SP
(s) = I
2
A
SP
s
( )
G
SP

=
s − M
q
( )
−M
α
− 1−
L
q
V
N
#
$
%
&
'
(
s +
L
α
V
N
( )
)
*
+
+
+
+
,

-
.
.
.
.
-1
M
δ
E
−L
δ
E
V
N
)
*
+
+
+
,
-
.
.
.

Δ

x
SP
=

Δ

q
Δ

α
#
$
%
%
&
'
(
(
≈
M
q
M
α
1 −
L
q
V
N
+
,
-
.
/
0

−
L
α
V
N
#
$
%
%
%
&
'
(
(
(
Δq
Δ
α
#
$
%
%
&
'
(
(
+
M
δ
E

−L
δ
E
V
N
#
$
%
%
%
&
'
(
(
(
Δ
δ
E
•  Dynamic Equation"
Transfer Function Matrix for
Short-Period Approximation "
•  Transfer Function Matrix (with H
x
= I, H
u
= 0)"
H
SP
(s ) = A
SP

s
( )
G
SP
=
s +
L
α
V
N
( )
M
α
1−
L
q
V
N
#
$
%
&
'
(
s − M
q
( )
)
*
+

+
+
+
,
-
.
.
.
.
M
δ
E
−L
δ
E
V
N
)
*
+
+
+
,
-
.
.
.
s − M
q
( )

s +
L
α
V
N
( )
− M
α
1−
L
q
V
N
#
$
%
&
'
(
Transfer Function Matrix for
Short-Period Approximation "
H
SP
(s) =
M
δ
E
s +
L
α

V
N
( )
−
L
δ
E
M
α
V
N
$
%
&
'
(
)
M
δ
E
1−
L
q
V
N
*
+
,
-
.

/
−
L
δ
E
V
N
( )
s − M
q
( )
$
%
&
'
(
)
$
%
&
&
&
&
&
'
(
)
)
)
)

)
s
2
+ −M
q
+
L
α
V
N
( )
s − M
α
1−
L
q
V
N
*
+
,
-
.
/
+ M
q
L
α
V
N

$
%
&
'
(
)
=
M
δ
E
s +
L
α
V
N
−
L
δ
E
M
α
V
N
M
δ
E
( )
$
%
&

'
(
)
−
L
δ
E
V
N
( )
s +
V
N
M
δ
E
L
δ
E
1−
L
q
V
N
*
+
,
-
.
/

− M
q
$
%
&
'
(
)
0
1
2
3
4
5
6
3
$
%
&
&
&
&
&
'
(
)
)
)
)
)

Δ
SP
s
( )
Transfer Function Matrix for
Short-Period Approximation "

H
SP
(s) 
k
q
n
δ
E
q
(s)
k
α
n
δ
E
α
(s)
#
$
%
%
&
'

(
(
s
2
+ 2
ζ
SP
ω
n
SP
s +
ω
n
SP
2
=
Δq(s)
Δ
δ
E(s)
Δ
α
(s)
Δ
δ
E(s)
#
$
%
%

%
%
%
&
'
(
(
(
(
(
dim = 2 x 1"
Scalar Transfer
Functions for Short-
Period Approximation "
Δq(s)
Δ
δ
E(s)
=
M
δ
E
s +
L
α
V
N
−
L
δ

E
M
α
V
N
M
δ
E
( )
%
&
'
(
)
*
s
2
+ −M
q
+
L
α
V
N
( )
s − M
α
1 −
L
q

V
N
+
,
-
.
/
0
+ M
q
L
α
V
N
%
&
'
(
)
*
=
k
q
s − z
q
( )
s
2
+ 2
ζ

SP
ω
n
SP
s +
ω
n
SP
2
Δq(s)
Δ
α
(s)
#
$
%
%
&
'
(
(
=
Δq(s)
Δ
δ
E(s)
Δ
α
(s)
Δ

δ
E(s)
#
$
%
%
%
%
&
'
(
(
(
(
Δ
δ
E(s)
Δ
α
(s)
Δ
δ
E(s)
=
−
L
δ
E
V
N

( )
s +
V
N
M
δ
E
L
δ
E
1 −
L
q
V
N
%
&
'
(
)
*
− M
q
+
,
-
.
/
0
1

2
3
4
3
5
6
3
7
3
s
2
+ −M
q
+
L
α
V
N
( )
s − M
α
1 −
L
q
V
N
%
&
'
(

)
*
+ M
q
L
α
V
N
+
,
-
.
/
0
=
k
α
s − z
α
( )
s
2
+ 2
ζ
SP
ω
n
SP
s +
ω

n
SP
2
•  Pitch Rate Transfer Function"
•  Angle of Attack Transfer Function"
Short-Period Frequency Response (s = j

)
Expressed as Amplitude Ratio and Phase
Angle"
Pitch-rate frequency response"
Angle-of-attack frequency
response"
Δq( j
ω
)
Δ
δ
E( j
ω
)
=
k
q
j
ω
− z
q
( )
−

ω
2
+ 2
ζ
SP
ω
n
SP
j
ω
+
ω
n
SP
2
= AR
q
(
ω
) e
j
φ
q
(
ω
)
Δ
α
( j
ω

)
Δ
δ
E( j
ω
)
=
k
α
j
ω
− z
α
( )
−
ω
2
+ 2
ζ
SP
ω
n
SP
j
ω
+
ω
n
SP
2

= AR
α
(
ω
) e
j
φ
α
(
ω
)
Bode Plot
(Frequency Response of a
Scalar Transfer Function)

Angle and Rate
Response of a
DC Motor over
Wide Input-
Frequency
Range "
!  Long-term response
of a dynamic system
to sinusoidal inputs
over a range of
frequencies"
!  Determine
experimentally or "
!  from the Bode plot
of the dynamic

system!
Very low damping!
Moderate damping!
High damping!
Bode Plot Portrays Response
to Sinusoidal Control Input"
•  Express amplitude ratio in decibels
"
€
AR(dB) = 20log
10
AR original units
( )
[ ]
20 dB = factor of 10!
€
Δq( j
ω
)
Δ
δ
E( j
ω
)
=
k
q
j
ω
− z

q
( )
−
ω
2
+ 2
ζ
SP
ω
n
SP
j
ω
+
ω
n
SP
2
= AR
q
(
ω
) e
j
φ
q
(
ω
)
•  Plot AR(dB) vs. log

10
(
ω
input
)"
•  Plot phase angle,
ϕ
(deg) vs. log
10
(
ω
input
)"
•  Asymptotes form “skeleton” of
response amplitude ratio"
•  Asymptotes change at poles and zeros"
Products in original units
are sums in decibels!
# zeros = 1!
# poles = 2"
Constant Gain Bode Plot"
€
H( j
ω
) = 1
€
H( j
ω
) = 10
€

H( j
ω
) = 100
y t
( )
= hu t
( )
Slope = 0dB / dec, Amplitude Ratio = constant
Phase Angle = 0°
Integrator Bode Plot"
€
H( j
ω
) =
1
j
ω
€
H( j
ω
) =
10
j
ω
y t
( )
= h u t
( )
dt
0

t
∫
Slope = −20dB / dec
Phase Angle = −90°
Differentiator Bode Plot"
H ( j
ω
) = j
ω
€
H( j
ω
) =10 j
ω
y t
( )
= h
du t
( )
dt
Slope = +20dB / dec
Phase Angle = +90°
Sign Change"
H ( j
ω
) = −
h
j
ω
y t

( )
= −h u t
( )
dt
0
t
∫
H ( j
ω
) = − j
ω
y t
( )
= −h
du t
( )
dt
Slope = −20dB / dec
Phase Angle = +90°
Slope = +20dB / dec
Phase Angle = −90°
Integral!
Derivative!
Multiple Integrators and
Differentiators"
H ( j
ω
) = h j
ω
( )

2
y t
( )
= h
d
2
u t
( )
dt
2
H ( j
ω
) =
h
j
ω
( )
2
y t
( )
= h u t
( )
dt
2
0
t
∫
0
t
∫

Slope = −40dB / dec
Phase Angle = −180°
Slope = +40dB / dec
Phase Angle = +180°
Double Integral!
Double Derivative!
Constant Gain, Integrator, and
Differentiator Bode Plots Form the
Asymptotes for More Complex
Transfer Functions"
+20 "
dB/dec"
+40 "
dB/dec"
0 "
dB/dec"
+20 "
dB/dec"
–20 "
dB/dec"
Bode Plots of First-Order Lags"
H
red
( j
ω
) =
10
j
ω
+10

( )
H
blue
( j
ω
) =
100
j
ω
+10
( )
H
green
( j
ω
) =
100
j
ω
+100
( )
Bode Plot Asymptotes, Departures,
and Phase Angles for First-Order Lags"
•  General shape of amplitude
ratio governed by
asymptotes"
•  Slope of asymptotes
changes by multiples of ±20
dB/dec at poles or zeros"
•  Actual AR departs from

asymptotes"
•  Phase angle of a real,
negative pole"
–  When
ω
= 0,
ϕ
= 0°"
–  When
ω
=
λ
,
ϕ
=–45°"
–  When ω -> ∞,
ϕ
-> –90°"
•  AR asymptotes of a real pole"
–  When
ω
= 0, slope = 0 dB/
dec"
–  When
ω
≥
λ
, slope = –20 dB/
dec"
Bode Plots of Second-Order Lags

(No Zeros)"
Effect of
Damping Ratio!
H
green
( j
ω
) =
10
2
j
ω
( )
2
+ 2 0.1
( )
10
( )
j
ω
( )
+10
2
H
blue
( j
ω
) =
10
2

j
ω
( )
2
+ 2 0.4
( )
10
( )
j
ω
( )
+10
2
H
red
( j
ω
) =
10
2
j
ω
( )
2
+ 2 0.707
( )
10
( )
j
ω

( )
+10
2
Bode Plots of Second-Order Lags
(No Zeros)"
H
red
( j
ω
) =
10
2
j
ω
( )
2
+ 2 0.1
( )
10
( )
j
ω
( )
+10
2
Effects of Gain and
Natural Frequency!
H
green
( j

ω
) =
10
3
j
ω
( )
2
+ 2 0.1
( )
10
( )
j
ω
( )
+10
2
H
blue
( j
ω
) =
100
2
j
ω
( )
2
+ 2 0.1
( )

100
( )
j
ω
( )
+100
2
Amplitude Ratio Asymptotes and Departures of
Second-Order Bode Plots (No Zeros)"
•  AR asymptotes of a
pair of complex poles"
–  When
ω
= 0, slope
= 0 dB/dec"
–  When
ω
≥
ω
n
,
slope = –40 dB/
dec"
•  Height of resonant
peak depends on
damping ratio"
Phase Angles of Second-Order
Bode Plots (No Zeros)"
•  Phase angle of a pair
of complex negative

poles"
–  When
ω
= 0,
ϕ
= 0°"
–  When
ω
=
ω
n
,
ϕ
=–
90°"
–  When
ω
-> ∞,
ϕ
-> –
180°"
•  Abruptness of phase
shift depends on
damping ratio"
MATLAB Bode Plot with asymp.m"
/> />2
nd
-Order Pitch Rate Frequency Response"
asymp.m"bode.m"
Frequency

Response AR
Departures in the
Vicinity of Poles"
•  Difference between
actual amplitude ratio
(dB) and asymptote =
departure (dB)"
•  Results for multiple roots
are additive"
•  from McRuer, Ashkenas,
and Graham, Aircraft
Dynamics and Automatic
Control, Princeton
University Press, 1973"
•  Zero departures have
opposite sign"
First- and Second-Order Departures
from Amplitude Ratio Asymptotes"
First- and Second-
Order Phase Angles"
Phase Angle
Variations in the
Vicinity of Poles"
•  Results for multiple roots
are additive"
•  from McRuer, Ashkenas,
and Graham, Aircraft
Dynamics and Automatic
Control, Princeton
University Press, 1973"

•  LHP zero variations have
opposite sign"
•  RHP zeros have same
sign"
Next Time:
Control Devices and Systems

Reading
Flight Dynamics, 214-234
Virtual Textbook, Part 16
Supplementary
Material
Bode Plots of 1
st
- and 2
nd
-Order Lags"
€
H
red
( j
ω
) =
10
j
ω
+ 10
( )
H
blue

( j
ω
) =
100
2
j
ω
( )
2
+ 2 0.1
( )
100
( )
j
ω
( )
+ 100
2
Bode Plots of 3
rd
-Order Lags"
€
H
blue
( j
ω
) =
10
j
ω

+ 10
( )
#
$
%
&
'
(
100
2
j
ω
( )
2
+ 2 0.1
( )
100
( )
j
ω
( )
+ 100
2
#
$
%
%
&
'
(

(
H
green
( j
ω
) =
10
2
j
ω
( )
2
+ 2 0.1
( )
10
( )
j
ω
( )
+ 10
2
#
$
%
%
&
'
(
(
100

j
ω
+ 100
( )
#
$
%
&
'
(
Bode Plot of a 4
th
-Order System with
No Zeros"
H ( j
ω
) =
1
2
j
ω
( )
2
+ 2 0.05
( )
1
( )
j
ω
( )

+ 1
2
"
#
$
$
%
&
'
'
100
2
j
ω
( )
2
+ 2 0.1
( )
100
( )
j
ω
( )
+ 100
2
"
#
$
$
%

&
'
'
•  Resonant peaks and
large phase shifts at
each natural frequency"
•  Additive AR slope shifts
at each natural
frequency"
# zeros = 0!
# poles = 4"
Left-Half-Plane Transfer Function Zero"
H ( j
ω
) = j
ω
+ 10
( )
•  Zeros are numerator singularities "
H (j
ω
) =
k j
ω
− z
1
( )
j
ω
− z

2
( )

j
ω
−
λ
1
( )
j
ω
−
λ
2
( )
j
ω
−
λ
n
( )
• Single zero in left half
plane"
• Introduces a +20 dB/
dec slope"
• Produces phase lead
in vicinity of zero"
Right-Half-Plane Transfer Function Zero"
H ( j
ω

) = − j
ω
− 10
( )
• Single zero in right half
plane"
• Introduces a +20 dB/dec
slope"
• Produces phase lag in
vicinity of zero"
Second-Order Transfer Function Zero"
€
H( j
ω
) = j
ω
− z
( )
j
ω
− z
*
( )
= j
ω
( )
2
+ 2 0.1
( )
100

( )
j
ω
( )
+ 100
2
[ ]
• Complex pair of
zeros produces an
amplitude ratio
notch at its
natural frequency"
4
th
-Order Transfer Function with
2
nd
-Order Zero"
€
H( j
ω
) =
j
ω
( )
2
+ 2 0.1
( )
10
( )

j
ω
( )
+ 10
2
[ ]
j
ω
( )
2
+ 2 0.05
( )
1
( )
j
ω
( )
+ 1
2
[ ]
j
ω
( )
2
+ 2 0.1
( )
100
( )
j
ω

( )
+ 100
2
[ ]
Elevator-to-
Normal-Velocity
Frequency
Response"
Δw(s)
Δ
δ
E(s)
=
n
δ
E
w
(s)
Δ
Lon
(s)
≈
M
δ
E
s
2
+ 2
ζω
n

s +
ω
n
2
( )
Approx Ph
s − z
3
( )
s
2
+ 2
ζω
n
s +
ω
n
2
( )
Ph
s
2
+ 2
ζω
n
s +
ω
n
2
( )

SP
0 dB/dec!
+40 dB/dec!
0 dB/dec!
–40 dB/dec!
–20 dB/dec!
• (n – q) = 1"
• Complex zero
almost (but
not quite)
cancels
phugoid
response
"
Short "
Period"
Phugoid"