Transfer Functions and
Frequency Response!
Robert Stengel, Aircraft Flight Dynamics!
MAE 331, 2014"
Learning Objectives!
•!
•!
•!
•!
Frequency domain view of initial condition response"
Response of dynamic systems to sinusoidal inputs"
Transfer functions"
Bode plots"
Reading:!
Flight Dynamics!
342-357!
Airplane Stability and Control!
Chapter 20!
Copyright 2014 by Robert Stengel. All rights reserved. For educational use only.!
http://www.princeton.edu/~stengel/MAE331.html!
http://www.princeton.edu/~stengel/FlightDynamics.html!
1!
2!
Fourier and Laplace
Transforms!
3!
Fourier Transform of a
Scalar Variable"
Transformation from time domain
to frequency domain
!
F [ !x(t)] = !x( j" ) =
$
% !x(t)e
# j" t
dt, " = frequency, rad / s
#$
j! : Imaginary operator, rad/s
!x(t) : real variable
!x( j" ) : complex variable
= a(" )+ jb(" )
A : amplitude
! : phase angle
= A(" )e j# (" )
4!
Fourier Transform of a
Scalar Variable"
!x(t)
!x( j" ) = a(" ) + jb(" )
5!
Laplace Transform of
a Scalar Variable"
Laplace transformation from time domain
to
frequency domain
!
#
L [ !x(t)] = !x(s) = $ !x(t)e" st dt
0
s = ! + j"
= Laplace (complex) operator, rad/s
!x(t) : real variable
!x(s) : complex variable
= a(s)+ jb(s)
= A(s)e j" (s )
6!
Laplace Transformation is a
Linear Operation"
Sum of Laplace transforms!
L [ !x1 (t)+ !x2 (t)] = L [ !x1 (t)] + L [ !x2 (t)] = !x1 (s)+ !x2 (s)
Multiplication by a constant!
L [ a!x(t)] = aL [ !x(t)] = a!x(s)
7!
Laplace Transforms of
Vectors and Matrices"
Laplace transform of a vector variable!
" !x1 (s) %
'
$
L [ !x(t)] = !x(s) = $ !x2 (s) '
$
... '&
#
Laplace transform of a matrix variable!
! f11 (s)
#
L [ F(t)] = F(s) = # f21 (s)
# ...
"
f12 (s) ... $
&
f22 (s) ... &
...
... &%
Laplace transform of a time-derivative!
L [ !!x(t)] = s!x(s) " !x(0)
8!
Laplace Transform of
a Dynamic System"
System equation!
!!x(t) = F !x(t) + G !u(t) + L!w(t)
dim(!x) = (n " 1)
dim(!u) = (m " 1)
dim(!w) = (s " 1)
Laplace transform of system equation!
s!x(s) " !x(0) = F !x(s) + G! u(s) + L!w(s)
9!
Laplace Transform of
a Dynamic System"
Rearrange Laplace transform of dynamic equation!
F to left, I.C. to right!
s!x(s) " F! x(s) = !x(0)+ G! u(s)+ L!w(s)
Combine terms!
[ sI ! F] "x(s) = "x(0)+ G" u(s)+ L"w(s)
Multiply both sides by inverse of (sI – F)!
!x(s) = [ sI " F]
"1
[!x(0)+ G !u(s)+ L!w(s)]
10!
Matrix Inverse"
Forward"
Inverse"
y = Ax; x = A !1y
[A]
!1
=
Adj( A ) Adj( A )
=
=
A
det A
dim(x) = dim(y) = (n ! 1)
dim(A) = (n ! n)
(n " n)
(1 " 1)
C
; C = matrix of cofactors
det A
T
Cofactors are signed
minors of A"
ijth minor of A is the
determinant of A with
the ith row and jth
column removed"
Numerator is a square matrix of cofactor transposes"
Denominator is a scalar"
11!
Matrix Inverse Examples"
dim(A) = (1 ! 1)
A = a; A !1 =
! a11 a12
A=#
#" a21 a22
T
! a22 'a21 $
! a22 'a12 $
&
&
#
#
'a12 a11 &
'a21 a11 &
$
#
#
%
%
"
"
& ; A '1 =
=
a11a22 ' a12 a21
a11a22 ' a12 a21
&%
T
dim(A) = (3 ! 3)
! a11 a12
#
A = # a21 a22
# a
a
" 31 32
1
a
dim(A) = (2 ! 2)
a13
a23
a33
! (a a ' a a ) ' (a a ' a a ) (a a ' a a ) $
22 33
23 32
21 33
23 31
21 32
22 31
&
#
# ' ( a12 a33 ' a13a32 ) ( a11a33 ' a13a31 ) ' ( a11a32 ' a12 a31 ) &
&
#
$
# ( a12 a23 ' a13a22 ) ' ( a11a23 ' a13a21 ) ( a11a22 ' a12 a21 ) &
&
%
"
'1
&; A =
a
a
a
+
a
a
a
+
a
a
a
'
a
a
a
'
a
a
a
'
a
a
a
11 22 33
12 23 31
13 21 32
13 22 31
12 21 33
11 23 32
&
%
! (a a ' a a ) ' (a a ' a a ) (a a ' a a ) $
22 33
23 32
12 33
13 32
12 23
13 22
&
#
# ' ( a21a33 ' a23a31 ) ( a11a33 ' a13a31 ) ' ( a11a23 ' a13a21 ) &
&
#
# ( a21a32 ' a22 a31 ) ' ( a11a32 ' a12 a31 ) ( a11a22 ' a12 a21 ) &
%
= "
a11a22 a33 + a12 a23a31 + a13a21a32 ' a13a22 a31 ' a12 a21a33 ' a11a23a32
12!
Matrix Inverse Examples"
A = 5; A !1 =
1
= 0.2
5
! 4 '2 $
#
&
! 1 2 $
1 $
" '3 1 % = ! '2
'1
;
A
A=#
=
&
#
&
'2
" 3 4 %
" 1.5 '0.5 %
! '30 18
4 $
#
&
# 20 '15 5 & ! '3 1.8
! 1 2 3 $
0.4 $
4 '2 %& #
#" 0
#
&
&
'1
A = # 4 6 7 &; A =
= # 2 '1.5 0.5 &
10
#" 8 12 9 &%
#" 0 0.4 '0.2 &%
13!
Characteristic Matrix Inverse"
Characteristic matrix"
(short-period model as example)!
[ sI ! F ]
SP
Inverse of characteristic matrix"
[ sI ! F ]
!1
SP
Adj ( sI ! FSP ) CTSP ( s )
=
=
sI ! FSP
" SP (s)
(2 # 2)
(1# 1)
Denominator is characteristic polynomial, a scalar!
sI ! FSP " # SP (s)
= s 2 + c1s + c0
14!
Numerator of the Characteristic
Matrix Inverse"
Numerator is an (n x n) matrix of polynomials!
# n q (s) n q (s) &
q
"
(
Adj ( sI ! FSP ) = % "
"
% nq (s) n" (s) (
$
'
For example,
nqq ( s ) = k ( s ! z )
15!
(sI – F)–1 Distributes and Shapes
the Effects of Initial Conditions"
# n q (s) n q (s)
"
% q
"
% nq (s) n"" (s)
[ sI ! FSP ]!1 = $ s 2 + c s + c
1
0
&
(
(
'
(2 ) 2)
(1) 1)
Denominator determines the modes of motion"
Numerator distributes each element of the initial
condition to each element of the state!
!x(s) =
Adj ( sI " FSP )
!x(0)
sI " FSP
( 2 # 1)
16!
Initial Condition
Response in
Frequency Domain "
!x(s) = [ sI " F] !x(0)
"1
Longitudinal dynamic model (time domain)!
# !q(t)
!
%
%$ !"! (t)
#
Mq
& %
( = % * Lq (' % ,+ 1) V /.
N
$
M"
)
L"
VN
&
( # !q(t) &
(,
(%
!
"
(t)
%
('
($
'
# !q(0) &
%
( given
%$ !" (0) ('
Longitudinal model (frequency domain)!
# !q(s) &
# !q(0) &
)1
%
( = [ sI ) FSP ] %
(
"
(s)
"
(0)
!
!
%$
('
%$
('
17!
Transfer Function Matrix"
•! Frequency-domain effect of all inputs
on all outputs"
•! Assume control effects do not appear
directly in the output: Hu = 0"
•! Transfer function matrix!
H (s) = H x [ sI ! F ] G
!1
( r ! n )( n ! n )( n ! m )
= (r ! m)
18!
1st-Order Transfer Function"
Scalar dynamic system!
x! ( t ) = fx ( t ) + gu ( t )
y ( t ) = hx ( t )
Scalar transfer function (= first-order lag)!
y ( s)
hg
!1
= H (s) = h [ s ! f ] g =
u ( s)
(s ! f )
(n = m = r = 1)
19!
2nd-Order Transfer Function"
Second-order dynamic system!
! x! ( t )
1
x! ( t ) = #
# x!2 ( t )
"
$ ! f
& = # 11
& #" f21
%
! y (t )
1
y (t ) = #
# y2 ( t )
"
f12 $ ! x1 ( t )
&#
f22 & # x2 ( t )
%"
$ ! h
h
& = # 11 12
& #" h21 h22
%
$ ! g $
& + # 1 & u (t )
& #" g2 &%
%
$ ! x1 ( t )
&#
&% #" x2 ( t )
$
& +"
&
%
Second-order transfer function matrix!
H(s) = H x ( sI ! F )
!1
" h
h
( s ) G = $ 11 12
$# h21 h22
( r ! n )( n ! n )( n ! m )
= (r ! m) = (2 ! 2)
" (s ! f )
11
adj $
$ ! f21
%
#
'
( (s ! f )
'&
11
det *
*) ! f21
! f12
( s ! f22 )
! f12
( s ! f22 )
%
'
'
&
+
-,
" g1 %
$
'
$# g2 '&
20!
Numerator and Denominator
of 2nd-Order (sI – F)–1"
" (s ! f )
11
adj $
$ ! f21
#
"
det $
$#
! f12
( s ! f22 )
( s ! f11 )
! f12
% " (s ! f )
22
'=$
' $
f21
& #
f12
( s ! f11 )
%
'
'
&
%
' = ( s ! f11 ) ( s ! f22 ) ! f12 f21
'&
( s ! f22 )
= s 2 ! ( f11 + f22 ) s + ( f11 f22 ! f12 f21 )
! f21
! s 2 + 2() n s + ) n2 ! * ( s )
21!
2nd-Order Transfer Function"
H(s) = H x ( sI ! F )
"
&
&
&
H(s) = #
"
&
&
&
=#
!1
" h11
( s)G = $
h12
$# h21 h22
" (s ! f )
f12
22
$
f21
( s ! f11 )
%$
#
'
s 2 + 2() n s + ) n2
'&
%
'
'" g %
&$ 1 '
$# g2 '&
"# h11 f12 + h12 ( s ! f11 ) $% $ " g1 $
'&
'
"# h21 ( s ! f22 ) + h22 f21 $% "# h21 f12 + h22 ( s ! f11 ) $% ' &# g2 '%
'%
2
2
s + 2() n s + ) n
"# h11 ( s ! f22 ) + h12 f21 $%
"# h11 ( s ! f22 ) + h12 f21 $% g1 + "# h11 f12 + h12 ( s ! f11 ) $% g2 $
'
"# h21 ( s ! f22 ) + h22 f21 $% g1 + "# h21 f12 + h22 ( s ! f11 ) $% g2 '
'%
2
2
s + 2() n s + ) n
22!
2nd-Order Transfer Function"
"
&
&
&
H(s) = #
"# h11 ( s ! f22 ) + h12 f21 $% g1 + "# h11 f12 + h12 ( s ! f11 ) $% g2 $
'
"# h21 ( s ! f22 ) + h22 f21 $% g1 + "# h21 f12 + h22 ( s ! f11 ) $% g2 '
'%
2
2
s + 2() n s + ) n
" k (s ! z ) %
1
$ 1
'
$ k 2 ( s ! z2 ) '
&
! #2
s + 2() n s + ) n2
23!
Transfer Function Matrix for
Short-Period Approximation"
Dynamic Equation"
# !q! ( t )
!!x SP ( t ) = %
% !"! ( t )
$
#
Mq
& %
()% + L
.
( % - 1* q V 0
'
,
N/
$
M"
*
L"
VN
&
( # !q ( t )
(%
( %$ !" ( t )
'
& # M1 E
( + % *L
1E
( %
VN
' %$
&
(
( !1 E ( t )
('
Transfer Function Matrix (with Hx = I, Hu = 0)"
H SP (s) = I 2 ( sI ! F )SP ( s ) G SP
!1
(
)
)
s ! Mq
+
=+ # L
&
+ ! % 1! q V (
N'
+* $
(
!M "
s+
L"
VN
)
,
.
.
.
.-
-1
) M/ E
+
+ !L/ E
VN
+*
24!
,
.
.
.-
Transfer Function Matrix for
Short-Period Approximation"
Transfer Function Matrix (with Hx = I, Hu = 0)"
(
H SP (s) = [ sI ! FLon ] G SP
!1
)
,
)
L"
M
s
+
. ) M/ E
+
"
VN
.+
+
. + !L/ E
+ # Lq &
VN
+ %$ 1! VN (' s ! M q . +*
=*
# L
L
&
s ! M q s + " V ! M " % 1! q V (
$
N
N'
(
)(
(
)
)
25!
Transfer Function Matrix for
Short-Period Approximation"
'
)
)
$
' )
* Lq - L! E
M
#
1#
s
#
M
,
/
(
)
& !E
q ) )
VN .
VN
+
%
( )(
H SP (s) =
$
'
* L
s 2 + #M q + L"
s # & M " ,1# q / + M q L" )
VN
VN .
VN (
+
%
$
&
&
&
&
&%
$
L"
L! E M " '
&% M ! E s + VN #
VN )(
(
(
=
)
( )
)
(
)
$
L
L M
$
'
M! E &s + " V # ! E " V M )
&
N
N
!E (
%
&
&
0 $ VN M ! E * Lq '4
& # L! E
s
+
1#
#
M
1
,
/
q )5
VN 3 & L
VN .
+
&
!E
%
( 63
2
%
7 SP ( s )
(
)
'
)
)
)
)
)
(
26!
,
.
.
.-
Transfer Function Matrix for
Short-Period Approximation"
# k n q (s) &
# +q(s)
(
% q !E
%
% k" n!"E (s) (
+! E(s)
'
$
%
H SP (s) ! 2
=
2
s + 2) SP* nSP s + * nSP % +" (s)
%
dim = 2 x 1"
%$ +! E(s)
&
(
(
(
(
('
27!
Scalar Transfer Functions for
Short-Period Approximation"
Pitch Rate Transfer Function"
(
)
)
L
L M
%
(
M"E 's + # V $ "E # V M *
k q s $ zq
!q(s)
N
N
"E )
&
=
= 2
2
Lq .
!" E(s)
%
+
L# ( s + 21 SP2 nSP s + 2 nSP
L#
2
s + $M q + V s $ ' M # - 1 $ V 0 + M q V *
,
N
N/
N)
&
(
(
)
Angle of Attack Transfer Function"
. 35
Lq (
31 + VN M # E %
2s + '& 1 $ V *) $ M q 0 6
V
N 3
N
k" ( s $ z" )
!" (s)
/ 73
4 , L# E
=
= 2
L
!# E(s)
+
. s + 28 SP9 nSP s + 9 n2SP
%
(
L
L
s 2 + $M q + " V s $ - M " ' 1 $ q V * + M q " V 0
&
N
N)
N/
,
$
(
(
L# E
)
)
28!
Relationship of (sI – F)–1 to
State Transition Matrix, !(t,0)"
Initial condition response!
Time !
Domain!
Frequency !
Domain!
!x(t) = " ( t, 0 ) !x(0)
!x(s) = [ sI " F ] !x(0) =
"1
"x(s) is the Laplace transform of "x(t)!
!x(s) = L [ !x(t)] = L #$ " ( t,0 ) !x(0) %& = L #$ " ( t,0 ) %& !x(0)
29!
Relationship of (sI – F)–1 to
State Transition Matrix, (t,0)"
Therefore,!
[ sI ! F ]!1 = L #$ " (t,0 )%&
= Laplace transform of the state transition matrix
30!
Initial Condition Response of a
Single State Element (Frequency
Domain)"
"1
!x(s) = [ sI " F ] !x(0)
"
$
$
$
$
$
#
" n (s) n (s) ! n (s)
12
1n
$ 11
$ n21 ( s ) n22 ( s ) ! n2n ( s )
$
!
!
!
!x1 ( s ) % $ !
'
!x2 ( s ) ' $# nn1 ( s ) nn2 ( s ) ! nn2 ( s )
'=
! (s)
! '
!xn ( s ) '
&
%
'
'
'"
%
' $ !x1 ( 0 ) '
' $ !x ( 0 ) '
&
2
'
$
!
'
$
$ !xn ( 0 ) '
&
#
31!
Initial Condition Response of a
Single State Element"
Initial condition response of "x2(s)!
!x2 (s) =
n21 ( s )
n (s)
n (s)
!x1 (0) + 22
!x2 (0) +!+ 2n
!xn (0)
! (s)
! (s)
! (s)
"
p2 ( s )
! (s)
32!
Partial Fraction Expansion of the
Initial Condition Response"
Scalar response can be expressed with n parts,
each containing a single mode!
!xi (s) =
pi ( s )
! (s)
$ d1
d2
dn '
=&
+
+!
, i = 1,n
)
s
"
#
s
"
#
s
"
#
( 2 ) ( n )( i
%(
1)
For each i, the coefficients are
(
d j = s ! "j
)
pi ( s )
,
# ( s ) s="
j = 1,n
j
33!
Partial Fraction Expansion of the
Initial Condition Response"
Time response is the inverse Laplace transform!
!xi (t) = L"1 [ !xi (s)]
$ d1
d2
dn '
= L"1 &
+
+!
)
s
"
#
s
"
#
s
"
#
(
)
(
)
(
)
1
2
n
%
(i
= ( d1e#1t + d2 e#2t +!+ dn e#nt )i , i = 1,n
Each element’s time response contains
every mode of the system (although
some coefficients may be negligible)"
34!
Longitudinal Motions Contain
Both Modes"
Phugoid (Long-Period) Mode"
Airspeed!
Flight Path Angle!
Pitch Rate!
Angle of Attack!
35!
Aircraft Modes of Motion!
36!
Characteristic Polynomial
of a LTI Dynamic System"
!x(s) = [ sI " F ]
"1
[ !x(0) + G !u(s) + L!w(s)]
Inverse of characteristic matrix!
[ sI ! F]
!1
=
Adj ( sI ! F )
(n x n)
sI ! F
•! Characteristic polynomial of the system "
–! is a scalar"
–! defines the system’s modes of motion!
sI ! F = det ( sI ! F ) " #(s)
= s n + cn!1s n!1 + ...+ c1s + c0
37!
Eigenvalues (or Roots) of
a Dynamic System"
Characteristic equation of the system!
!(s) = sI " F = s n + cn"1s n"1 + ...+ c1s + c0 = 0
= ( s " #1 ) ( s " #2 ) (...) ( s " #n ) = 0
... where !i are the eigenvalues of F or the
roots of the characteristic polynomial!
38!
Eigenvalues (or Roots) of
a Dynamic System"
Eigenvalues are real or complex numbers
that can be plotted in the s plane!
•! Real root!
!i = " i
•! Complex roots occur in
conjugate pairs!
!i = " i + j# i
! = " i # j$ i
*
i
s Plane!
Positive real part
indicates instability"
39!
Roots of the Aircraft Dynamics
Characteristic Equation"
•!
•!
•!
•!
12th-order system of LTI equations"
12 eigenvalues of the stability matrix, F"
12 roots of the characteristic equation"
Characteristic equation of the system !
!(s) = s12 + c11s11 + ...+ c1s + c0 = 0
= ( s " #1 ) ( s " #2 ) (...) ( s " #12 ) = 0
Up to 12 modes of motion!
In steady, level flight, longitudinal and lateral-directional
LTI perturbation models are uncoupled!
!(s) = $%( s " #1 )!( s " #6 )&'long $%( s " #1 )!( s " #6 )&'lat"dir = 0
40!
Lateral-Directional Modes of
Motion in Steady, Level Flight"
!!x Lat"Dir (t) =
FLat"Dir !x Lat"Dir (t) + G Lat"Dir !u Lat"Dir (t) + L Lat"Dir !w Lat"Dir (t)
Roots of the lateral-directional characteristic equation!
! LD (s) = ( s " #1 ) ( s " #2 ) (...) ( s " #6 ) = 0
= ( s " #CR ) ( s " # Head ) ( s " #S ) ( s " # R ) $%( s " # DR ) ( s " # * DR ) &'
5 modes of motion (typical)!
(
)
! LD (s) = ( s " #CR ) ( s " # Head ) ( s " #S ) ( s " # R ) s 2 + 2$ DR% nDR s + % n2DR = 0
Crossrange"
Heading"
Spiral"
Roll"
Dutch Roll"
41!
Longitudinal Modes of Motion in
Steady, Level Flight"
!!x Lon (t) = FLon !x Lon (t) + G Lon !u Lon (t) + L Lon !w Lon (t)
6 roots of the longitudinal characteristic equation!
! Lon (s) = ( s " #1 ) ( s " #2 ) (...) ( s " #6 ) = 0
= ( s " # R ) ( s " # H ) $%( s " # P ) ( s " # *P ) &' $%( s " #SP ) ( s " # *SP ) &'
Real"
Real"
Complex"
Complex"
Complex"
4 modes of motion (typical)!
(
)(
Complex"
)
! Lon (s) = ( s " # R ) ( s " # H ) s 2 + 2$ P% nP s + % nP 2 s 2 + 2$ SP% nSP s + % n2SP = 0
Range"
Height"
Phugoid"
Short Period"
42!
Complex Conjugate Roots Form a
Single Oscillatory Mode of Motion"
Phugoid Roots"
( s ! "P )( s ! " *P )
= %& s ! (# P + j$ P ) '( %& s ! (# P ! j$ P ) '(
(
= s 2 + 2) P$ nP s + $ n2P
! n : Natural frequency, rad/s
)
! : Damping ratio, -
Short Period Roots"
( s ! "SP )( s ! " *SP )
= %& s ! (# SP + j$ SP ) '( %& s ! (# SP ! j$ SP ) '(
(
= s 2 + 2) SP$ nSP s + $ n2SP
)
43!
Response to a Control Input"
Neglect initial condition"
State response to control"
s!x(s) = F!x(s)+ G!u(s)+ !x(0), !x(0) ! 0
!x(s) = [ sI " F] G !u(s)
"1
Output response to control"
!y(s) = H x !x(s) + H u !u(s)
= H x [ sI " F ] G !u(s) + H u !u(s)
"1
{
}
= H x [ sI " F ] G + H u !u(s)
"1
44!
Longitudinal Transfer
Function Matrix"
•! With Hx = I, and assuming"
–! Elevator produces only a pitching moment"
–! Throttle affects only the rate of change of velocity"
–! Flaps produce only lift!
HLon (s) = H x Lon [ sI ! FLon ] G Lon
!1
"
$
$
$
$#
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
=
"
%$
'$
'$
'$
'& $
$
#
Douglas AD-1 Skyraider!
nVV (s) n(V (s) nqV (s) n)V (s) % " 0
'
$
(
(
(
(
nV (s) n( (s) nq (s) n) (s) ' $ 0
'
$
nVq (s) n(q (s) nqq (s) n)q (s) ' $ M * E
'
nV) (s) n() (s) nq) (s) n)) (s) ' $# 0
&
T* T
+ Lon ( s )
0
0
L* F / VN
0
!L* F / VN
0
%
'
'
'
'
'
&
0
45!
Longitudinal Transfer
Function Matrix"
•! There are 4 outputs and 3 inputs!
$ nV (s) nV (s) nV (s) '
!T
!F
)
& !E
"
"
"
& n! E (s) n! T (s) n! F (s) )
)
& q
q
q
& n! E (s) n! T (s) n! F (s) )
)
& #
#
#
n
(s)
n
(s)
n
(s)
!T
!F
)(
&% ! E
HLon (s) = 2
s + 2* P+ nP s + + n2P s 2 + 2* SP+ nSP s + + n2SP
(
)(
)
46!
Longitudinal Transfer
Function Matrix"
•! Input-output relationship!
$
&
&
&
&
&%
!V (s) '
$ !* E(s) '
)
!" (s) )
)
&
= HLon (s) & !* T (s) )
)
!q(s)
& !* F(s) )
)
(
%
!# (s) )(
47!
Forssman bomber (?)"
Westland P.12 Lysander"
48!
•!
AEA Cygnet II, Alexander
Graham Bell, Glenn
Curtiss, 1909"
•!
Hargrave quadraplane
(model), 1889"
•!
DEquevillery, 1908"
49!
•!
Phillips, 1904"
•!
Wight
Quadraplane,
1916 "
Phillips, 1907"
•!
•!
Pemberton-Billings
Nighthawk, 1916"
•!
•!
Vedo Villi, 1911"
John Septaplane, 1919"
50!
•!
Caproni Ca 60, 1920"
Miraculously, this machine DID fly
the first time in 1921- it reached a
height of 60 feet, collapsed, and
plummeted toward the lake just after
take off, killing both pilots."
Wings derived from
Ca.42 bomber"
51!
•!
Farman 3-engine Jabiru"
•!
Heinkel 5-engine He111Z"
Tarrant 6-engine Tabor, 1919"
•!
•!
Farman 4-engine Jabiru, 1923"
52!
Scalar Transfer Function
from "uj to "y"i
•! Just one element of the matrix, H(s)"
•! Each numerator term is a polynomial with q zeros,
where q varies from term to term and # n – 1!
(
q
q"1
nij (s) kij s + bq"1s + ...+ b1s + b0
H ij (s) =
=
!(s)
( s n + cn"1s n"1 + ...+ c1s + c0 )
)
•! Denominator polynomial contains n roots"
( s ! z1 )ij ( s ! z2 )ij ...( s ! zq )ij
= kij
( s ! "1 )( s ! "2 )...( s ! "n )
# zeros = q!
# poles = n"
53!
Control Response of a Single State
Element"
nij (s)
!yi ( s ) = kij
!u j ( s )
!(s)
54!
Bode Plot!
(Frequency Response of a
Scalar Transfer Function)!
55!
Scalar Frequency Response Function"
Substitute: s = j"!
H ij (j! ) =
kij ( j! " z1 )ij ( j! " z2 )ij ... j! " zq
(
( j! " #1 )( j! " #2 )...( j! " #n )
)
ij
= a(! )+ jb(! ) " AR(! ) e j# (! )
•!Frequency response is a complex function of
input frequency, ""
–! Real and imaginary parts, or"
–! ** Amplitude ratio and phase angle **!
56!
Short-Period Frequency Response (s = j")
Expressed as Amplitude Ratio and Phase
Angle"
Pitch-rate frequency response"
kq ( j" $ zq )
!q( j" )
=
2
!# E( j" ) $" + 2% SP" nSP j" + " n2SP
= ARq (" ) e
j&q (" )
Angle-of-attack frequency
response"
k" ( j# % z" )
!" ( j# )
=
!$ E( j# ) %# 2 + 2& SP# nSP j# + # n2SP
= AR" (# ) e j'" (# )
57!
Bode Plot Portrays Response
to Sinusoidal Control Input"
kq ( j# % zq )
"q( j# )
j' (# )
=
= ARq (# ) e q
2
2
"$E( j# ) %# + 2& SP# n SP j# + # n SP
Express amplitude ratio in
decibels"
AR(dB) =
20 log10 !" AR ( original units ) #$
# zeros = 1!
# poles = 2"
20 dB = factor of 10!
Products in original units
are sums in decibels!
58!
Bode Plot Portrays Response
to Sinusoidal Control Input"
# zeros = 1!
# poles = 2"
Plot AR(dB) vs. log10("input)"
Plot phase angle, #(deg) vs. log10("input)"
Asymptotes form “skeleton” of response amplitude ratio"
59!
Constant Gain Bode Plot"
y ( t ) = hu ( t )
H( j" ) = 1
H( j" ) = 10
H( j" ) = 100
Slope = 0dB / dec, Amplitude Ratio = constant
Phase Angle = 0°
60!
Integrator Bode Plot"
t
y ( t ) = h ! u ( t ) dt
0
H( j" ) =
1
j"
H( j" ) =
10
j"
Slope = !20dB / dec
Phase Angle = !90°
61!
Differentiator Bode Plot"
y (t ) = h
du ( t )
dt
H ( j! ) = j!
H( j" ) = 10 j"
Slope = +20dB / dec
Phase Angle = +90°
62!
Sign Change"
Integral!
t
y (t ) = !h " u (t ) dt
0
H ( j! ) = "
Slope = !20dB / dec
Phase Angle = +90°
h
j!
Derivative!
y (t ) = !h
du (t )
dt
Slope = +20dB / dec
H ( j! ) = " j!
Phase Angle = !90°
63!
Multiple Integrators and
Differentiators"
Double Integral!
t
y (t ) = h !
t
! u (t ) dt
0 0
2
H ( j! ) =
Slope = !40dB / dec
Phase Angle = !180°
h
2
( j! )
Double Derivative!
d 2 u (t )
y (t ) = h
dt 2
H ( j! ) = h ( j! )
2
Slope = +40dB / dec
Phase Angle = +180°
64!
Why Plot Vertical Lines
where " = z and "n?"
AR Asymptotes change at
frequencies corresponding to
poles and zeros"
(
)
kq j" $ zq
!q( j" )
=
2
!# E( j" ) $" + 2% SP" nSP j" + " nSP 2
(
)
When ! = "zq for negative zq ,
(
)
kq j! " zq = kq zq ( " j " 1) = "kq zq ( j + 1) = kq zq e+45°
When ! = ! nSP , " ! n2SP + 2# SP j! n2SP + ! n2SP = j2# SP! n2SP
=
1
j2# SP!
2
nSP
=
"j
1
=
e –90° for positive # SP
2
2
2# SP! nSP 2# SP! nSP
65!
Bode Plots of First-Order Lags"
H red ( j! ) =
10
( j! +10)
H blue ( j! ) =
100
( j! +10)
H green ( j! ) =
100
( j! +100)
66!
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"
•! AR asymptotes of a real pole"
–! When " = 0, slope = 0 dB/
dec"
–! When " % !, slope = –20 dB/
dec"
•! Phase angle of a real,
negative pole"
–! When " = 0, # = 0°"
–! When " = !, # =–45°"
–! When # -> $, # -> –90°"
67!
Bode Plots of Second-Order Lags
(No Zeros)"
H green ( j! ) =
10 2
2
( j! ) + 2 (0.1) (10) ( j! ) +10 2
Effect of
Damping Ratio!
H blue ( j! ) =
10 2
( j! ) + 2 (0.4 ) (10) ( j! ) +10 2
2
H red ( j! ) =
10 2
( j! ) + 2 (0.707) (10) ( j! ) +10 2
2
68!
Bode Plots of Second-Order Lags
(No Zeros)"
H red ( j! ) =
10 2
2
( j! ) + 2 (0.1) (10) ( j! ) +10 2
H green ( j! ) =
10 3
2
( j! ) + 2 (0.1) (10) ( j! ) +10 2
Effects of Gain and
Natural Frequency!
H blue ( j! ) =
100 2
( j! ) + 2 (0.1) (100) ( j! ) +100 2
2
69!
Amplitude Ratio Asymptotes and Departures
of Second-Order Bode Plots (No Zeros)"
•! AR asymptotes of a
pair of complex poles"
–! When " = 0, slope
= 0 dB/dec"
–! When " % "n,
slope = –40 dB/
dec"
•! Height of resonant
peak depends on
damping ratio"
70!
Phase Angles of Second-Order
Bode Plots (No Zeros)"
•! Phase angle of a pair
of complex negative
poles"
–! When " = 0, # = 0°"
–! When " = "n, # =–
90°"
–! When " -> $, # -> –
180°"
•! Abruptness of phase
shift depends on
damping ratio"
71!
MATLAB Bode Plot with asymp.m"
http://www.mathworks.com/matlabcentral/"
http://www.mathworks.com/matlabcentral/fileexchange/10183-bode-plot-with-asymptotes"
2nd-Order Pitch Rate Frequency Response"
bode.m"
asymp.m"
72!
Constant Gain, Integrator, and Differentiator
Bode Plots Form Asymptotes for More
Complex Transfer Functions"
+20 "
dB/dec"
+40 "
dB/dec"
0"
dB/dec"
+20 "
dB/dec"
–20 "
dB/dec"
73!
First- and Second-Order Departures
from Amplitude Ratio Asymptotes"
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"
•! Zero departures have
opposite sign"
McRuer, Ashkenas, and
Graham, Aircraft Dynamics
and Automatic Control,
Princeton University Press,
1973"
74!
First- and SecondOrder Phase Angles"
Phase Angle
Variations in the
Vicinity of Poles"
•! Results for multiple
roots are additive"
•! LHP zero variations
have opposite sign"
•! RHP zeros have same
sign"
McRuer, Ashkenas, and
Graham, Aircraft Dynamics
and Automatic Control"
75!
Curtiss Autocar, 1917"
Stout Skycar, 1931"
Waterman Aerobile, 1935"
ConsolidatedVultee 111,
1940s"
76!
ConvAIRCAR 116 (w/
Crosley auto), 1940s"
Taylor AirCar, 1950s"
Hallock Road Wing , 1957"
77!
Mitzar SkyMaster Pinto, 1970s"
Haynes Skyblazer, concept, 2004"
Lotus Elise Aerocar, concept, 2002"
Aeromobil, 2014"
78!
Terrafugia Transition"
Terrafugia TF-X, concept"
… or, for the same price"
Cessna Skycatcher 162!
Jaguar F Type!
PLUS"
79!
Next Time:!
Root Locus Analysis!
Reading:!
Flight Dynamics!
357-361, 465-467, 488-490,
509-514!
80!
Supplementary
Material
81!
Longitudinal Modes of Motion"
•! Eigenvalues determine the damping and natural
frequencies of the linear systems modes of motion"
!ran : range mode " 0
!hgt : height mode " 0
•! Longitudinal characteristic
equation has 6 eigenvalues"
–! 4 eigenvalues normally
appear as 2 complex pairs"
–! Range and height modes
usually inconsequential"
(#
(#
SP
P
, $ nP ) : phugoid mode
, $ nSP ) : short - period mode
82!
Simplified Longitudinal Modes of
Motion"
Short-Period Mode"
•!
Note change in
time scale"
Airspeed!
Flight Path Angle!
Pitch Rate!
Angle of Attack!
83!
Lateral-Directional Modes of Motion"
•! Lateral-directional characteristic
equation has 6 eigenvalues"
–! 2 eigenvalues normally appear as
a complex pair"
–! Crossrange and heading modes
usually inconsequential"
!cr : crossrange mode " 0
!head : heading mode " 0
!S : spiral mode
! R : roll mode
(#
DR
, $ nDR ) : Dutch roll mode
84!
Simplified Lateral Modes of Motion"
Dutch-Roll Mode"
Sideslip Angle!
Yaw Rate!
85!
Simplified Lateral Modes of Motion"
Roll and Spiral Modes"
Roll Rate!
Roll Angle!
86!
Bode Plots of 1st- and 2nd-Order Lags"
H red ( j" ) =
H blue ( j" ) =
10
( j" + 10)
( j" )
2
100 2
+ 2(0.1)(100)( j" ) + 100 2
87!
Bode Plots of 3rd-Order Lags"
&
# 10 &#
100 2
(
H blue ( j" ) = %
(%
2
2
$ ( j" + 10) '%$ ( j" ) + 2(0.1)(100)( j" ) + 100 ('
#
&# 100 &
10 2
(%
H green ( j" ) = %
(
2
2
%$( j" ) + 2(0.1)(10)( j" ) + 10 ('$( j" + 100) '
88!
Bode Plot of a 4th-Order System
with No Zeros"
%"
%
"
100 2
12
H ( j! ) = $
2
2
2 '$
2 '
$# ( j! ) + 2 ( 0.05 ) (1) ( j! ) + 1 '& $# ( j! ) + 2 ( 0.1) (100 ) ( j! ) + 100 '&
# zeros = 0!
# poles = 4"
•! Resonant peaks and
large phase shifts at
each natural frequency"
•! Additive AR slope shifts
at each natural
frequency"
89!
Left-Half-Plane Transfer Function
Zero"
Zeros are numerator singularities"
H ( j! ) = ( j! + 10 )
H (j! ) =
k ( j! " z1 ) ( j! " z2 ) ...
( j! " #1 )( j! " #2 )...( j! " #n )
•!Single zero in left half
plane"
•!Introduces a +20 dB/
dec slope"
•!Produces phase lead
in vicinity of zero"
90!
Right-Half-Plane Transfer
Function Zero"
H ( j! ) = " ( j! " 10 )
•!Single zero in right half
plane"
•!Introduces a +20 dB/dec
slope"
•!Produces phase lag in
vicinity of zero"
91!
Second-Order Transfer Function Zero"
H( j" ) = ( j" # z)( j" # z* ) =
[( j" )
2
+ 2(0.1)(100)( j" ) + 100 2
]
•!Complex pair of zeros
produces an
amplitude ratio
notch
at its natural
frequency
"
92!
4th-Order Transfer Function
with 2nd-Order Zero"
[( j" ) + 2(0.1)(10)( j" ) + 10 ]
H( j" ) =
[( j" ) + 2(0.05)(1)( j" ) + 1 ][( j" ) + 2(0.1)(100)( j" ) + 100 ]
2
2
2
2
2
2
93!
Elevator-toNormal-Velocity
Frequency
Response"
M " E s 2 + 2$% n s + % n2 Approx Ph ( s & z3 )
!w(s)
n"wE (s)
=
#
!" E(s) ! Lon (s) s 2 + 2$% n s + % n2
s 2 + 2$% n s + % n2 SP
Ph
(
)
(
) (
)
0 dB/dec!
0 dB/dec!
+40 dB/dec!
•!(n – q) = 1"
•!Complex zero
almost (but
not quite)
cancels
phugoid
response"
–40 dB/dec!
Phugoid"
Short "
Period"
–20 dB/dec!
94!