Transfer Functions and
Frequency Response!
Robert Stengel, Aircraft Flight Dynamics!
MAE 331, 2016
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!
Copyright 2016 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
Review Questions!
!! How is the initial-condition response of an
LTI system calculated?!
!! What is a “discrete-time model”?!
!! What is a “sampled-data model”?!
!! What is “step response”, and how is it
calculated?!
!! What is the “superposition property” of LTI
systems?!
!! How is the equilibrium response of an LTI
system calculated?!
!! What is a “phase plane plot”?!
!! What is “frequency response”?!
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 frequency 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 frequency) 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)
Cofactors are signed
minors of A
(n " n)
(1 " 1)
ijth minor of A is the
determinant of A with
the ith row and jth
column removed
C
; C = matrix of cofactors
det A
T
Numerator is a square matrix of cofactor transposes
Denominator is a scalar
11
Matrix Inverse Examples
dim(A) = (1 ! 1)
A = a; A !1 =
1
a
dim(A) = (2 ! 2)
T
! a11 a12
A=#
#" a21 a22
! a22 'a21 $
! a22 'a12 $
#
&
#
&
'a
a
'a
a
$
#
&%
#
&%
12
11
21
11
& ; A '1 = "
="
a11a22 ' a12 a21
a11a22 ' a12 a21
&%
12
Matrix Inverse Examples
dim(A) = (3 ! 3)
! a11 a12
#
A = # a21 a22
# a
a
" 31 32
a13 $
&
a23 & ;
a33 &
%
T
! (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 ) &
%
A '1 = "
a11a22 a33 + a12 a23a31 + a13a21a32 ' a13a22 a31 ' a12 a21a33 ' a11a23a32
" (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
13
Characteristic Matrix Inverse
Characteristic matrix
(2nd-order example)
[ sI ! F ]
Inverse of characteristic matrix
[ sI ! F ]
!1
Adj ( sI ! F ) CT ( s )
=
=
sI ! F
"(s)
(2 # 2)
(1# 1)
Denominator is the characteristic polynomial, a scalar
sI ! F " #(s)
= s 2 + c1s + c0
Roots of "(s) are eigenvalues of F
14
Numerator of the Characteristic
Matrix Inverse
Numerator is an (n x n) matrix of polynomials
" n1 (s) n1 (s) %
1
2
'
Adj ( sI ! F ) = $ 2
2
$ n1 (s) n2 (s) '
#
&
For example,
n11 ( s ) = k ( s ! z )
15
Initial Condition Response of a
2nd-Order System
!x(s) = [ sI " F] !x(0)
"1
Time-domain model
" !!x1 (t) % " f11
$
'=$
!!
x
(t)
$#
'& $# f21
2
f12 % " !x1 (t) %
',
'$
f22 ' $ !x2 (t) '
&
&#
" !x1 (0) %
$
' given
!x
(0)
$#
'&
2
Frequency-domain model
" !x1 (s) %
" !x1 (0) %
(1
$
' = [ sI ( F ] $
'
!x
!x
(s)
(0)
$#
'&
$#
'&
2
2
16
(sI – F)–1 Distributes and Shapes
the Effects of Initial Conditions
Numerator distributes each
element of the initial condition to
each element of the state
--------------------------------------------Denominator determines the
common modes of motion
" n1 (s) n1 (s)
2
$ 1
2
$ n1 (s) n22 (s)
!1
[ sI ! F ] = # s 2 + c s + c
1
0
%
'
'
&
(2 ( 2)
(1( 1)
n11 (s)!x1 (0) + n12 (s)!x2 (0)
!x1 (s) =
s 2 + c1s + c0
n12 (s)!x1 (0) + n22 (s)!x2 (0)
!x2 (s) =
s 2 + c1s + c0
17
Initial Condition Response of a
Single State Element (Frequency
Domain)
"1
!x(s) = [ sI " F ] !x(0)
"" n (s)
n11 ( s )
$ $ 11
$ $ nn21 ( s( )s )
21
" " !x ( s ) % % $ $ !
$ $ !x1 1 ( s ) ' ' $ $ !
$ nn1 ( s )
$ $ !x
!x2 (2 s( )s ) ' '= # $# nn1 ( s )
'' =
$$
!
$$ ! ''
$ $ !x
(s) '
# # !xn n ( s ) & '&
n12
n12( s( )s )
nn22 ( s( )s )
22
!
!
nnn2 ( s( )s )
n2
!
!
!
!
!
!
!
!
!!( s )s
( )
%
n1n
n1n( s( )s ) ' %
'
nn2n ( s( )s ) ' '
2n
' '"
%
!
! ' '$ " !x
!x1 (1 0( 0) ) ' %
nnn2 ( s( )s ) ' '$ $ !x ( 0 ) ' '
& & $ !x2 ( 0 ) '
n2
2
''
$$
!
$$ ! ''
$ $ !x
(0) '
# # !xn n ( 0 ) & '&
18
Initial Condition Response of a
Single State Element
(Frequency Domain)
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)
n21 ( s ) !x1 (0) + n22 ( s ) !x2 (0) +!+ n2n ( s ) !xn (0)
! (s)
"
p2 ( s )
! (s)
19
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 " #n ) )( i
% ( s " #1 ) ( s " #2 )
For each eigenvalue, !i , the coefficients are
(
dj = s " !j
)
(
)
) ( )
pi ! j
pi ( s )
= s " !j
# ( s ) s=!
# !j
(
j
j = 1,n
20
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
!xi (t) = ( 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)
Forssman Tri-Plane (?)
21
Westland P.12 Lysander
22
AEA Cygnet II, Alexander
Graham Bell, Glenn Curtiss,
1909 (3,393 tetrahedral cells)
Hargrave quadraplane
(model), 1889
DEquevillery, 1908
23
Phillips, 1904
Wight Quadraplane,
1916
Phillips, 1907
Pemberton-Billings
Nighthawk, 1916
Vedo Villi, 1911
John Septaplane, 1919
24
•!
Farman 3-engine Jabiru
•!
Heinkel 5-engine He111Z
Tarrant 6-engine Tabor, 1919
•!
•!
Farman 4-engine Jabiru, 1923
25
Caproni Ca 60, 1920
Nemeth Parasol, 1934
Miles Libellula, 1943
Lippisch Aerodyne, 1950
26
Control Response in
Frequency Domain
State Response
" !x1 (s) %
" !u1 (s) %
(1
$
' = [ sI ( F ] G $
'
!x
!u
(s)
(s)
$#
'&
$#
'&
2
2
Output Response (Hu = 0)
" !y1 (s) %
" !u1 (s) %
(1
$
' = H x [ sI ( F ] G $
'
!y
!u
(s)
(s)
$#
$#
'&
'&
2
2
" !u1 (s) %
'
! H (s) $
!u
(s)
$#
'&
2
27
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)
28
2nd-Order Transfer Function
2nd-order dynamic system
! x! ( t )
1
x! ( t ) = #
# x!2 ( t )
"
$ ! f
& = # 11
& #" f21
%
f12 $ ! x1 ( t )
&#
f22 & # x2 ( t )
%"
$ ! g
& + # 11
& #" g21
%
g12 $ ! u1 ( t )
&#
g22 & # u2 ( t )
%"
! y (t )
1
y (t ) = #
# y2 ( t )
"
$ ! h
h
& = # 11 12
& #" h21 h22
%
$ ! x1 ( t )
&#
&% #" x2 ( t )
$
& +"
&
%
$
&
&
%
2nd-order transfer function matrix
!1
H (s) = H x ( sI ! F ) ( s ) G
" (s ! f )
11
adj $
$ ! f21
" h11 h12 %
#
'
=$
( (s ! f )
$# h21 h22 '&
11
det *
* ! f21
( r ! n )( n ! n )( n ! m ) )
! f12
( s ! f22 )
! f12
( s ! f22 )
%
'
'
&
+
-,
" g11
$
$# g21
g12 %
'
g22 '
&
= (r ! m) = (2 ! 2)
29
Eigenvalues of a Dynamic System
!(s) = ( s " #1 ) ( s " #2 ) (...) ( s " #n ) = 0
Eigenvalues are real or complex numbers
that can be plotted in the s plane
•! Real root
!i = " i
•! Complex roots occur in
conjugate pairs
!i = " i + j# i
! *i = " i # j$ i
s Plane
Positive real part
indicates instability
30
Aircraft Modes of Motion!
31
Longitudinal and Lateral-Directional
Roots of the Characteristic Equation
•! 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) = ! Long (s)! Lat"Dir (s)
= $%( s " #1 )!( s " #6 ) &' Long $%( s " #1 )!( s " #6 ) &' Lat"Dir = 0
32
Longitudinal Modes of Motion in
Steady, Level Flight
Longitudinal characteristic equation has 6 roots
! Lon (s) = ( s " # R ) ( s " # H ) $%( s " # P ) ( s " # *P ) &' $%( s " #SP ) ( s " # *SP ) &'
Real
Real
Complex
Complex
Complex
Complex
4 modes of motion (typical)
(
)(
)
! Lon (s) = #$ s " ( 0 ) %& #$ s " ( ~ 0 ) %& s 2 + 2' P( nP s + ( nP 2 s 2 + 2' SP( nSP s + ( n2SP = 0
Range
Height
Phugoid
Short Period
33
Each Complex Conjugate Pair Forms
One Oscillatory Mode of Motion
Phugoid Roots
( s ! "P )( s ! " *P )
= %& s ! (# P + j$ P ) '( %& s ! (# P ! j$ P ) '( = s 2 ! 2# P s + (# P 2 + $ P 2 )
(
! 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 s + (# SP 2 + $ SP 2 )
(
! s 2 + 2) SP$ nSP s + $ n2SP
)
34
Lateral-Directional Modes of
Motion in Steady, Level Flight
Lateral-directional characteristic equation has 6 roots
! LD (s) = ( s " #CR ) ( s " # Head ) ( s " #S ) ( s " # R ) $%( s " # DR ) ( s " # * DR ) &'
5 modes of motion (typical)
(
)
! LD (s) = #$ s " ( 0 ) %& #$ s " ( 0 ) %& ( s " 'S ) ( s " ' R ) s 2 + 2( DR) nDR s + ) n2DR = 0
Crossrange
Heading
Spiral
Roll
Dutch Roll
35
Bode Plot!
(Frequency Response of a
Scalar Transfer Function)!
36
Scalar (Single-Input/Single-Output)
Frequency Response Function
Substitute: s = j!
H ij (j! ) = H x th
[ sI " F ]"1 G j column
kij ( j! " z1 )ij ( j! " z2 )ij ...( j! " zq )
ij
=
( j! " #1 )( j! " #2 )...( j! " #n )
i
th
row
= a(! )+ jb(! ) " AR(! ) e j# (! )
•!Frequency response is a complex function of
input frequency, !
–! Real and imaginary parts, or
–! ** Amplitude ratio and phase angle **
37
Bode Plots of 1st-Order System
H red ( j! ) =
10
( j! + 10 )
H
blue ( j! ) =
100
100
H green ( j! ) =
( j! + 100 )
( j! + 10 )
38
Amplitude Asymptotes and
Departures for 1st-Order Bode Plot
•! General shape governed by
asymptotes
•! Slopes changes by 20 dB/
dec at poles
–! When ! = 0, slope = 0 dB/
dec
–! When ! " ", slope = –20 dB/
dec
•! Actual AR departs from
asymptotes
39
Phase Angles for
1st-Order Bode Plot
•!Phase angle of a
real, negative pole
–!When ! = 0, # = 0°
–!When ! = ", # =–45°
–!When " -> #, # -> –90°
40
Bode Plots of 2nd-Order System
(No Zeros)
H green ( j! ) =
H blue ( j! ) =
H red ( j! ) =
( j! )
( j! )
( j! )
2
2
2
10 2
+ 2 ( 0.1) (10 ) ( j! ) + 10 2
Effect of
Damping Ratio
10 2
+ 2 ( 0.4 ) (10 ) ( j! ) + 10 2
10 2
+ 2 ( 0.707 ) (10 ) ( j! ) + 10 2
41
Amplitude Asymptotes and Departures of 2ndOrder 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 (see
Supplemental
Material)
42
Phase Angles of 2nd-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 (see
Supplemental Material)
43
Transfer Function Matrix for
Short-Period Approximation
( Hx = I2 , Hu = 0 )
# k n q (s) &
# +q(s)
% q !E
(
%
% k" n!"E (s) (
+! E(s)
$
'
%
H SP (s) ! 2
=
2
% +" (s)
s + 2) SP* nSP s + * nSP
%
%$ +! E(s)
&
(
(
(
(
('
44
Components of Approximate
Short-Period Transfer Function
L! E = 0
Pitch Rate Transfer Function
( )
L
$
'
M" E &s + # V )
N
%
(
$
'
+ Lq .
L
L#
2
s + *M q + V s * & M # - 1* V 0 + M q # V )
,
/
N
N
N
%
(
!q(s)
=
!" E(s)
(
)
!
(
k q s * zq
)
s + 21 SP2 nSP s + 2 nSP 2
2
Angle of Attack Transfer Function
!" (s)
=
!# E(s)
M# E
+
.
% L
L
L
(
s 2 + $M q + " V s $ - M " ' 1$ q V * + M q " V 0
&
)
N
N
N/
,
k"
! 2
s + 21 SP2 nSP s + 2 nSP 2
(
)
45
Short-Period Frequency Response,
Amplitude Ratio and Phase Angle
s ! j!
Pitch-rate frequency response
kq ( j" $ zq )
!q( j" )
=
!# E( j" ) $" 2 + 2% SP" nSP j" + " n2SP
= ARq (" ) e
j&q (" )
Angle-of-attack frequency
response
!" ( j# )
k"
=
2
!$ E( j# ) %# + 2& SP# nSP j# + # nSP 2
= AR" (# ) e j'" (# )
46
Bode Plot Portrays Response
to Sinusoidal Control Input
kq ( j# % zq )
"q( j# )
j' (# )
=
= ARq (# ) e q
2
2
"$E( j# ) %# + 2& SP# n SP j# + # n SP
Express amplitude ratio in
decibels
AR(dB) =
# zeros = 1
# poles = 2
20 log10 !" AR ( original units ) #$
20 dB = factor of 10
Products in original units
are sums in decibels
47
Why Plot Vertical Lines
where ! = z and !n?
Asymptotes change at frequencies
corresponding to poles and zeros
(
)
kq j" $ zq
!q( j" )
=
2
!# E( j" ) $" + 2% SP" nSP j" + " nSP 2
(
)
When input frequency, ! = "zq for negative zq ,
(
)
kq j! " zq = kq zq ( " j " 1) = "kq zq ( j + 1) = kq zq e+45°
When ! = ! nSP (" SP > 0)
# ! n2SP + 2" SP j! n2SP + ! n2SP = j2" SP! n2SP
=
1
j2" SP!
2
nSP
=
#j
1
=
e –90°
2
2
2" SP! nSP 2" SP! nSP
48
Pole and Zero Effects are Additive for
Amplitude Ratio (dB) and Phase Angle (deg)
49
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
50
Asymptotes form Complex Bode Plots for
Transfer Functions
4 th -order model of !q ( j" ) !# E ( j" )
+20
dB/dec
+40
dB/dec
0
dB/dec
+20
dB/dec
–20
dB/dec
51
Next Time:!
Root Locus Analysis!
Reading:!
Flight Dynamics!
357-361, 465-467, 488-490,
509-514!
Learning Objectives
Effects of system parameter variations on modes of motion
Root locus analysis
Evanss rules for construction
Application to longitudinal dynamic models
52
Supplementary
Material
53
Matrix Inverse Examples
A = 5; A !1 =
1
= 0.2
5
! 4 '2 $
#
&
'3 1 % ! '2
! 1 2 $
1 $
"
'1
A=#
=#
&; A =
&
'2
" 3 4 %
" 1.5 '0.5 %
! '30 18
4
#
20
'15
5
#
! 1 2 3 $
4 '2
# 0
#
&
A = # 4 6 7 & ; A '1 = "
10
#" 8 12 9 &%
$
&
& ! '3 1.8
0.4 $
&% #
&
= # 2 '1.5 0.5 &
#" 0 0.4 '0.2 %&
54
Longitudinal Modes of Motion
Eigenvalues determine the damping and natural
frequencies of the linear systems modes of motion
•! 6 eigenvalues
–! 4 eigenvalues normally appear as 2 complex pairs
–! Range and height modes usually inconsequential
!ran : range mode " 0
!hgt : height mode " 0
(#
(#
SP
P
, $ nP ) : phugoid mode
, $ nSP ) : short - period mode
55
Phugoid (Long-Period) Mode
Airspeed
Flight Path Angle
Pitch Rate
Angle of Attack
56
Short-Period Mode
Note change in time scale
Airspeed
Flight Path Angle
Pitch Rate
Angle of Attack
57
Lateral-Directional Modes of Motion
•! 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
58
Dutch-Roll Mode
Sideslip Angle
Yaw Rate
59
Roll and Spiral Modes
Roll Rate
Roll Angle
60
Transfer Function Matrix
Frequency-domain effect of all inputs on
all outputs (Hu = 0)
H (s) = H x [ sI ! F ] G
!1
( r ! n )( n ! n )( n ! m )
= (r ! m)
61
Characteristic Polynomial
of a LTI Dynamic System
Response to initial condition, control, and disturbance
!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
sI ! F = det ( sI ! F ) " #(s)
= s n + cn!1s n!1 + ...+ c1s + c0
62
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
63
Scalar Transfer Function
from !uj to !yi
•! 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
64
Control Response of a Single State
Element
nij (s)
!yi ( s ) = kij
!u j ( s )
!(s)
65
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)
66
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
67
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
68
Constant Gain Bode Plot
y ( t ) = hu ( t )
H ( j! ) = 1
H ( j! ) = 10
H ( j! ) = 100
Slope = 0 dB/dec, Amplitude Ratio = constant
Phase Angle = 0°
69
Integrator Bode Plot
t
y ( t ) = h ! u ( t ) dt
0
H ( j! ) =
1
j!
H ( j! ) =
10
j!
Slope = !20 dB/dec
Phase Angle = !90°
70
Differentiator Bode Plot
y (t ) = h
du ( t )
dt
H ( j! ) = j!
H ( j! ) = 10 j!
Slope = +20 dB/dec
Phase Angle = +90°
71
Sign Change
Integral
t
y (t ) = !h " u (t ) dt
0
H ( j! ) = "
h
j!
Slope = !20 dB/dec
Phase Angle = +90°
Derivative
y (t ) = !h
du (t )
dt
H ( j! ) = " j!
Slope = +20 dB/dec
Phase Angle = !90°
72
Multiple Integrators and
Differentiators
Double Integral
t
y (t ) = h !
t
! u (t ) dt
2
0 0
H ( j! ) =
Slope = !40 dB/dec
Phase Angle = !180°
h
( j! )2
Double Derivative
y (t ) = h
d 2 u (t )
dt 2
H ( j! ) = h ( j! )
2
Slope = +40 dB/dec
Phase Angle = +180°
73
Bode Plots of 1st- and 2nd-Order Lags
H red ( j! ) =
H blue ( j! ) =
( j! )
2
10
( j! + 10 )
100 2
+ 2 ( 0.1) (100 ) ( j! ) + 100 2
74
Numerator and Denominator
of 2nd-Order (sI – F)–1
Numerator
" (s ! f )
11
adj $
$ ! f21
#
"
det $
$#
! f12
( s ! f22 )
( s ! f11 )
% " (s ! f )
22
'=$
' $
f21
& #
f12
( s ! f11 )
%
'
'
&
Denominator
! f12
%
' = ( 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 )
75
2nd-Order Transfer Function
Include H and G
H (s) = H x ( sI ! F )
!1
" h11 h12
( s)G = $
$# h21 h22
" (s ! f )
f12
22
$
f21
( s ! f11 )
%$
#
'
s 2 + 2() n s + ) n2
'&
%
'
'" g %
&$ 1 '
$# g2 '&
"
&
&
&
H (s) = #
"# 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 $% 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
"# h11 ( s ! f22 ) + h12 f21 $%
$
'
'%
76
2nd-Order Transfer Function
"
&
&
&
H (s) = #
Collect Terms
"# 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
77
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
78
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
79
2nd-Order Transfer Function Zero
2
H ( j! ) = ( j! " z ) ( j! " z * ) = #$( j! ) + 2 ( 0.1) (100 ) ( j! ) + 100 2 %&
•!Complex pair of zeros
produces an
amplitude ratio
notch
at its natural
frequency
80
Bode Plots of 2nd-Order Lags
(No Zeros)
H red ( j! ) =
10 2
( j! )2 + 2 ( 0.1)(10 )( j! ) + 10 2
H green ( j! ) =
10 3
( j! )2 + 2 ( 0.1)(10 )( j! ) + 10 2
Effects of Gain and
Natural Frequency
H blue ( j! ) =
100 2
( j! )2 + 2 ( 0.1)(100 )( j! ) + 100 2
81
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
!
+
100
(
)
j
!
+
2
0.1
10
j
!
+
10
(
)
(
)
(
)
(
)
#
&
#
&
82
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
83
Longitudinal Transfer
Function Matrix
•! With Hx = I, Hu = 0, 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 ! FLon ] G Lon
!1
"
$
$
$
$#
=
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
"
%$
'$
'$
'$
'& $
$
#
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
&
+ Lon ( s )
T* T
0
0
L* F / VN
0
!L* F / VN
0
0
84
%
'
'
'
'
'
&
Douglas AD-1 Skyraider
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
H Lon (s) = 2
s + 2* P+ nP s + + nP 2 s 2 + 2* SP+ nSP s + + nSP 2
(
)(
85
Longitudinal Transfer Function Matrix
Relates All Inputs to All Outputs
•! Input-output relationship
$
&
&
&
&
&%
!V (s) '
$ !* E(s) '
)
!" (s) )
)
&
= H Lon (s) & !* T (s) )
)
!q(s)
& !* F(s) )
)
(
%
!# (s) )(
86
)
Transfer Function Matrix for 2ndOrder 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
+*
,
.
.
.-
87
Transfer Function Matrix for
Short-Period Approximation
Transfer Function Matrix (with Hx = I, Hu = 0)
Expand Inverse
(
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'
(
(
)(
)
)
88
,
.
.
.-
Transfer Function Matrix for
Short-Period Approximation
Expand Products
(
)
( )(
'
$
L"
L! E M " '
$
M
s
+
#
)
&
&% ! E
VN
VN )(
)
&
& $
' )
* L
- L
& & M ! E , 1# q V / # ! E V s # M q ) )
+
N.
N
( )(
&% %
H SP (s) =
$
'
* L
L
L
s 2 + #M q + " V s # & M " , 1# q V / + M q " V )
+
N
N.
N(
%
(
=
)
(
Collect
)
)
$
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
VN .
L! E +
&
%
( 36
2
%
7 SP ( s )
(
)
'
)
)
)
)
)
(
89
4th-Order Transfer Function
with 2nd-Order Zero
"( j! )2 + 2 ( 0.1) (10 ) ( j! ) + 10 2 $
#
%
H ( j! ) =
2
2
2
"( j! ) + 2 ( 0.05 ) (1) ( j! ) + 1 $ "( j! ) + 2 ( 0.1) (100 ) ( j! ) + 100 2 $
#
%#
%
90
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
Short
Period
Phugoid
–20 dB/dec
91
Response to a Control Input
Neglect initial condition and disturbance
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
92
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
93
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
94