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Transfer Functions and Frequency Response !
Robert Stengel, Aircraft Flight Dynamics!MAE 331, 2016
•! Frequency domain view of initial condition response•! Response of dynamic systems to sinusoidal inputs•! Transfer functions•! Bode plots
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
Reading:!Flight Dynamics!
342-357!
Learning Objectives
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
!x(t) : real variable!x( j") : complex variable
= a(")+ jb(")= A(")e j# (" )
A : amplitude! : phase angle
j! : Imaginary frequency operator, rad/s
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
0
#
$ dt
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
L !x1(t)+ !x2 (t)[ ] = L !x1(t)[ ]+L !x2 (t)[ ]= !x1(s)+ !x2 (s)
L a!x(t)[ ] = aL !x(t)[ ] = a!x(s)
Sum of Laplace transforms
Multiplication by a constant
7
Laplace Transforms of Vectors and Matrices
Laplace transform of a vector variable
L !x(t)[ ] = !x(s) =!x1(s)!x2 (s)...
"
#
$$$
%
&
'''
Laplace transform of a matrix variable
L F(t)[ ] = F(s) =f11(s) f12 (s) ...f21(s) f22 (s) ...... ... ...
!
"
###
$
%
&&&
Laplace transform of a time-derivative
L !!x(t)[ ] = s!x(s)" !x(0)8
Laplace Transform of a Dynamic System
!!x(t) = F!x(t) +G!u(t) + L!w(t)System equation
Laplace transform of system equation
s!x(s) " !x(0) = F!x(s) +G!u(s) + L!w(s)
dim(!x) = (n "1)dim(!u) = (m "1)dim(!w) = (s "1)
9
Laplace Transform of a Dynamic System
Rearrange Laplace transform of dynamic equation
s!x(s)"F!x(s)= !x(0)+G!u(s)+L!w(s)
sI!F[ ]"x(s)= "x(0)+G"u(s)+L"w(s)
!x(s)= sI"F[ ]"1 !x(0)+G!u(s)+L!w(s)[ ]
F to left, I.C. to right
Combine terms
Multiply both sides by inverse of (sI – F)
10
Matrix Inverse
A[ ]!1 = Adj A( )A
=Adj A( )detA
(n " n)(1" 1)
= CT
detA; C = matrix of cofactors
Cofactors are signed minors of A
ijth minor of A is the determinant of A with the ith row and jth column removed
y = Ax; x = A!1y dim(x) = dim(y) = (n !1)dim(A) = (n ! n)
Forward Inverse
Numerator is a square matrix of cofactor transposesDenominator is a scalar
11
Matrix Inverse Examples
A =a11 a12a21 a22
!
"##
$
%&&; A'1 =
a22 'a21'a12 a11
!
"##
$
%&&
T
a11a22 ' a12a21=
a22 'a12'a21 a11
!
"##
$
%&&
a11a22 ' a12a21
dim(A) = (2 ! 2)
A = a; A!1 =1a
dim(A) = (1!1)
12
Matrix Inverse Examples
A =a11 a12 a13a21 a22 a23a31 a32 a33
!
"
###
$
%
&&&;
A'1 =
a22a33 ' a23a32( ) ' a21a33 ' a23a31( ) a21a32 ' a22a31( )' a12a33 ' a13a32( ) a11a33 ' a13a31( ) ' a11a32 ' a12a31( )a12a23 ' a13a22( ) ' a11a23 ' a13a21( ) a11a22 ' a12a21( )
!
"
####
$
%
&&&&
T
a11a22a33 + a12a23a31 + a13a21a32 ' a13a22a31 ' a12a21a33 ' a11a23a32
dim(A) = (3! 3)
13=
a22a33 ! a23a32( ) ! a12a33 ! a13a32( ) a12a23 ! a13a22( )! a21a33 ! a23a31( ) a11a33 ! a13a31( ) ! a11a23 ! a13a21( )a21a32 ! a22a31( ) ! a11a32 ! a12a31( ) a11a22 ! a12a21( )
"
#
$$$$
%
&
''''
a11a22a33 + a12a23a31 + a13a21a32 ! a13a22a31 ! a12a21a33 ! a11a23a32
Characteristic Matrix Inverse
sI! F " #(s)= s2 + c1s + c0
sI! F[ ]!1 = Adj sI! F( )sI! F
=CT s( )"(s)
(2 # 2)1#1( )
Denominator is the characteristic polynomial, a scalar
14
sI! F[ ]
Characteristic matrix (2nd-order example)
Inverse of characteristic matrix
Roots of "(s) are eigenvalues of F
Numerator of the Characteristic Matrix Inverse
Adj sI! F( ) = n11(s) n2
1 (s)
n12 (s) n2
2 (s)
"
#$$
%
&''
Numerator is an (n x n) matrix of polynomials
15
For example,n1
1 s( ) = k s ! z( )
Initial Condition Response of a 2nd-Order System
Time-domain model
!x(s)= sI"F[ ]"1!x(0)
!!x1(t)!!x2 (t)
"
#$$
%
&''=
f11 f12f21 f22
"
#$$
%
&''
!x1(t)!x2 (t)
"
#$$
%
&'',
!x1(0)!x2 (0)
"
#$$
%
&''given
Frequency-domain model
!x1(s)!x2 (s)
"
#$$
%
&''= sI( F[ ](1 !x1(0)
!x2 (0)
"
#$$
%
&''
16
(sI – F)–1 Distributes and Shapes the Effects of Initial Conditions
sI! F[ ]!1 =
n11(s) n2
1 (s)
n12 (s) n2
2 (s)
"
#$$
%
&''
s2 + c1s + c0(2 ( 2)1(1( )
Numerator distributes each element of the initial condition to
each element of the state---------------------------------------------
Denominator determines the common modes of motion
17
!x1(s) =n11(s)!x1(0)+ n2
1 (s)!x2 (0)s2 + c1s + c0
!x2 (s) =n12 (s)!x1(0)+ n2
2 (s)!x2 (0)s2 + c1s + c0
Initial Condition Response of a Single State Element (Frequency
Domain)
18
!x(s) = sI" F[ ]"1!x(0)
!x1 s( )!x2 s( )!
!xn s( )
"
#
$$$$$
%
&
'''''
=
n11 s( ) n12 s( ) ! n1n s( )n21 s( ) n22 s( ) ! n2n s( )! ! ! !
nn1 s( ) nn2 s( ) ! nn2 s( )
"
#
$$$$$
%
&
'''''
! s( )
!x1 0( )!x2 0( )!
!xn 0( )
"
#
$$$$$
%
&
'''''
!x1 s( )!x2 s( )!
!xn s( )
"
#
$$$$$
%
&
'''''
=
n11 s( ) n12 s( ) ! n1n s( )n21 s( ) n22 s( ) ! n2n s( )! ! ! !
nn1 s( ) nn2 s( ) ! nn2 s( )
"
#
$$$$$
%
&
'''''
! s( )
!x1 0( )!x2 0( )!
!xn 0( )
"
#
$$$$$
%
&
'''''
Initial Condition Response of a Single State Element(Frequency Domain)
!x2 (s) =n21 s( )! s( ) !x1(0)+
n22 s( )! s( ) !x2 (0)+!+
n2n s( )! s( ) !xn (0)
=n21 s( )!x1(0)+ n22 s( )!x2 (0)+!+ n2n s( )!xn (0)
! s( )
"p2 s( )! s( )
Initial condition response of !!x2(s)
19
Partial Fraction Expansion of the Initial Condition Response
Scalar response can be expressed with n parts, each containing a single mode
!xi (s) =pi s( )! s( )
= d1s " #1( ) +
d2s " #2( ) +!
dns " #n( )
$
%&'
() i, i = 1,n
For each eigenvalue, !i , the coefficients are
d j = s " ! j( ) pi s( )# s( ) s=! j
= s " ! j( ) pi ! j( )# ! j( ) j = 1,n
20
Partial Fraction Expansion of the Initial Condition Response
Time response is the inverse Laplace transform
!xi (t) = L "1 !xi (s)[ ]
= L "1 d1s " #1( ) +
d2s " #2( ) +!
dns " #n( )
$
%&
'
()i
Each element’s time response contains every mode of the system (although some coefficients may be negligible)
21
!xi (t) = d1e
"1t + d2e"2t +!+ dne
"nt( )i , i = 1,n
Westland P.12 LysanderForssman Tri-Plane (?)
22
AEA Cygnet II, Alexander Graham Bell, Glenn Curtiss, 1909 (3,393 tetrahedral cells)
D Equevillery, 1908
Hargrave quadraplane (model), 1889
23
Phillips, 1907
John Septaplane, 1919Wight Quadraplane,
1916
Phillips, 1904 Vedo Villi, 1911
Pemberton-Billings Nighthawk, 1916
24
•! Farman 3-engine Jabiru
•! Heinkel 5-engine He111Z
•! Tarrant 6-engine Tabor, 1919
•! Farman 4-engine Jabiru, 1923
25
Caproni Ca 60, 1920
26
Nemeth Parasol, 1934Lippisch Aerodyne, 1950
Miles Libellula, 1943
Control Response in Frequency Domain
State Response
!x1(s)!x2 (s)
"
#$$
%
&''= sI( F[ ](1G !u1(s)
!u2 (s)
"
#$$
%
&''
27
Output Response (Hu = 0)
!y1(s)!y2 (s)
"
#$$
%
&''= Hx sI( F[ ](1G !u1(s)
!u2 (s)
"
#$$
%
&''
! HH (s)!u1(s)!u2 (s)
"
#$$
%
&''
1st-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
28
2nd-Order Transfer Function
H (s) = Hx sI! F( )!1 s( )G
=h11 h12h21 h22
"
#$$
%
&''
adjs ! f11( ) ! f12! f21 s ! f22( )
"
#
$$
%
&
''
dets ! f11( ) ! f12! f21 s ! f22( )
(
)**
+
,--
g11 g12g21 g22
"
#$$
%
&''
2nd-order transfer function matrix
r ! n( ) n ! n( ) n ! m( )= r ! m( ) = 2 ! 2( )
!x t( ) =!x1 t( )!x2 t( )
!
"##
$
%&&=
f11 f12f21 f22
!
"##
$
%&&
x1 t( )x2 t( )
!
"##
$
%&&+
g11 g12g21 g22
!
"##
$
%&&
u1 t( )u2 t( )
!
"##
$
%&&
y t( ) =y1 t( )y2 t( )
!
"##
$
%&&=
h11 h12h21 h22
!
"##
$
%&&
x1 t( )x2 t( )
!
"##
$
%&&+"
2nd-order dynamic system
29
Eigenvalues of a Dynamic System
Eigenvalues are real or complex numbers that can be plotted in the s plane
s Plane
!i = " i + j# i
•! Real root
!*i = " i # j$ i
Positive real part indicates instability
•! Complex roots occur in conjugate pairs
!i =" i
30
!(s) = s " #1( ) s " #2( ) ...( ) s " #n( ) = 0
Aircraft Modes of Motion!
31
Longitudinal and Lateral-Directional Roots of the Characteristic Equation
!(s) = s12 + c11s11 + ...+ c1s + c0 = 0
= s " #1( ) s " #2( ) ...( ) s " #12( ) = 0
•! 12 roots of the characteristic equation•! Characteristic equation of the system
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 = 032
Longitudinal Modes of Motion in Steady, Level Flight
!Lon (s) = s " #R( ) s " #H( ) s " #P( ) s " #*P( )$% &' s " #SP( ) s " #*SP( )$% &'
!Lon (s) = s " 0( )#$ %& s " ~ 0( )#$ %& s2 + 2' P( nPs +( nP
2( ) s2 + 2' SP( nSPs +( nSP
2( ) = 0
Longitudinal characteristic equation has 6 roots
4 modes of motion (typical)
Real Complex
Range Height Phugoid Short Period
Real Complex Complex Complex
33
Each Complex Conjugate Pair Forms One Oscillatory Mode of Motion
s ! "P( ) s ! "*P( )= s ! # P + j$ P( )%& '( s ! # P ! j$ P( )%& '( = s
2 ! 2# Ps + # P2 +$ P
2( )! s2 + 2) P$ nP
s +$ nP2( )
s ! "SP( ) s ! "*SP( )= s ! # SP + j$ SP( )%& '( s ! # SP ! j$ SP( )%& '( = s
2 ! 2# SPs + # SP2 +$ SP
2( )! s2 + 2) SP$ nSP
s +$ nSP2( )
Short Period Roots
Phugoid Roots
!n : Natural frequency, rad/s ! : Damping ratio, -
34
Lateral-Directional Modes of Motion in Steady, Level Flight
!LD (s) = s " #CR( ) s " #Head( ) s " #S( ) s " #R( ) s " #DR( ) s " #*DR( )$% &'
!LD (s) = s " 0( )#$ %& s " 0( )#$ %& s " 'S( ) s " 'R( ) s2 + 2( DR) nDRs +) nDR
2( ) = 0
Lateral-directional characteristic equation has 6 roots
5 modes of motion (typical)
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
H ij (j! ) = Hxi th rowsI" F[ ]"1G jthcolumn
=kij j! " z1( )ij j! " z2( )ij ... j! " zq( )ij
j! " #1( ) j! " #2( )... j! " #n( )
Substitute: s = j!!
•!Frequency response is a complex function of input frequency, !!
–! Real and imaginary parts, or–! ** Amplitude ratio and phase angle **
= a(!)+ jb(!)" AR(!) e j# (! )
37
Bode Plots of 1st-Order System
Hred ( j! ) =
10j! +10( )
H blue( j! ) =100j! +10( )
Hgreen ( j! ) =100
j! +100( )
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)
Effect of Damping Ratio
Hgreen ( j! ) =
102
j!( )2 + 2 0.1( ) 10( ) j!( ) +102
Hblue( j! ) =
102
j!( )2 + 2 0.4( ) 10( ) j!( ) +102
Hred ( j! ) =
102
j!( )2 + 2 0.707( ) 10( ) j!( ) +102
41
•!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
Amplitude Asymptotes and Departures of 2nd-Order Bode Plots (No Zeros)
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
HH SP (s) !
kqn!Eq (s)
k"n!E" (s)
#
$
%%
&
'
((
s2 + 2) SP* nSPs +* nSP
2 =
+q(s)+!E(s)+" (s)+!E(s)
#
$
%%%%%
&
'
(((((
44
Hx = I2, Hu = 0( )
Components of Approximate Short-Period Transfer Function
!q(s)!"E(s)
=M"E s + L#
VN( )$%&
'()
s2 + *Mq +L#VN( )s * M# 1* Lq VN
+,-
./0 +Mq
L#VN
$%&
'()
!kq s * zq( )
s2 + 21 SP2 nSPs +2 nSP
2
!" (s)!#E(s)
= M#E
s2 + $Mq +L"VN( )s $ M" 1$ Lq VN
%&'
()* +Mq
L"VN
+,-
./0
!k"
s2 + 21 SP2 nSPs +2 nSP
2
Pitch Rate Transfer Function
Angle of Attack Transfer Function
45
L!E = 0
Short-Period Frequency Response, Amplitude Ratio and Phase Angle
Pitch-rate frequency response
Angle-of-attack frequency response
!q( j")!#E( j")
=kq j" $ zq( )
$" 2 +2%SP"nSPj" +"nSP
2
= ARq (") ej&q (" )
!" ( j# )!$E( j# )
= k"%# 2 + 2& SP# nSP
j# +# nSP2
= AR" (# ) ej'" (# )
46
s ! j!
Bode Plot Portrays Response to Sinusoidal Control Input
Express amplitude ratio in decibelsAR(dB) =
20 log10 AR original units( )!" #$
20 dB = factor of 10
"q( j#)"$E( j#)
=kq j# % zq( )
%# 2 + 2& SP#nSPj# +#nSP
2 = ARq (#) ej'q (# )
Products in original units are sums in decibels
# zeros = 1# poles = 2
47
Why Plot Vertical Lines where ! = z and !n?
48
Asymptotes change at frequencies corresponding to poles and zeros
!q( j" )!#E( j" )
=kq j" $ zq( )
$" 2 + 2% SP" nSPj" +" nSP
2
When ! =! nSP(" SP > 0)
#! nSP2 + 2" SP j! nSP
2 +! nSP2 = j2" SP! nSP
2
= 1j2" SP! nSP
2 = # j2" SP! nSP
2 = 12" SP! nSP
2 e–90°
When input frequency, ! = "zq for negative zq( ),kq j! " zq( ) = kqzq " j "1( ) = "kqzq j +1( ) = kq zq e+45°
Pole and Zero Effects are Additive for Amplitude Ratio (dB) and Phase Angle (deg)
49
MATLAB Bode Plot with asymp.mhttp://www.mathworks.com/matlabcentral/
http://www.mathworks.com/matlabcentral/fileexchange/10183-bode-plot-with-asymptotes
2nd-Order Pitch Rate Frequency Response
asymp.mbode.m
50
Asymptotes form Complex Bode Plots for Transfer Functions
+20 dB/dec
+40 dB/dec
0 dB/dec
+20 dB/dec
–20 dB/dec
51
4 th -order model of !q j"( ) !#E j"( )
Next Time:!Root Locus Analysis!
Reading:!Flight Dynamics!
357-361, 465-467, 488-490, 509-514!
52
Effects of system parameter variations on modes of motionRoot locus analysis
Evans s rules for constructionApplication to longitudinal dynamic models
Learning Objectives
Supplementary Material
53
Matrix Inverse Examples
A = 1 23 4
!
"#
$
%&; A'1 =
4 '2'3 1
!
"#
$
%&
'2= '2 1
1.5 '0.5!
"#
$
%&
A =1 2 34 6 78 12 9
!
"
###
$
%
&&&; A'1 =
'30 18 420 '15 50 4 '2
!
"
###
$
%
&&&
10=
'3 1.8 0.42 '1.5 0.50 0.4 '0.2
!
"
###
$
%
&&&
A = 5; A!1 = 15= 0.2
54
Longitudinal Modes of MotionEigenvalues determine the damping and natural
frequencies of the linear system s modes of motion
!ran :range mode " 0!hgt :height mode " 0
#P ,$nP( ) : phugoid mode#SP ,$nSP( ) : short - period mode
•! 6 eigenvalues–! 4 eigenvalues normally appear as 2 complex pairs–! Range and height modes usually inconsequential
55
Phugoid (Long-Period) Mode
Airspeed Flight Path Angle
Pitch Rate Angle of Attack
56
Short-Period Mode
Airspeed Flight Path Angle
Pitch Rate Angle of Attack
Note change in time scale
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
Yaw Rate Sideslip Angle
59
Roll and Spiral Modes
Roll Rate Roll Angle
60
Transfer Function MatrixFrequency-domain effect of all inputs on
all outputs (Hu = 0)
HH (s) = Hx sI! F[ ]!1Gr ! n( ) n ! n( ) n ! m( )
= r ! m( )
61
Characteristic Polynomial of a LTI Dynamic System
sI!F[ ]!1 =Adj sI!F( )sI!F
(n x n)
Characteristic polynomial
sI! F = det sI! F( ) " #(s)
= sn + cn!1sn!1 + ...+ c1s + c0
!x(s) = sI " F[ ]"1 !x(0) +G!u(s) + L!w(s)[ ]Inverse of characteristic matrix
62
Response to initial condition, control, and disturbance
Eigenvalues (or Roots) of a Dynamic System
!(s) = sI" F = sn + cn"1sn"1 + ...+ c1s + c0 = 0
= s " #1( ) s " #2( ) ...( ) s " #n( ) = 0
Characteristic equation of the system
... where ""i are the eigenvalues of F or the roots of the characteristic polynomial
63
Scalar Transfer Function from !!uj to !!yi
H ij (s) =
nij (s)!(s)
=kij s
q + bq"1sq"1 + ...+ b1s + b0( )
sn + cn"1sn"1 + ...+ c1s + c0( )
# zeros = q# poles = n
•! 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
= kijs ! z1( )ij s ! z2( )ij ... s ! zq( )ijs ! "1( ) s ! "2( )... s ! "n( )
64
Denominator polynomial contains n roots
Control Response of a Single State Element
65
!yi s( ) = kijnij (s)!(s)
!u j s( )
Relationship of (sI – F)–1 to State Transition Matrix, ##(t,0)
Initial condition response
!x(s) = sI" F[ ]"1!x(0) =
!x(t) = "" t,0( )!x(0)Time Domain
Frequency Domain
!!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)
sI! F[ ]!1 = L "" t,0( )#$ %&= Laplace transform of the state transition matrix
Therefore,
67
Bode Plot Portrays Response to Sinusoidal Control Input
# zeros = 1# poles = 2
68
Plot AR(dB) vs. log10(!!input)Plot phase angle, ##(deg) vs. log10(!!input)
Asymptotes form “skeleton” of response amplitude ratio
Constant Gain Bode Plot
H ( j! ) = 1
H ( j! ) = 10
H ( j! ) = 100
y t( ) = hu t( )
Slope = 0 dB/dec, Amplitude Ratio = constantPhase Angle = 0°
69
Integrator Bode Plot
H ( j! ) = 1
j!
H ( j! ) = 10
j!
y t( ) = h u t( )dt0
t
!
Slope = !20 dB/decPhase Angle = !90°
70
Differentiator Bode Plot
H ( j! ) = j!
H ( j! ) = 10 j!
y t( ) = h du t( )dt
Slope = +20 dB/decPhase Angle = +90°
71
Sign Change
H ( j! ) = " h
j!y t( ) = !h u t( )dt
0
t
"
H ( j! ) = " j!y t( ) = !hdu t( )dt
Slope = !20 dB/decPhase Angle = +90°
Slope = +20 dB/decPhase Angle = !90°
Integral
Derivative
72
Multiple Integrators and Differentiators
H ( j! ) = h j!( )2y t( ) = hd 2u t( )dt 2
H ( j! ) = h
j!( )2y t( ) = h u t( )dt 2
0
t
!0
t
!Slope = !40 dB/decPhase Angle = !180°
Double Integral
Double Derivative
73
Slope = +40 dB/decPhase Angle = +180°
Bode Plots of 1st- and 2nd-Order Lags
Hred ( j! ) =10
j! +10( )
Hblue( j! ) =1002
j!( )2 + 2 0.1( ) 100( ) j!( ) +1002 74
Numerator and Denominator of 2nd-Order (sI – F)–1
adjs ! f11( ) ! f12! f21 s ! f22( )
"
#
$$
%
&
''=
s ! f22( ) f12f21 s ! f11( )
"
#
$$
%
&
''
75
dets ! f11( ) ! f12! f21 s ! f22( )
"
#$$
%
&''= s ! f11( ) s ! f22( )! f12 f21
= s2 ! f11 + f22( )s + f11 f22 ! f12 f21( )! s2 + 2() ns +) n
2 ! * s( )
Numerator
Denominator
2nd-Order Transfer Function
HH (s) = Hx sI! F( )!1 s( )G =
h11 h12h21 h22
"
#$$
%
&''
s ! f22( ) f12f21 s ! f11( )
"
#
$$
%
&
''
s2 + 2() ns +) n2
g1g2
"
#$$
%
&''
76
H (s) =
h11 s ! f22( ) + h12 f21"# $% h11 f12 + h12 s ! f11( )"# $%h21 s ! f22( ) + h22 f21"# $% h21 f12 + h22 s ! f11( )"# $%
"
#
&&&
$
%
'''
g1g2
"
#&&
$
%''
s2 + 2() ns +) n2
=
h11 s ! f22( ) + h12 f21"# $%g1 + h11 f12 + h12 s ! f11( )"# $%g2h21 s ! f22( ) + h22 f21"# $%g1 + h21 f12 + h22 s ! f11( )"# $%g2
"
#
&&&
$
%
'''
s2 + 2() ns +) n2
Include H and G
2nd-Order Transfer Function
77
H (s) =
h11 s ! f22( ) + h12 f21"# $%g1 + h11 f12 + h12 s ! f11( )"# $%g2h21 s ! f22( ) + h22 f21"# $%g1 + h21 f12 + h22 s ! f11( )"# $%g2
"
#
&&&
$
%
'''
s2 + 2() ns +) n2
!
k1 s ! z1( )k2 s ! z2( )
"
#
$$
%
&
''
s2 + 2() ns +) n2
Collect Terms
Left-Half-Plane Transfer Function Zero
H ( j! ) = j! +10( )
Zeros are numerator singularities
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
H ( j! ) = j! " z( ) j! " z*( ) = j!( )2 + 2 0.1( ) 100( ) j!( ) +1002#$ %&
•!Complex pair of zeros produces an amplitude ratio notch at its natural
frequency
80
Bode Plots of 2nd-Order Lags (No Zeros)
Hred ( j! ) =
102
j!( )2 + 2 0.1( ) 10( ) j!( ) +102
Effects of Gain and Natural Frequency
Hgreen ( j! ) =
103
j!( )2 + 2 0.1( ) 10( ) j!( ) +102
Hblue( j! ) =
1002
j!( )2 + 2 0.1( ) 100( ) j!( ) +1002 81
Bode Plots of 3rd-Order Lags
Hblue( j! ) =10
j! +10( )"
#$
%
&'
1002
j!( )2 + 2 0.1( ) 100( ) j!( ) +1002"
#$
%
&'
Hgreen ( j! ) =102
j!( )2 + 2 0.1( ) 10( ) j!( ) +102"
#$
%
&'
100j! +100( )
"
#$
%
&'
82
Bode Plot of a 4th-Order System with No Zeros
H ( j! ) = 12
j!( )2 + 2 0.05( ) 1( ) j!( ) +12"
#$
%
&'
1002
j!( )2 + 2 0.1( ) 100( ) j!( ) +1002"
#$
%
&'
•!Resonant peaks and large phase shifts at each natural frequency
•!Additive AR slope shifts at each natural frequency
# zeros = 0# poles = 4
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
HLon (s) = HxLonsI! FLon[ ]!1GLon
=
1 0 0 00 1 0 00 0 1 00 0 0 1
"
#
$$$$
%
&
''''
nVV (s) n(
V (s) nqV (s) n)
V (s)
nV( (s) n(
( (s) nq( (s) n)
( (s)
nVq (s) n(
q (s) nqq (s) n)
q (s)
nV) (s) n(
) (s) nq) (s) n)
) (s)
"
#
$$$$$$
%
&
''''''
0 T*T 00 0 L*F /VNM*E 0 00 0 !L*F /VN
"
#
$$$$$
%
&
'''''
+Lon s( )84
Longitudinal Transfer Function Matrix
HLon (s) =
n!EV (s) n!T
V (s) n!FV (s)
n!E" (s) n!T
" (s) n!F" (s)
n!Eq (s) n!T
q (s) n!Fq (s)
n!E# (s) n!T
# (s) n!F# (s)
$
%
&&&&&&
'
(
))))))
s2 + 2* P+ nPs ++ nP2( ) s2 + 2* SP+ nSP
s ++ nSP2( )
•! There are 4 outputs and 3 inputs
Douglas AD-1 Skyraider
85
Longitudinal Transfer Function Matrix Relates All Inputs to All Outputs
!V (s)!" (s)!q(s)!# (s)
$
%
&&&&&
'
(
)))))
=HLon (s)!*E(s)!*T (s)!*F(s)
$
%
&&&
'
(
)))
•! Input-output relationship
86
Transfer Function Matrix for 2nd-Order Short-Period Approximation
Transfer Function Matrix (with Hx = I, Hu = 0)
HSP (s) = I2 sI! F( )SP!1 s( )GSP =
s !Mq( ) !M"
! 1! Lq VN#$%
&'( s + L"
VN( ))
*
++++
,
-
.
.
.
.
--11
M/E
!L/EVN
)
*
+++
,
-
.
.
.
!!xSP t( ) =! !q t( )! !" t( )
#
$%%
&
'(()
Mq M"
1* Lq VN+,-
./0 * L"
VN
#
$
%%%
&
'
(((
!q t( )!" t( )
#
$%%
&
'((+
M1E
*L1EVN
#
$
%%%
&
'
(((!1E t( )
Dynamic Equation
87
Transfer Function Matrix for Short-Period Approximation
Transfer Function Matrix (with Hx = I, Hu = 0)
HSP (s) = sI! FLon[ ]!1GSP =
s + L"VN( ) M"
1! Lq VN#$%
&'( s !Mq( )
)
*
++++
,
-
.
.
.
.
M/E
!L/EVN
)
*
+++
,
-
.
.
.
s !Mq( ) s + L"VN( )!M" 1! Lq VN
#$%
&'(
88
Expand Inverse
Transfer Function Matrix for Short-Period Approximation
HSP (s) =
M!E s + L"VN( )# L!EM"
VN$%&
'()
M!E 1# Lq VN*+,
-./ #
L!EVN( ) s #Mq( )$
%&'()
$
%
&&&&&
'
(
)))))
s2 + #Mq +L"VN( )s # M" 1# Lq VN
*+,
-./ +Mq
L"VN
$%&
'()
==
M!E s + L"VN
# L!EM"VNM!E( )$
%&'()
# L!EVN( ) s + VNM!E
L!E
1# Lq VN*+,
-./ #Mq
$
%&
'
()
0123
4563
$
%
&&&&&
'
(
)))))
7SP s( ) 89
Expand Products
Collect
4th-Order Transfer Function with 2nd-Order Zero
H ( j! ) =j!( )2 + 2 0.1( ) 10( ) j!( ) +102"# $%
j!( )2 + 2 0.05( ) 1( ) j!( ) +12"# $% j!( )2 + 2 0.1( ) 100( ) j!( ) +1002"# $%
90
Elevator-to-Normal-Velocity Frequency Response
!w(s)!"E(s)
=n"Ew (s)
!Lon (s)#
M"E s2 + 2$%ns +%n2( )Approx Ph s & z3( )
s2 + 2$%ns +%n2( )Ph s2 + 2$%ns +%n
2( )SP
0 dB/dec
+40 dB/dec0 dB/dec
–40 dB/dec
–20 dB/dec
•!(n – q) = 1•!Complex zero almost (but not quite) cancels phugoid response
Short PeriodPhugoid
91
Response to a Control InputNeglect 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[ ]"1G!u(s)
Output response to control
!y(s) = Hx!x(s)+Hu!u(s)
= Hx sI" F[ ]"1G!u(s)+Hu!u(s)
= Hx sI" F[ ]"1G +Hu{ }!u(s)92
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
First- and Second-Order Departures from Amplitude Ratio Asymptotes
93
McRuer, Ashkenas, and Graham, Aircraft Dynamics
and Automatic Control, Princeton University Press,
1973
First- and Second-Order 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
94
McRuer, Ashkenas, and Graham, Aircraft Dynamics
and Automatic Control