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