Transfer Functions andFrequency Response

Robert Stengel, Aircraft Flight Dynamics

MAE 331, 2010

• Frequency domain view of initial

condition response

• Response of dynamic systems

to sinusoidal inputs

• Transfer functions

• Bode plots

Copyright 2010 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE331.html

http://www.princeton.edu/~stengel/FlightDynamics.html

Laplace Transform ofInitial Condition Response

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)

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)[ ]Initial

Condition

Control/Disturbance

Input

4th-Order Initial

Condition Response

• Longitudinal dynamic model (time domain)

!x(s) = sI " F[ ]"1!x(0)

! !V (t)

! !" (t)

! !q(t)

! !#(t)

$

%

&&&&&

'

(

)))))

=

*DV *g *Dq *D#

LVVN

0Lq

VN

L#VN

MV 0 Mq M#

* LVVN

0 1 * L#VN

$

%

&&&&&&&&

'

(

))))))))

!V (t)

!" (t)

!q(t)

!#(t)

$

%

&&&&&

'

(

)))))

,

!V (0)

!" (0)

!q(0)

!#(0)

$

%

&&&&&

'

(

)))))

given

• Longitudinal model (frequency domain)

!V (s)

!" (s)

!q(s)

!#(s)

$

%

&&&&&

'

(

)))))

= sI * FLon[ ]

*1

!V (0)

!" (0)

!q(0)

!#(0)

$

%

&&&&&

'

(

)))))

Elements of the CharacteristicMatrix Inverse

sI ! FLon

" #Lon(s)

= s4+ a

3s3+ a

2s2+ a

1s + a

0

Adj sI ! FLon( ) =

nVV(s) n"

V(s) nq

V(s) n#

V(s)

nV"(s) n"

"(s) nq

"(s) n#

"(s)

nVq(s) n"

q(s) nq

q(s) n#

q(s)

nV#(s) nV

#(s) nV

#(s) nV

#(s)

$

%

&&&&&&

'

(

))))))

sI ! FLon[ ]!1

=Adj sI ! FLon( )sI ! FLon

=Adj sI ! FLon( )

"Lon (s)(4 x 4)

• Denominator is scalar

• Numerator is an (n x n) matrix

Characteristic Matrix InverseDistributes Effects of Initial Condition

sI ! FLon[ ]

!1=

nV

V(s) n"

V(s) n

q

V(s) n#

V(s)

nV

"(s) n"

"(s) n

q

"(s) n#

"(s)

nV

q(s) n"

q(s) n

q

q(s) n#

q(s)

nV

#(s) n

V

#(s) n

V

#(s) n

V

#(s)

$

%

&&&&&&

'

(

))))))

s4+ a

3s3+ a

2s2+ a

1s + a

0

(4 x 4)

• Denominator determines the modes of motion

• Numerator distributes the initial condition to eachelement of the state

!x(s) =Adj sI " FLon( )sI " FLon

!x(0)

Relationship to State

Transition Matrix• 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) = sI " F[ ]"1!x(0) = L !x(t)[ ] = L # t,0( )!x(0)$% &'

sI ! F[ ]!1= L " t,0( )#$ %&

• Therefore,

Initial Condition Response of aSingle State Element

• Define A s( ) ! sI ! F[ ]

!1

• Typical (ijth) element of A(s)

aij (s) =kijnij (s)

!(s)=kij s

q+ bq"1s

q"1+ ...+ b

1s + b

0( )sn+ an"1s

n"1+ ...+ a

1s + a

0( )

=kij s " z1( )

ijs " z

2( )ij... s " zq( )

ij

s " #1( ) s " #

2( )... s " #n( )

Initial Condition Response of aSingle State Element

pi (s) = ki1 sq1 + ...+ b

0( )1!x

1(0) + ki2 s

q2 + ...+ b0( )2!x

1(0) +!+ kin s

qn + ...+ b0( )

n!xn (0)

= kpi sqmax + ...+ b

0( )

• Terms sum to produce a single numerator polynomial

!xi(s) = a

i1s( )!x1(0) + ai2 s( )!x1(0) +!+ a

ins( )!xn (0) "

pis( )

! s( )

• Initial condition response of !xi(s)

Partial Fraction Expansion ofthe Initial Condition Response

• Scalar response can be expressed with n parts,each containing a single mode

!xi (s) =pi s( )

! s( )=

d1

s " #1( )+

d2

s " #2( )

+!dn

s " #n( )

$

%&'

()i

, i = 1,n

where, for each i, the (possibly complex) coefficients are

d j = s ! " j( )pi s( )

# s( )s=" j

, j = 1,n

Partial Fraction Expansion ofthe Initial Condition Response

• Time response is the inverse Laplace transform

!xi(t) = L

"1 !xi(s)[ ] = L"1

d1

s " #1( )+

d2

s " #2( )

+!dn

s " #n( )

$

%&

'

()i

= d1e#1t + d

2e#2 t +!+ d

ne#nt( )

i, i = 1,n

... and the time response contains every mode of the system

Scalar and MatrixTransfer Functions

Response to a Control Input

• 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)

Transfer Function Matrix

• Frequency-domain effect of all inputs on alloutputs

• Assume control effects do not appear directlyin the output: Hu = 0

• Transfer function matrix

H (s) = HxsI ! F[ ]

!1G =

HxAdj sI ! F( )G

sI ! F

r ! n( ) n ! n( ) n ! m( )

= r ! m( )

First-OrderTransfer Function

y s( )

u s( )= H (s) = h s ! f[ ]

!1g =

hg

s ! f( )(n = m = r = 1)

• Scalar transfer function(= first-order lag)

!x t( ) = fx t( ) + gu t( )

y t( ) = hx t( )

• Scalar dynamic system

Second-Order

Transfer Function

H(s) = HxsI ! F[ ]

!1G =

h11 h12

h21 h22

"

#$$

%

&''

adjs ! f11 ! f12! f21 s ! f22

"

#$$

%

&''

dets ! f11 ! f12! f21 s ! f22

(

)**

+

,--

g11 g12

g21 f22

"

#$$

%

&''

(n = m = r = 2)

• Second-order transfer function matrix

r ! n( ) n ! n( ) n ! m( )

= r ! m( ) = 2 ! 2( )

!x t( ) =!x1t( )

!x2t( )

!

"

##

$

%

&&=

f11

f12

f21

f22

!

"

##

$

%

&&

x1t( )

x2t( )

!

"

##

$

%

&&+

g11

g12

g21

f22

!

"

##

$

%

&&

u1t( )

u2t( )

!

"

##

$

%

&&

y t( ) =y1t( )

y2t( )

!

"

##

$

%

&&=

h11

h12

h21

h22

!

"

##

$

%

&&

x1t( )

x2t( )

!

"

##

$

%

&&

• Second-order dynamic system

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

xLonsI ! F

Lon[ ]!1G

Lon

=

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

"

#

$$$$

%

&

''''

nV

V(s) n(

V(s) n

q

V(s) n)

V(s)

nV

((s) n(

((s) n

q

((s) n)

((s)

nV

q(s) n(

q(s) n

q

q(s) n)

q(s)

nV

)(s) n(

)(s) n

q

)(s) n)

)(s)

"

#

$$$$$$

%

&

''''''

0 T*T 0

0 0 L*F /VN

M*E 0 0

0 0 !L*F /VN

"

#

$$$$$

%

&

'''''

+Lon

s( )

Pilatus TurboPorter PC-6

Longitudinal TransferFunction Matrix

HLon (s) =

n!EV(s) n!T

V(s) n!F

V(s)

n!E"(s) n!T

"(s) n!F

"(s)

n!Eq(s) n!T

q(s) n!F

q(s)

n!E#(s) n!T

#(s) n!F

#(s)

$

%

&&&&&&

'

(

))))))

s2+ 2*P+nPs ++nP

2( ) s2 + 2*SP+nSPs ++nSP

2( )

• There are 4 outputs and 3 inputs

!V (s)

!" (s)

!q(s)

!#(s)

$

%

&&&&&

'

(

)))))

= HLon(s)

!*E(s)

!*T (s)

!*F(s)

$

%

&&&

'

(

)))

• Input-output relationship

Douglas AD-1 Skyraider

Scalar Transfer Function from!uj to !yi

Hij (s) =kijnij (s)

!(s)=kij s

q+ bq"1s

q"1+ ...+ b

1s + b

0( )sn+ an"1s

n"1+ ...+ a

1s + a

0( )

# zeros = q

# poles = n

• Denominator polynomial contains n roots

• Each numerator term is a polynomial with q zeros,where q varies from term to term and ! n – 1

=kij s ! z1( )

ijs ! z

2( )ij... s ! zq( )

ij

s ! "1( ) s ! "

2( )... s ! "n( )

Scalar Frequency ResponseFunction

Hij (j! ) =kij j! " z

1( )ijj! " z

2( )ij... j! " zq( )

ij

j! " #1( ) j! " #

2( )... j! " #n( )

• Substitute: s = j!

• Frequency response is a complex function ofinput frequency, !

– Real and imaginary parts

– Amplitude ratio and phase angle

= a(! ) + jb(! )" AR(! ) e j# (! )

Transfer Function Matrix forShort-Period Approximation

• Transfer Function Matrix (with Hx = I, Hu = 0)

HSP (s) = HxsI ! F[ ]

!1G =

s ! Mq( ) !M"

! 1!LqVN

#$%

&'(

s +L"VN( )

)

*

++++

,

-

.

.

.

.

!1

M/E

!L/EVN

)

*

+++

,

-

.

.

.

= HxsI ! F[ ]

!1{ }G =

s +L"VN( ) M"

1!LqVN

#$%

&'(

s ! Mq( )

)

*

+++++

,

-

.

.

.

.

.

s ! Mq( ) s + L"VN( ) ! M" 1!

LqVN

#$%

&'(

M/E

!L/EVN

)

*

+++

,

-

.

.

.

!!xSP =! !q! !"

#

$%%

&

'(()

Mq M"

1*LqVN

+,-

./0

* L"VN

#

$

%%%

&

'

(((

!q!"

#

$%%

&

'((+

M1E

*L1EVN

#

$

%%%

&

'

(((!1E

• Dynamic equation

Transfer Function Matrix forShort-Period Approximation

HSP (s) =

M!E s +L"VN( ) # L!EM"

VN

$%&

'()

M!E 1#LqVN

*+,

-./# L!E

VN( ) s # Mq( )$%&

'()

$

%

&&&&&

'

(

)))))

s2+ #Mq +

L"VN( )s # M" 1#

LqVN

*+,

-./+ Mq

L"VN

$%&

'()

=

M!E s +L"VN

# L!EM"VNM!E

( )$%&

'()

# L!EVN( ) s +

VNM!E

L!E

1#LqVN

*+,

-./# Mq

$

%&

'

()

012

32

452

62

$

%

&&&&&

'

(

)))))

7SP s( )

HSP (s) !

kqn!Eq(s)

k"n!E"(s)

#

$

%%

&

'

((

s2+ 2)SP*nSP

s +*nSP

2=

+q(s)+!E(s)

+"(s)+!E(s)

#

$

%%%%%

&

'

(((((

• In this case, transfer function matrix dimension is (2 x 1)

Short-Period Motions

Dornier Do-128 Demonstrationhttp://www.youtube.com/watch?v=3hdLXE0rc9Q

Simulator Demonstration of Short Period Response to Elevator Deflection

http://www.youtube.com/watch?v=1O7ZqBS0_B8

Dornier Do-128D

• Rapid damping

• Pitch angle and rate response

• Flight path angle reoriented by differencebetween pitch angle and angle of attack

Scalar Transfer

Functions for Short-

Period Approximation

!q(s)!"E(s)

=

M"E s +L#VN

$ L"EM#VNM"E

( )%&'

()*

s2+ $Mq +

L#VN( )s $ M# 1$

LqVN

+,-

./0+ Mq

L#VN

%

&'(

)*

=kq s $ zq( )

s2+ 21SP2nSP

s +2nSP

2

!q(s)

!"(s)

#

$%%

&

'((=

!q(s)!)E(s)

!"(s)!)E(s)

#

$

%%%%

&

'

((((

!)E(s)

!"(s)!#E(s)

=

$ L#EVN( ) s +

VNM#E

L#E

1$LqVN

%&'

()*$ Mq

+

,-

.

/0

123

43

563

73

s2+ $Mq +

L"VN( )s $ M" 1$

LqVN

%&'

()*+ Mq

L"VN

+,-

./0

=k" s $ z"( )

s2+ 28SP9nSP

s +9nSP

2

• Pitch Rate Transfer Function

• Angle of Attack Transfer Function

Short Period

Frequency Response

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" j# % z"( )%# 2

+ 2&SP#nSPj# +#nSP

2

= AR" (# ) ej'" (# )

Bode Plot

Angle and RateResponse of a

DC Motor overWide Input-FrequencyRange

! Long-term responseof a dynamic systemto sinusoidal inputsover a range offrequencies

! Determineexperimentally or

! from the Bode plotof the dynamicsystem

Very low damping

Moderate damping

High damping

Bode Plot Portrays Response

to Sinusoidal Control Input

• Express amplitude ratio in decibels

AR(dB) = 20log10 AR original units( )[ ]

20 dB = factor of 10

!q( j")

!#E( j")=

kq j" $ zq( )$" 2

+ 2% SP"nSPj" + "nSP

2= ARq (") e

j&q (" )

• Plot AR(dB) against log10(!input)

• Plot phase angle, " (deg) againstlog10(!input)

Products in original units

are sums in decibels

# zeros = 1

# poles = 2

Constant Gain Bode Plot

H( j!) =1

H( j!) =10

H( j!) =100

• Amplitude ratio, AR(dB)= constant

• Phase angle, " (deg) = 0

y t( ) = hu t( )

Integrator Bode Plot

H( j!) =1

j!

H( j!) =10

j!

• Slope of AR(dB)= –20 dB/decade

• " (deg) = –90°

y t( ) = h u t( )dt0

t

!

Differentiator Bode Plot

H( j!) = j!

H( j!) =10 j!

• Slope of AR(dB)= +20 dB/decade

• " (deg) = +90°

y t( ) = hdu t( )

dt

Bode Plots of First-Order Lags

H( j!) =10

j! +10( )

H( j!) =100

j! +10( )

H( j!) =100

j! +100( )

Bode Plots of First-Order Lags

• General shape of amplituderatio governed byasymptotes

• Slope of asymptoteschanges by multiples of ±20dB/dec at poles or zeros

• Phase angle of a real,negative pole

– When ! = 0, " = 0°

– When ! = #, " =–45°

– When " -> ", " -> –90°

• AR asymptotes of a real pole– When ! = 0, slope = 0

dB/dec

– When ! # #, slope = –20dB/dec

Bode Plots of Second-Order Lags

Effect of

Damping Ratio

Hgreen ( j!) =10

2

j!( )2

+ 2 0.1( ) 10( ) j!( ) +102

Hblue ( j!) =10

2

j!( )2

+ 2 0.4( ) 10( ) j!( ) +102

Hred ( j!) =10

2

j!( )2

+ 2 0.707( ) 10( ) j!( ) +102

Bode Plots of Second-Order Lags

Hred ( j! ) =10

2

j!( )2+ 2 0.1( ) 10( ) j!( ) +102

Hgreen ( j! ) =10

3

j!( )2+ 2 0.1( ) 10( ) j!( ) +102

Hblue( j! ) =100

2

j!( )2+ 2 0.1( ) 100( ) j!( ) +1002

Effects of Gain and Natural Frequency

Amplitude Ratio Asymptotes

of Second-Order Bode Plots

• AR asymptotes of apair of complex poles

– When ! = 0, slope= 0 dB/dec

– When ! # !n,slope = –40dB/dec

• Height of resonantpeak depends ondamping ratio

Phase Angles of Second-Order

Bode Plots

• Phase angle of a pairof complex negativepoles

– When ! = 0, " = 0°

– When ! = !n, " =–90°

– When " -> ", " ->–180°

• Abruptness of phaseshift depends ondamping ratio

FrequencyResponse AR

Departures in theVicinity of Poles

• Difference betweenactual amplitude ratio(dB) and asymptote =departure (dB)

• Results for multipleroots are additive

• from McRuer, Ashkenas,and Graham, AircraftDynamics and AutomaticControl, PrincetonUniversity Press, 1973

• Zero departures haveopposite sign

First- and Second-Order Departures

from Amplitude Ratio Asymptotes

First- and Second-

Order Phase Angles

Phase Angle

Variations in theVicinity of Poles

• Results for multipleroots are additive

• from McRuer, Ashkenas,and Graham, AircraftDynamics and AutomaticControl, PrincetonUniversity Press, 1973

• LHP zero variations haveopposite sign

• RHP zeros have samesign

Next Time:Configuration and Power Effects

on Flight StabilitySupplementary

Material

Bode Plots of 1st- and 2nd-Order Lags

Hred ( j!) =10

j! +10( )

Hblue ( j!) =100

2

j!( )2

+ 2 0.1( ) 100( ) j!( ) +1002

Bode Plots of 3rd-Order Lags

Hblue ( j!) =10

j! +10( )

"

# $

%

& '

1002

j!( )2

+ 2 0.1( ) 100( ) j!( ) +1002

"

#

$ $

%

&

' '

Hgreen ( j!) =10

2

j!( )2

+ 2 0.1( ) 10( ) j!( ) +102

"

#

$ $

%

&

' '

100

j! +100( )

"

# $

%

& '

Bode Plot of a 4th-Order Systemwith No Zeros

H ( j! ) =12

j!( )2+ 2 0.05( ) 1( ) j!( ) +12

"

#$$

%

&''

1002

j!( )2+ 2 0.1( ) 100( ) j!( ) +1002

"

#$$

%

&''

• Resonant peaks andlarge phase shifts ateach natural frequency

• Additive AR slope shiftsat each naturalfrequency

# zeros = 0

# poles = 4

Left-Half-Plane Transfer Function Zero

H ( j! ) = j! +10( )

• Zeros are numerator singularities H (j! ) =k j! " z

1( ) j! " z2( )...

j! " #1( ) j! " #

2( )... j! " #n( )

• Single zero in left halfplane

• Introduces a +20dB/dec slope

• Produces phase leadin vicinity of zero

Right-Half-Plane Transfer Function Zero

H ( j! ) = " j! "10( )

• Single zero in right halfplane

• Introduces a +20 dB/decslope

• Produces phase lag invicinity of zero

Second-Order Transfer Function Zero

H( j!) = j! " z( ) j! " z*( ) = j!( )

2+ 2 0.1( ) 100( ) j!( ) +100

2[ ]

• Complex pair ofzeros produces anamplitude ratio“notch” at its“natural frequency”

4th-Order Transfer Function with

2nd-Order Zero

H( j!) =j!( )

2+ 2 0.1( ) 10( ) j!( ) +10

2[ ]j!( )

2+ 2 0.05( ) 1( ) j!( ) +1

2[ ] j!( )2

+ 2 0.1( ) 100( ) j!( ) +1002[ ]

Elevator-to-Normal-VelocityFrequencyResponse

!w(s)

!"E(s)=n"Ew(s)

!Lon (s)#

M"E s2+ 2$%ns +%n

2( )Approx Ph

s & z3( )

s2+ 2$%ns +%n

2( )Ph

s2+ 2$%ns +%n

2( )SP

0 dB/dec

+40 dB/dec

0 dB/dec

–40 dB/dec

–20 dB/dec

• (n – q) = 1

• Complex zeroalmost (butnot quite)cancelsphugoidresponse

Short PeriodPhugoid