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( )ij s " z2( )ij ... s " zq( )ij

s " #1( ) s " #2( )... s " #n( )

Initial Condition Response of aSingle State Element

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

0( )1 !x1(0) + ki2 sq2 + ...+ b

0( )2 !x1(0) +!+ kin sqn + ...+ b

0( )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) +!+ ain s( )!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+nSP s ++nSP2( )

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( )ij s ! z2( )ij ... s ! zq( )ij

s ! "1( ) s ! "2( )... s ! "n( )

Scalar Frequency ResponseFunction

Hij (j! ) =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 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) = Hx sI ! 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!E1#

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#E1$

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"nSP j" +"nSP

2

= ARq (" ) ej&q (" )

!"( j# )

!$E( j# )=

k" j# % z"( )%# 2 + 2&SP#nSP j# +#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"nSP j" +"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!( ) +1002[ ]

Complex pair ofzeros produces anamplitude rationotch at itsnatural frequency

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[ ]

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