8/11/2019 Mae 331 Lecture 18

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Advanced Problems ofLongitudinal Dynamics

Robert Stengel, Aircraft Flight DynamicsMAE 331, 2012

Angle-of-attack-rate aero effects

Fourth-order dynamics

Steady-state response to control

Transfer functions

Frequency response

Root locus analysis of parametervariations

Numerical solution for trimmedflight condition

Nichols chart

Pilot-aircraft interactions

Copyright 2012 by Robert Stengel. A ll rights reserved. For education al use only.http://www.princeton.edu/~stengel/MAE331.html

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

Distinction Between Angle-of-Attack Rate and Pitch Rate

!! = q

!! " 0 " q; q =0

! With no vertical motion of thec.m., pitch rate and angle-of-

attack rate are the same

! With no pitching, vertical heaving(or plunging) motion of the c.m.,producesangle-of-attack rate butno pitch rate

Vertical velocitydistribution induced

by pitch rate

Angle-of-Attack RateHas Two Effects! Pressure variations at wing

convect downstream,

arriving at tail tseclater!Lag of the downwash

!Delayed tail-lift/pitch-moment effect

! Vertical force opposed by amass of air (apparentmass) as well as airplanemass

! Vertical accelerationproduces added lift andmoment

Flight Dynamics, pp. 204-206, 284-285

! !q = Mq!q+M"!"+M#E!#E+M!"!!"

!!"= 1$Lq

VN

%

&'

(

)*!q$

L"

VN( )!"$ L

#E

VN( )!#E$ L

!"

VN( )!!"

! !q"M!#!!#= Mq!q+M#!#+M$E!$E

!!#+ L !#VN( )!!#= 1"

LqVN

%

&'

(

)*!q"

L#

VN( )!#" L

$E

VN( )!$E

1 !M!"

0 1+L

!"

VN( )#

$%

&

'(

#

$

%%

%

&

'

((

(

) !q

)!"

#

$

%

%

&

'

(

(

=

Mq M"

1

!

Lq

VN

*

+,

-

./ !

L"

VN( )

#

$

%%

%

&

'

((

(

)q

)"

#

$

%

%

&

'

(

(

+

M0E

!

L0E

VN( )

#

$

%%

%

&

'

((

(

)0E

Angle-of-Attack-Rate Effects PrincipallyAffect the Short-Period Mode

! Lift and pitching moment proportional to angle-of-attack rate

! Bring effects to left side

! Vector-matrix form

8/11/2019 Mae 331 Lecture 18

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1 !M!"

0 1+L

!"

VN

( )#$%&

'(

#

$

%%%

&

'

(((

!1

=

1+L

!"

VN

( )#$%&

'( M

!"

0 1

#

$

%%%

&

'

(((

1+L

!"

VN

( )#$%&

'(

! !q

!!"

#

$

%

%

&

'

(

(

=

1 )M!"

0 1+L

!"VN( )

#$%

&'(

#

$

%%%

&

'

(((

)1M

q M

"

1)L

qVN

*

+, -

./ ) L"V

N( )

#

$

%%%

&

'

(((

!q

!"

#

$

%

%

&

'

(

(

+

M0E

) L

0EVN( )

#

$

%%%

&

'

(((

!0E

1

233

433

5

633

733

Angle-of-Attack-Rate Effects

!Inverseof the apparent mass matrix

!Pre-multiply both sides by inverse

! !q

!!"

#

$

%

%

&

'

(

(

=

1+L

!"

VN

( )#$%&

'( M

!"

0 1

#

$

%%%

&

'

(((

1+L

!"

VN( )#

$%&

'(

Mq

M"

1

)

Lq

VN

*

+,

-

./ )

L"

VN

#

$

%%

%

&

'

((

(

!q

!"

#

$

%

%

&

'

(

(

+"

0

122

322

4

522

622

Angle-of-Attack-Rate Effects

! !q

!!"

#

$%%

&

'((=

1

1+L

!"

VN

( )#$%&'(

1+L

!"

VN

( )#$%&'(M

q+ M

!" 1)

Lq

VN

*

+,

-

./

012

345

1+L

!"

VN

( )#$%&'(M

") M

!"

L"

VN

( )012345

1)L

q

VN

*

+,

-

./ )

L"

VN

#

$

%%%%%

&

'

(((((

!q

!"

#

$%%

&

'((

+

1+L

!"

VN

( )#$%&'(M

6E) M

!"

L6E

VN

( ))L

6E

VN

#

$

%%%%

&

'

((((

!6E

0

1

77777

2

77777

3

4

77777

5

77777

!Multiply matrices

!Substitute

Simplification of Angle-of-Attack-Rate Effects

! !q

! !"

#

$%%

&

'(("

Mq+ M

!"{ } M") M !" L"VN

( ){ }1 )

L"

VN

( )

#

$

%%%%%

&

'

(((((

!q

!"

#

$%%

&

'((+

M*E) M

!"

L*E

VN

( )) L

*E

VN

( )

#

$

%%%%

&

'

((((

!*E

! Typically Lq and L !! have small effects for large aircraft*

Mq and M!! are same order of magnitudeand have more significant effects

!Neglecting Lqand L

!!

* but not for small aircraft, e.g.,R/C models and micro-UAVs

2nd-Degree Characteristic Polynomial with

!Short-period characteristic polynomial

!Damping is increased

!Natural frequency is unaffected

! s( ) =s " M

q+ M

!#( )$% &' " M#"M!# L#VN( )$%( &')

"1 s+L

#

VN

( )$%(&

')

= s" Mq+ M

!#( )$% &' s +L

#

VN

( )$%(&')" M

#" M

!#

L#

VN

( )$%(&')

! s( ) = s2 + L"VN

( )# Mq + M!"( )$%&'()s+ M"#Mq

L"

VN

( )$%&'()

*+,

-./

= s2

+ 201ns+1n2

= 0

Lq and L !! " 0

= s2+

L!

VN

( )" Mq + M!!( )#$%&

'(s+ M

!" M

q+ M

!!( ) L

!

VN

( )#$%&

'(+M

!!

L!

VN

( ))*+,-.

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Linear, Time-Invariant Fourth-OrderLongitudinal Model

(Neglecting angle-of-attack-rate aero)

!!V(t)

!!"(t)

! !q(t)

!!#(t)

$

%

&&&&&

'

(

)))))

=

*DV

*g 0 *D#

LVVN

0 0 L#

VN

MV

0 Mq M#

*LVVN

0 1 *L#VN

$

%

&&&&&&&

'

(

)))))))

!V(t)

!"(t)

!q(t)

!#(t)

$

%

&&&&&

'

(

)))))

+

0 T+T 0

0 0 L+F/ VN

M+E 0 0

0 0 *L+F/ VN

$

%

&&&&&

'

(

)))))

!+E(t)

!+T(t)

!+F(t)

$

%

&&&

'

(

)))

Stability and control derivatives are defined at atrimmed (equilibrium) flight condition

Perturbations to the Trimmed Condition

Initial pitch rate [q(0)] = 0.1 rad/s Elevator step input [ E(0)] = 1 deg

Small linear and nonlinearperturbations are virtually identical

Steady-State Response

Steady-State Response of the 4th-OrderLTI Longitudinal Model

!VSS!"SS!qSS!#SS

$

%

&&&&&

'

(

)))))

=*

*DV *g 0 *D#LV

VN0 0

L#VN

MV 0 Mq M#

*LVVN

0 1 *L#VN

$

%

&&&&&&&

'

(

)))))))

*1

0 T+T 0

0 0 L+F/VN

M+E 0 0

0 0 *L+F/VN

$

%

&&&&&

'

(

)))))

!+ESS

!+TSS

!+FSS

$

%

&&&

'

(

)))

!xSS

= "F"1G!u

SS

How do we calculate the equilibrium response to control?

!!x(t) = F!x(t)+G!u(t)

For the longitudinal model

8/11/2019 Mae 331 Lecture 18

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Algebraic Equation forEquilibrium Response

!VSS

!"SS

!qSS

!#SS

$

%

&&&&&

'

(

)))))

=

*gM+EL#

VN

$%&

'()

0 gM#L+F/ VN[ ]

DV

L#

VN*D

#

LV

VN( )M

+E

$

%&

'

() M

V

L#

VN*M

#

LV

VN( )T+T

$

%&

'

() D

#M

V

*DV

M#( )

L+F

/ VN

$%

'(

0 0 0

*gM+EL

V

VN

$%&

'()

0 L+F/ VN[ ]

$

%

&&&

&&&&&

'

(

)))

)))))

g MVL#

VN*M#

LV

VN( )

!+ESS

!+TSS

!+FSS

$

%

&&&

'

(

)))

!VSS

!"SS

!qSS

!#SS

$

%

&&&&&

'

(

)))))

=

a 0 b

c d e

0 0 0

f 0 g

$

%

&&&&

'

(

))))

!*ESS

!*TSS

!*FSS

$

%

&&&

'

(

)))

Roles of stability and controlderivatives identified

Result is a simple equation relatinginput and output

4th-Order Steady-State Response MayBe Counterintuitive!VSS =a!"ESS + 0( )!"TSS + b!"FSS!#SS =c!"ESS +d!"TSS+ e!"FSS

!qSS = 0( )!ESS + 0( )!"TSS + 0( )!"FSS!$SS = f!"ESS+ 0( )!"TSS+ g!"FSS

Observations Thrust command

Elevator and flap commands

Steady-state pitch rate is zero

4th-order model neglects air density gradient effects

!"SS = !#SS + !$SS = c + f( )!%ESS +d!%TSS + e +g( )!%FSS

Steady-state pitch angle

Effects of Stability DerivativeVariations on 4th-Order

Longitudinal Modes

Primary and Coupling Blocks of theFourth-Order Longitudinal Model

FLon =

!DV !g 0 !D"

LVVN 0 0 L"

VN

MV 0 Mq M"

!LVVN

0 1 !L"VN

#

$

%

%%%%%%

&

'

(

((((((

=

FPh FSPPh

FPhSP

FSP

#

$

%%

&

'

((

Some stability derivatives appear only in primaryblocks (DV, Mq, M) Effects are well-described by 2nd-order models

Some stability derivatives appear only in couplingblocks (MV, D) Effects are ignored by 2nd-order models

Some stability derivatives appear in both (LV, L)

Require 4th

-order modeling

8/11/2019 Mae 331 Lecture 18

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M

Effect on 4th-Order Roots

ShortPeriod

ShortPeriod

Phugoid

Phugoid

!Lon

(s)= s4+ D V+

L"

VN#M

q( ) s3

+ g#D"

( )LV VN+D

V

L"

VN#M

q( )#Mq L" VN #M"o$%&

'()

s2

+ Mq

D"# g( )LV

VN#D

V

L"

VN

$

%&

'

()+D"MV# DVM"o

{ }s

+ g MVL

"

VN

#M"o

LV

VN

( )#!M" s2 +DVs+ gLV VN

( )* d(s)+ kn(s)

Group all terms multiplied by M

to form numerator for M #

M

Effect on 4th-Order RootsPrimary effect: The same as in the approximate short-

period model

Numerator zeros The same as the approximate phugoid mode characteristic

polynomial

Effect of M

variation on phugoid mode is small#

ShortPeriod

ShortPeriod

Phugoid

Phugoid

Direct Thrust Effect on SpeedStability, TV

In steady, level flight, nominal thrust balances nominal drag

!T

!V=

0

TN! D

N = C

TN

1

2

"VN

2S! C

DN

1

2

"VN

2S = 0

Effect of velocity change

Small velocity perturbation grows if Therefore

propelleris stabilizing for velocity change

turbojethas neutral effect

ramjetis destabilizing

Pitching Moment Due to Thrust, MV Thrust line above or below center of mass induces a pitching moment

Aerodynamic and thrust pitching moments sensitive to velocityperturbation

Couples phugoid and short-period modes

Martin XB-51

McDonnell Douglas MD-11

Consolidated PBY

Fairchild-Republic A-10

8/11/2019 Mae 331 Lecture 18

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Negative "M/"V(Pitch-down effect) tends to increase velocity

Positive "M/"V(Pitch-up effect) tends to decrease velocity

With propeller thrust line above the c.m., increased velocitydecreases thrust, producing a pitch-up moment

Tilting the thrust line can have benefits

Up: Lake Amphibian, MD-11

Down: F6F, F8F, AD-1!M

thrust

!V"

!T

!V# Moment Arm

Douglas AD-1

Lake Amphibian

Grumman F8F

McDonnell DouglasMD-11

Pitching Moment Due to Thrust, MVMVEffect on 4

th-OrderRoots

Large positive value producesoscillatory phugoid instability

Large negative value producesreal phugoid divergence

!Lon

(s)= s4+ D

V+L

"

VN#M

q

( )s3

+ g#D"

( )L

V

VN

+DV

L"

VN

#Mq( )#Mq L"V

N

#M"

$

%&'

()s2

+ Mq D

"# g( )

LV

VN

#DV

L"

VN

$%&

'()+D

"M

V#D

VM

"{ }sgM

"

LV

VN

+MV D

"s+g

L"

VN

( ) = 0

D"=0

ShortPeriod

Phugoid

L

/VNand LV/VNEffects onFourth-Order Roots

LV/VN: Damped naturalfrequency of the phugoid

Negligible effect on the short-period

L

/VN: Increased dampingof the short-period

Small effect on the phugoidmode

ShortPeriod

Phugoid

Pitch and ThrustControl Effects

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8/11/2019 Mae 331 Lecture 18

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Elevator-to-Normal-VelocityTransfer Function

!w(s)

!"E(s)=

n"Ew

(s)

!Lon(s)=

M"E s2 +2#$ns +$n2( )Approx Ph s %z3( )s2 +2#$ns +$n

2( )Ph

s2 +2#$ns +$n2( )

SP

Normal velocity transfer function is analogousto angle of attack transfer function(! #!w/VN)

z3often neglected due to high frequency

Elevator-to-Normal-VelocityFrequency

Response

!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 (but notquite) cancelsphugoidresponse

Elevator-to-Pitch-RateNumerator and Transfer Function

HxAdj sI ! FLon( )G = 0 0 1 0"# $%

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) nV'

(s) nV'

(s) nV'

(s)

"

#

((((((

$

%

))))))

00

M*E

0

"

#

((((

$

%

))))

=n*Eq

(s)

!q(s)

!"E(s)=

n"Eq

(s)

!Lon

(s)#

M"Es s $z1( ) s $z2( )s2+2%&ns +&n

2( )Ph

s2+2%&ns +&n

2( )SP

Free s

in numerator differentiatespitch angle transfer function

Elevator-to-Pitch-Rate FrequencyResponse

+20 dB/dec

+20 dB/dec+40 dB/dec

0 dB/dec20 dB/dec

!qSS= 0

( )!"E

SS+ 0

( )!"T

SS+ 0

( )!"F

SS

(n q) = 1Negligible low-

frequency response,except at phugoidnatural frequency

High-frequencyresponse wellpredicted by 2nd-order model

!q(s)

!"E(s)=

n"E

q(s)

!Lon

(s)#

M"E

s s $z1( ) s $z2( )s

2+2%&ns +&n

2( )Ph

s2+2%&n s +&n

2( )SP

8/11/2019 Mae 331 Lecture 18

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Transfer Functions of Elevator Inputto Angle Output*

!"(s)

!#E(s)=

n#E

"(s)

!Lon

(s); n

#E

"(s)= M

#E s+ 1

T"1

$

%&

'

() s+ 1

T"2

$

%&

'

()

!"(s)

!#E(s)=

n#E"

(s)

!Lon(s); n#E

"(s)=M#E s

2+2$%ns+%n

2( )Approx Ph

!"(s)

!#E

(s)

=

n#E"

(s)

!Lon(s)

; n#E"

(s)= M#EL$

VN

s+ 1T"

1

%

&

'(

)

*

Elevator-to-Flight Path Angle transfer function

Elevator-to-Angle of Attack transfer function

Elevator-to-Pitch Angle transfer function

* Flying qualities notation for zero time constants

Frequency Response ofAngles to Elevator Input

Pitch angle frequencyresponse (!%= !$+ !)

Similar to flight pathangle near phugoidnatural frequency

Similar to angle ofattack near short-period naturalfrequency

!"SS =c!#ESS

!$SS = f!#ESS

!%SS = c & f( )!#ESS

Transfer Functions of ThrustInput to Angle Output

!"(s)

!#T(s)=

n#T

" (s)

!Lon

(s); n

#T

" (s) = T#T

s + 1T

"T

$%&

'()

!"(s)

!#T(s)=

n#T

"(s)

!Lon

(s); n

#T

"(s) =T

#Ts s + 1

T"T

$%&

'()

!"(s)

!#T(s)=

n#T" (s)

!Lon

(s); n#T

"(s) =T#T

LV

VNs

2+2$%

ns +%

n

2( )Approx SP

Thrust-to-Flight Path Angle transfer function

Thrust-to-Angle of Attack transfer function

Thrust-to-Pitch Angle transfer function

Frequency Response of Anglesto Thrust Input

Primarily effects flight path angle and low-frequency pitch angle

8/11/2019 Mae 331 Lecture 18

10/16

Gain and Phase Margins:The Nichols Chart

Nichols Chart:Gain vs. Phase Angle

Bode Plot

Two plots

Open-Loop Gain (dB) vs. log10&

Open-Loop Phase Angle vs. log10

Nichols Chart

Single crossplot; inputfrequency not shown

Open-Loop Gain (dB) vs. Open-

Loop Phase Angle

Gain and Phase Margins

Gain Margin At the input frequency, &, for which '(j&) = 180

Difference between 0 dBand transfer function magnitude,

20 log10AR(j&)

Phase Margin At the input frequency, &, for which 20 log10AR(j&) = 0 dB

Difference between the phase angle '(j&), and180

Axis intercepts on the Nichols Chart identify GMand PM

Examples of Gain and Phase Margins

Bode Plot# Nichols Chart

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!

( )+10

2

"

#

$

$

%

&

'

'

100

j!+ 100

( )

"

#

$%

&

'

8/11/2019 Mae 331 Lecture 18

11/16

Gain and Phase Margins inPitch-Tracking Task

Elevator-to-Pitch-AngleBode Plot

Elevator-to-Pitch-AngleNichols Chart

Amplitude Ratio vs. Phase Angle

Gain Margin:Amplitude ratio below 0 dBwhen phase angle = 180

Phase Margin:Phase angle above 180when amplitude ratio = 0 dB

Pilot-Vehicle Interactions

Pilot Inputs to Control

* p. 421-425, Flight Dynamics

Effect of Pilot Dynamics onPitch-Angle Control Task

Pilot Transfer Function =!u s( )!"

s( )

= KP1/TP

s+1/

TP

= KP1/ 0.25

s+1/ 0.25

Pilot introduces neuromuscular lag while closing the control loop

Example

Model the lag by a 1st-order time constant, TP, of 0.25 s

Pilots gain, KP, is either 1 or 2

8/11/2019 Mae 331 Lecture 18

12/16

Open-Loop Pilot-AircraftTransfer Function

H(s) = KP

1 /TP

s +1 /TP( )

!

"#

$

%&

M'E s +

1T(1

)*+

,-. s + 1

T(2

)*+

,-.

s2+2/0

ns +0

n

2( )Ph

s2+2/0

ns +0

n

2( )SP

!

"

####

$

%

&&&&

Effect of PilotDynamics onPitch-Angle

Control Task

Gain and phasemargins becomenegative for pilotgain between 1and 2

Then, pilotdestabilizes thesystem (PIO)

Effect of Pilot Dynamics on Elevator/Pitch-Angle Control Root Locus

Pilot transfer function changes asymptotes of the root locus

Configuration Effects

8/11/2019 Mae 331 Lecture 18

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Pitch Up Crossplot CLvs. Cm

Positive break in Cmdue to forward movementof center of pressure, decreasing static margin

F-100 crash due to tip stallhttp://www.youtube.com/watch?v=NyJkKcXYqSU&feature=related

North American F-100

Sweep Effect on Pitch MomentCoefficient, CLvs. Cm

c/4= 0

Low center of pressure(c.p.) in front of the quarterchord

Stable break at stall (c.p.moves aft)

c/4= 15

Low c.p. aft of the quarter-chord

Stable break at stall (c.p.moves aft)

c/4= 30

Low c.p. aft of the quarter-chord

Unstable break at stall (c.p.moves forward)

Outboard wing stalls beforeinboard wing (tip stall)

Cm()

CL()

Stall

NACA TR-1339

StableBreak

Stall UnstableBreak

Pitch Up and Deep Stall, Cmvs.

Possibility of 2stable equilibrium(trim) points with

same controlsetting

Low

High #

High-angle trim iscalled deep stall Low lift

High drag

Large controlmoment required toregain low-angletrim

Shortal-MagginLongitudinal

Stability Boundaryfor Swept Wings

Stable or unstable pitchbreak at the stall

Stability boundary isexpressed as a function of

Aspect ratio

Sweep angle of thequarter chord

Taper ratio

AR

c/4NACA TR-1339

8/11/2019 Mae 331 Lecture 18

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8/11/2019 Mae 331 Lecture 18

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Numerical Solution to Estimate theTrimmed Condition for Level Flight

Specify desired altitude and airspeed, hNand VN

Guess starting values for the trim parameters, T0, E0, and %0#

Calculate starting values of f1, f2, and f3

f1, f2, and f3= 0 in equilibrium, but not for arbitrary T0, E0, and %0

Define a scalar, positive-definite trim error cost function, e.g.,

f1=

1

mT !T,!E,",h,V( )cos #+ i( )$D !T,!E,",h,V( )%& '(

f2 =

1

mVNT !T,!E,",h,V( )sin #+ i( )+L !T,!E,",h,V( )$mg%& '(

f3= M !T,!E,",h,V( ) /Iyy

J !T,!E,"( ) = a f12( )+b f22( )+ c f32( )

Minimize the Cost Function withRespect to the Trim Parameters

Cost is minimized at bottom of bowl, i.e., when

!J

!"T

!J

!"E

!J

!#

$

%&

'

()= 0

Error cost is bowl-shaped

Search to find the minimum value of J

J !T,!E,"( ) = a f12( )+b f22( )+ c f32( )

Example of Search for TrimmedCondition (Fig. 3.6-9, Flight Dynamics)

In MATLAB, usefminsearch[Nelder-Mead Downhill Simplex Method]to find trim settings

!T*,!E*,"*( ) = fminsearch J, !T,!E,"( )#$ %&

Airspeed Frequency Response toElevator and Thrust Inputs

Response is primarily through the lightly dampedphugoid mode

8/11/2019 Mae 331 Lecture 18

16/16

Altitude Frequency Responseto Elevator and Thrust Inputs

Altitude perturbation:Integral of the flight path angle perturbation

!z(t) ="VN

!#($)d$0

t

%

!z(s)

!"E(s)

=# V

N

s

$

%&

'

()

!*(s)

!"E(s)

!z(s)

!"T(s)

=# V

N

s

$

%&

'

()

!*(s)

!"T(s)

High- and Low-Frequency Limits ofFrequency Response Function

Hij(j"# 0)#kij j"( )

q+ b

q$1 j"( )q$1

+ ...+ b1 j"( )+ b0[ ]j"( )

n+a

n$1 j"( )n$1

+ ...+a1 j"( )+a0[ ]

#

kijb0

a0

, b0%0

kijj(0)b

1

a0

, b0= 0,b

1%0,etc.

&

'((

)((

Hij(j") =AR(")ej#(")

Hij(j"#$)#

kij j"( )q+bq%1 j"( )

q%1+...+b

1 j"( ) +b0[ ]

j"( )n+an%1 j"( )

n%1+...+a1 j"( ) +a0[ ]

#kij

j"( )n%q

Feedback Control: Angles to Elevator

Variations incontrol gain

Principal effect ison short-periodroots