mae 331 lecture 18
TRANSCRIPT
-
8/11/2019 Mae 331 Lecture 18
1/16
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
2/16
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
( ))*+,-.
-
8/11/2019 Mae 331 Lecture 18
3/16
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
4/16
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
5/16
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
6/16
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
$%&
'()+D
"M
V#D
VM
"{ }sgM
"
LV
VN
+MV D
"s+g
L"
VN
( ) = 0
D"=0
ShortPeriod
Phugoid
L
/VNand LV/VNEffects onFourth-Order Roots
LV/VN: Damped naturalfrequency of the phugoid
Negligible effect on the short-period
L
/VN: Increased dampingof the short-period
Small effect on the phugoidmode
ShortPeriod
Phugoid
Pitch and ThrustControl Effects
-
8/11/2019 Mae 331 Lecture 18
7/16
-
8/11/2019 Mae 331 Lecture 18
8/16
Elevator-to-Normal-VelocityTransfer Function
!w(s)
!"E(s)=
n"Ew
(s)
!Lon(s)=
M"E s2 +2#$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
9/16
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
13/16
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
14/16
-
8/11/2019 Mae 331 Lecture 18
15/16
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