Advanced Problems of
Longitudinal DynamicsRobert Stengel, Aircraft Flight Dynamics
MAE 331, 2010
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 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
Linear, Time-Invariant Fourth-Order
Longitudinal Model
! !V (t)
! !" (t)
! !q(t)
! !#(t)
$
%
&&&&&
'
(
)))))
=
*DV
*g 0 *D#LV
VN
0 0L#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
Initial-Condition and Step-
Input Responses of aBusiness Jet Aircraft
Initial pitch rate [!q(0)] = 0.1 rad/s Elevator step input [!!E(0)] = 1 deg
Trimmed Solution of theEquations of Motion
Flight Conditions for
Steady, Level Flight
!V = f1=1
mT cos ! + i( ) " D " mgsin#$% &'
!# = f2=1
mVT sin ! + i( ) + L " mgcos#$% &'
!q = f3= M / Iyy
!! = f4= !( " !# = q "
1
mVT sin ! + i( ) + L " mgcos#$% &'
Nonlinear longitudinal model
Nonlinear longitudinal model must be in equilibrium
0 = f1=1
mT cos ! + i( ) " D " mgsin#$% &'
0 = f2=1
mVT sin ! + i( ) + L " mgcos#$% &'
0 = f3= M / Iyy
0 = f4= !( " !# = q "
1
mVT sin ! + i( ) + L " mgcos#$% &'
In level flight, q = 0, and " = #
Numerical Solution to Estimate
the Trimmed Condition Specify desired altitude and airspeed, hN and VN Guess starting values for the trim parameters throttle
setting, elevator angle, and pitch angle (= angle of attack),!T0, !E0, and #0
Calculate f1, f2, and f3 (f4 has same value when q = 0)
Define a scalar,positive-definite trim cost function, e.g.,
f1=1
mT !T
0,!E
0,"
0,hN ,VN( )cos " + i( ) # D !T0 ,!E0 ,"0 ,hN ,VN( )$% &' ( 0
f2=
1
mVNT !T
0,!E
0,"
0,hN ,VN( )sin " + i( ) + L !T0 ,!E0 ,"0 ,hN ,VN( ) # mg$% &' ( 0
f3= M !T
0,!E
0,"
0,hN ,VN( ) / Iyy ( 0
J !Tk ,!Ek ," k( ) = a f1
2( )k+ b f
2
2( )k+ c f
3
2( )k, k = 0,1,!,K
Minimize the Cost Function withRespect to the Trim Parameters
Cost is minimized at the bottom of the bowl, i.e., when
J !Tk ,!Ek ," k( ) ! J uk( ) = a f1
2( )k+ b f
2
2( )k+ c f
3
2( )k, k = 0,1,",K
!J
!uu=u*
=!J
!u1
!J
!u2
!J
!u3
"
#$$
%
&''u=u*
= 0
Cost is bowl-shaped
Search in direction perpendicular to contours of equal cost
Example of Search for TrimmedCondition (Fig. 3.6-9, Flight Dynamics)
In MATLAB, use fminsearch [Nelder-Mead Downhill Simplex Method]
u = fminsearch J,u0( )
Steady-State Response
Steady-State Response of the
Fourth-Order Longitudinal Model
!VSS!" SS!qSS!#SS
$
%
&&&&&
'
(
)))))
= *
*DV *g 0 *D#LVVN
0 0L#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
Algebraic Equation for
Equilibrium Response
!VSS!" SS!qSS!#SS
$
%
&&&&&
'
(
)))))
=
*gM+EL#VN
$%&
'()
0 gM#L+F /VN[ ]
DVL#VN
* D#LVVN( )M+E
$%&
'()
MVL#VN
* M#LVVN( )T+T
$%&
'()
D#MV * DVM#( )L+F /VN$% '(
0 0 0
*gM+ELVVN
$%&
'()
0 L+F /VN[ ]
$
%
&&&&&&&&&
'
(
)))))))))
g MV
L#VN
* M#LVVN( )
!+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 ResponseMay Be 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
Pitch angle response
Effects of StabilityDerivative Variations on 4th-Order Longitudinal Modes
Primary and Coupling Blocks of the
Fourth-Order Longitudinal Model
FLon =
!DV !g 0 !D"
LVVN
0 0L"VN
MV 0 Mq M"
!LVVN
0 1 !L"VN
#
$
%%%%%%%
&
'
(((((((
=FPh FSP
Ph
FPhSP
FSP
#
$
%%
&
'
((
Some stability derivatives appear only in primary blocks (DV, Mq, M")
Effects are well-described by 2nd-order models
Some stability derivatives appear only in coupling blocks (MV, D")
Effects are ignored by 2nd-order models
Some stability derivatives appear in both (LV, L")
May require 4th-order modeling
$M" Effect on Fourth-Order Roots
!Lon (s) = s4+ DV +
L"VN
# Mq( )s3
+ g # D"( )LVVN
+ DVL"VN
# Mq( ) # Mq L" VN # M"o$
%&'
()s2
+ Mq D" # g( )LVVN
# DVL"VN
$%&
'()+ D"MV # DVM"o{ }s
+ g MVL"VN
# M"oLVVN( ) # !M" s
2+ DVs + g
LVVN( )
* d(s) + kn(s)
Group all terms multiplied byM" to form numerator for $M"
Primary effect: The same as in theapproximate short-period model
Numerator zeros The same as the approximate phugoid mode
characteristic polynomial
Consequently, effect of M" variation onphugoid mode is small
sI !
!DV !g
LVVN
0
"
#
$$$
%
&
'''
= (Ph (s) = s2+ DVs + g
LVVN
Short
Period
Short
Period
Phugoid
Phugoid
MV Effect on Fourth-
Order Roots
Large positive value producesoscillatory phugoid instability
Large negative value producesreal phugoid divergence
!Lon(s) = s
4+ D
V+L"VN
# Mq( )s3
+ g # D"( )LV
VN
+ DV
L"VN
# Mq( ) # Mq L" V
N
# M"$
%&'
()s2
+ Mq
D" # g( )LV
VN
# DV
L"VN
$%&
'()+ D"MV # DVM"{ }s
gM"LV
VN
+ MVD"s + g
L"VN
( ) = 0
D!
= 0
Short
Period
Phugoid
L" /VN and LV /VN Effects onFourth-Order Roots
LV /VN: Damped natural frequency of thephugoid
Negligible effect on the short-period
L" /VN: Increased damping of the short-period
Small effect on the phugoid mode
Pitch and ThrustControl Effects
Longitudinal Model
Transfer Function Matrix With Hx = I
HLon(s) = H
xLonsI ! F
Lon[ ]!1G
Lon=
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
"
#
$$$$
%
&
''''
nV
V(s) n(
V(s) n
q
V(s) n)
V(s)
nV
((s) n(
((s) n
q
((s) n)
((s)
nV
q(s) n(
q(s) n
q
q(s) n)
q(s)
nV
)(s) n(
)(s) n
q
)(s) n)
)(s)
"
#
$$$$$$
%
&
''''''
0 T*T 0
0 0 L*F /VN
M*E 0 0
0 0 !L*F /VN
"
#
$$$$$
%
&
'''''
s4+ a
3s3+ a
2s2+ a
1s + a
0
HLon (s) =
n!EV(s) n!T
V(s) n!F
V(s)
n!E"(s) n!T
"(s) n!F
"(s)
n!Eq(s) n!T
q(s) n!F
q(s)
n!E#(s) n!T
#(s) n!F
#(s)
$
%
&&&&&&
'
(
))))))
s2+ 2*P+nP s ++nP
2( ) s2 + 2*SP+nSP s ++nSP2( )
There are 4 outputs and 3 inputs
!V (s)
!" (s)
!q(s)
!#(s)
$
%
&&&&&
'
(
)))))
= HLon(s)
!*E(s)
!*T (s)
!*F(s)
$
%
&&&
'
(
)))
A Little More About
Output Matrices With Hu = 0, suppose the output equals the state:
!y = !x = Hx!x; then Hx = I4and
!y1
!y2
!y3
!y4
"
#
$$$$$
%
&
'''''
=
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
"
#
$$$$
%
&
''''
!x1
!x2
!x3
!x4
"
#
$$$$$
%
&
'''''
!
!V!(
!q
!)
"
#
$$$$$
%
&
'''''
Suppose the only output is !V; then
!y = !V = 1 0 0 0"# $%
!V!&
!q
!'
"
#
(((((
$
%
)))))
Suppose !V and !" are measured; then
!y =!y
1
!y2
"
#$$
%
&''=
!V!(
"
#$
%
&' =
1 0 0 0
0 0 0 1
"
#$
%
&'
!V!)
!q
!(
"
#
$$$$$
%
&
'''''
A Little More About
Output Matrices Output (measurement) of body-axis velocity and pitch
rate and angle
Transformation from [!V, !%, !q, !"] to [!u, !w, !q, !#]
!u!w!q
!"
#
$
%%%%
&
'
((((
=
cos)N
0 0 *VNsin)
N
sin)N
0 0 VNcos)
N
0 0 1 0
0 1 0 1
#
$
%%%%%
&
'
(((((
!V!+
!q
!)
#
$
%%%%%
&
'
(((((
Separate measurement of state and control perturbations
!y =!x
!u
"
#$
%
&' = Hx!x +Hu!u
!y1
!y2
!y3
!y4
!y5
!y6
"
#
$$$$$$$$
%
&
''''''''
=
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
0 0 0 0
0 0 0 0
"
#
$$$$$$$
%
&
'''''''
!V!(
!q
!)
"
#
$$$$$
%
&
'''''
+
0 0
0 0
0 0
0 0
1 0
0 1
"
#
$$$$$$$
%
&
'''''''
!*E!*T
"
#$
%
&'
Elevator-to-Normal-
Velocity Numerator
HxAdj sI ! FLon( )G = sin"N 0 0 VN cos"N#$
%&
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) n'
"(s) nq
"(s) n"
"(s)
#
$
((((((
%
&
))))))
0
0
M*E
0
#
$
((((
%
&
))))
= n*Ew(s)
Transform though "N back to body axes
n!Ew(s) = sin"
N0 0 V
Ncos"
N#$
%&
nq
V(s)
nq
'(s)
nq
q(s)
nq
"(s)
#
$
((((((
%
&
))))))
M!E = M!E sin"N( )nqV(s) + V
Ncos"
N( )nq"(s)#$ %&
Transfer function numerator
Elevator-to-Normal-Velocity
Transfer Function
!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
Normal velocity transfer function is analogous to angle ofattack transfer function ($" ! $w/VN)
z3 often neglected due to high frequency
Elevator-to-Normal-Velocity FrequencyResponse
!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) cancelsphugoid response
Elevator-to-Pitch-RateNumerator and Transfer Function
HxAdj sI ! FLon( )G = 0 0 1 0"# $%
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)
"
#
((((((
$
%
))))))
0
0
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 differentiates pitch angle transfer function
Elevator-to-Pitch-
Rate FrequencyResponse
+20 dB/dec
+20 dB/dec+40 dB/dec
0 dB/dec20 dB/dec
!qSS = 0( )!ESS + 0( )!"TSS + 0( )!"FSS
(n q) = 1
Negligible low-frequency response,except at phugoidnatural frequency
High-frequencyresponse wellpredicted by 2nd-ordermodel
!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
Transfer Functions of
Angles from Elevator Input
!"(s)!#E(s)
=n#E"(s)
!Lon(s); n#E
"(s) = M#E s +
1T"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#E
L$
VN
s + 1T" 1
%&'
()*
Elevator-to-Flight Path Angle transfer function
Elevator-to-Angle of Attack transfer function
Elevator-to-Pitch Angle transfer function
Frequency Response ofAngles to Elevator Input
Pitch anglefrequency response($# = $% + $")
Similar to flightpath angle nearphugoid naturalfrequency
Similar to angleof attack nearshort-periodnaturalfrequency
!" SS = c!#ESS
!$SS = f!#ESS
!%SS = c & f( )!#ESS
Transfer Functions ofAngles from Thrust Input
!"(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#T s s +
1T"
T
$%&
'()
!" (s)
!#T (s)=n#T"(s)
!Lon (s); n#T
"(s) = T#T
LV
VNs2+ 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 Angles
to Thrust Input Thrust primarily effects flight path angle and low-frequency pitch angle
Feedback Control: Angles to Elevator
Variations in control gain
Principal effect is onshort-period roots
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 dB and transfer function magnitude,
20 log10 AR(j&)
Phase Margin
At the input frequency, &, for which 20 log10 AR(j&) = 0 dB
Difference between the phase angle '(j&), and 180
Axis intercepts on the Nichols Chart identify GM
and 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!( ) +102
"
#$$
%
&''
100
j! +100( )
"
#$
%
&'
Gain and Phase Margins in
Pitch-Tracking TaskElevator-to-Pitch-Angle
Bode Plot
Elevator-to-Pitch-Angle
Nichols Chart
Amplitude Ratio vs. Phase Angle
Gain Margin: Amplitude ratio below 0 dB
when 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 on
Pitch-Angle Control Task
Pilot Transfer Function = KP1 /TP
s +1 /TP= KP
1 / 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
Pilot"s gain, KP, is either 1 or 2
Open-Loop Pilot-Aircraft
Transfer 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 Pilot
Dynamics on
Pitch-Angle
Control Task
Gain and phase
margins become
negative for pilot
gain between 1
and 2
Effect of Pilot Dynamics on Elevator/Pitch-
Angle Control Root Locus
Pilot transfer function changes asymptotes of
the root locus
Next Time:Fourth-Order Lateral-Directional Dynamics
Supplemental
Material
Airspeed Frequency Response toElevator and Thrust Inputs
Response is primarily through the lightly dampedphugoid mode
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)
= #VN
s
$%&
'()
!* (s)!"E(s)
!z(s)!"T (s)
= #VN
s
$%&
'()
!* (s)!"T (s)
High- and Low-Frequency Limits of
Frequency Response Function
Hij ( j! " 0)"kij j!( )
q+ bq#1 j!( )
q#1+ ...+ b
1j!( ) + b0[ ]
j!( )n
+ an#1 j!( )n#1
+ ...+ a1j!( ) + a0[ ]
"
kijb0
a0
, b0$ 0
kij j(0)b1
a0
, b0
= 0, b1$ 0, etc.
%
& ' '
( ' '
Hij ( j!) = AR(!) ej" (! )
Hij ( j! "#)"kij j!( )
q+ bq$1 j!( )
q$1+ ...+ b
1j!( ) + b0[ ]
j!( )n
+ an$1 j!( )n$1
+ ...+ a1j!( ) + a0[ ]
"kij
j!( )n$q
