Advanced Problems of Longitudinal Dynamics Robert Stengel, Aircraft Flight Dynamics MAE 331, 2010 Fourth-order dynamics Steady-state response to control Transfer functions Frequency response Root locus analysis of parameter variations Numerical solution for trimmed ﬂight 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 ) \$ % & & & & & ( ) ) ) ) ) = * D V *g 0 * D # L V V N 0 0 L # V N M V 0 M q M # * L V V N 0 1 * L # V N \$ % & & & & & & & ( ) ) ) ) ) ) ) !V (t ) !" (t ) !q(t ) !# (t ) \$ % & & & & & ( ) ) ) ) ) + 0 T +T 0 0 0 L + F / V N M + E 0 0 0 0 * L + F / V N \$ % & & & & & ( ) ) ) ) ) !+ E(t ) !+T (t ) !+ F(t ) \$ % & & & ( ) ) ) Stability and control derivatives are deﬁned at a trimmed (equilibrium) ﬂight condition Initial-Condition and Step- Input Responses of a Business Jet Aircraft Initial pitch rate [!q(0)] = 0.1 rad/s Elevator step input [!!E(0)] = 1 deg Trimmed Solution of the Equations of Motion
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Longitudinal DynamicsRobert Stengel, Aircraft Flight Dynamics

MAE 331, 2010

Fourth-order dynamics

Transfer functions

Frequency response

Root locus analysis of parametervariations

Numerical solution for trimmedflight condition

Nichols chart

Pilot-aircraft interactions

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

!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( )

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

!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

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)

\$

%

&&&

'

(

)))

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

!(

"

#

\$\$\$\$\$

%

&

'''''

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

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

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

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

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