Boeing 747 Longitudinal-directional flight
control system design
Group members
Tianxiong chen
Yang Gao
In this project we will discuss:
How to build the longitudinal-directional flight control
system of aircrafts. We take Boeing 747 as an example.
 Linearization
of aircraft longitudinal equations of
motion;
 Determine the system transfer function;
 Frequency response and time response curves analysis;
 Altitude hold mode control system design;
 Some studies on the combined system control.
Figure:Boeing 747
 Dimensions:
 wing area:5500 ft 2
 mean chord:27.3 ft
 wingspan:196 ft
 Flight condition: high cruise
 altitude:20000 ft
 speed:738 km/h
 weight:636636 lb
Longitudin al Equations of Motion
W  U 1q   g sin1  Z uU  Z    Z    Z q q  Z ee.......... ......... 1
q  M uU  M TuU  M    M T   M    M q q  M ee.......... ...... 2
Here we find parameters of boeing 747 in Table 1, put them into calculated ,
Also, for small perturbations,

w
 w  U1 and w  U1，finally we got
U1
d
 0.525  q  0.038e.......... .......... .......... .......... .......... ......... 3
dt
dq
d
 1.303  0.106
 0.542q  1.694e.......... .......... .......... ...4
dt
dt
Transforming the linearized EOM to the Laplace Domain
S . ( S )  0.525 ( S )  q( S )  0.038e( S )......... .......... .......... ...... 5
S .q( S )  1.303 ( S )  0.106S . ( S )  0.542q( S )  1.694e( S )...6
from formula 5， we know that
q ( S ) ( S  0.525) ( S )  0.038e( S ).......................................................7
from formula 6 and 7， we can get the transferfunction Ge ( s )
S
1)
 (S )

45.132
G (s) 

..................................................8
e
e( S ) S 2 1.173S 1.588
1.715(
Figure1:nyquist
matlab source code:
numG=[1.715/45.132 1.715];
denG=[1 1.173 1.588];
grid
nyquist(num,den)
title('G(s)=[1.715(s/45.132+1)]/(s^2+1.173
s+1.588)')
Figure2:Unit step response
curve
matlab source code:
GH=tf([-1.715/45.132 -1.715],[1 1.173 1.588]);
step(GH)
title('Unit step response curve');
grid
Figure3:Impulse response
curve
matlab source code:
GH=tf([-1.715/45.132 -1.715],[1 1.173
1.588]);
Impulse(GH)
title(‘Impulse response curve');
grid
Open - Loop system transferfunction
S
1.715(
1)
45.132  K ............9
G ( s ) H ( s )  K Ge ( s ) 
S 2 1.173S 1.588
The Open-Loop characteristic equation is
S
2
 1.173S  1.588  0..................................................10
Closed - Loop system transfer function is
G(s)
( s)
...............................................................................11
1 G ( s ) H ( s )
S
1.715(
1)
Ge ( s )
45
.
132
(s) 

.......12

2
1 K Ge ( s ) S  (1.173 0.038K ) S  (1.5881.715K )
S
1.715(
1)
Ge ( s )
45.132
(s) 

.................12

2
1 K Ge ( s ) S  (1.1730.038K ) S  (1.5881.715K )
The Closed - Loop characteristic equation is
S 2  (1.173  0.038K ) S  (1.588  1.715K )  0.......... .......... ..13
According to Routh - Hourwitz criterion,the Closed - Loop
system is stability when
(1.173  0.038K )  0 and (1.588  1.715K )  0.......... .......... .14
so, we should have k < 0.983
Now the question is : How to determine " K"
Flying quality requirements
An aircraft’s compliance to the dynamic stability
requirements of MIL-8785C is defined in terms of
three flying quality levels.(Table 7.6)
b) MIL-8785C requires that the short period mode
meet both a damping ratio (Table 7.7) and natural
frequency requirement.(Figure 7.20)
c) It also requires that the short period natural
frequency fall within an upper and lower limit as a
function of the aircraft’s n/a ratio and flight phase
category. (Table 7.5)
a)
Table7.5 MIL-F-8785C flight phase categories
Table7.6 MIL-F-8785C flying quality levels
Table7.5 MIL-F-8785C flight phase categories
MIL-F-8785C short period natural frequency
requirements---Category B flight phases
 For n/a,this parameter can be estimated for an aircraft
using
z


g
n
.......... ......... 15
 n/a can be thought of as a load factor (n) sensitivity
parameter. It increases with increasea in( CL ) and
wing area(S), and it decreases as weight (m) increases .
 The MIL-STD-1797A short period requirements are
recast in terms of the control anticipation
parameter(CAP) over an acceptable range of damping
ratios.
 The CAP is estimated as
n
CAP 
.......... .......... ......... 16
n /
2
sp
From Table 1,we know
Z  353 .52
z
 353.52



 10.98......17

g
32.2
n
because flight condition is high cruise,
category B,
Level 1
from Table 7.7 and Figure 7.20,we know the range of
CAP , ω n sp , and 
we have already known the closed - loop transfer function
(s) 
Ge ( s )
1.715 (
1 K Ge ( s )

S
45.132
1)
S 2  (1.173  0.038 K ) S  (1.588 1.715 K )
The generalize d transfer function for a second- order system is
G(s) 
X (s)
Y (s)

2
n
2
2
S  2n S   n
............................18
Here we make :
  (1.588  1.715K )......... .......... .......... .... 19
2 n  (1.173  0.038K )......... .......... ......... 20
2
n
from Figure 7.20 , we have 0.086  CAP  3.6
from Figure 7.20 , we have 1.0  ω
n sp
 6.0
from Table7.7 , we have 0.301    2.0
2
ωn
CAP 
n/α
z
 353.52


 10.98

g
32.2
n
 n2  (1.588  1.715K )
2  n  (1.173  0.038K )
n/α andCAP   n2  K  
with different CAP we got different groups of result .
after comparition, we chose one group of:
CAP=0.2
ω n sp=1.4819
k=-0.3545
=0.4003
go back to
S


1
.
715
(
1)
Ge ( s )
45
.
132
( s) 


1 K Ge ( s ) S 2  (1.1730.038K ) S (1.5881.715K )
 We finally get the open-loop system transfer function
as
S
1)

45.132
G(s) H (s)  K  G (s) 
.......21
e
S 2 1.173S 1.588
0.608(
 the closed-loop system transfer function is
S
1.715(
1)
Ge ( s )
45.132
( s) 

...............22
1 K Ge ( s ) S 2 1.162S  2.196
Figure4:Closed loop Unit
step response curve
matlab source code:
GH=tf([-1.715/45.132 -1.715],[1
1.162 2.196]);
step(GH)
title('Closed loop Unit step
response curve');
Figure5:Closed loop
impulse response curve
matlab source code:
GH=tf([-1.715/45.132 -1.715],[1
1.162 2.196]);
step(GH)
title('Closed loop impulse
response curve');
grid
With the same step, we got
 (S )
3.4( S  0.521 )

(s) 

e
e ( S )
S 3 1.605 S 2  0.906 S
G
Now we derving the h / e transfer function :
First, we found that the rate of climb (ROC) was
 U sin  U 
ROC  h
1
1
Taking the Laplace transform,
s.h ( s )  U  ( s )
1
And forming the desired transfer function
h( s )
U1

e( s )
S
Because
   
  ( s) 

 e( s ) 



 The altitude to elevator transfer function becomes:
h( s )
U 1   (s)
 (s) 





e( s )
S  e( s ) e( s ) 

 (S )
3.4( S  0.521)

G (s) 

e
e ( S )
S 3 1.605S 2  0.906S
S
1)

(
S
)

45.132
G (s) 

e
e ( S )
S 2 1.173S 1.588
1.715(
Altitude hold block diagram
Attitude and Lift Equation
s
 L=½pV²CL
w
 There are three parameters could change.
 Change velocity----increase or decrease the engine thrust
 Change lift coefficient----adjust the airfoil( leading edge, flap, spoiler)
or change the AOA
 Change the wing surface----adjust the airfoil
Conclusion
There are two ways to change the lift force, one is change the engine
thrust and another is act the control surfaces. Through change the lift
force value, we also can get the respect attitude.
Autotri
m
Input
Control
Surfaces
Attitude
Stable
FCC
Auto
Throttle
Sensor
Engine
Flight Control Computer
 The FCC is the primary computing component for the AFDS(Autopilot/Flight
Director System). It provides Autopilot (A/P) commands to the servos that move
the control surfaces, Flight Director (F/D) commands to the PFD to direct manual
pilot control, and system monitoring and fault detection for the CMCS(Central
Maintenance Computers).
 The left, center, and right FCCs are identical and contain all logic and signal
handling circuitry for direct pitch axis control (elevators and stabilizer trim), roll
axis control (ailerons), and yaw axis control (rudders). Yaw control is active only
during a multi-channel approach and rollout.
 Each FCC provides autopilot command output signals which drive hydraulic servos
that deflect the control surfaces. Each FCC also provides flight director data to the
Electronics Interface Units (EIUs) for steering commands and mode annunciation
display on the PFDs.
 FCC program pin connections are permanently wired at the shelf connector. The pin
connections provide inputs to the FCC, which allow the selection of the
configuration options.
Automatic Stabilizer Trim System
The Automatic Stabilizer Trim
(AST) System provides automatic
trim
control of the horizontal
stabilizer position to reduce the
load on the
elevator.
load on the
elevator
Speed Trim System
 General
 The Speed Trim mode of the stabilizer trim system provides incremental
horizontal stabilizer positioning in response to changes in computed airspeed
(Vc). The mode is only engaged when the airplane is in the air and the manual
and autotrim functions are not engaged.
 The alternate electric trim switches, located on the aisle stand, are not
activated.
 Airplane is in the air and airspeed is below 220 Knots.
 During speed trim operation, the stabilizer position is related
 to Vc. ( Speed trim control law)
Speed Trim Control Law Schedule
The speed trim control law is
synchronized to the stabilizer
position as the speed trim mode
is engaged, and the incremental
control of the stabilizer position
is provided as the function of Vc.
Speed trim is inhibited when Vc
is greater than 220 Knots. The
allowable trim threshold is less
than or equal to -0.075 degrees.
After trimming, the final
stabilizer
position
change
produced is approximately
equal to the position change
corresponding to the ideal
characteristic shown on the left.
Especially appreciate Prof. Zhang's
patient guidance
Thank you all !!!