Lecture #16 Topics
• Linear Model Assembly – Lockheed C5A Galaxy
• Aircraft modes
• Develop approximate models for aircraft modes
• Connection between static and dynamic stability
Lockheed C-5A Galaxy
Longitudinal State-Space Description
***************************************************
Longitudinal State Space Model
Alon =
-0.1407
-0.0002
-0.0000
0
Blon =
0
-0.0378
-1.5616
0
17.0992
0 -32.1763
-0.6076 0.9591 -0.0020
-3.1928 -0.6971 0.0002
0 1.0000
0
Lateral-Directional State-Space Description
***************************************************
Lateral/Directional State Space Model
AlatD =
-0.0208 0.0000 -0.9942 0.0517
0
-1.8718 -0.6576 0.0396
0
0
0.8650 -0.0383 -0.0163
0
0
0 1.0000 0.0384
0
0
0
0 1.0007
0
0
BlatD =
0.0016
0.6076
0.0266
0
0
0.0010
-0.0168
-0.6031
0
0
Approximate Models
Longitudinal Model (Straight & Level)
 u  X    X u
U  Z    

 0
   Zu
 q  M   M u

 


 
 X Pu
X
Xq
 Z Pu
Z
Zq U 0
 M Pu
M   M P
Mq
0
1
0
 g cos 0   u   XT
 g sin  0     ZT
 
0
  q  MT
  
0
    0
X E 
Z  E  T 

M  E   E 

0 
Rearrange the left hand side
1
0

0

0
X
U 0  Z 
M 
0
0 0   u   Xu
 
0 0     Z u


1 0  q  M u
  
0 1    
 X Pu
X
Xq
 Z Pu
Z
Zq U 0
 M Pu
M   M P
Mq
0
1
0
 g cos 0   u   XT
 g sin  0     ZT
 
0
  q  MT
  
0
    0
X E 
Z  E  T 

M  E   E 

0 
Short Period Approximation
The approximation is based on the following observations:
• Dominant in angle-of-attack and pitch rate dynamics
• Velocity of the aircraft has no components of the short-period mode;
i.e., velocity is virtually constant when the airplane is excited in the
short-period mode
Thus, short-period approximation is obtained by removing the velocity and pitchattitude dynamics (deleting 𝑢 𝑎𝑛𝑑 𝜃)
Z
U 0  Z  0    

  q   M  M
1    
P
 M 
where
C Mq  C L H
 X Ref  X ACH 
U 0c W
2
qH SH
q  SW
Z q  U 0     ZT




M q   q  MT
Z  E  T 
M  E   E 
Short Period Approximation:
Characteristic Equation
(Z q  U 0 )
Z


 
 M
 Mq  
U 0  Z
 U 0  Z

 M q   Zq  U 0 
Z 

 M   M P  0
 U 0  Z   U 0  Z 
2


Which implies
(Z q  U 0 )
Z


2 SP SP   
 M
 Mq 
U 0  Z
 U 0  Z

  Zq  U 0 

 M   M P
U

Z
U

Z
   0
 
 0

 2SP  Z  
Mq

 Z

     M  Mq 
 U0


 Mq
 Z 
 U0

  M   M P



Root-Locus of SP eigenvalues (Navion)
Augmentation System
Pitch Damper
(Augmentation System)
• Used to increase short period damping
• From the characteristic equation damping can be
increased by increasing the effective pitch damping
derivative
(Z q  U 0 )
Z


2 SP SP   
 M
 Mq 
U 0  Z
 U 0  Z

 Z

     M  Mq 
 U0

• But pitch damping derivative is effectively pitching
moment due to pitch rate. Thus, to provide
additional damping we need to provide additional
pitching moment proportional to pitch rate.
Designing Pitch Damper
Begin from longitudinal state space form
X
0 0   u   X u  X Pu
X
1
0 U  Z  0 0     Z  Z
Z
0

Pu

    u
M 
1 0   q   M u  M Pu M   M P
0

  
0
0
0
1
0
0

   
Xq
Zq U 0
Mq
1
 g cos 0   u   XT
 g sin  0     ZT
 
0
  q  MT
  
0
    0
X E 
Z  E  T 

M  E   E 

0 
And use the perturbation control input as 𝛿𝑒 = 𝐾𝑞 𝑞 = 𝐾𝑞 𝜃
1
0

0

0
X
U 0  Z 
M 
0
0 0   u   Xu
 
0 0     Z u


1 0  q  M u
  
0 1    
 X Pu
X
Xq  X e K q
 Z Pu
Z
Zq U 0  Z EK q
 M Pu
M   M P
M q  M  EK q
0
1
0
 g cos 0   u   XT 
 g sin  0     ZT  T 
 

0
  q   M T   
  

0
    0 
Vary the feedback gain Kq until desired damping is achieved.
IMPORTANT: Feedback gain has units. Larger the gain, larger the commanded signal.
Implementation:
Stability Augmentation System
Without augmentation
𝛿𝑒0
Aircraft
(Full Nonlinear System)
Pilot command
to maintain trim
Reference Condition
+ Perturbation Response
With augmentation
𝛿𝑒0
Pilot command
to maintain trim
Aircraft
(Full Nonlinear System)
𝛿𝐸
Perturbation
input
Kq
Reference Condition
+ Perturbation Response
q
Perturbation
pitch-rate