regulation quad parbackstepping

1. NONLINEAR BACKSTEPPING CONTROL WITH
OBSERVER DESIGN FOR A 4 ROTORS HELICOPTER
L. Mederreg, F. Diaz and N. K. M’sirdi
LRV
Laboratoire de Robotique de Versailles,
Université de Versailles Saint Quentin en Yvelines,
10, avenue de l’Europe 78140, Vélizy, France.

2. 1 Introduction.
2 4 rotors Helicopter model Presentation
3 Back stepping controller synthesis
4 Back stepping controller synthesis with observer
5 Simulation and results
6 Conclusion.
OUTLINE

3. Introduction
• Thanks to its special configuration, the 4 rotor helicopter allows to
achieve many tasks in different fields.
 Symmetry of the platform geometry
 Low weight
Low cost
•Autonomous flight  Non linear control law Synthesis.
 Complexity of the dynamical system
 Presence of Perturbations due to the wind
 Unavailability of some state variables

4. 4 rotors Helicopter model Presentation

5. 0 0 0
( , , )T
u v w Absolute velocities / Earth frame
( , , )T
   Orientation angels: Yaw, Roll, Pitch.
State vector:
0 0 0 0 0 0
( , , , , , , , , , , , )T
x x y z u v w p q r
  

Gravity center coordinates
0 0 0
( , , )T
x y z
( , , )T
p q r Angular velocities / Helicopter frame
( , , )T
x y z
A A A Aero dynamical forces
( , , )T
p q z
A A A Aero dynamical Momentums

6. 0
0
0
0
0
x
y
z
x u
w
p
q
r





 
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
The state representation is given by:
0
0
0
1
2
3
4
sin sec cos sec
cos sin
sin tan cos tan
1
( )(cos cos sin sin sin )
1
( )(cos sin sin cos sin )
( ) 1
( )( , , )
( )
( )
( ) 1
y z
x x
z x
y y
x y
y y
u
v
w
q r
q r
p q r
m
m
F x
g
m
u
qr I I d
u
I I
pr I I d
u
I I
qp I I
u
I I
   
 
   
     
     
   





 

 

 

 
























 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

( )
x F x


7. 2
( )
1
2
d
E y y
V E
 

System of 4 equations 4 unknowns
0
i j i j i j i j j
a u bu cu d u h
     , :1 4
i j 
System outputs: 0 0 0
( , , , )
y x y z 

Desired outputs: ( , , , )
d d d d
y x y z 

Control laws: 1 2 3 4
( , , , )
u u u u u

Back stepping controller synthesis
We consider that all the state vector is measurable

8. SIMULINK bloc diagram of the controller

9. 2
( )
1
2
d
E y y
V E
 

• We include in the expression of V the observing errors to be
cancelled
Back stepping controller synthesis with
observer
• We shall observe the absolute velocity vector
0 0 0
ˆ ˆ ˆ
( , , )T
u v w : Difficult to measure
• We consider that all the other parameters are measurable
Where V is a LYAPUNOV
candidate function
System 4 equations
4 unknowns
Convergence of the tracking errors
Convergence of observing errors

10. Simulation and results
Simulation of a vertical helix trajectory flight in presence of
perturbations (7 newton front wind blowing)
 The controller gains are adjusted by doing intensive simulations
cos( )
sin( )
2
10
3
d
d
d
d
x t
t
y
t
z




 

Tracking Trajectory : Initiales positions:
0
0
0
0
0
0
0
x
y
z






11. 3D Tracking trajectory

12. Tracking errors for the BACKSTEPPING controller

13. Observation Errors for the BACKSTEPPING Observer

14. Tracking Errors for the BACKSTEPPING controller
with Observer

15. Conclusion :
This approach has shown :
 Good robustness of the Controller
 Good convergence of the couple controller observer
 allows to decrease the number of the required
sensors