NONLINEAR BACKSTEPPING CONTROL WITH OBSERVER DESIGN FOR A 4 ROTORS HELICOPTER

L. Mederreg, F. Diaz and N. K. M’sirdiLRV

Laboratoire de Robotique de Versailles,Université de Versailles Saint Quentin en Yvelines,

10, avenue de l’Europe 78140, Vélizy, France.

11 Introduction.Introduction.

22 4 rotors Helicopter model Presentation4 rotors Helicopter model Presentation

33 Back stepping controller synthesisBack stepping controller synthesis

44 Back stepping controller synthesis with observerBack stepping controller synthesis with observer

55 Simulation and resultsSimulation and results

66 Conclusion.Conclusion.

OUTLINEOUTLINE

Introduction

• Thanks to its special configuration, the 4 rotor helicopter allows to achieve many tasks in different fields.

Symmetry of the platform geometry Low weightLow 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 rotors Helicopter model Presentation

0 0 0( , , )Tu v w Absolute velocities / Earth frame

( , , )T Orientation angels: Yaw, Roll, Pitch.

State vector:

0 0 0 0 0 0( , , , , , , , , , , , )Tx x y z u v w p q r

Gravity center coordinates0 0 0( , , )Tx y z

( , , )Tp q r Angular velocities / Helicopter frame

( , , )Tx y zA A A Aero dynamical forces

( , , )Tp q zA A A Aero dynamical Momentums

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

mF x

gm

u

qr I I du

I I

pr I I du

I I

qp I Iu

I I

( )x F x

2

( )

12

dE y y

V E

System of 4 equations 4 unknowns

0i j i j i j i j ja u bu c u d u h , :1 4i j

System outputs: 0 0 0

( , , , )y x y z

Desired outputs: ( , , , )d d d dy 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

SIMULINK bloc diagram of the controller

2

( )

12

dE 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 0ˆ ˆ ˆ( , , )Tu 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

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

ty

tz

Tracking Trajectory : Initiales positions:

0

0

0

0

0

0

0

x

y

z

3D Tracking trajectory

Tracking errors for the BACKSTEPPING controller

Observation Errors for the BACKSTEPPING Observer

Tracking Errors for the BACKSTEPPING controller with Observer

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