ONE WAY TO DESIGN THE CONTROL LAW OF A MINI-UAV Projet c o l e - - PowerPoint PPT Presentation

É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

ONE WAY TO DESIGN THE CONTROL LAW OF A MINI-UAV

Projet

É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

 Plan

• Introduction
• Model of the μDrone
• Control design of the MAV
• Choice of the adjustment parameters
• Conclusion

É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

 Introduction

• The ENSMM μDrone

This MAV uses six propellers to fly :

• Two are counter-rotating and provide the lift
• Two provide the trim stabilization
• Two are used for the propulsion
• Approximation of the MAV behaviour

The model is a MIMO linear time-invariant system

• Control design of the MAV

The LQ state feedback regulator design is applied to a « standard model » To compute the weighting matrices of quadratic criterions we use a « partial observability gramian »

The great advantage of this method is due to the use of only three scalars to synthesize a robust control law.

É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

 Model of the μDrone

The μDrone is a « gyrodyne » with six propellers We worked on the assumption that it can be described as seven solids in interaction

• Model of the actuators

Let denote the thrust produced by the ith actuator and the associated

• moment. If denotes the control input of the actuator, we assume that
• Global model

With this assumption and after linearization of the mecanical model in the vicinity of a horizontal trim, the following MIMO linear time-invariant system is an approximation of the MAV behaviour :

i

F

i

M

i

u

É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

 Control design of the MAV

• Standard model

We introduce a constant perturbation vector : Initial model is then augmented :

• Extraction of an observable subsystem

É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

• k
• State estimator
• Gain matrix is chosen as the solution of the LQ problem :

A positive scalar is to be designed so that and the weighting matrices where is a partial observability gramian

• k

É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

Partial observability gramian :

where

State estimator is then :

É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

• Uncontrollable modes influence rejection
• Controllability staircase form of

uncontrollable modes perturb controllable ones

• Rejection with a linear transformation

leading to :

nc

x

É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

It is often possible to find such as : with these conditions :

A state feedback is then possible…

É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

• Control feedback
• New LQ problem

choice of the weighting matrices

define another positive scalar , a positive control horizon such as : and the partial observability gramian :

c

k

c

T

É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

• Feedback control elaboration

with this feedback control the system is : and we want at permanent rate : If that matrix is invertible, the solution is :

É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

Then, the control vector is : and the use of transformation matrix leads to

• Regulator equations

with

• c

T

É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

 Choice of the adjustment parameters

• First step

Assign a fixed value to

,

Adjust filtering horizon so that is a stability matrix.

• Second step (quantify the stability robustness)

The regulator, without reference, is converted to a transfer matrix with

• T

reg

A

É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

• Similarly, initial model of the MAV is converted to
• Now define
• Loop transfers
• Static margin
• Dynamic margin
• When is a stability matrix, feedback is robust in relation to :

and

reg

A

É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

• Last step

Now it is not difficult to compute static and dynamic margins for different values of the three parameters :

• Filtering horizon
• Shape factor
• Recovery factor

The reasons why this strategy is often efficient is due to :

• The natural robustness of a LQ problem solution
• The notion of loop transfer recovery (LTR)
• T
• k

c

k

É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

• Implementation on the μDrone

Static margin : Dynamic margin :

1.2

• T

s

1.2

• k

3.2

c

k

42.2%

st

M 32.7

dyn

M ms

É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

É c o l e N a t i o n a l e S u p é r i e u r e d e M é c a n i q u e e t d e s M i c r o t e c h n i q u e s

 Conclusion

• The proposed method is quite easy to use

Only three positive scalars have to be adjusted

• Tricky problems

Rejection of uncontrollable modes is not always possible Internal stability of regulator is sometimes difficult to reach

• Philippe de Larminat’s recent book in which this method is

explained in detail is of great interest.