adaptiv e p ar ameter identiÞcation for simpliÞed...

Adaptive Parameter Identification for Simplified 3D-Motion Model

of ‘LAAS Helicopter Benchmark’

Sylvain Le Gac§ & Dimitri PEAUCELLE† & Boris ANDRIEVSKY‡

§ SEDITEC

† LAAS-CNRS - Universite de Toulouse, FRANCE

‡ IPME-RAS - St Petersburg, RUSSIA

CNRS-RAS cooperative research project

”Robust and adaptive control of complex systems: Theory and applications”

Introduction

CNRS-RAS cooperation objectives

! Investigate robustness issues of adaptive algorithms for control

both theoretically and through experiments

! Adaptive Identification (CCA’07, ALCOSP’07)

! Direct adaptive control (ROCOND’06, ALCOSP’07, ACC’07, ACA’07)

!State-estimation in limited-band communication channel

Other cooperations

! Also part of ECO-NET project ”Polynomial optimization for complex systems”,

funded by French Ministry of Foreign Affairs, and handled by Egide.

Concerned countries : Czech Republic, France, Russian Federation, Slovakia.

! Submitted a PICS project ”Robust and adaptive control of complex systems”

(funded by CNRS and RFBR).

& 1 IFAC ALCOSP’07, August 2007, St. Petersburg

Introduction

”Helicopter” Benchmark by Quanser at LAAS-CNRS

! Purpose : demonstration of research results & educational

! Simplified model needed with identified parameters

! Identification via adaptive algorithms

! Outline : Theory / Experiments


MISO LTI systems

LTI system: order n with m inputs

y(n)(t)+ . . .+a1y(t)+a0y(t) =m!

i=1

binu(n)i (t)+ . . .+ bi1ui(t)+ bi0ui(t).

Define the following vectors

Xy(t) =

"

####$

y(n!1)(t)...

y(t)

%

&&&&', Xui(t) =

"

####$

u(n!1)i (t)

...

ui(t)

%

&&&&',

!T (t) =(

XTy (t) u(n)

1 (t) XTu1(t) · · · u(n)

m (t) XTum(t)

)

!T =(

an!1 . . . a0 b1n . . . b10 . . . bmn . . . bm0

)

System compact model: y(n)(t) = !T (t)!.

Identification: least square estimation of ! assumed constant.


Filters D(s): avoid derivation of y(t) and ui(t)

" Only y(t) and ui(t) are measured, numerical time-derivatives amplify noise

# Let an order n Hurwitz polynomial D(s) = sn + . . . + d1s + d0 then

y(n)(t) = !T (t)! ! yn(t) = !T (t)!

where yn(t) = D!1(s)y(n)(s) and !(s) = D!1(s)!(s) obtained by:

!T =*

XTy u1n XT

u1 · · · umn XTum

+

and for all z = y, u1, . . . um :

"

#$˙Xz(t)

zn(t)

%

&' =

,

-----------.

0 1 0. . . . . .

0 0 1

"d0 "d1 · · · "dn!1

"d0 "d1 · · · "dn!1

/

000000000001

Xz(t) +

,

-----------.

0...

0

1

1

/

000000000001

z(t)


Kalman filtering for yn(t) = !T (t)!

Estimator of

Estimate !" = !(t#$) where !(t) solution of adaptive algorithm

!(t) = ""(t)!(t)(!T (t)!(t)" yn(t)

)

"(t) = ""(t)!(t)!T (t)"(t)+""(t)

For " = 0: guaranteed convergence if permanent excitation on ui(t).

" > 0 small: forgetting factor, to be used for slowly time varying parameters.


Implementation for ’helicopter’ identification

Simplified model of 3D-Motion of ’helicopter’ benchmark

#(t) + a!1#(t) + a!

0 sin(#(t)" #0) = b!0µd(t)

$(t) + a"1$(t) + a"

0 sin($(t)" $0) + c#!%(t)#(t) = b"0µs(t) cos #(t)

%(t) + a#1 %(t) = b#

0µs(t) sin #(t)


Identification of the pitch motion

MISO model of the non-linear dynamics

#(t) + a!1#(t) + a!

0 sin(#(t)" #0) = b!0µd(t)

%#(t) + a!

1#(t) = "a!0 s(t)

2345sin(!(t)!!0)

+b!0µd(t)

! #0 = "7.8o measured as the equilibrium for µd = 0.

! D(s) = s2 + 2&d'ds + &2 = s2 + 1.4s + s2

! Permanent excitation: square + chirp

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4


Pitch identification results

$ " & [0, 0.001]: good convergence (else oscillations appear)

$ No major dependency w.r.t. initial guess !(0)

! " # $ % &! &" &# &$ &% "!!&'(

!&

!!'(

!

!'(

&

&'(

$ "(0) ' 1031 for quicker convergence

! "! #! $! %! &! '! (! )! *! "!!!#

!"+&

!"

!!+&

!

!+&

"

"+&

#


Pitch identification results

$ For different experimental conditions (various choices of the excitation signal,

disturbances...) the identified parameters are close but slightly different.

$ Obtained values are uncertain in intervals

b!0 & [0.25, 0.3] , a!

0 & [0.58, 0.67] , a!1 & [0.058, 0.068]

$ A PID controller is designed for the median values of identified parameters

$ Error in closed-loop behavior of non-linear model and system is satisfying

!" !# !! !$ !% &" &# &!

!&

!!

!'

!#

!(

"

(


Identification of elevation and travel axis

$ Both axes identified simultaneously because

! Both excited by µs(t), the sum of propeller forces

! Have coupled dynamics

$ Identification done with PID control on µd(t), the difference of propeller forces

! Identification for various references #ref on the pitch

! #ref (= 0 for travel to be exited

$ Results give about 20% variation on parameter between experiments

! Median values are given by

b"0 = 0.16 , c#! = 0.026 , a"

0 = 2.59 , a"1 = 0.032

b#0 = "0.112 , a#

1 = 0.114


Work done since the final paper - Conclusions

Closed-loop 3D-motion experiments

# Good behavior of the model for some simple and slow moves

! Instability for quick changes of reference signal

" Errors in transient behavior of the model for low propeller speed

" Need to improve the model

Identification with other filter D(s)

" Algorithm converges to other values of parameters

" Need to clarify the dependency of results w.r.t. excitation signal and D(s)


adaptiv e p ar ameter identiÞcation for simpliÞed...

Documents