Adaptiv e P ar ameter IdentiÞcation for SimpliÞed...
Transcript of 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
& 2 IFAC ALCOSP’07, August 2007, St. Petersburg
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.
& 3 IFAC ALCOSP’07, August 2007, St. Petersburg
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 :
"
#$˙Xz(t)
zn(t)
%
&' =
,
-----------.
0 1 0. . . . . .
0 0 1
"d0 "d1 · · · "dn!1
"d0 "d1 · · · "dn!1
/
000000000001
Xz(t) +
,
-----------.
0...
0
1
1
/
000000000001
z(t)
& 4 IFAC ALCOSP’07, August 2007, St. Petersburg
Kalman filtering for yn(t) = !T (t)!
Estimator of
Estimate !" = !(t#$) 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.
& 5 IFAC ALCOSP’07, August 2007, St. Petersburg
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)
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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
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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
! "! #! $! %! &! '! (! )! *! "!!!#
!"+&
!"
!!+&
!
!+&
"
"+&
#
& 8 IFAC ALCOSP’07, August 2007, St. Petersburg
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
!" !# !! !$ !% &" &# &!
!&
!!
!'
!#
!(
"
(
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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
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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)
& 11 IFAC ALCOSP’07, August 2007, St. Petersburg