y ( n ) ( t )+ . . . + a 1 y ( t )+ a 0 y ( t ) = i ( t )+ . . . + b - - PowerPoint PPT Presentation

Adaptive Parameter Identification for Simplified 3D-Motion Model

• f ‘LAAS Helicopter Benchmark’

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

§ SEDITEC † LAAS-CNRS - Universit´

e 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)+. . .+a1 ˙ y(t)+a0y(t) =

m

• i=1

binu(n)

i (t)+. . .+bi1 ˙

ui(t)+bi0ui(t).

Define the following vectors

Xy(t) =

     

y(n−1)(t)

. . .

y(t)

     

, Xui(t) =

     

u(n−1)

i

(t)

. . .

ui(t)

     

, φT (t) =

• XT

y (t)

u(n)

1 (t)

XT

u1(t)

· · · u(n)

m (t)

XT

um(t)

• ΩT =
• an−1

. . . a0 b1n . . . b10 . . . bmn . . . bm0

• System compact model: y(n)(t) = φT(t)Ω.

Identification: least square estimation of Ω 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) = φT(t)Ω ⇒ ˜ yn(t) = ˜ φT(t)Ω

where ˜

yn(t) = D−1(s)y(n)(s) and ˜ φ(s) = D−1(s)φ(s) obtained by: ˜ φT =

• ˜

XT

y

˜ u1n ˜ XT

u1

· · · ˜ umn ˜ XT

um

• and for all z = y, u1, . . . um :

  

˙ ˜ Xz(t) ˜ zn(t)

   =             

1

... ...

1 −d0 −d1 · · · −dn−1 −d0 −d1 · · · −dn−1

            

˜ Xz(t) +

             . . .

1 1

            

z(t)

& 4 IFAC ALCOSP’07, August 2007, St. Petersburg

Kalman filtering for ˜

yn(t) = ˜ φT(t)Ω

Estimator of

Estimate Ω∗ = Ω(t → ∞) where Ω(t) solution of adaptive algorithm

˙ Ω(t) = −Γ(t)˜ φ(t)

˜

φT(t)Ω(t) − ˜ yn(t)

• ˙

Γ(t) = −Γ(t)˜ φ(t)˜ φT(t)Γ(t)+αΓ(t)

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

α > 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

¨ θ(t) + aθ

1 ˙

θ(t) + aθ

0 sin(θ(t) − θ0) = bθ 0µd(t)

¨ ǫ(t) + aǫ

1 ˙

ǫ(t) + aǫ

0 sin(ǫ(t) − ǫ0) + cλθ ˙

λ(t) ˙ θ(t) = bǫ

0µs(t) cos θ(t)

¨ λ(t) + aλ

1 ˙

λ(t) = bλ

0µs(t) sin θ(t) & 6 IFAC ALCOSP’07, August 2007, St. Petersburg

Identification of the pitch motion

MISO model of the non-linear dynamics

¨ θ(t) + aθ

1 ˙

θ(t) + aθ

0 sin(θ(t) − θ0) = bθ 0µd(t)

⇓ ¨ θ(t) + aθ

1 ˙

θ(t) = −aθ s(t)

• sin(θ(t)−θ0)

+bθ

0µd(t)

➙ θ0 = −7.8o measured as the equilibrium for µd = 0. ➙ D(s) = s2 + 2ωdρds + ω2 = s2 + 1.4s + s2 ➙ Permanent excitation: square + chirp

5 10 15 20 25 30 0.2 0.4 0.6 0.8 1 1.2 1.4

& 7 IFAC ALCOSP’07, August 2007, St. Petersburg

Pitch identification results ✪ α ∈ [0, 0.001]: good convergence (else oscillations appear) ✪ No major dependency w.r.t. initial guess Ω(0)

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

✪ Γ(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 bθ

0 ∈ [0.25, 0.3] ,

aθ

0 ∈ [0.58, 0.67] ,

aθ

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

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

& 9 IFAC ALCOSP’07, August 2007, St. Petersburg

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 θref on the pitch ➙ θref = 0 for travel to be exited ✪ Results give about 20% variation on parameter between experiments ➙ Median values are given by bǫ

0 = 0.16 ,

cλθ = 0.026 , aǫ

0 = 2.59 ,

aǫ

1 = 0.032

bλ

0 = −0.112 ,

aλ

1 = 0.114 & 10 IFAC ALCOSP’07, August 2007, St. Petersburg

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