Statistical detection approach to flutter monitoring Laurent M - - PowerPoint PPT Presentation

Statistical detection approach to flutter monitoring

Laurent M´ evel, Mich` ele Basseville, Albert Benveniste, IRISA (CNRS & INRIA, Rennes, France) basseville@irisa.fr - http://www.irisa.fr/sigma2/

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Contents

• Motivation and modelling
• Eigenstructure monitoring: subspace-based residual
• Using the residual for flutter monitoring

– Local approach and quadratic tests (GLR) – Other approximation and linear tests (CUSUM)

• Numerical results - Ariane flight 501

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In-flight modal analysis and aircraft stability

• Ensuring aircraft stability:

in-flight tests with increasing altitude and airspeed

• Limited choices for measured excitation inputs,

natural excitation input (turbulence): not controlled, not measured, and nonstationary

• Flutter: monitoring critical damping coefficients:

accuracy and real-time issues in identification

• Idea: detection algorithms (shorter response time)

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Modelling - Eigenstructure problem FE model:

                  

M ¨ Z(s) + C ˙ Z(s) + K Z(s) = ν(s) Y (s) = L Z(s) (Mµ2 + Cµ + K) Ψµ = 0 , ψµ = L Ψµ

State space:

                  

Xk+1 = F Xk + Vk Yk = H Xk F ϕλ = λ ϕλ , φλ

∆

= H ϕλ

Parameter:

eδµ = λ

• modes

, ψµ = φλ

• mode shapes

; θ ∆ =

    

Λ vec Φ

    

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Output-only covariance-based subspace identification

Ri

∆

= E

 Yk Y T

k−i

 

• k if stationary !

, H = Hank(Ri) , Ri = H F i G G ∆ = E

 Xk Y T

k

  ,

O ∆ =

        

H HF HF 2

. . .

         , C ∆

=

• G F G F 2G . . .
• H = O C , H −

→ O − → (H, F ) − → (λ, ϕλ)

Implementation:

ˆ Ri

∆

= 1/n

n

• k=1 Yk Y T

k−i

• k when nonstationary !

, ˆ H = Hank( ˆ Ri)

SVD( ˆ

H) + truncation − → ˆ O − → ( ˆ H, ˆ F ) − → (ˆ λ, ˆ ϕλ)

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Eigenstructure monitoring

θ0 : reference parameter, known (or identified) Yk: n-size sample of new measurements

System parameter characterization:

Hp+1,q and Op+1(θ) have the same left kernel. ∃U, UT U = Is, UT Op+1(θ0) = 0;

say U(θ0)

θ0 ↔ (R0

i )i

characterized by:

UT (θ0) ˆ H0

p+1,q = 0

Subspace-based residual for eigenstructure monitoring

ζn(θ0) ∆ = √n vec(UT (θ0) ˆ Hp+1,q)

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The residual is asymptly Gaussian (local approach) Mean sensitivity (Jacobian) J (θ0) and covariance Σ(θ0)

ζn(θ0) →

                        

N ( 0, Σ(θ0))

under

Pθ0 N ( J (θ0) δθ, Σ(θ0))

under

Pθ0+ δθ

√n

(GLR) χ2-test for modal monitoring

ζT

n Σ−1 J (J T Σ−1 J )−1 J T Σ−1 ζn

≥ h

(GLR) Directional χ2-test for modal diagnosis

ζT

n Σ−1 Ji (J T i

Σ−1 Ji)−1 J T

i

Σ−1 ζn ≥ h

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Flutter monitoring with residual ζ - Quadratic tests

• χ2 focussed on damping ρ : BUT ρ = ρ0 irrelevant;
• Hypothesis of interest: ρ < ρc
• 1. ρc = ρ0: Use GLR test for (local) δρ ≥ 0 against δρ < 0:

l(θ0) = − sign(ζ) · χ2, χ2 ∆ = ζ T Σ −1ζ

• 2. ρc < ρ0: non local hypotheses ρ ≥ ρc against ρ < ρc:

˜ θ0 ∆ = θ0 except that ρ0 ← ρc ;

use l(˜

θ0) : BUT biased

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Flutter monitoring with residual ζ - Linear tests

• Another approximation for ζ:

ζ(˜ θ0) =

n

• k=1 Zk(˜

θ0)/√n, Zk(˜ θ0) ∆ = U(˜ θ0)T Y+

k,p+1 Y− k,q n

• k=1 Zk/√n → N (0, ·)

under ρ = ρc

⇒ Zk → N (0, ·), and the Zk’s are iid

• Idea: for overcoming a bias, handle deviations !

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Flutter monitoring - Linear tests (Contd.) Use CUSUM test for ρ = ρc + ǫ against ρ = ρc − ǫ:

Sn(˜ θ0)

∆

=

n

• k=1 Zk(˜

θ0), Mn(˜ θ0) ∆ = max

1≤k≤n Sk(˜

θ0) gn(˜ θ0)

∆

= Mn(˜ θ0) − Sn(˜ θ0) ≥ 0

under ρ = ρc + ǫ :

gn(˜ θ0) ≈ 0

under ρ = ρc − ǫ :

gn(˜ θ0) > 0

if ρ decreases,

gn(˜ θ0) increases

if ρ increases,

gn(˜ θ0) decreases

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Numerical results - Ariane

• Identification: 2000-size blocks;

Flutter monitoring: sample by sample

• Flutter monitoring: 2 reference data sets:
• ne at the beginning → ρc = 1.5% ,
• ne at the middle → ρc = 0.5%
• Test without and with filtering
• Of interest: increase/decrease of the test gn

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−1.0 18.2 37.3 56.5 75.7 94.8 114.0 20.17 23.40 27.04 30.27 33.51 37.14 40.38 44.01 47.24 (Test value) (Sec) −1.0 2.8 6.7 10.5 14.3 18.2 22.0 20.17 23.40 27.04 30.27 33.51 37.14 40.38 44.01 47.24 (Test value) (Sec)

¨

Damping & frequency, CUSUM w/o filtering: ρc = 1.5% and 0.5%

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−1 54 108 163 218 272 327 20.17 23.40 27.04 30.27 33.51 37.14 40.38 44.01 47.24 (Test value) (Sec) −1.0 2.5 6.0 9.5 13.0 16.5 20.0 20.17 23.40 27.04 30.27 33.51 37.14 40.38 44.01 47.24 (Test value) (Sec)

Damping & frequency, CUSUM with filtering: ρc = 1.5% and 0.5%

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