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SLIDE 1

An adaptive statistical approach to flutter detection

Rafik Zouari, Laurent Mevel, Mich` ele Basseville IRISA (INRIA & CNRS), Rennes, France Eurˆ eka project no 3341 FliTE2 michele.basseville@irisa.fr -- http://www.irisa.fr/sisthem/

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Content

Introduction Subspace-based residual for modal monitoring CUSUM test for monitoring a scalar instability index Using and tuning the CUSUM test A moving reference version Experimental results Conclusion

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Introduction - (1)

• Flutter: critical aircraft instability phenomenon

unfavorable interaction of aerodynamic, elastic and inertial forces; may cause major failures

• Flight flutter testing, very expensive and time consuming :

Design the flutter free flight envelope

• Flutter clearance techniques:

In-flight identification: output-only, or using input excitations Data processing: time-frequency, wavelet, envelope function Flutter prediction based on model-based approaches: flutterometer (µ-robustness), physical model updating

• Some challenges:

Real time on-board monitoring, robustness to noise and uncertainties

• Our approach:

Statistical detection for monitoring instability indicators

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Introduction - (2)

• Aim of in-flight online flutter monitoring:

Early detection of a deviation in the aircraft modal parameters before it develops into flutter.

• Change-point detection: natural approach
• For a scalar instability criterion ψ and a critical value ψc,
• nline hypotheses testing:

H0 : ψ > ψc and H1 : ψ ≤ ψc

• CUSUM test as an approximation to the optimal test
• A moving reference version

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SLIDE 2

Subspace-based residual for modal monitoring

                

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

∆

= H φλ Ri

∆

= E

• Yk Y T

k−i

• ,

H

∆

=

           

R0 R1 R2 . . . R1 R2 R3 . . . R2 R3 R4 . . . . . . . . . ... . . .

           

Ri = H F i G = ⇒ H = O C

O ∆ =

        

H HF HF 2 . . .

        

, C ∆ =

G

F G F 2G . . .

• G ∆

= E

• Xk Y T

k

• Output-only covariance-driven subspace identification

SVD of H − → O − → (H, F ) − → (λ, ϕλ)

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Canonical parameter : θ ∆ =

   

Λ vec Φ

   

modes mode shapes

Observability in modal basis : Op+1(θ) =

        

Φ Φ∆ . . . Φ∆p

        

Given:

• a reference parameter θ⋆, by SVD of ˆ

H⋆

p+1,q (reference data)

U(θ⋆)T ˆ H⋆

p+1,q = 0

parameter estimating function U(θ⋆)T Op+1(θ⋆) = 0 , U(θ⋆)T U(θ⋆) = I

• a n-size sample of new data; ˆ

Hp+1,q For testing θ = θ⋆, statistics (residual) : ζn(θ⋆) ∆ = √n vec

• U(θ⋆)T

ˆ Hp+1,q

• 6

Local approach to testing

• H0 : θ = θ⋆

and

• H1 : θ = θ⋆ + Υ/√n

Mean sensitivity and covariance matrices: Jn(θ⋆, θ) ∆ = 1/√n ∂/∂ ˜ θ Eθ ζn(˜ θ)

• ˜

θ=θ⋆ , Σn(θ⋆, θ) ∆

= Eθ

• ζn(θ⋆) ζn(θ⋆)T
• If Σn(θ⋆, θ) is positive definite, and for all Υ, under both hypoth:

Σn(θ⋆, θ)−1/2 (ζn(θ⋆) − Jn(θ⋆, θ) Υ) n → ∞ → N (0, I) Normalized residual: ζn(θ⋆) ∆ = Kn(θ⋆, θ) ζn(θ⋆)

Kn(θ⋆, θ) ∆ = Σ−1/2

n

J T

n Σ−1 n

, Σn(θ⋆, θ) ∆ = J T

n Σ−1 n Jn

• ζn(θ⋆) − Σn(θ⋆, θ)1/2 Υ
• n → ∞

→ N (0, I)

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Data-driven computation for online detection

ζn(θ⋆) ≈

n−p

• k=q Zk(θ⋆)/√n

Zk(θ⋆) ∆ = Kk(θ⋆, θ) vec

• U(θ⋆)

T Y+ k,p+1 Y−T k,q

• Another approximation

For n large enough, and k = 1, . . . , n, Zk(θ⋆) ≈ Gaussian i.i.d., mean 0 before change and = 0 after. Monitoring any function ψ(θ) Replace Jn(θ⋆, θ) with Jn(θ⋆, θ) J ⋆

θψ, where J ⋆ θψ = ∂θ/∂ψ|θ=θ⋆.

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SLIDE 3

CUSUM test for monitoring a scalar index

The crossing of a critical ψc by ψ is reflected into a change with the same sign in the mean ν of the i.i.d. Gaussian Zk(θ⋆). The CUSUM test may be used for testing between: H0 : ν > 0 and H1 : ν ≤ 0 Procedure for unknown sign and magnitude of change in ψ i) Set a min. change magnitude νm > 0, and test between: H0 : ν > νm/2 and H1 : ν ≤ −νm/2 Sn(θ⋆) ∆ =

n−p

• k=q (Zk(θ⋆) + νm), Tn(θ⋆) ∆

= max

k=q,...,n−p Sk(θ⋆)

gn(θ⋆) ∆ = Tn(θ⋆) − Sn(θ⋆)

H1

> <

H0

̺ threshold ii) Run 2 tests in parallel, for decreasing and increasing ψ; iii) Make a decision from the first test which fires; iv) Reset all sums and extrema to 0, switch to the other test.

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Using and tuning the CUSUM test

For detecting aircraft instability precursors, select: a) An instability criterion ψ and a critical value ψc; b) A left kernel matrix U(.); c) Estimates of Jn(θ⋆, θ) and Σn(θ⋆, θ); d) A min. change magnitude νm and a threshold ̺. Two solutions for b)-c):

• 1. θ⋆ ∆

= θ0 identified on reference data for the stable system; U(θ⋆) computed, Jn, Σn estimated once for all with those data.

• 2. U(.) ∆

= ˆ Un estimated on test data; Jn, Σn estimated recursively with those test data.

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The moving reference version

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Example - Aeroelastic Hancock wing model

Rigid wing with constant chord; 2 d.o.f. in bending and torsion. Matrix F , and eigenvalues λ: functions of airspeed V . Flutter airspeed: Vf = 88.5m/s. Stability indicator: Damping coefficient

10 20 30 40 50 60 70 80 90 7 8 9 10 11 12 13 Bending mode Torsional mode F(Hz) Modal frequencies variation with airspeed V(m/s) 10 20 30 40 50 60 70 80 88.5 −0.05 0.05 0.1 0.15 0.2 Bending mode Torsional mode Modal damping coefficients variation with airspeed Damping coefficient V(m/s)

Frequency Damping coefficient 20700-size 2D-samples simulated (300 for each V =20:1:88m/s).

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SLIDE 4

Example - Numerical results

CUSUM test run with νm = 0.1, ̺ = 100, and the damping as ψ. Solution 1. with θ⋆ = θ0 at V = 20m/s and fixed J , Σ. Solution 2. with online ˆ Un, ˆ Jn, ˆ Σn. Solution 1. θ⋆ far from instability, too early alarm at V =69m/s. The test detects that torsional damping decreases under the predefined threshold. Solution 2. Alarm at V = 82m/s much closer to flutter. The test detects that the torsional damping decreases abruptly. Both algorithms do what they are intended to do. Only Solution 2 is a flutter detection algorithm.

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20 30 40 50 60 70 80 85 88 20 40 60 80 100

CUSUM test for the bending damping coefficient V(m/s) (300 samples for each 1m/s)

20 30 40 50 60 70 80 85 88 20 40 60 80 100

CUSUM test for the torsional damping coefficient V(m/s) (300 samples for each 1m/s)

Solution 1: Bending mode Solution 1: Torsion mode

20 30 40 50 60 70 80 85 88 20 40 60 80 100

CUSUM test with moving reference for the bending damping coefficient V(m/s) (300 samples for each 1m/s)

20 30 40 50 60 70 80 85 88 20 40 60 80 100

V(m/s) (300 samples for each 1m/s) CUSUM test with moving reference for the torsional damping coefficient

Solution 2: Bending mode Solution 2: Torsion mode

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Conclusion

Online detection of instability precursors Model-free subspace statistics, local approach, CUSUM Analytical model for flutter prediction Recursive computation of covariance matrix Relevance on a small simulated structure Limitations: cost of online kernel and covariance computation Major issues: dimension of θ, large number of correlated criteria

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