Content Variations on CUSUM tests for flutter monitoring - - PowerPoint PPT Presentation

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Variations on CUSUM tests for flutter monitoring

Mich` ele Basseville, Laurent Mevel, Rafik Zouari IRISA (CNRS & INRIA), 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 Variations on the CUSUM test 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, Handling transients between steady flight test points

• 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
• Variations on the CUSUM test

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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
• H⋆

p+1,q (reference data)

U(θ⋆)T

• 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;
• Hp+1,q

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

• U(θ⋆)T
• Hp+1,q
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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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Variations on the CUSUM test - (1)

For detecting aircraft instability precursors, select: a) An instability criterion ψ and a critical value ψc; b) A left kernel matrix U(.); c) A reference θ⋆ for estimating Jn(θ⋆) and Σ−1

n (θ⋆);

d) A min. change magnitude νm and a threshold ̺.

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Variations on the CUSUM test - (2)

Three solutions for b)-c):

• 1. θ⋆ ∆

= θ0 identified on reference data for the stable system; U(θ⋆) computed, Jn(θ0), Σ−1

n (θ0) estimated recursively with the test data.

• 2. θ⋆ ∆

= θc, critical parameter closer to instability, computed at each flight point using θ0 and an aeroelastic model; U(θ⋆) computed, Jn(θc), Σ−1

n (θc) estimated recursively with the test data.

• 3. U(.) ∆

=

• Un estimated on test data,

Jn(θ0), Σ−1

n (θ0) estimated recursively with the test data.

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Solution 2. - Details

i) Compute the critical eigenvalues λc at flight point t using identified modal signatures (θ1, . . . , θt) and extrapolation of the characteristic polynomial associated with the quasi-steady aeroelastic model M ¨ q + (D + V B) ˙ q + (K + V 2C)q = 0; ii) Build the critical modal signature θc from λc and the mode-shapes ϕλ identified at flight point t; iii) Compute the Zk(θc)’s and Sn(θc); Compute ˆ Jn(θc) and ˆ Σ−1

n (θc) with the test data;

iv) Run the CUSUM test between flight points t and t + 1; v) Repeat these steps for flight point t + 1: modal identification of θt+1 to update the prediction of θc, computation of ˆ Jn(θc) and ˆ Σ−1

n (θc),

CUSUM test between t + 1 and t + 2.

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Solution 3. - Details

i) Initialization For an initial airspeed: Estimate a reference θ0 and compute the constant terms in ˆ Jn; Select data sample size L, lag τ, block size K, νm, and ̺; Compute ˆ Σ−1

L+τ and

ˆ JL+τ with the first L + τ samples; Compute ˆ UL+τ with (Y1, ..., YL); Compute the Zk’s and SL+τ. ii) Recursive loop Running the CUSUM test: for each n ≥ L + τ: Compute recursively

• Un with (Yn−τ−L+1, ..., Yn−τ);

Use

• Un with
• Σ−1

n

and

• Jn to compute Sn and gn until gn ≥ ̺.

Update recursively

• Σ−1

n

and

• Jn every K samples.

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Solution 3. - Details (Contd.)

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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 ψ: 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)

Frequencies Damping coefficients Bending & torsion modes 20700-size 2D-samples simulated (300 for each V =20:1:88m/s).

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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, online recursive ˆ Jn, ˆ Σn. Solution 2. with θ⋆ = θc at V = 85m/s, online recursive ˆ Jn, ˆ Σn. Solution 3. with online recursive ˆ Un, ˆ Jn, ˆ Σn. Alarm onset times depend on threshold; ˆ Vf is more important. Solution 1. θ⋆ far from instability, alarm at V =67m/s, ˆ Vf=65m/s. The test detects that torsional damping decreases under ψc. Solution 2. θ⋆ close to instability, alarm at V =88m/s, ˆ Vf=85m/s. The test detects that flutter is happening between two steady points, and confirms the flutter prediction. Solution 3. Alarm at V =88m/s, ˆ Vf=78m/s much closer to flutter. Good behavior for light damping decrease before alarm. Detection (before flutter) of torsional damping drop.

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SLIDE 5

20 30 40 50 60 70 80 88 10 20 30 40 50 60 70 80 90 100 Airspeed (m/s) CUSUM test for the flutter monitoring for the flexional damping mode using V=20m/s reference 20 30 40 50 60 65 67 70 80 90 10 20 30 40 50 60 70 80 90 100 CUSUM test for flutter monitoring for the torsional damping coefficient using V=20m/s reference Airspeed (m/s)

Solution 1: Bending mode Solution 1: Torsion mode

30 40 50 60 70 80 82 85 88 50 100 150 200 250 300 350 400 450 500 Airspeed (m/s) CUSUM test with moving reference for the bending damping coefficient 30 40 50 60 70 80 82 85 88 50 100 150 200 250 300 350 400 450 500 Airspeed (m/s) CUSUM test with moving reference for the torsional damping coefficient

Solution 3: Bending mode Solution 3: Torsion mode No alarm Alarm of Sol.3 closer to flutter

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20 40 60 80 100 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 Modal Frequencies (Hz) Airspeed (m/s) 20 40 60 80 100 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Modal Damping Coefficients Airspeed (m/s) Torsion mode Flexion mode Torsion mode Flexion mode 41 42 43 44 45 46 47 48 49 50 50 100 150 200 250 300 350 400 450 500 CUSUM test between V=41m/s and V=50m/s Airspeed (m/s)

θ’s estimated up to V = 40m/s,

• Sol. 2 between flight points,

θc predicted V = 40m/s and 50m/s

20 40 60 80 100 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 Modal Frequencies (Hz) Airspeed (m/s) 20 40 60 80 100 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Modal Damping Coefficients Airspeed (m/s) Torsion mode Flexion mode Flexion mode Torsion mode 41 50 60 70 8081 85 86 87 88 50 100 150 200 250 300 350 400 450 500 CUSUM test for flutter detection Airspeed (m/s) Test results between V=41m/s and V=80m/s Test results between V=81m/s and V= 88m/s

θ’s estimated up to V = 80m/s

• Sol. 2 between flight points

θc predicted V = 40m/s and 80m/s and V = 81m/s and 90m/s Drop of the torsion mode damping

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Conclusion

Online detection for flutter monitoring Model-free subspace statistics, local approach, CUSUM Analytical model for flutter prediction Recursive computation of Jacobian and covariance matrices Three variants of CUSUM Algo 1: detection of ψ ≤ ψ0 Algo 2: flutter detection Algo 3: detection of abrupt drop in ψ Relevance on a small simulated structure Limitations: cost of online covariance computation Availability of flutter prediction model in real cases Major issues: dimension of θ, large number of correlated criteria

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