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/ 1

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 2

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) 3

Modelling - Eigenstructure problem ¨ ˙  Z ( s ) + C Z ( s ) + K Z ( s ) = ν ( s ) M          FE model:     Y ( s ) = L Z ( s )      ( Mµ 2 + Cµ + K ) Ψ µ = 0 , ψ µ = L Ψ µ  X k +1 = F X k + V k         State space:      Y k = H X k      ∆ F ϕ λ = λ ϕ λ , φ λ = H ϕ λ   Λ e δµ = λ ; θ ∆ Parameter: , ψ µ = φ λ =       vec Φ   � �� � � �� � modes mode shapes 4

Output-only covariance-based subspace identification R i = H F i G ∆    Y k Y T H = Hank( R i ) , R i = E , k − i  � �� � ok if stationary !   H   G ∆   O ∆  , C ∆  X k Y T  , HF � G F G F 2 G . . . �   = E = =     k HF 2       .  . . H = O C , H − → O − → ( H, F ) − → ( λ, ϕ λ ) n ∆ k =1 Y k Y T ˆ H = Hank( ˆ ˆ Implementation: R i = 1 /n , R i ) � k − i � �� � ok when nonstationary ! → (ˆ SVD ( ˆ → ˆ → ( ˆ H, ˆ H ) + truncation − O − F ) − λ, ˆ ϕ λ ) 5

Eigenstructure monitoring θ 0 : reference parameter, known (or identified) Y k : n -size sample of new measurements System parameter characterization: H p +1 ,q and O p +1 ( θ ) have the same left kernel. U T O p +1 ( θ 0 ) = 0; U T U = I s , ∃ U, say U ( θ 0 ) U T ( θ 0 ) ˆ θ 0 ↔ ( R 0 H 0 i ) i characterized by: p +1 ,q = 0 Subspace-based residual for eigenstructure monitoring = √ n vec( U T ( θ 0 ) ˆ ζ n ( θ 0 ) ∆ H p +1 ,q ) 6

The residual is asymptly Gaussian (local approach) Mean sensitivity (Jacobian) J ( θ 0 ) and covariance Σ( θ 0 )  N ( 0 , Σ( θ 0 )) under P θ 0            ζ n ( θ 0 ) →   N ( J ( θ 0 ) δθ, Σ( θ 0 )) P θ 0 + δθ under      √ n       (GLR) χ 2 -test for modal monitoring n Σ − 1 J ( J T Σ − 1 J ) − 1 J T Σ − 1 ζ n ζ T ≥ h (GLR) Directional χ 2 -test for modal diagnosis n Σ − 1 J i ( J T Σ − 1 J i ) − 1 J T Σ − 1 ζ n ζ T ≥ h i i 7

Flutter monitoring with residual ζ - Quadratic tests • χ 2 focussed on damping ρ : BUT ρ � = ρ 0 irrelevant; • Hypothesis of interest: ρ < ρ c 1. ρ c = ρ 0 : Use GLR test for (local) δρ ≥ 0 against δρ < 0 : = ζ T Σ − 1 ζ χ 2 ∆ l ( θ 0 ) = − sign( ζ ) · χ 2 , 2. ρ c < ρ 0 : non local hypotheses ρ ≥ ρ c against ρ < ρ c : θ 0 ∆ ˜ use l (˜ = θ 0 except that ρ 0 ← ρ c ; θ 0 ) : BUT biased 8

Flutter monitoring with residual ζ - Linear tests • Another approximation for ζ : θ 0 ) / √ n, Z k (˜ n θ 0 ) T Y + k,p +1 Y − θ 0 ) ∆ ζ (˜ k =1 Z k (˜ = U (˜ θ 0 ) = � k,q k =1 Z k / √ n → N (0 , · ) n under ρ = ρ c � ⇒ Z k → N (0 , · ) , and the Z k ’s are iid • Idea: for overcoming a bias, handle deviations ! 9

Flutter monitoring - Linear tests (Contd.) Use CUSUM test for ρ = ρ c + ǫ against ρ = ρ c − ǫ : n ∆ θ 0 ) ∆ S n (˜ k =1 Z k (˜ M n (˜ 1 ≤ k ≤ n S k (˜ θ 0 ) = θ 0 ) , = max θ 0 ) � ∆ g n (˜ = M n (˜ θ 0 ) − S n (˜ θ 0 ) ≥ 0 θ 0 ) g n (˜ θ 0 ) ≈ 0 under ρ = ρ c + ǫ : g n (˜ under ρ = ρ c − ǫ : θ 0 ) > 0 g n (˜ if ρ decreases , θ 0 ) increases g n (˜ if ρ increases , θ 0 ) decreases 10

Numerical results - Ariane • Identification: 2000-size blocks; Flutter monitoring: sample by sample • Flutter monitoring: 2 reference data sets: one at the beginning → ρ c = 1 . 5% , one at the middle → ρ c = 0 . 5% • Test without and with filtering • Of interest: increase/decrease of the test g n 11

(Test value) 114.0 94.8 75.7 56.5 37.3 18.2 (Sec) ¨ −1.0 20.17 23.40 27.04 30.27 33.51 37.14 40.38 44.01 47.24 (Test value) 22.0 18.2 14.3 10.5 6.7 2.8 (Sec) −1.0 20.17 23.40 27.04 30.27 33.51 37.14 40.38 44.01 47.24 Damping & frequency, CUSUM w/o filtering: ρ c = 1.5% and 0.5% 12

(Test value) 327 272 218 163 108 54 (Sec) −1 20.17 23.40 27.04 30.27 33.51 37.14 40.38 44.01 47.24 (Test value) 20.0 16.5 13.0 9.5 6.0 2.5 (Sec) −1.0 20.17 23.40 27.04 30.27 33.51 37.14 40.38 44.01 47.24 Damping & frequency, CUSUM with filtering: ρ c = 1.5% and 0.5% 13