Lessons learned from the theory and practice of. change detection. Introduction. Content. Simulated data - One change (Signal and spectral densities)

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1 Lessons learned from the theory and practice of change detection Simulated data - One change (Signal and spectral densities) - Michèle Basseville IRISA / CNRS, Rennes, France michele.basseville@irisa.fr Introduction Detection of changes Content Stochastic models (static, dynamic) uncertainties Parameterized models (physical interpretation, diagnostics) Damage deviation in the parameter vector Changes in the mean Changes in the spectrum Many changes of interest are small Changes in the system dynamics and vibration-based SHM Early detection of (small) deviations is useful Key issue (): which function of the data should be handled? Example: flutter monitoring Key issue (): how to make the decision? 3 4

2 Changes in the mean Hypothesis testing Key concepts - Independent data Likelihood p θ (y i ) Log-likelihood ratio s i = ln p θ (y i ) p θ (y i ) E θ (s i ) < E θ (s i ) > p θ (Y N Likelihood ratio Λ N = ) p θ (Y N) = i p θ (y i ) i p θ (y i ) Log-likelihood ratio S N = ln ΛN = N i= s i Hypotheses H H Simple θ θ Known parameter values Composite Θ Θ Unknown parameter values Simple hypotheses : Likelihood ratio test If Λ N λ/s N h : decide H, H otherwise Composite hypotheses : Generalized likelihood ratio (GLR) Λ N = sup θ Θ p θ (Y N) sup θ Θ p θ (Y N) = p θ (Y N ) p θ (Y N ) Maximize the likelihoods w.r.t. unknown values of θ and θ 6 5 On-line change detection, unknown onset time Simple case: Known θ - CUSUM algorithm θ o t o θ? n Ratio of likelihoods under H and H : t i= p θ (y i ) ki= p θ (y i ) ki=t p θ (y i ) = ki=t p θ (y i ) ki=t p θ (y i ) = Λk t Hypothesis H θ = θ known ( i k) Hypothesis H t s.t. θ = θ ( i<t ) θ (t i k) Alarm time t a : t a =min{k :g k h} decision function Maximize over the unknown onset time t : ( t ) k = arg max j k = arg max j k = arg max j k j i= p θ (y i ) Λ k j S k j, k i=j p θ (y i ) Sk j =ln Λk j Estimated onset time: t g k = max j k Sk j = lnλ kˆt 7 8

3 CUSUM algorithm (Contd.) Lesson CUSUM algorithm (Contd.) - Gaussian example g k = max j k Sk j = S k min j k Sj = Sk m k, m k = min j k Sj N (µ, σ ), θ = µ, p θ (y) = σ π exp (y i µ) σ s i = ln p µ (y i ) p µ (y i ) t a = min{k :S k m k + h} Adaptative threshold = ( (yi σ µ ) (y i µ ) ) g k = (g k + s k ) + g k = ( S k k N k +) +, Nk = Nk I(g k )+ = ν σ y i µ ν, ν = µ µ ( t ) k = t a N ta + Sliding window with adaptive size 9 S k involves k i= y i : Integrator (with adaptive threshold) CUSUM algorithm - Gaussian example (Contd.) Composite case: Unknown θ Modified CUSUM algorithms y k Minimum magnitude of change Weighted CUSUM GLR algorithm -4 S k Double maximization g k = max j k sup θ S k j (θ ) g k Gaussian case, additive faults: second maximization explicit.

4 Unknown θ - Gaussian example (Contd.) Changes in the spectrum Minimum magnitude of change ν m Lesson Key concepts - Dependent data Decreasing mean T k = k i= y i µ + ν m Increasing mean U k = k i= y i µ ν m Conditional likelihood p θ (y i Y i ) Log-likelihood ratio s i = ln p θ (y i Y i ) p θ (y i Y i ) E θ (s i ) < M k = max j k T j m k = min j k U j t a =min{k :M k T k h} t a =min{k :U k m k h} E θ (s i ) > p θ (Y N Likelihood ratio Λ N = ) p θ (Y N) = i p θ (y i Y i ) i p θ (y i Y i ) Log-likelihood ratio S N = ln ΛN = N i= s i 3 4 Key concepts - Dependent data (Contd.) Key concepts - Dependent data (Contd.) θ : reference parameter, known (or identified) Which residuals? Lesson 3 Y k : N-size sample of new measurements Likelihood ratio computationally complex Build a residual ζ significantly non zero when damage Efficient score : likelihood sensitivity w.r.t. parameter vector Other estimating fonctions Residual Estimating function ζ N (θ, Y N ) Characterized by: E θ ζ N (θ, Y N )= θ = θ Innovation not sufficient for monitoring the dynamics. Mean sensitivity J (θ ) and covariance Σ(θ ) of ζ N (θ ) 5 6

5 Key concepts - Dependent data (Contd.) Changes in the dynamics and vibration-based SHM The residual is asymptotically Gaussian ζ N (θ ) N (, Σ(θ )) if P θ N ( J (θ ) δθ, Σ(θ )) if P θ + δθ N (On-board) χ -test in the residual small deviations Structural monitoring : Eigenstructure monitoring X k+ = F X k + V k F Φ λ = λ Φ λ Y k = H X k ϕ λ = H Φλ ζ T N Σ J (J T Σ J ) J T Σ ζ N h Canonical parameter : θ = Λ vec Φ modes mode shapes Noises and uncertainty on θ taken into account. 7 8 Example: flutter monitoring Detecting structural changes Monitoring a damping coefficient Reference data covariances Hankel matrix H Left null space S s.t. S T H = (S T O(θ )=) Fresh data covariances Hankel matrix H Check if ζ = S T H ζ asympt. Gaussian, test: χ in ζ Test ρ ρ c against ρ<ρ c Write the subspace-based residual ζ as a cumulative sum Introduce a minimum change magnitude (actual change magnitude unknown) 9 Run two CUSUM tests in parallel (actual change direction unknown)

6 Flutter monitoring (Contd) Application to real datasets Ariane booster launcher during a launch scenario on the ground (not during a real flight test) Running the test with critical values With Laurent Mével, Maurice Goursat, Albert Benveniste Free COSMAD Toolbox to used with Scilab Automatic identification. Each symbol: processing 5 sec. data. Conclusion Advanced statistical signal processing mandatory for SHM On-line test for ρ c = ρ = ρ () c. A statistical framework enlightens the meaning and increases the power of a number of familiar operations integration, averaging, sensitivity, adaptive thresholds & windows Change detection useful for (vibration-based) SHM On-line test for ρ c = ρ = ρ () c <ρ () c. Bottom: g n reflects ρ<ρ c. Top: g + n reflects ρ>ρ c. Current investigations flutter monitoring (with Dassault, Airbus), handling the temperature effect on civil structures (with LCPC) 3 4