Motivations Monitoring (the eigenstructure of) a (linear) system: - PowerPoint PPT Presentation

Motivations – Monitoring (the eigenstructure of) a (linear) system: On sensors positioning identification, damage detection and localization. for Structural Health Monitoring – For given monitoring requirements: How to achieve optimum sensors positioning ? Mich` ele Basseville – For a given sensors set: IRISA / CNRS, Rennes, France Which damages can be efficiently monitored ? basseville@irisa.fr - http://www.irisa.fr/sisthem/ – Criteria to assess sensors sets, handling eigenstructure, monitoring requirements, noise and uncertainties, excitation type. 1 2 Contents Sensors positioning issues for SHM – Sensors positioning issues for SHM System models and parameters – Sensors positioning criteria Invariant parameterizations – Scalar functions of a matrix Different numbers of sensors – Exploiting a distance between two matrices Frequency content and geometry of the excitation – A criterion: the power of a damage detection test 3 4

System models and invariant parameterizations Structural monitoring and sensors positioning problems statement ¨ ˙  M Z ( s ) + C Z ( s ) + K Z ( s ) = ε ( s )      FEM:   Y ( s ) = L Z ( s )    Structural monitoring   ( M µ 2 + C µ + K ) Ψ µ = 0 , ψ µ = L Ψ µ For a given sensor positioning L : monitor the modes and modeshapes ( λ, ϕ λ ).  X k +1 = F X k + V k       State space:  Y k = H X k      Sensors positioning e δµ = λ ∆ F Φ λ = λ Φ λ , ϕ λ = H Φ λ , , ψ µ = ϕ λ � �� � � �� � For a given excitation level and profile: modes mode − shapes p − 1 p Optimize an objective function w.r.t. matrix L : ARMA: Y k = i =1 A i Y k − i + j =0 B j W k − j � � – Using a parameterization invariant w.r.t. L ! A p λ p + ... + A 1 λ − I � � – Handling different numbers of sensors. ϕ λ = 0 5 6 Sensors positioning criteria Scalar functions of common use Matrix criteria For a q -dimensional matrix M and z < 0 : Observability, controllability, estimation error covariance, 1 /z Trace ( M z )     c z =   MAC matrix, Fisher information, ...     q   Scalar functions of a matrix Determinant, trace, extremal eigenvalue: Determinant, trace, extremal eigenvalues, q minimizing off-diagonal terms (e.g. of MAC), ... lim z ր 0 c z = | M | 1 /q , c − 1 = � M − 1 � Trace Invariance properties lim z ց−∞ c z = λ min ( M ) Measurements scaling, mode-shapes normalization, ... 7 8

Kullback distance between two symmetric matrices 2 K( M 1 , M 2 ) ∆ = Trace( M 1 M − 1 − I q ) − ln | M 1 M − 1 Exploiting a distance between matrices | 2 2 Invariance: K( A M 1 A T , A M 2 A T ) = K( M 1 , M 2 ) A distance between two matrices Distance to a diagonal matrix Distance to a diagonal matrix m ii q q i =1 ln δ i − ln | M | − q 2 K( M, ∆ q ) = + � � δ i i =1 Scalar functions of potential interest Approximation F ≈ � M − 1 − I q � 2 4 K( M, I q ) ≈ � M − I q � 2 F 9 10 Useful criterion: power of a damage detection test Scalar functions of potential interest, with different invariance properties q -dimensional Gaussian residual ζ Sensor set L reflected into matrices J , Σ in: ∆ = K( M, I q ) = (Trace M − ln | M | − q ) / 2 C 1 ( M )  H 0 : Υ = 0       ∆  A M A T     ζ ∼ N ( J Υ , Σ) : C 2 ( M ) = min δ> 0 K( M, δ I q ) = AA T = I C 0 max       H 1 : Υ � = 0 damage      ∆ � � C 3 ( M ) = ∆ q > 0 K min M, ∆ q = C 0 ( M ) How to compare different sensor sets, possibly with different numbers, thus different q ? q   ∆ = − 1 / 2 ln  | M | / C 0 ( M ) i =1 M ii mutual info  �     Use the test power 11 12

The power criterion Compensating for q χ 2 -test: For q large and small Υ (damage): χ 2 = ζ T Σ − 1 J ( J T Σ − 1 J ) − 1 J T Σ − 1 ζ (level) P 0 ( χ 2 < λ ) ≤ α , β = P 1 ( χ 2 ≥ λ ) (power) Noncentrality parameter: e − δ 2 / 2 β ≈ α + γ 2 δ = φ − 1 (1 − α ) √ 2 πq , Γ = J T Σ − 1 J γ 2 (Υ) = Υ T Γ Υ , 2 Fisher info δ does not depend upon q . Hence use: Test power : function of γ 2 only. Hence the criterion: ( β − α ) e δ 2 / 2 = γ 2 / 2 √ 2 πq Υ ∈ R m , � Υ � 2 =1 γ 2 (Υ) d Υ = Area( S m ) � Trace(Γ) Trace(Γ) m Integrating over unit sphere: √ q But, for a fixed false alarms rate, the test power de- pends on q ∆ = dim( ζ ) ! (implemented within the COSMAD toolbox) 13 14 Conclusion • Trade-off between instrumentation costs and information and efficiency of SHM algorithms • Criteria to be optimized for sensors positioning • Relevance of a given sensors set (number, positions) often summarized in a matrix • Invariance properties • Scalar functions of matrices : distances, test power 15