Uncertainty Quantification for Stochastic Subspace Identification of Multi-Setup Measurements

In Operational Modal Analysis, the modal parameters (natural frequencies, damping ratios and mode shapes) obtained from Stochastic Subspace Identification (SSI) of a structure, are afflicted with statistical uncertainty. For evaluating the quality of the obtained results it is essential to know the appropriate confidence intervals of these figures. In this paper we present algorithms that automatically compute the confidence intervals of modal parameters obtained from covariance-driven and data-driven SSI of a structure based on vibration measurements. These algorithms are adapted to handle data from different measurements of a structure, where roving sensors are moved from one measurement setup to another, while some reference sensors stay fixed throughout all the measurements. In this case, the different ambient excitations of the structure between the measurements have to be taken into account. With these new algorithms, confidence intervals of the modal parameters of some relevant industrial example are computed. 2 IOMAC'11 – 4 International Operational Modal Analysis Conference estimated with the data-driven UPC algorithm was computed. In this paper, the uncertainty quantification is generalized to stochastic subspace identification using multi-setup measurements. 2 MULTI-SETUP STOCHASTIC SUBSPACE IDENTIFICATION 2.1 Models and Parameters The behaviour of a mechanical system is assumed to be described by a stationary linear dynamical system ( ) ( ) ( ) ( ), ( ) ( ) MZ t CZ t KZ t v t Y t LZ t + + = = ɺɺ , (1) where t denotes continous time, M, C and K are the mass, damping and stiffness matrices, highdimensional vector Z collects the displacements of the degrees of freedom of the structure, the non-measured external force v modelled as non-stationary Gaussian white noise, the measurments are collected in the vector Y and matrix L indicates the sensor locations. The eigenstructure of (1) with the modes μ and mode shapes φμ is a solution of 2 2 det( ) 0, ( ) 0, M C K M C K L μ μ μ μ μ μ μ φ φ φ + + = + + = = . (2) Sampling model (1) at some rate 1/τ yields the discrete model in state-space form 1 1, k k k k k X FX V Y HX + + = + = , (3) whose eigenstructure is given by det( ) 0, ( ) 0, F I F I H λ λ λ λ λ φ φ φ − = − = = . (4) Then, the eigenstructure of the continous system (1) is related to the eigenstructure of the discrete system (3) by e , τμ μ λ λ φ φ = = . (5) The collection of modes and mode shapes (λ,φλ) is a canonical parameterization of system (3). From the eigenvalues μ the natural frequencies f and damping ratios d with f = Im(μ)/(2π), d = –Re(μ)/|μ|. (6) are retrieved. 2.2 Single-Setup Stochastic Subspace Identification To obtain the modal parameters (frequencies, damping ratios and mode shapes) from measurements (Yk)k = 1,...,N, the covariance-driven output-only subspace identification algorithm (Benveniste and Fuchs 1985, Peeters and De Roeck 1999) and the data-driven Unweighted Principal Component algorithm (Van Overschee and De Moor 1996, Peeters and De Roeck 1999) are used. They only differ in the computation the so-called subspace matrix H. In the covariance-driven SSI, a block Hankel matrix H is filled with the correlation lags Ri = E(YkYk-i ) of the output data