Polyreference Frequency-Domain Least-Squares Estimation with Confidence Intervals

The PolyMAX estimator is used intensively in modal analysis applications nowadays. The main advantages are its speed and the very clear stabilization diagrams it yields. Recently, an algorithm allowing a fast calculation of confidence intervals has been derived for frequency-domain least-squares estimators based on a common-denominator model. In this contribution, the approach is extended to the PolyMAX estimator, i.e. a polyreference frequency-domain least-squares estimator, based on a right matrix-fraction description. If the coherences of the measured frequency response functions (FRFs) are available, the covariance matrix of the estimated model parameters can be obtained without major additional calculations. The confidence intervals can then be calculated in a second step. The correctness of the approach is verified by means of Monte Carlo simulations. NOMENCLATURE Hm(Ω f ) FRF right matrix-fraction model H(Ω f ) measured FRF N(Ω f ) numerator matrix polynomial D(Ω f ) denominator matrix polynomial No(Ω f ) Numerator matrix polynomial for output o Ω f polynomial basis function θ coefficients matrix θD denominator coefficients matrix θNo numerator coefficients matrix ε(Ω f , θ,H) error equation W(ω f ) weighting matrix in error equation J jacobian matrix Γo jacobian submatrix corresponding to numerator coefficients Υ jacobian submatrix corresponding to denominator coefficients (.)T matrix transpose M reduced normal equations Ro reduced normal equations submatrix So reduced normal equations submatrix To reduced normal equations submatrix I identity matrix , by definition ∆H perturbation on the FRF ∆θD perturbation on the denominator coefficients ∆M perturbation on the reduced normal equations ∆Υ perturbation on the Υ jacobian submatrix Ξ covariance matrix submatrix Cov(.) covariance matrix δi j kronecker delta (1 if i = j, 0 elsewhere) Re(.) taking real part Im(.) taking imaginary part Dcom companion matrix Vr eigenvector of companion matrix λr eigenvalue of companion matrix Lr modal participation vector P weighting matrix for covariance matrix .m mth iteration γ2 multiple coherence function Syy autopower spectrum of outputs S f f autopower spectrum of outputs