8 Iterative Pole-Zero Model Updating Using Multiple Frequency Response Functions 67 Now assume that m-FRF measurements from different actuator/sensor configurations are available. In IPZ model updating using multiple FRFs, the quadratic pole-zero error functional i. / D 0 BB B@ 2 66 64 p;e z1;e : : : zm;e 3 77 75 2 66 64 p;n. / z1;n. / : : : zm;n. / 3 77 75 i 1 CC CA H W 0 BB B@ 2 66 64 p;e z1;e : : : zm;e 3 77 75 2 66 64 p;n. / z1;n. / : : : zm;n. / 3 77 75 i 1 CC CA ; (8.7) is iteratively minimized by updating the generic parameters T D Œ T p; T z1 ; : : :; T zm , where the subscripts e and n indicate experimentally and numerically obtained quantities, respectively. z1; : : :; zm stand for the set of zeros from the first up to the mth-FRF measurement, respectively. The diagonal weighting matrix W 0 is applied in order to have equal contributions of the relative errors from each pole and zero. In a linear time invariant (LTI) system, pole locations are identical in any FRF measurement of the system. Therefore, in (8.7), pole residuals is used once, whereas the zero residuals are repeated for each FRF measurement since zero locations vary per actuator/sensor configuration. Replacing the entries p;n. / and z1;n. / up to zm;n. / in (8.7) by their first-order Taylor series approximations around i results in the approximation of the error functional, or i . / D r Hi . /W ri. /; (8.8) where ri. / D 2 66 64 p;e z1;e : : : zm;e 3 77 75 2 66 64 p;n. / z1;n. / : : : zm;n. / 3 77 75 i „ ƒ‚ … i 2 66 66 4 @ p;n. / @ p 0 : : : 0 0 @ z 1 ;n. / @ z 1 : : : 0 : : : : : : : : : : : : 0 0 : : : @ zm;n. / @ zm 3 77 77 5i „ ƒ‚ … Si 2 66 64 p z1 : : : zm 3 77 75 i „ ƒ‚ … i : (8.9) Si in (8.9) represents the sensitivity matrix which includes the sensitivity of each pole denoted by @ p;n. p/ @ p D p T @Kr @ p p 2 p;n p TMr p C p TBr p ; (8.10) together with the sensitivity of each zero @ z;n. z/ @ z D z T @Ks @ z z 2 z;n z TMs z C z TBs z : (8.11) The number of design parameters is generally equal or lower than the number of measured poles and zeros. Therefore, the optimization problem is (over)determined; hence, a least squares approach is applied. Minimizing (8.8) with regard to i, and requiring that @ i=@ i D0, leads to the following equation [9]: Re SHi W Si i DRe S Hi W i : (8.12) The iteration process will be terminated using the following stop criterion: j i i 1j ı; (8.13) where ı is a sufficiently small number. In summary, the proposed IPZ model updating algorithm using multiple FRFs will be given as follows: 1. Preparation (a) Choose the frequency range of interest; identify the experimental values of the poles and zeros in this range.
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