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Advanced Sensors in Pr ecision Manufacturing
                             The method was developed for use under the following practical   301
                          constraints:

                              •  The structure cannot be characterized in advance with enough
                                 accuracy for purposes of control.
                              •  The dynamics of the structure can change in service.
                              •  The numbers, types, placements, and frequency responses of
                                 sensors that measure the motions and actuators that control
                                 them are limited.
                              •  Time available during service for characterization of the
                                 dynamics is limited.
                              •  The dynamics are dominated by a resonant mode at low fre-
                                 quency.
                              •  In-service measurements of the dynamics are supervised by a
                                 digital computer and are taken at a low rate of sampling, con-
                                 sistent with the low characteristic frequencies of the control
                                 system.
                              •  The system must operate under little or no human
                                 supervision.
                             The method is based on extracting the desired model and control-
                          design data from the response of the structure to known vibrational
                          excitations (Fig. 6.20). Initially, wideband stochastic excitations are
                          used to obtain the general characteristics of the structure. Narrow-
                          band stochastic and piece-wise-constant (consistent with sample-
                          and-hold discretizations) approximations to sinusoidal excitations
                          are used to investigate specific frequency bands in more detail.
                             The relationships between the responses and excitations are first
                          computed nonparametrically—by spectral estimation in the case of
                          stochastic excitations and by estimation of gains and phases in the
                          case of approximately sinusoidal excitations. In anticipation of the
                          parametric curve fitting to follow, the order of a mathematical model
                          of the dynamics of the structure is estimated by use of a  product
                          moment matrix (PMM). Next, the parameters of this model are identi-
                          fied by a least-squares fit of transfer-function coefficients to the non-
                          parametric data. The fit is performed by an iterative reweighting
                          technique to remove high-frequency emphasis and assure minimum-
                          variance estimation of the transfer-function coefficient. The order of
                          the model starts at the PMM estimate and is determined more pre-
                          cisely thereafter by successively adjusting a number of modes in the
                          fit at each iteration until an adequately small output-error profile is
                          observed.
                             In the analysis of the output error, the additive uncertainty is
                          estimated to characterize the quality of the parametric estimate of
                          the transfer function and for later use in the analysis and design of
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