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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