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Experiment

A first order system unit step response is described by the equation

  equation53

where A is the gain and tex2html_wrap_inline219 is the time constant of the system. The objective of this experiment is to estimate the values of A and tex2html_wrap_inline223 recursively from the sampled data. That is, the estimates of A and tex2html_wrap_inline223 are improved in real-time as each new data point is measured.

Following the approach presented in [2] and [3], we obtain a first-order Taylor series expansion of (gif) about a point tex2html_wrap_inline229 . Then we minimize the sum of squared differences between the measured data tex2html_wrap_inline231 and the model in (gif) with respect to the parameters A and tex2html_wrap_inline223 . This process leads to the recursive algorithm described next.

The recursive algorithm begins with an initial estimate of tex2html_wrap_inline237 and tex2html_wrap_inline239 . The next values of A and tex2html_wrap_inline223 are computed by solving for the updates tex2html_wrap_inline245 and tex2html_wrap_inline247 in (gif)

  equation69

where

  equation104

and

  equation107

The first order system used in the experiment is the RC circuit shown in Figure gif. The objective is to estimate the parameters A and tex2html_wrap_inline223 by observing the output voltage tex2html_wrap_inline253 when the input voltage tex2html_wrap_inline255 is a unit step function. In this circuit, tex2html_wrap_inline257 and tex2html_wrap_inline259 . The parameter estimates are updated in real-time as each data point is measured. Data collection can be stopped when the estimates have converged.

   figure113
Figure: RC circuit.



Kozick Rich
Fri Jun 21 11:03:19 EDT 1996