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


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)






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.

Figure: RC circuit.

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