Physiological Modeling - Introduction to Biomedical Engineering - page 920

any frequency content above the highest frequency of the signal, and sample at a rate of at
least 2.5 times the highest frequency. The transfer function is then estimated.

13.10.2 Identification of a Linear First-Order System
The technique for solving for the parameters of a first-order system is presented here.
Assume that the system of interest is a first-order system that is excited with a step input.
The response to the input is

t
t

y ð t Þ¼ y ss þ Ke u ð t Þ ð 13 : 70 Þ

where the response is

63 percent of the way from the initial to the steady-state value. Similarly, at

K ¼ y ss y ð ð 0 Þ Þ and is shown in Figure 13.64. Note that at t ¼ t ,

t ¼ 4 t, the

response is 98 percent of the way from the initial to the steady-state value.
Suppose step input data are collected from an unknown first-order system shown in
Figure 13.64. To fully describe the system, the parameters of Eq. (13.70) need to be esti-
mated. One way to estimate the system time constant is from the initial slope of the
response and a smoothed steady-state value (via averaging). That is, the time constant t
is found from

t
t

t
t

1
t

¼ y ss y

ðÞ 0

y ð t Þ¼

ð y ss y ð 0 Þ Þ e ! t e

y ð t Þ

¼ y ss y

ðÞ 0

At t ¼ 0, t , where y ð 0 Þ is the initial slope of the response. The equation for

y ð 0 Þ

estimating t is nothing more than the equation of a straight line. This technique is illustrated
in Figure 13.65.

y ss

y ( )

y (0)

0

Time

FIGURE 13.64 First-order system response to a step input.