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A New Do11ain and More Process Models 91

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F1aURE 4-1.5 Bounded growth when phase lag is 170°.

Alternatively, we should design a controller such that the open-loop
phase lag is less than 180° when the open-loop gain is unity. The com-
bination of open-loop amplitudes greater than unity and phase lags
greater than 180° spells trouble.
With these concepts in mind, let's take an alternative look at our
first-order process under feedback control.

4-3 The First-Order Process with Feedback Control
in the Frequency Domain
Back in Sec. 3-2-3 the first-order process under feedback control
was studied. With proportional-only control, the differential equation
describing the controlled system was first order and suggested that the
response was bounded under all conditions. When integral control
was added the order of the describing differential equation jumped
to two but the response still appeared to be bounded although at high
integral control gains there could be underdamped behavior.
Before we can use the results of this section for actual design we
need to discuss some process models that have more phase lag than
the simple first-order system. These models are not only more com-
plicated but they also are able to describe important characteristics of
real processes. But for the time being, we will stick with the first-
order process.
When the Laplace transform was applied to this first-order pro-
cess under PI (proportional-integral) control in Otap. 3, the behavior