Page 82 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

Basic Concepts in Process Analysis 57

It also showed that proportional control alone will not drive the pro-
cess variable all the way to set point. The response, although inade-
quate because of the offset between the process output and set point,
was smooth and without oscillations.
When the integral component was added, the process output was
driven to set point. Aggressive integral control could cause some
overshoot. Excessively aggressive integral control could cause sus-
tained oscillations.
This might be considered a logical point to end the chapter but I
choose not to for the simple reason that I need you to quickly move
on from the time domain to the Laplace domain before you forget the
above results and insights.

3-3 The Laplace Transform
In the last section we had a little trouble with the second-order differen-
tial equation. In this section we introduce a tool, the Laplace transform,
which will remove some of the problems associated with differential
equations but with the cost of having to learn a new concept. The theory
of the Laplace transform is dealt with in App. F so we will start with a
simple recipe for applying the tool to the first-order differential equation.
The first-order model in the time domain is
dY
T-+Y=gU (3-31)
dt
To move to the Laplace transform domain, the derivative operator is
simply replaced by s, the SO<alled Laplace transform operator, and wig-
gles are placed over the symbols Y and U since they are in a new domain
(3-32)

Before dealing with Eq. (3-32) consider some Laplace transform
transition rules in the box:

d
dt => s

Y(t) => Y(s)
U(t) => U(s)

c
C=>-
s
J
Y(u)du => Y(s)
0 s

lims-+O sY(s) = Y(oo)