Page 99 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

74 Chapter Three

The example process was quite simple but the idea that proportional-
only control leaves an offset between the set point and the process
variable is general. That the addition of integral control can remove
the offset but can cause underdamped behavior if applied too aggres-
sively is another general concept.
Perhaps the reader could see that the mathematics required to
describe the behavior of anything more complicated than PI control
applied to a first-order process was going to get messy quite quickly.
This set the scene for the introduction of the Laplace transform which
allowed us to move away from differential equations and get back to
algebra. The recipe for using the s (or Heaviside) operator was intro-
duced and shown to be useful in gaining insight into the differential
equations that described the dynamic behavior of processes. The
Laplace transform also facilitated the introduction of the block dia-
gram and the associated block algebra.
Coupled with the appendices the reader saw that Laplace trans-
forms could often be inverted by use of partial fractions. From the
simple examples, the reader saw that poles of the s-domain transfer
function are related to exponential terms in the time domain.
At this point it appears as though the Laplace transform is mostly
useful in solving differential equations. Later on, we will see that the
Laplace transform can be used to gain significant amounts of insight
in other ways that do not involve inversion.
We have broken the ice and are ready to dive into the cold, deep
water. First, we will move into yet one more domain, the frequency
domain. Then a couple of processes more sophisticated than the sim-
ple liquid tank will be introduced before we look at controlled sys-
tems in the three domains of time, Laplace, and frequency.