Page 71 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

4& Chapter Three

A little rearrangement, which the reader should verify, will yield

(3-18)

This is the differential equation that describes the closed-loop sys-
tem containing the process under simple proportional control. It has
the same form at Eq. (3-10) except for the following replacements

1 gk
1 U=>S
=> 1+gk g=> 1+gk

Therefore, by observation, we can obtain the solution to a process
under simple proportional feedback control subject to a step in the set
point (from 0 to S) at time zero. That is, Eq. (3-11), with the above
substitutions, becomes

(3-19)

Faster Response
Since both g and k are positive, the new effective time constant is less
than the original one (where there was no control) by a factor of
1/ (1 + gk). As the control gain k increases, the effective time constant
decreases. This is something we would hope for since the effect of
adding control should be to speed things up.

Offset from Set Point
Look at what Eq. (3-19) yields when t ~ (which drives the expo-
oo
nential terms to zero):

So, the proportional control does not ultimately drive the
process output all the way to the set point. In fact, the process
output settles out at a fraction, namely, gk I (1 + gk), of the set
point S c· If the controller gain k is quite large, as in the case of
an aggressive controller, this fraction will be nearly unity. Rais-
ing the control gain k so as to decrease the offset is risky because
our model is an idealization and in real life a high control gain
might cause some problems that would lead to instability. Also,