Page 73 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

48 Chapter Three

3·2·4 The First-Order Model and Proportional-Integral
Control
The "keep on going" control feature can be obtained if the controller
algorithm is modified to be

I
U(t) = ke(t) +I J due(u) (3-20)
0

where we have added a second term that is proportional to the inte-
gral of the error with I being the proportionality constant. (The inte-
gral is reviewed in App. A.) In the face of a step in the set point,
assume that the error does not go to zero and remains, say, positive.
The second term in Eq. (3-20), because e(t) is positive, will increase
and continue to do so. This will cause U to increase until either the
error is driven to zero or until U runs out of room.
To get a better feel of how Eq. (3-20) behaves let's simply set the
error equal to a constant at time zero, namely,
e(t) = 0 t < 0
e(t) = C t ~ 0

That is, let's assume that some sort of disturbance is taking place
that keeps the process variable away from the set point by a con-
stant amount. Don't worry about how this could actually happen.
Eq. (3-20) becomes
t
U(t) = kC+I jduC
0
= kC+ICt

which says that, in the face of a constant error C that starts at t = 0, the
controller output makes an initial jump of kC (the proportional com-
ponent) at time zero and then ramps up at a rate of IC. (The integral
of a constant is reviewed in App. A.) The continual increase in the
controller output, due to the term ICt, is the integral action and it is
telling us that, as long as the error is constant and as long as the pro-
cess does not respond to the controller output, the controller output
will continue to increase. This is the feature that we needed to reduce
the offset between the process output and the set point when propor-
tional-only control was used.

Showing That There Is No Offset
Can we prove this contention of zero offset with our simple mathe-
matics? Well, sort of. Let's combine Eqs. (3-20) and (3-10):