Page 69 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

44 Chapter Three

The simple model equation now has the form that we will use
when we get to state space formulations where Y and U will be
vectors and a and b will be matrices. We know intuitively that, in
the face of steps in U, the response of Y will be bounded or that Y
will behave stably. Equation (3-14) suggests that as long as a is
negative (which it has to be for our simple example), stability will
result. The reader should quickly convince himself that when a is
positive (physically unrealistic for this model), the response of Y
will be unbounded or unstable.

Question 3-1 Do you understand the comments about "instability"?

Answer Look at Eq. (3-11) with a replacing -1/r.

We know that, by definition, a is negative but if it were not, note that Y would
increase without bound in the face of a positive value of u.., that is, there would
be instability. For this simple case of a first-order model there is no question
about the sign of a but later on when the models get more sophisticated this
will not always be the case and the "sign" of whatever replaces a will give us
insight into stability.
This temporarily concludes our development of the simple first-
order model where we have really beaten a couple of elementary
equations to death. As things get more complicated we will repeat-
edly come back to this model.

3-2-3 The First-Order Model and Proportional Control
Although optional, it would be quite helpful if the reader is able to
follow the math in App. E used to arrive at the solution of the dif-
ferential equation for the first-order model. We will now take a little
side trip and see what can be learned from this model from the con-
trol point of view.
Let's tack a simple "proportional" controller onto our model
and see if we can control the process output to a desired set point.
Our starting point is the first-order model for the process to be
controlled

dY
r-+Y=gU (3-15)
dt
Our goal is to try to keep the process output Y "acceptably near"
the set pointS by adjusting U in some fashion. The simplest "fashion"
is to form an error
e=S-Y (3-16)