Page 83 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

58 Chapter Three

Most control books have extensive tables giving the transforms
for a wide variety of time functions. Note the following comments
about the contents of the box given in Sec. 3-3.
1. All initial values must be zero. (Later on, nonzero initial
conditions will be covered.
2. The differential operator d/dt is replaced with s.
3. The integral operator J; ... du is replaced with 1/s.
4. The quantity C is a constant.
5. The last equation in the box is really not a transform rule.
Rather it is the final value theorem and it shows how one can
find the final value in the time domain if one has the Laplace
transform. The basis for these rules and the final value
theorem are given in App. F.

For the case of the water tank, Y had units of length or m, U had units
3
of volume per unit time or m I sec, and time thad unit of sec. In the new
domain, s has units of reciprocal time or sec- , Y has units of m-sec, and
1
U has units of m • It's not obvious why Y and U have those units-that
3
should be apparent from the discussion in App. F-but it may make sense
that s has units of sec- by looking at the appearance of -rs in Eq. (3-21)
1
and realizing that it would be nice to have this product be unitless.
In any case, Eq. (3-32) does not contain ~erivatives-in fact, it is
an algebraic equation and can be solved for Y:
Y=-g-U=GU
'rS + 1 p

G =-g- (3-33)
P 'rS + 1

This equation gives the Laplace transform of Y in terms of the
Laplace transform of U and a factor G, that is called the process trans-
fer function.
Equation (3-33) will be solved for Y(t) later on in Sec. 3-3-2 but
for the time being let's comment on the big picture. We have to take
two steps. First, the Laplace transform for U must be found. In the
examples so far U has been a step change, so the Laplace transform
for the step-change function must be developed. App. F gives the
derivation of the Laplace transform of a step change. As a tempo-
rary alternative, consider the step in U at time zero as a constant Uc
that had zero value for t < 0 . In this case, using the fourth entry in
the box given in Sec. 3-3, ~e Laplace transform of U is U/s.
Second, after replacing y with its transform in terms of s, the modi-
fied algebraic equation for Y must be inverted, that is, transformed back
to the time domain. There are a variety of ways of doing this but the