Page 68 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

Basic Concepts in Process Analysis 43

With Y = 0 there is only one term left in Eq. (3-11), namely,
0

(3-12)

At time zero, this equation yields Y = 0 and then, as time increases,
Y finally asymptotically approaches gU,. Second, by taking the deriv-
ative of Eq. (3-11), or by solving Eq. (3-10) for the derivative, the rate
of change of Y can be obtained as

Therefore, when Y = 0 the rate at time zero (when the step in U
0
is first applied) can be obtained as

Y(O)= gU, (3-13)
f
Since Y and its rate of change were zero before the step in U was
applied at time zero, Eq. (3-13) tells us that, at the time of the step in
U, the rate of change of Y experiences a discontinuity, jumping from
zero to gUc/f. Examination of Fig. 3-2 should support this conten-
tion. We know that in real life there are few process quantities that
experience true discontinuities, that is, "Mother Nature abhors dis-
continuities." Therefore, this model is, among other things, an ideal-
ization, albeit useful (as we will see).
Equation (3-13) also tells us that the initial rise rate of Y is directly
proportional to the strength of the step in U, directly proportional to
the gain of the process, and inversely proportional to the process time
constant. It is comforting that common sense is supported by simple
mathematics, no?
Third, note that when time equals the value of the time constant,
that is, when t = f, and when Y = 0, Eq. (3-12) yields
0
r
1
Y = gU,(1- e -r) = gU,(1- e- ) = gU,(1- 0.36788) = 0.63212gU,
Therefore, at t = f, Y equals 63% of its ultimate final value gU;
hence the basis for the above definition of the time constant.
For future reference, Eq. (3-10) can be rewritten as

dY = (-.!.)y + .R..u
dt f f
(3-14)
dY
-=aY+bU
dt