Page 76 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

Basic Concepts in Process Analysis 51

Question 3-3 Does any of this logic based on assuming derivatives go to zero
as t -+ oo bother you?

Answer Actually, it should. What if, somehow, the integral gain was mistakenly
set to a negative number? Using the tools of the next section you should be
able to show that a negative integral gain will cause instability and that the
derivatives will definitely not settle out to zero.
Trying a Partial Solution for the Transient Part
As in App. E, a solution of the form

Y, =Ce" 1 (3-24)

is tried. When Eq. (3-24) is inserted into Eq. (3-23), the following qua-
dratic equation results (the reader should try this, verify it, and then
perhaps check App. E)

1
1
1
2
-rCa e" + (1 + gk)Cae" + g1Ce" = 0 (3-25)
or, after cancelling Ct!"
-ra2 + (1 + gk)a + gl = 0

1his quadratic equation can be solved for a (the root of the equation),
yielding two values, a and a • The roots of a quadratic equation can be
2
1
found from the famous quadratic equation root solver (see App. B):

(3-26)

Critical Damping
Eq. (3-26) shows that the roots will have two parts and that if

2
(1 + gk) = 4-rgl
or
I= (1+gk)2
4-rg

then both roots will be the same