Page 101 - Dynamics and Control of Nuclear Reactors

8.3 Basic control theory 95

de
f D tðÞ ¼ K d (8.4)
dt
8.3.6 Combined controllers
Controllers employing more than one mode of control are often used. The most com-
mon is proportional plus integral (PI control) [1, 2]. The designer of a PI control sys-
tem must identify and implement proper selection of the coefficients, K p and K i . The
PI control action is expressed as.
Z t
ftðÞ ¼ K p etðÞ + K i evðÞdv (8.5)
o

8.3.7 An example of proportional and integral controller for a first
order system
To illustrate the specific properties of proportional and integral control actions, the
closed-loop system shown in Fig. 8.2 is considered here. The process G p (s) is
selected for this example is a first order system given by.
1
G p sðÞ ¼ (8.6)
s + a
8.3.7.1 Proportional controller
In this type of controller, the control action is proportional to the error. Thus,
G c sðÞ ¼ K p ,K p > 0 andFsðÞ ¼ K p EsðÞ
As shown in Appendix D, Section D.5, the closed-loop transfer function is.
YsðÞ G c sðÞG p sðÞ
(8.7)
¼
XsðÞ 1+ G c sðÞG p sðÞ
For our example, Eq. (8.7) becomes
YsðÞ K p
, K p > 0 (8.8)
¼
XsðÞ s + a + K p
The Laplace transform of the error, e(t), is given by (See Section 3.7).
XsðÞ
(8.9)
EsðÞ ¼
1+ G c sðÞG p sðÞ
For our example, E(s) becomes.
ð s + aÞXsðÞ
(8.10)
EsðÞ ¼
s + a + K p
For a unit step input, X(s)¼1/s, and
ð s + aÞ
EsðÞ ¼
ss + a + K p