Page 102 - Dynamics and Control of Nuclear Reactors

96 CHAPTER 8 Reactor control

The final, steady-state value of the error, e(t), when x(t) is a unit step, is given by.
lim a
etðÞ ¼ sE sðÞ ¼ (8.11)
s ! 0 a + K p
Note that the steady-state error is not equal to zero, and may be decreased by
increasing K p , the proportional gain. For a¼0.02 and K p ¼1, the steady-state
error is e(∞)¼1/51¼0.0196. If K p is increased to 10, then the steady-state error
is e(∞)¼0.002.

Remarks
1. As the proportional gain constant K p increases, the steady-state error decreases. However, the
steady-state error is never zero.
2. As the gain K p increases, the time constant of the system decreases, and the rate of system
response increases. For the open-loop system in the above example (a¼0.02), the time constant,
τ¼50s. For the closed-loop system with a proportional gain K p , the time constant is given by
1
(8.12)
τ ¼
a + K p
Thus, for K p ¼0.1, the time constant decreases to τ¼8.33s.
3. An increased K p results in a faster response of the control system. Since there are limitations on
the achievable responses of control devices, K p should not be increased arbitrarily. Furthermore,
an increase in K p above a certain value can make the system unstable. The stability margin will
decrease with increasing values of K p .

All these points must be considered in choosing the proportional gain. The pro-
portional controller increases the speed of response of a system. The steady-state
error is nonzero.

8.3.7.2 Integral controller
An integral control is expressed as:
Z t
ftðÞ ¼ K i evðÞdv
o
In the Laplace domain, the control action has the form.

K i EsðÞ
FsðÞ ¼ (8.13)
s
A very important caution to be exercised in using integral controllers is the consid-
eration of actuator limits. When an actuator reaches its lower or upper limit (because
of actuator saturation), the error may remain large and the integral action will con-
tinuously increase the error term. This characteristic is called integral windup and
must be avoided by changing the sign of the error using appropriate logic. A large
transient may be observed if the actuator reaches its limit.