A First Course in FUZZY and NEURAL CONTROL

2.7. PROPORTIONAL-INTEGRAL-DERIVATIVE CONTROL 63

and results in a high average power consumption. It is for this reason that we
need to consider a controller that provides the best control actions that result
in the least power consumption.

Proportional control A proportional controller attempts to perform better
than the ON-OFF type by applying powerW, to the heater in proportion to
the difference in temperature between the oven and the set-point

W=P(Ts−To)

whereP is known as the ìproportional gainî of the controller. As its gain
is increased, the system responds faster to changes in set-point, but becomes
progressively underdamped and eventually unstable. As shown in Figure 2.28,
thefinal oven temperature lies below the set-point for this system because some
difference is required to keep the heater supplying power. The heater power
must always lie between zero and the maximum because it can only act as a
heat source, and not act as a heat sink.

Figure 2.28. Response to proportional control

Proportional + derivative control The stability and overshoot problems
that arise when a proportional controller is used at high gain can be mitigated
by adding a term proportional to the time-derivative of the error signal,

W=P(Ts−To)+D

d

dt

(Ts−To)

This technique is known as PD control. The value of the damping constant,
D, can be adjusted to achieve a critically damped response to changes in the
set-point temperature, as shown in Figure 2.29.
It is easy to recognize that too little damping results in overshoot and ringing
(oscillation in the oven temperature characterized as underdamped); too much
damping causes an unnecessarily slow response characterized as overdamped.
As such, critical damping gives the fastest response with little or no oscillation
in the oven temperature.