A First Course in FUZZY and NEURAL CONTROL

62 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL

ter time constant is usually very small, so its effects will be assumed to be
negligible during much of the following discussion.
We discussed earlier that conventional control requires the use of a system
model for the development of a controller. In this context, we develop a state-
variable model as follows

∑ ̇

Th

T ̇o

∏

=





−Rho^1 Ch Rho^1 Ch

1

RhoCo −

≥

1

RhoCo+

1

RoCo

¥





"

Th

To

+

" 1

Ch

0

W

[To]=

£

01

§

∑

Th

To

∏

that represents the mathematical model of the system. The heat loss to the
environmentTeis a component that cannot be modeled and can only be com-
pensated by supplying sufficient heat by the controlling elements. The set of
differential equations (state equations) are driven by the inputWthat the con-
troller provides.Tois the oven temperature which is a sensed parameter. The
error ñ that is, the difference between the desired temperature and the sensed
temperature, acts as input to the controller. Such a model can very easily be
modeled inMatlabas demonstrated in the following simulations of the system
behavior.

Figure 2.27. Fluctuations in temperature

It is important to mention at the outset that the simplest form of control that
can be adopted is the ON-OFF control used by almost all domestic thermostats.
When the oven is cooler than the set-point temperature, the heater is turned on
at maximum power, and once the oven is hotter than the set-point temperature,
the heater is switched offcompletely. The turn-on and turn-offtemperatures are
deliberately made to differ by a small amount, known ashysteresis,toprevent
noise from rapidly and unnecessarily switching the heater when the temperature
is near the set-point. Thefluctuations in temperature shown in the graph in
Figure 2.27 are significantly larger than the hysteresis, due to the significant heat
capacity of the heating element. Naturally, the ON-OFF control is inefficient