A First Course in FUZZY and NEURAL CONTROL

2.7. PROPORTIONAL-INTEGRAL-DERIVATIVE CONTROL 71

integral gain so that the steady-state error is within a tolerable margin, typically
within 2%. Figure 2.38 illustrates the response with afinal set of PID control
parameters that adequately meet some typical design performance characteris-
tics.
In this section we have very systematically shown the trial and error approach
of developing a PID controller for a plant. It is clear that the controller design
is facilitated by the use of a linear model. The main problem with such a design
is that any major nonlinearity in the plant makes it impossible to design a
controller with this approach. As such, there is a need to consider a model-free
approach to designing controllers.

2.7.3 Example: controlling dynamics of a servomotor

A DC servomotor is a commonly used actuator in control systems. It provides
rotary motion as well as transitional motion. In this example we demonstrate
how a controller can be developed to control the dynamics of the servomotor
effectively. We will extend this example in Chapter 4 to incorporate a fuzzy
controller.

For a classical controller design, we need a model of the system. The objec-

Figure 2.39. DC servomotor

tive in this modeling is to obtain a relationship between the angular position of
the rotor and the applied voltage. The electric circuit of the armature and the
free-body diagram of the rotor are shown in Figure 2.39.

System equations The motor torqueTis related to the armature currenti
by a constant factorKtas

T=Kti

The back electromotive force (emf)eis related to the rotational velocityθ ̇by a
constant factorKeas

e=Keθ ̇