A First Course in FUZZY and NEURAL CONTROL

2.7. PROPORTIONAL-INTEGRAL-DERIVATIVE CONTROL 65

Proportional + integral control Sometimes derivative action can cause the
heater power tofluctuate wildly. This can happen when the sensor measuring
the oven temperature is susceptible to noise or other electrical interference. In
these circumstances, it is often sensibleto use a PI controller or set the derivative
action of a PID controller to zero.

Third-order systems Systems controlled using an integral action controller
are almost always at least third-order. Unlike second-order systems, third-order
systems are fairly uncommon in physics, but the methods of control theory
make the analysis quite straightforward. For instance, there is a systematic
way of classifying the complex roots of the auxiliary equation for the model,
known as the Routh-Hurwitz stability criterion. Provided the integral gain is
kept sufficiently small, parameter values can be found to give an acceptably
damped response, with the error temperature eventually tending to zero, if the
set-point is changed by a step or linear ramp in time. Whereas derivative control
improved the system damping, integral control eliminates steady-state error at
the expense of stability margin.

Using MATLABfor PID controller design The transfer function of a
PID controller looks like the following

Kp+

Ki

s

+sKd=

Kds^2 +Kps+Ki

s

whereKpis the proportional gain,Kiis the integral gain, andKdis the deriv-
ative gain. First, Let us take a look at the effect of a PID controller on the
closed-loop system using the schematic in Figure 2.31. To begin, the variable

Figure 2.31. Overview of controller and plant

eis the tracking error or the difference between the desired reference valueR
and the actual outputY. The controller takes this error signal and computes
both its derivative and its integral. The signaluthat is sent to the actuator is
now equal to the proportional gainKptimes the magnitude of the error, plus
the integral gainKitimes the integral of the error, plus the derivative gainKd
times the derivative of the error.
Generally speaking, for an open-loop transfer function that has the canonical
second-order form
1
s^2 +2ζωRs+ω^2 R