A First Course in FUZZY and NEURAL CONTROL

64 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL

Figure 2.29. Response to PD control

Proportional + integral + derivative control Although PD control deals
well with the overshoot and ringing problems associated with proportional con-
trol, it does not solve the problem with the steady-state error. Fortunately, it is

Figure 2.30. Response to PID control

possible to eliminate this while using relatively low gain by adding an integral
term to the control function which becomes

W=P(Ts−To)+D

d

dt

(Ts−To)+I

Z

(Ts−To)dt

whereI,theintegral gain parameteris sometimes known as thecontroller reset
level. This form of function is known as PID control. The effect of the inte-
gral term is to change the heater poweruntil the time-averaged value of the
temperature error is zero. The method works quite well but complicates the
mathematical analysis slightly because the system is now a third-order system.
Figure 2.30 shows that, as expected, adding the integral term has eliminated
the steady-state error. The slight undershoot in the power suggests that there
may be scope for further tweaking the PID parameters.