Modern Control Engineering

Section 8–2 / Ziegler–Nichols Rules for Tuning PID Controllers 575

Unit-Step Response

Amplitude

0

0.6

1.2

0.8

0.4

0.2

1

Time (sec)

012 3 4 5 6 7

Figure 8–9
Unit-step response of
the system shown in
Figure 8–6 with PID
controller having
parameters
and
Td=0.7692.

Ti=3.077,

Kp=18,

Amplitude

1.4

0.8

0.4

0

1

1.2

0.6

0.2

Unit-Step Response

Time (sec)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Figure 8–10
Unit-step response of
the system shown in
Figure 8–6 with PID
controller having
parameters

and
Td=0.7692.

Ti=3.077,

Kp=39.42,

then the speed of response is increased, but the maximum overshoot is also increased to approxi-

mately 28%, as shown in Figure 8–10. Since the maximum overshoot in this case is fairly close to 25%

and the response is faster than the system with given by Equation (8–1), we may consider

as given by Equation (8–2) as acceptable. Then the tuned values of and become

It is interesting to observe that these values respectively are approximately twice the values sug-

gested by the second method of the Ziegler–Nichols tuning rule. The important thing to note here

is that the Ziegler–Nichols tuning rule has provided a starting point for fine tuning.

It is instructive to note that, for the case where the double zero is located at s=–1.4235,in-

creasing the value of Kpincreases the speed of response, but as far as the percentage maximum

overshoot is concerned, varying gain Kphas very little effect. The reason for this may be seen from

Kp=39.42, Ti=3.077, Td=0.7692

Kp ,Ti , Td

Gc(s) Gc(s)