Modern Control Engineering

Section 8–2 / Ziegler–Nichols Rules for Tuning PID Controllers 571

Pcr

0 t

c(t)

Figure 8–5
Sustained oscillation
with period
( is measured in
sec.)

Pcr

Pcr.

Type of

Controller

P 0.5Kcr q 0

PI 0.45Kcr 0

PID 0.6Kcr 0.5Pcr 0.125Pcr

1

1.2

Pcr

Kp Ti Td

Table 8–2 Ziegler–Nichols Tuning Rule Based on Critical Gain

Kcrand Critical Period Pcr(Second Method)

determined (see Figure 8–5). Ziegler and Nichols suggested that we set the values of

the parameters Kp ,Ti ,and Tdaccording to the formula shown in Table 8–2.

Notice that the PID controller tuned by the second method of Ziegler–Nichols rules

gives

Thus, the PID controller has a pole at the origin and double zeros at

Note that if the system has a known mathematical model (such as the transfer func-

tion), then we can use the root-locus method to find the critical gain Kcrand the fre-

quency of the sustained oscillations vcr, where These values can be found

from the crossing points of the root-locus branches with the jvaxis. (Obviously, if the

root-locus branches do not cross the jvaxis, this method does not apply.)

2 pvcr=Pcr.

s=- 4 Pcr.

=0.075Kcr Pcr

as+

4

Pcr

b

2

s

=0.6Kcra 1 +

1

0.5Pcr s

+0.125Pcr sb

Gc(s)=Kpa 1 +

1

Ti s

+Td sb