Modern Control Engineering

Section 8–2 / Ziegler–Nichols Rules for Tuning PID Controllers 573

PID controller

1

s(s+ 1)(s+ 5)

6.3223 (s+ 1.4235)^2

s

R(s) C(s)

+–

Figure 8–7
Block diagram of the
system with PID
controller designed
by use of the
Ziegler–Nichols
tuning rule (second
method).

Examining the coefficients of the first column of the Routh table, we find that sustained oscilla-

tion will occur if Thus, the critical gain Kcris

Kcr= 30

With gain Kpset equal to the characteristic equation becomes

s^3 +6s^2 +5s+30=0

To find the frequency of the sustained oscillation, we substitute s=jvinto this characteristic

equation as follows:

(jv)^3 +6(jv)^2 +5(jv)+30=0

or

6 A5-v^2 B+jvA5-v^2 B=0

from which we find the frequency of the sustained oscillation to be or Hence, the

period of sustained oscillation is

Referring to Table 8–2, we determine and as follows:

The transfer function of the PID controller is thus

The PID controller has a pole at the origin and double zero at s=–1.4235.A block diagram of

the control system with the designed PID controller is shown in Figure 8–7.

=

6.3223(s+1.4235)^2

s

= 18 a 1 +

1

1.405s

+0.35124sb

Gc(s)=Kpa 1 +

1

Ti s

+Td sb

Td=0.125Pcr=0.35124

Ti=0.5Pcr=1.405

Kp=0.6Kcr= 18

Kp ,Ti , Td

Pcr=

2 p

v

=

2 p

15

=2.8099

v^2 = 5 v= 15.

Kcr(= 30 ),

Kp=30.