Modern Control Engineering

570 Chapter 8 / PID Controllers and Modified PID Controllers

Kp Plant

r(t) u(t) c(t)

Figure 8–4 +–

Closed-loop system

with a proportional

controller.

functionC(s)/U(s)may then be approximated by a first-order system with a transport

lag as follows:

Ziegler and Nichols suggested to set the values of and according to the formula

shown in Table 8–1.

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

gives

Thus, the PID controller has a pole at the origin and double zeros at s=–1/L.

Second Method. In the second method, we first set and Using the

proportional control action only (see Figure 8–4), increase Kpfrom 0 to a critical value

Kcrat which the output first exhibits sustained oscillations. (If the output does not ex-

hibit sustained oscillations for whatever value Kpmay take, then this method does not

apply.) Thus, the critical gain Kcrand the corresponding period Pcrare experimentally

Ti=q Td=0.

=0.6T

as+

1

L

b

2

s

=1.2

T

L

a 1 +

1

2Ls

+0.5Lsb

Gc(s)=Kpa 1 +

1

Ti s

+Td sb

Kp ,Ti , Td

C(s)

U(s)

=

Ke-Ls

Ts+ 1

Type of

Controller

P q 0

PI 0

PID 1.2 2L 0.5L

T

L

L

0.3

0.9

T

L

T

L

Kp Ti Td

Table 8–1 Ziegler–Nichols Tuning Rule Based on Step Response

of Plant (First Method)

Openmirrors.com