Modern Control Engineering

Section 2–3 / Automatic Control Systems 25

or the transfer function of the controller is

where is called the integral time.

Proportional-Plus-Derivative Control Action. The control action of a proportional-

plus-derivative controller is defined by

and the transfer function is

where is called the derivative time.

Proportional-Plus-Integral-Plus-Derivative Control Action. The combination of

proportional control action, integral control action, and derivative control action is

termed proportional-plus-integral-plus-derivative control action. It has the advantages

of each of the three individual control actions. The equation of a controller with this

combined action is given by

or the transfer function is

whereKpis the proportional gain, is the integral time, and is the derivative time.

The block diagram of a proportional-plus-integral-plus-derivative controller is shown in

Figure 2–10.

Ti Td

U(s)

E(s)

=Kpa 1 +

1

Ti s

+Td sb

u(t)=Kp e(t)+

Kp

Ti 3

t

0

e(t)dt+Kp Td

de(t)

dt

Td

U(s)

E(s)

=KpA 1 +Td sB

u(t)=Kp e(t)+Kp Td

de(t)

dt

Ti

U(s)

E(s)

=Kpa 1 +

1

Ti s

b

+–

E(s) Kp(1+Tis+TiTds (^2) ) U(s)
Tis
Figure 2–10
Block diagram of a
proportional-plus-
integral-plus-
derivative controller.

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