Modern Control Engineering

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Section 5–7 / Effects of Integral and Derivative Control Actions on System Performance 219

Proportional Control of Systems. We shall show that the proportional control

of a system without an integrator will result in a steady-state error with a step input. We

shall then show that such an error can be eliminated if integral control action is included

in the controller.

Consider the system shown in Figure 5–37. Let us obtain the steady-state error in the

unit-step response of the system. Define

Since

the error E(s)is given by

For the unit-step input R(s)=1/s, we have

The steady-state error is

Such a system without an integrator in the feedforward path always has a steady-state

error in the step response. Such a steady-state error is called an offset. Figure 5–38 shows

the unit-step response and the offset.

ess=tlimSqe(t)=limsS 0 sE(s)=slimS 0

Ts+ 1

Ts+ 1 +K

=

1

K+ 1

E(s)=

Ts+ 1

Ts+ 1 +K

1

s

E(s)=

1

1 +G(s)

R(s)=

1

1 +

K

Ts+ 1

R(s)

E(s)

R(s)

=

R(s)-C(s)

R(s)

= 1 -

C(s)

R(s)

=

1

1 +G(s)

G(s)=

K

Ts+ 1

1

Ts+ 1

+

• 
  

R(s) E(s) C(s)

K

Proportional

controller

Figure 5–37 Plant
Proportional control
system.

c(t)

1

0 t

Offset

Figure 5–38
Unit-step response
and offset.