Modern Control Engineering

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220 Chapter 5 / Transient and Steady-State Response Analyses

Integral Control of Systems. Consider the system shown in Figure 5–39. The

controller is an integral controller. The closed-loop transfer function of the system is

Hence

Since the system is stable, the steady-state error for the unit-step response can be

obtained by applying the final-value theorem, as follows:

Integral control of the system thus eliminates the steady-state error in the response to

the step input. This is an important improvement over the proportional control alone,

which gives offset.

Response to Torque Disturbances (Proportional Control). Let us investigate

the effect of a torque disturbance occurring at the load element. Consider the system

shown in Figure 5–40. The proportional controller delivers torque Tto position the load

element, which consists of moment of inertia and viscous friction. Torque disturbance is

denoted by D.

Assuming that the reference input is zero or R(s)=0, the transfer function between

C(s)andD(s)is given by

C(s)

D(s)

=

1

Js^2 +bs+Kp

= 0

=lim

sS 0

s^2 (Ts+1)

Ts^2 +s+K

1

s

ess=lim

sS 0

sE(s)

E(s)

R(s)

=

R(s)-C(s)

R(s)

=

s(Ts+1)

s(Ts+1)+K

C(s)

R(s)

=

K

s(Ts+1)+K

1

+– Ts+ 1

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

Figure 5–39 s

Integral control

system.

+– +

R +

D

ET C

Kp s(Js^1 +b)

Figure 5–40

Control system with

a torque disturbance.

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