Modern Control Engineering

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Section 5–7 / Effects of Integral and Derivative Control Actions on System Performance 221

Hence

The steady-state error due to a step disturbance torque of magnitude is given by

At steady state, the proportional controller provides the torque which is equal in

magnitude but opposite in sign to the disturbance torque The steady-state output due

to the step disturbance torque is

The steady-state error can be reduced by increasing the value of the gain Kp. Increasing

this value, however, will cause the system response to be more oscillatory.

Response to Torque Disturbances (Proportional-Plus-Integral Control). To

eliminate offset due to torque disturbance, the proportional controller may be replaced

by a proportional-plus-integral controller.

If integral control action is added to the controller, then, as long as there is an error

signal, a torque is developed by the controller to reduce this error, provided the control

system is a stable one.

Figure 5–41 shows the proportional-plus-integral control of the load element,

consisting of moment of inertia and viscous friction.

The closed-loop transfer function between C(s)andD(s)is

In the absence of the reference input, or r(t)=0, the error signal is obtained from

E(s)=-

s

Js^3 +bs^2 +Kp s+

Kp

Ti

D(s)

C(s)

D(s)

=

s

Js^3 +bs^2 +Kp s+

Kp

Ti

css=-ess=

Td

Kp

Td.

- Td ,

=-

Td

Kp

=limsS 0

- s

Js^2 +bs+Kp

Td

s

ess=limsS 0 sE(s)

Td

E(s)

D(s)

=-

C(s)

D(s)

=-

1

Js^2 +bs+Kp

++

E C

D

R= 0 T

Kp(1+^1

Tis

)^1

s(Js+b)

+

• 
  

Figure 5–41
Proportional-plus-
integral control of a
load element
consisting of moment
of inertia and viscous
friction.