Modern Control Engineering

Example Problems and Solutions 629

Solution.The closed-loop transfer functionC(s)/R(s)of the I-PD-controlled system is

The closed-loop transfer functionC(s)/R(s)of the PID-controlled system with input filter

shown in Figure 8–60(b) is

The closed-loop transfer functions of both systems are the same. Thus, the two systems are equivalent.

A–8–9. The basic idea of the I-PD control is to avoid large control signals (which will cause a saturation

phenomenon) within the system. By bringing the proportional and derivative control actions to

the feedback path, it is possible to choose larger values for than those possible by the

PID control scheme.

Compare, qualitatively, the responses of the PID-controlled system and I-PD-controlled system

to the disturbance input and to the reference input.

Solution.Consider first the response of the I-PD-controlled system to the disturbance input.

Since, in the I-PD control of a plant, it is possible to select larger values for than those

of the PID-controlled case, the I-PD-controlled system will attenuate the effect of disturbance

faster than the PID-controlled case.

Next, consider the response of the I-PD-controlled system to a reference input. Since the

I-PD-controlled system is equivalent to the PID-controlled system with input filter (refer to Prob-

lemA–8–8), the PID-controlled system will have faster responses than the corresponding I-PD-con-

trolled system, provided a saturation phenomenon does not occur in the PID-controlled system.

Kp and Td

Kp and Td

=

Kp

Ti s

Gp(s)

1 +Kpa 1 +

1

Ti s

+Td sbGp(s)

C(s)

R(s)

=

1

1 +Ti s+Ti Td s^2

Kpa 1 +

1

Ti s

+Td sbGp(s)

1 +Kpa 1 +

1

Ti s

+Td sbGp(s)

C(s)

R(s)

=

Kp

Ti s

Gp(s)

1 +Kpa 1 +

1

Ti s

+Td sbGp(s)

(b)

Gp(s)

R(s) C(s)

Kp(1++^1 Tds)

Tis

1

1 +Tis+TiTds^2

+–

(a)

Kp

Tis Gp(s)

R(s) C(s)

Kp(1+Tds)

+

+

Figure 8–60
(a) I-PD-controlled
system;
(b) PID-controlled
system with input
filter.