Modern Control Engineering

Section 8–6 / Two-Degrees-of-Freedom Control 593

For this system, three closed-loop transfer functions Y(s)/R(s)=Gyr,

Y(s)/D(s)=Gyd,andY(s)/N(s)=Gynmay be derived. They are

[In deriving Y(s)/R(s),we assumed D(s)=0andN(s)=0.Similar comments apply

to the derivations of Y(s)/D(s)andY(s)/N(s).] The degrees of freedom of the control

system refers to how many of these closed-loop transfer functions are independent. In

the present case, we have

Among the three closed-loop transfer functions Gyr,Gyn,andGyd, if one of them is

given, the remaining two are fixed. This means that the system shown in Figure 8–28 is

a one-degree-of-freedom control system.

Next consider the system shown in Figure 8–29, where is the transfer function

of the plant. For this system, closed-loop transfer functions Gyr,Gyn,andGydare given,

respectively, by

Gyn=

Y(s)

N(s)

=-

AGc1+Gc2BGp

1 +AGc1+Gc2BGp

Gyd=

Y(s)

D(s)

=

Gp

1 +AGc1+Gc2BGp

Gyr=

Y(s)

R(s)

=

Gc1 Gp

1 +AGc1+Gc2BGp

Gp(s)

Gyn=

Gyd-Gp

Gp

Gyr=

Gp-Gyd

Gp

Gyn=

Y(s)

N(s)

=-

Gc Gp

1 +Gc Gp

Gyd=

Y(s)

D(s)

=

Gp

1 +Gc Gp

Gyr=

Y(s)

R(s)

=

Gc Gp

1 +Gc Gp

Gp(s)

Y(s)

N(s)

R(s)

B(s)

U(s)

D(s)

+– Gc(s) +

+

++

Figure 8–28
One-degree-of-
freedom control
system.