which is the same as Equation (8–15). Hence, we have shown that the systems shown in Figures

8–61(a) and (b) are equivalent.

It is noted that the system shown in Figure 8–61(b) has a feedforward controller In

such a case, does not affect the stability of the closed-loop portion of the system.

A–8–11. A closed-loop system has the characteristic that the closed-loop transfer function is nearly equal

to the inverse of the feedback transfer function whenever the open-loop gain is much greater

than unity.

The open-loop characteristic may be modified by adding an internal feedback loop with a

characteristic equal to the inverse of the desired open-loop characteristic. Suppose that a

unity-feedback system has the open-loop transfer function

Determine the transfer functionH(s)of the element in the internal feedback loop so that the inner

loop becomes ineffective at both low and high frequencies.

Solution.Figure 8–62(a) shows the original system. Figure 8–62(b) shows the addition of the in-

ternal feedback loop aroundG(s).Since

if the gain around the inner loop is large compared with unity, then is

approximately equal to unity, and the transfer functionC(s)/E(s)is approximately equal to 1/H(s).

On the other hand, if the gain is much less than unity, the inner loop becomes

ineffective andC(s)/E(s)becomes approximately equal toG(s).

To make the inner loop ineffective at both the low- and high-frequency ranges, we require that

Since, in this problem,

G(jv)=

K

A 1 +jvT 1 BA 1 +jvT 2 B

|G(jv)H(jv)|1, for v 1 and v 1

@G(s)H(s)@

G(s)H(s)C 1 +G(s)H(s)D

C(s)

E(s)

=

G(s)

1 +G(s)H(s)

=

1

H(s)

G(s)H(s)

1 +G(s)H(s)

G(s)=

K

AT 1 s+ 1 BAT 2 s+ 1 B

Gd(s)

Gd(s).

Example Problems and Solutions 631

(a)

(b)

G(s)

R C

G(s)

H(s)

R E C

(^1) GH(s)
H(s)
R E C

Figure 8–62 +– +– +– +–
(a) Control system;
(b) addition of the
internal feedback
loop to modify the
closed-loop
characteristic.