Modern Control Engineering

Example Problems and Solutions 47

G 1 G 2

R(s) C(s)

G 2

R(s) C(s)

G 1 + 1

R(s) C(s)

G 1 G 2 +G 2 + 1

(a)

(b)

(c)

+

+

+

+

+

+

Figure 2–20
Reduction of the
block diagram shown
in Figure 2–19.

Solution.The block diagram of Figure 2–19 can be modified to that shown in Figure 2–20(a).

Eliminating the minor feedforward path, we obtain Figure 2–20(b), which can be simplified to

Figure 2–20(c). The transfer function C(s)/R(s)is thus given by

The same result can also be obtained by proceeding as follows: Since signal X(s)is the sum

of two signals G 1 R(s)andR(s),we have

The output signal C(s)is the sum of G 2 X(s)andR(s).Hence

And so we have the same result as before:

A–2–3. Simplify the block diagram shown in Figure 2–21. Then obtain the closed-loop transfer function

C(s)/R(s).

C(s)

R(s)

=G 1 G 2 +G 2 + 1

C(s)=G 2 X(s)+R(s)=G 2 CG 1 R(s)+R(s)D+R(s)

X(s)=G 1 R(s)+R(s)

C(s)

R(s)

=G 1 G 2 +G 2 + 1

G 1 G 2

H 3

G 3 G 4

H 1 H 2

+– ++ +–

R(s) C(s)

Figure 2–21
Block diagram of a
system.