Modern Control Engineering

28 Chapter 2 / Mathematical Modeling of Control Systems

R

G 1

H 1

H 2

G 2 G 3

C

R

G 1

H 1

G 2 G 3

C

R

G 3

C

RC

RC

(a)

(b)

(c)

(d)

(e)

H 2

G 1

H 2

G 1

G 1 G 2

1 – G 1 G 2 H 1

G 1 G 2 G 3

1 – G 1 G 2 H 1 +G 2 G 3 H 2

G 1 G 2 G 3

1 – G 1 G 2 H 1 +G 2 G 3 H 2 +G 1 G 2 G 3

+– +

• ++

+– +

• ++

+– +

• 
  

+–

Figure 2–13

(a) Multiple-loop

system;

(b)–(e) successive

reductions of the

block diagram shown

in (a).

A complicated block diagram involving many feedback loops can be simplified by

a step-by-step rearrangement. Simplification of the block diagram by rearrangements

considerably reduces the labor needed for subsequent mathematical analysis. It should

be noted, however, that as the block diagram is simplified, the transfer functions in new

blocks become more complex because new poles and new zeros are generated.

EXAMPLE 2–1 Consider the system shown in Figure 2–13(a). Simplify this diagram.

By moving the summing point of the negative feedback loop containing H 2 outside the posi-

tive feedback loop containing H 1 , we obtain Figure 2–13(b). Eliminating the positive feedback loop,

we have Figure 2–13(c). The elimination of the loop containing H 2 /G 1 gives Figure 2–13(d). Finally,

eliminating the feedback loop results in Figure 2–13(e).

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