Modern Control Engineering

Section 6–9 / Parallel Compensation 343

The closed-loop transfer function for the system with parallel compensation

[Figure 6–60(b)] is

The characteristic equation is

By dividing this characteristic equation by the sum of the terms that do not involve Gc,

we obtain

(6–25)

If we define

then Equation (6–25) becomes

SinceGfis a fixed transfer function, the design of Gcbecomes the same as the case of

series compensation. Hence the same design approach applies to the parallel compen-

sated system.

Velocity Feedback Systems. A velocity feedback system (tachometer feedback

system) is an example of parallel compensated systems. The controller (or compensator)

in such a system is a gain element. The gain of the feedback element in a minor loop must

be determined properly so that the entire system satisfies the given design specifica-

tions. The characteristic of such a velocity feedback system is that the variable parame-

ter does not appear as a multiplying factor in the open-loop transfer function, so that

direct application of the root-locus design technique is not possible. However, by rewrit-

ing the characteristic equation such that the variable parameter appears as a multiply-

ing factor, then the root-locus approach to the design is possible.

An example of control system design using parallel compensation technique is pre-

sented in Example 6–10.

EXAMPLE 6–10 Consider the system shown in Figure 6–61. Draw a root-locus diagram. Then determine the value

ofksuch that the damping ratio of the dominant closed-loop poles is 0.4.

Here the system involves velocity feedback. The open-loop transfer function is

Open-loop transfer function=

20

s(s+1)(s+4)+20ks

1 +Gc Gf= 0

Gf=

G 2

1 +G 1 G 2 H

1 +

Gc G 2

1 +G 1 G 2 H

= 0

1 +G 1 G 2 H+G 2 Gc= 0

C

R

=

G 1 G 2

1 +G 2 Gc+G 1 G 2 H

R(s) 20 C(s)

(s+1) (s+ 4)

1

s

+

• +–

k
Figure 6–61
Control system.