Modern Control Engineering

324 Chapter 6 / Control Systems Analysis and Design by the Root-Locus Method

EXAMPLE 6–7 Consider the system shown in Figure 6–48(a). The feedforward transfer function is

The root-locus plot for the system is shown in Figure 6–48(b). The closed-loop transfer function

becomes

The dominant closed-loop poles are

The damping ratio of the dominant closed-loop poles is The undamped natural

frequency of the dominant closed-loop poles is 0.673 radsec. The static velocity error constant is

0.53 sec–1.

It is desired to increase the static velocity error constant Kvto about 5 sec–1without appreciably

changing the location of the dominant closed-loop poles.

To meet this specification, let us insert a lag compensator as given by Equation (6–19) in

cascade with the given feedforward transfer function. To increase the static velocity error con-

stant by a factor of about 10, let us choose b=10and place the zero and pole of the lag com-

pensator at s=–0.05ands=–0.005, respectively. The transfer function of the lag compensator

becomes

Gc(s)=Kˆc

s+0.05

s+0.005

z=0.491.

s=-0.3307;j0.5864

=

1.06

(s+0.3307-j0.5864)(s+0.3307+j0.5864)(s+2.3386)

C(s)

R(s)

=

1.06

s(s+1)(s+2)+1.06

G(s)=

1.06

s(s+1)(s+2)

1.06

s(s+ 1) (s+ 2)

Closed-loop poles

j 1

• j 2


• j 1


• 31 – 2 – 1 0

jv

s

(a) (b)

j 2

+

Figure 6–48

(a) Control system;

(b) root-locus plot.

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