Modern Control Engineering

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

The entire compensator for the system becomes

The value of can be determined from the magnitude condition. Since the open-loop transfer

function is

the magnitude condition becomes

Hence,

Thus the compensator becomes

The open-loop transfer function is given by

A root-locus plot for the compensated system is shown in Figure 6–96. The closed-loop poles for

the compensated system are indicated in the plot. The closed-loop poles, the roots of the charac-

teristic equation

(s+9.9158)^2 sAs^2 +0.1s+ 4 B+88.0227(s+2)^2 (s+4)= 0

Gc(s)G(s)=

88.0227(s+2)^2 (s+4)

(s+9.9158)^2 sAs^2 +0.1s+ 4 B

Gc(s)=88.0227

(s+2)^2 (s+4)

(s+9.9158)^2 (2s+0.1)

Gc(s)

=88.0227

Kc=^2

(s+9.9158)^2 sAs^2 +0.1s+ 4 B

(s+2)^2 (s+4)

2

s=- 2 +j2 13

(^2) Kc
(s+2)^2 (s+4)
(s+9.9158)^2 sAs^2 +0.1s+ 4 B
2
s=- 2 +j2 13

= 1

Gc(s)G(s)=Kc

(s+2)^2 (s+4)

(s+9.9158)^2 sAs^2 +0.1s+ 4 B

Kc

Gc(s)=Gˆc(s)

s+ 4

2s+0.1

=Kc

(s+2)^2

(s+9.9158)^2

s+ 4

2s+0.1

Gc(s)

Real Axis

• 15 – 10 – 5 0 5 10 15

Imag Axis

• 10

0

10

• 5

5

15

• 15

Root-Locus Plot of Compensated System

Closed-loop poles

Figure 6–96

Root-locus plot of

the compensated

system.

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