Modern Control Engineering

Example Problems and Solutions 621

Real Axis

–5 –4 –3 –2 –1 0 1

Imag Axis

–2

0

3

–3

2

–1

1

Root-Locus Plot of Gc(s)G(s)

Figure 8–50
Root-locus plot of
the compensated
system.

R(s) C(s)

PID controller Plant

Gc(s) G(s)

(s+a ) (s+b)

K s

1

s^2 + 1

+

Figure 8–49
PID-controlled
system.

the sum of the angles at one of the desired closed-loop poles, from the zero at

s=–1and poles at s=0, s=j,ands=–jis

90 °-143.794°-120°-110.104°=–283.898°

Hence the zero ats=–bmust contribute 103.898°. This requires that the zero be located at

b=0.5714

The gain constantKcan be determined from the magnitude condition.

or

K=2.3333

Then the compensator can be written as follows:

The open-loop transfer function becomes

From this equation a root-locus plot for the compensated system can be drawn. Figure 8–50 is a

root-locus plot.

Gc(s)G(s)=

2.3333(s+1)(s+0.5714)

s

1

s^2 + 1

Gc(s)=2.3333

(s+1)(s+0.5714)

s

(^2) K
(s+1)(s+0.5714)
s

1

s^2 + 1

2

s=- 1 +j 13

= 1

s=- 1 +j 13 ,