Modern Control Engineering

Example Problems and Solutions 537

Real Axis

• 2 – 1.5 – 1 – 0.5 0 0.5 1 1.5 2

Imag Axis

• 0.4

0.2

• 1

1

• 0.2

0.4

• 0.6


• 0.8

0

0.6

0.8

Nyquist Plots of G(s) and –G(s)

• G(s) G(s)

Use this Nyquist

plot for positive

feedback system

Use this Nyquist
plot for negative
Figure 7–131 feedback system
Nyquist plots for
positive-feedback
system and negative-
feedback system.

A–7–15. Consider the control system shown in Figure 7–60. (Refer to Example 7–19.) Using the inverse

polar plot, determine the range of gain Kfor stability.

Solution. Since

we have

Hence, the inverse of the feedforward transfer function is

Notice that 1/G(s)has a pole at s=–0.5. It does not have any pole in the right-half splane.

Therefore, the Nyquist stability equation

reduces to Z=NsinceP=0. The reduced equation states that the number Zof the zeros of

1+C1/G(s)Din the right-half splane is equal to N, the number of clockwise encirclements of

the–1+j0point. For stability,Nmust be equal to zero, or there should be no encirclement. Fig-

ure 7–132 shows the Nyquist plot or polar plot of K/G(jv).

Notice that since

=

0.5-0.5v^2 - v^4 +jvA- 1 +0.5v^2 B

0.25+v^2

K

G(jv)

=c

(jv)^3 +(jv)^2 + 1

jv+0.5

da

0.5-jv

0.5-jv

b

Z=N+P

1

G(s)

=

s^3 +s^2 + 1

K(s+0.5)

G(s)=G 1 (s)G 2 (s)=

K(s+0.5)

s^3 +s^2 + 1

G 2 (s)=

1

s^3 +s^2 + 1