Modern Control Engineering

528 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method

(^0) Re
(a)
Im
v=

• 1
  v= 0 Re

Im

• 1

G Plane

v=

v= 0 +

v= 0 –



(b)

Figure 7–123

(a) Nyquist plot;

(b) complete Nyquist

plot in the Gplane.

Thus, we have the relationship

This proves the theorem.

Note that by this mapping theorem, the exact numbers of zeros and of poles cannot be found—

only their difference. Note also that, from Figures 7–122(a) and (b), we see that if udoes not

change through 2prad, then the origin of the F(s)plane cannot be encircled.

A–7–7. The Nyquist plot (polar plot) of the open-loop frequency response of a unity-feedback control

system is shown in Figure 7–123(a). Assuming that the Nyquist path in the splane encloses the

entire right-half splane, draw a complete Nyquist plot in the Gplane. Then answer the following

questions:

(a) If the open-loop transfer function has no poles in the right-half splane, is the closed-loop

system stable?

(b) If the open-loop transfer function has one pole and no zeros in right-half splane, is the closed-

loop system stable?

(c) If the open-loop transfer function has one zero and no poles in the right-half splane, is the

closed-loop system stable?

N=Z-P

Re

Im

u 1

u u^2

2

Origin encircled

u 2 – u 1 = 2 p

Origin not encircled

u 2 – u 1 = 0

F(s) Plane F(s) Plane

0

(a) (b)

Re

Im

0

u 1

Figure 7–122

Determination of

encirclement of the

origin of F(s)plane.

Openmirrors.com