Modern Control Engineering

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

Re

Im

Re Re

Im Im

• 1 – 1

G Plane

G Plane G Plane

(Stable) (Unstable)

(a)

(b)

v=

v= v=

v=–

v=– v=–

v= 0

v= 0 v=^0

P= 1

N=– 1

Z= 0

P= 1

N= 0

K 1 K 1

Z= 1

K

2

K

2

Figure 7–125

(a) Polar plot of

K/(jv-1);

(b) polar plots of

K/(jv-1)for

stable and unstable

cases.

A–7–9. Consider the closed-loop system shown in Figure 7–124. Determine the critical value of Kfor

stability by the use of the Nyquist stability criterion.

Solution.The polar plot of

is a circle with center at –K/ 2on the negative real axis and radius K/ 2, as shown in Figure

7–125(a). As vis increased from –qtoq, the G(jv)locus makes a counterclockwise rotation.

In this system,P=1because there is one pole of G(s)in the right-half splane. For the closed-

loop system to be stable,Zmust be equal to zero. Therefore,N=Z-Pmust be equal to –1,or

there must be one counterclockwise encirclement of the –1+j0point for stability. (If there is no

encirclement of the –1+j0point, the system is unstable.) Thus, for stability,Kmust be greater

than unity, and K=1gives the stability limit. Figure 7–125(b) shows both stable and unstable cases

ofG(jv)plots.

G(jv)=

K

jv- 1

R(s) K C(s)

+– s– 1

Figure 7–124

Closed-loop system.

Openmirrors.com