Modern Control Engineering

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

R(s) C(s)

G(s)

G 1 (s) G 2 (s)

H 1 (s)

H 2 (s)

+– +–

Figure 7–59

Multiple-loop

system.

Conditionally Stable Systems. Figure 7–58 shows an example of a G(jv)H(jv)

locus for which the closed-loop system can be made unstable by varying the open-loop

gain. If the open-loop gain is increased sufficiently, the G(jv)H(jv)locus encloses the

–1+j0point twice, and the system becomes unstable. If the open-loop gain is decreased

sufficiently, again the G(jv)H(jv)locus encloses the –1+j0point twice. For stable

operation of the system considered here, the critical point –1+j0must not be located

in the regions between OAandBCshown in Figure 7–58. Such a system that is stable

only for limited ranges of values of the open-loop gain for which the –1+j0point is

completely outside the G(jv)H(jv)locus is a conditionally stable system.

A conditionally stable system is stable for the value of the open-loop gain lying be-

tween critical values, but it is unstable if the open-loop gain is either increased or de-

creased sufficiently. Such a system becomes unstable when large input signals are applied,

since a large signal may cause saturation, which in turn reduces the open-loop gain of

the system. It is advisable to avoid such a situation.

Multiple-Loop System. Consider the system shown in Figure 7–59. This is a mul-

tiple-loop system. The inner loop has the transfer function

G(s)=

G 2 (s)

1 +G 2 (s)H 2 (s)

Im

Re

GH Plane

0

0

C B A

v =`

Figure 7–58 v

Polar plot of a

conditionally stable

system.

Openmirrors.com