Modern Control Engineering

Section 7–6 / Stability Analysis 455

Im

• 1 Re

GH Plane

G(jv)H(jv)

v = –` v =^0

v =`

Figure 7–53
Polar plot of
G(jv)H(jv)
considered in
Example 7–14.

Im

– (^1) Re Re

• 1

GH Plane

v = 0 –

v = 0 +

Im GH Plane

v = 0–

v = 0 +

SmallK LargeK

(Stable) (Unstable)

P= 0 P= 0

N= 0

Z= 0

v =–

v =

v =–

v =

N= 2

Z= 2

Figure 7–54
Polar plots of the
system considered in
Example 7–15.

EXAMPLE 7–14 Consider a closed-loop system whose open-loop transfer function is given by

Examine the stability of the system.

A plot of G(jv)H(jv)is shown in Figure 7–53. Since G(s)H(s)does not have any poles in

the right-half splane and the –1+j0point is not encircled by the G(jv)H(jv)locus, this system

is stable for any positive values of K,T 1 ,and T 2.

G(s)H(s)=

K

AT 1 s+ 1 BAT 2 s+ 1 B

EXAMPLE 7–15 Consider the system with the following open-loop transfer function:

Determine the stability of the system for two cases: (1) the gain Kis small and (2)Kis large.

The Nyquist plots of the open-loop transfer function with a small value of Kand a large value

ofKare shown in Figure 7–54. The number of poles of G(s)H(s)in the right-half splane is zero.

G(s)H(s)=

K

sAT 1 s+ 1 BAT 2 s+ 1 B