Modern Control Engineering

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

Im

Re

GH Plane

v = 0 –

v = 0 +

v =–

v =

Im

Re

GH Plane

v = 0 –

v = 0 +

v =–

v =

Im

Re

GH Plane

v = 0 +

v = 0 –

v =

v =–

T 1 T 2

(Stable)

T 1 = T 2

G(jv)H(jv) locus

passes through the

• 1 +j0 point

T 1  T 2

(Unstable)

Figure 7–55

Polar plots of the

system considered in

Example 7–16.

Therefore, for this system to be stable, it is necessary that N=Z=0or that the G(s)H(s)locus

not encircle the –1+j0point.

For small values of K, there is no encirclement of the –1+j0point. Hence, the system is sta-

ble for small values of K. For large values of K, the locus of G(s)H(s)encircles the –1+j0point

twice in the clockwise direction, indicating two closed-loop poles in the right-half splane, and the

system is unstable. (For good accuracy,Kshould be large. From the stability viewpoint, however,

a large value of Kcauses poor stability or even instability. To compromise between accuracy and

stability, it is necessary to insert a compensation network into the system. Compensating tech-

niques in the frequency domain are discussed in Sections 7–11 through 7–13.)

EXAMPLE 7–16 The stability of a closed-loop system with the following open-loop transfer function

depends on the relative magnitudes of and Draw Nyquist plots and determine the stability

of the system.

Plots of the locus G(s)H(s)for three cases, and are shown

in Figure 7–55. For the locus of G(s)H(s)does not encircle the –1+j0point,

and the closed-loop system is stable. For , the G(s)H(s)locus passes through

the–1+j0point, which indicates that the closed-loop poles are located on the jvaxis. For

the locus of G(s)H(s)encircles the –1+j0point twice in the clockwise direction.

Thus, the closed-loop system has two closed-loop poles in the right-half splane, and the system

is unstable.

T 17 T 2 ,

T 1 =T 2

T 1 6 T 2 ,

T 16 T 2 ,T 1 =T 2 , T 17 T 2 ,

T 1 T 2.

G(s)H(s)=

KAT 2 s+ 1 B

s^2 AT 1 s+ 1 B

EXAMPLE 7–17 Consider the closed-loop system having the following open-loop transfer function:

Determine the stability of the system.

G(s)H(s)=

K

s(Ts-1)

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