Modern Control Engineering

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

Figure 7–132

Polar plot of

K/G(jv).

theK/G(jv)locus crosses the negative real axis at and the crossing point at the nega-

tive real axis is –2.

From Figure 7–132, we see that if the critical point lies in the region between –2and–q, then

the critical point is not encircled. Hence, for stability, we require

Thus, the range of gain Kfor stability is

2<K

which is the same result as we obtained in Example 7–19.

A–7–16. Figure 7–133 shows a block diagram of a space-vehicle control system. Determine the gain Ksuch

that the phase margin is 50°. What is the gain margin in this case?

Solution.Since

we have

The requirement that the phase margin be 50° means that must be equal to –130°, where

vcis the gain crossover frequency, or

/GAjvcB=-^130 °

/GAjvcB

/G(jv)=/jv+^2 -^2 /jv=tan-^1

v

2

- 180 °

G(jv)=

K(jv+2)

(jv)^2

- 16

- 2

K

v= 12 ,

Im

Re

K

K GPlane

GLocus





• 
  
  
  • 20 2
  
  
  
  

v = 0

v = 2

v

v

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