Modern Control Engineering

Example Problems and Solutions 541

40

20

• 20
  dB

0

• 40


• 60


• 80

0.1 0.2 0.4 0.6 1 2 4 6 10 20 4060 100

• 270 °


• 180 °


• 90 °

0 °

v in rad/sec

Figure 7–135
Bode diagram of the
open-loop transfer
function of a unity-
feedback control
system.

Noting that there is another corner frequency at v=0.5radsec and the slope of the magnitude

curve in the low-frequency region is –40dBdecade,G(jv)can be tentatively determined as

follows:

Since, from Figure 7–135 we find @G(j0.1)@=40dB, the gain value Kcan be determined to be

unity. Also, the calculated phase curve, versus v, agrees with the given phase curve. Hence,

the transfer function G(s)can be determined to be

A–7–19. A closed-loop control system may include an unstable element within the loop. When the Nyquist

stability criterion is to be applied to such a system, the frequency-response curves for the unsta-

ble element must be obtained.

How can we obtain experimentally the frequency-response curves for such an unstable ele-

ment? Suggest a possible approach to the experimental determination of the frequency response

of an unstable linear element.

Solution.One possible approach is to measure the frequency-response characteristics of the un-

stable element by using it as a part of a stable system.

G(s)=

4(2s+1)

s^2 As^2 +0.4s+ 4 B

/G(jv)

G(jv)=

Ka

jv

0.5

+ 1 b

(jv)^2 ca

jv

2

b

2

+0.1(jv)+ 1 d