Modern Control Engineering

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

The phase and gain margins can easily be obtained from the Bode diagram. A Bode diagram of

the given open-loop transfer function with K=10is shown in Figure 7–70(a). The phase and gain

margins for K=10are

Phase margin=21°, Gain margin=8dB

Therefore, the system gain may be increased by 8 dB before the instability occurs.

Increasing the gain from K=10toK=100shifts the 0-dB axis down by 20 dB, as shown in

Figure 7–70(b). The phase and gain margins are

Phase margin=–30°, Gain margin=–12dB

Thus, the system is stable for K=10,but unstable for K=100.

Notice that one of the very convenient aspects of the Bode diagram approach is the ease with

which the effects of gain changes can be evaluated. Note that to obtain satisfactory performance, we

must increase the phase margin to 30°~60°. This can be done by decreasing the gain K. Decreas-

ingKis not desirable, however, since a small value of Kwill yield a large error for the ramp input.

This suggests that reshaping of the open-loop frequency-response curve by adding compensation may

be necessary. Compensation techniques are discussed in detail in Sections 7–11 through 7–13.

Obtaining Gain Margin, Phase Margin, Phase-Crossover Frequency, and Gain-

Crossover Frequency with MATLAB. The gain margin, phase margin, phase-crossover

frequency, and gain-crossover frequency can be obtained easily with MATLAB. The com-

mand to be used is

[Gm,pm,wcp,wcg] = margin(sys)

30

20

10

0

• 30


• 20


• 10

0 °

• 90 °


• 180 °


• 270 °

|G

| in dB

|G

| in dB

G

0.2 0.4 0.6 0.8 1 2 4 6 8 10

0 °

• 90 °
  
  
  • 30 °
  
  
  
  


• 180 °


• 270 °

G

0.2 0.4 0.6 0.8 1 2 4 6 8 10

K= 10 K= 100

+ 8 dB (Gain margin)

(Phase margin) + 21 °

vv

30

20

10

0

• 10

50

40

(Gain margin) –12 dB

(Phase margin)

(a) (b)

Figure 7–70

Bode diagrams of the system shown in Figure 7–69; (a) with K=10and (b) with K=100.

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