Modern Control Engineering

The magnitude curve and phase-angle curve of G 1 (jv)are shown in Figure 7–104. From this plot,

the phase margin is found to be –20°, which means that the gain-adjusted but uncompensated

system is unstable.

Noting that the addition of a lag compensator modifies the phase curve of the Bode diagram, we

must allow 5° to 12° to the specified phase margin to compensate for the modification of the phase

curve. Since the frequency corresponding to a phase margin of 40° is 0.7 radsec, the new gain crossover

frequency (of the compensated system) must be chosen near this value. To avoid overly large time

constants for the lag compensator, we shall choose the corner frequency v=1/T(which corresponds

to the zero of the lag compensator) to be 0.1 radsec. Since this corner frequency is not too far below

the new gain crossover frequency, the modification in the phase curve may not be small. Hence, we add

about 12° to the given phase margin as an allowance to account for the lag angle introduced by the lag

compensator. The required phase margin is now 52°. The phase angle of the uncompensated open-loop

transfer function is –128° at about v=0.5radsec. So we choose the new gain crossover frequency

to be 0.5 radsec. To bring the magnitude curve down to 0 dB at this new gain crossover frequency, the

lag compensator must give the necessary attenuation, which in this case is –20dB. Hence,

or

The other corner frequency v=1(bT), which corresponds to the pole of the lag compen-

sator, is then determined as

1

bT

=0.01 radsec

b= 10

20 log

1

b

=- 20

11 dB

dB 0

0 °

• 90 °


• 180 °


• 270 °

v in rad/sec

0.004 0.02

G 1

G 1

GcG

Gc

K

Gc

40

20

• 20


• 40

40 °

0.01 0.04 0.1 0.2 0.40.6 1 2 4

GcG

Figure 7–104

Bode diagrams for

G 1 (gain-adjusted but

uncompensated

open-loop transfer

function),Gc

(compensator), and

GcG(compensated

open-loop transfer

function).

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

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