Modern Control Engineering

lead compensator shifts the gain crossover frequency to the right and decreases the

phase margin.

4.Determine the attenuation factor aby use of Equation (7–25). Determine the

frequency where the magnitude of the uncompensated system G 1 (jv)is equal to

Select this frequency as the new gain crossover frequency. This

frequency corresponds to and the maximum phase shift fmoccurs

at this frequency.

5.Determine the corner frequencies of the lead compensator as follows:

Zero of lead compensator:

Pole of lead compensator:

6.Using the value of Kdetermined in step 1 and that of adetermined in step 4,

calculate constant Kcfrom

7.Check the gain margin to be sure it is satisfactory. If not, repeat the design process

by modifying the pole–zero location of the compensator until a satisfactory result

is obtained.

EXAMPLE 7–26 Consider the system shown in Figure 7–94. The open-loop transfer function is

It is desired to design a compensator for the system so that the static velocity error constant Kv

is 20 sec–1,the phase margin is at least 50°, and the gain margin is at least 10 dB.

We shall use a lead compensator of the form

The compensated system will have the open-loop transfer function Gc(s)G(s).

Define

whereK=Kca.

G 1 (s)=KG(s)=

4K

s(s+2)

Gc(s)=Kc a

Ts+ 1

aTs+ 1

=Kc

s+

1

T

s+

1

aT

G(s)=

4

s(s+2)

Kc=

K

a

v=

1

aT

v=

1

T

vm= 1 A 1 aTB,

- 20 logA 1  1 aB.

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

4

+– s(s+ 2)

Figure 7–94

Control system.

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