Modern Control Engineering

fm=38° corresponds to a=0.24.Once the attenuation factor ahas been determined on the

basis of the required phase-lead angle, the next step is to determine the corner frequencies v=1/T

andv=1/(aT)of the lead compensator. To do so, we first note that the maximum phase-lead

anglefmoccurs at the geometric mean of the two corner frequencies, or [See Equa-

tion (7–26).] The amount of the modification in the magnitude curve at due to the

inclusion of the term (Ts+1)/(aTs+1)is

Note that

and@G 1 (jv)@=–6.2dB corresponds to v=9radsec. We shall select this frequency to be the new

gain crossover frequency vc. Noting that this frequency corresponds to or

we obtain

and

The lead compensator thus determined is

where the value of Kcis determined as

Thus, the transfer function of the compensator becomes

Note that

The magnitude curve and phase-angle curve for Gc(jv)/10are shown in Figure 7–96. The

compensated system has the following open-loop transfer function:

Gc(s)G(s)=41.7

s+4.41

s+18.4

4

s(s+2)

Gc(s)

K

G 1 (s)=

Gc(s)

10

10G(s)=Gc(s)G(s)

Gc(s)=41.7

s+4.41

s+18.4

= 10

0.227s+ 1

0.054s+ 1

Kc=

K

a

=

10

0.24

=41.7

Gc(s)=Kc

s+4.41

s+18.4

=Kc a

0.227s+ 1

0.054s+ 1

1

aT

=

vc

1 a

=18.4

1

T

= 1 avc=4.41

vc= 1 A 1 aTB,

1 A 1 aTB,

1

1 a

=

1

1 0.24

=

1

0.49

=6.2 dB

2

1 +jvT

1 +jvaT

2

v= 1 A 1 aTB

= 4

1 +j

1

1 a

1 +ja

1

1 a

4 =

1

1 a

v= 1 A 1 aTB

v= 1 A 1 aTB.

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

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