Modern Control Engineering

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/Tandv=1/(aT)of the lead

compensator. Notice that the maximum phase-lead angle fmoccurs at the geometric mean of the

two corner frequencies, or

The amount of the modification in the magnitude curve at due to the inclusion

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

Note that

We need to find the frequency point where, when the lead compensator is added, the total mag-

nitude becomes 0 dB.

From Figure 7–145 we see that the frequency point where the magnitude of G 1 (jv)is

–6.7778dB occurs between v=1and 10 radsec. Hence, we plot a new Bode diagram of

G 1 (jv)in the frequency range between v=1and 10 to locate the exact point where

G 1 (jv)=–6.7778dB. MATLAB Program 7–25 produces the Bode diagram in this frequency

range, which is shown in Figure 7–146. From this diagram, we find the frequency point where

occurs at v=6.5686radsec. Let us select this frequency to be the new

gain crossover frequency, or vc=6.5686radsec. Noting that this frequency corresponds to

or

we obtain

and

1

aT

=

vc

1 a

=

6.5686

1 0.21

=14.3339

1

T

=vc 1 a=6.5686 1 0.21=3.0101

vc=

1

1 aT

1 A 1 aTB,

@G 1 (jv)@=-6.7778 dB

1

1 a

=

1

1 0.21

=6.7778 dB

2

1 +jvT

1 +jvaT

2

v= 11 aT

= 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.

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

MATLAB Program 7–25

num = [20];

den = [1 1 0];

w = logspace(0,1,100);

bode(num,den,w)

title('Bode Diagram of G1(s) = 20/[s(s + 1)]')

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