Modern Control Engineering

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

MATLAB Program 7–22

num = [10 1];

den = [1 1.5 0.5 0];

bode(num,den)

title('Bode Diagram of G(s) = (10s + 1)/[s(s + 0.5)(s + 1)]')

Let us plot the Bode diagram of G(s)whenK=1. MATLAB Program 7–22 may be used for this

purpose. Figure 7–139 shows the Bode diagram produced by this program. From this diagram the

required phase margin of 60° occurs at the frequency v=1.15radsec. The magnitude of G(jv)

at this frequency is found to be 14.5 dB. Then gain Kmust satisfy the following equation:

or

K=0.188

20 logK=-14.5 dB

Frequency (rad/sec)

Bode Diagram of G(s) = (10s + 1)/[s(s + 0.5)(s + 1)]

− 200

− 150

− 50

− 100

− 50

Phase (deg); Magnitude (dB)

0

100

50

10 −^310 −^210 −^1100101

Figure 7–139

Bode diagram of

G(s)=

10s+ 1

s(s+0.5)(s+1)

.

Thus, we have determined the value of gain K. Since the angle curve does not cross the –180° line,

the gain margin is ±qdB.

To verify the results, let us draw a Nyquist plot of Gfor the frequency range

w = 0.5:0.01:1.15

The end point of the locus (v=1.15radsec)will be on a unit circle in the Nyquist plane. To check

the phase margin, it is convenient to draw the Nyquist plot on a polar diagram, using polar grids.

To draw the Nyquist plot on a polar diagram, first define a complex vector zby

z = re + i*im = reiu

whererandu(theta) are given by

r = abs(z)

theta = angle(z)

The absmeans the square root of the sum of the real part squared and imaginary part squared;

anglemeans tan–1(imaginary part/real part).

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