Modern Control Engineering

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

A–7–11. Consider a unity-feedback system with the following open-loop transfer function:

Draw a Nyquist plot with MATLAB and examine the stability of the closed-loop system.

Solution.MATLAB Program 7–17 produces the Nyquist diagram shown in Figure 7–127. From

this figure, we see that the Nyquist plot does not encircle the –1+j0point. Hence,N=0in the

Nyquist stability criterion. Since no open-loop poles lie in the right-half splane,P=0.Therefore,

Z=N+P=0.The closed-loop system is stable.

G(s)=

20 As^2 +s+0.5B

s(s+1)(s+10)

A–7–12. Consider the same system as discussed in Problem A–7–11. Draw the Nyquist plot for only the

positive-frequency region.

Solution.Drawing a Nyquist plot for only the positive-frequency region can be done by the use

of the following command:

[re,im,w] = nyquist(num,den,w)

The frequency region may be divided into several subregions by using different increments. For

example, the frequency region of interest may be divided into three subregions as follows:

w1 = 0.1:0.1:10;

w2 = 10:2:100;

w3 = 100:10:500;

w = [w1 w2 w3]

MATLAB Program 7–17

num = [20 20 10];

den = [1 11 10 0];

nyquist(num,den)

v = [-2 3 -3 3]; axis(v)

grid

Real Axis

− 23 −1.5 − 1 −0.5 0 0.5 1 1.5 2 2.5

Imaginary Axis

− 3

3

2

1

− 2

− 1

0

Nyquist Diagram

Figure 7–127

Nyquist plot of

G(s)=

20 As^2 +s+0.5B

s(s+1)(s+10)

.

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