Modern Control Engineering

Example Problems and Solutions 535

MATLAB Program 7–20

num = [-1 -4 -6];

den = [1 5 4];

nyquist(num,den);

grid

title('Nyquist Plot of G(s) = -(s^2 + 4s + 6)/(s^2 + 5s + 4)')

A–7–14. Consider a unity-feedback, positive-feedback system with the following open-loop transfer

function:

Draw a Nyquist plot.

Solution.The Nyquist plot of the positive-feedback system can be obtained by defining numand

denas

num = [-1 -4 -6]

den = [1 5 4]

and using the command nyquist(num,den). MATLAB Program 7–20 produces the Nyquist plot,

as shown in Figure 7–130.

This system is unstable, because the –1+j0point is encircled once clockwise. Note that this

is a special case where the Nyquist plot passes through –1+j0point and also encircles this point

once clockwise. This means that the closed-loop system is degenerate; the system behaves as if it

were an unstable first-order system. See the following closed-loop transfer function of the positive-

feedback system:

=

s^2 +4s+ 6

s- 2

C(s)

R(s)

=

s^2 +4s+ 6

s^2 +5s+ 4 - As^2 +4s+ 6 B

G(s)=

s^2 +4s+ 6

s^2 +5s+ 4

Real Axis

− 23 −1.5 − 1 −0.5 0 0.5 1 1.5 2 2.5

Imaginary Axis

− 5

0

−0.5

− 1

−2.5

− 3

−3.5

− 4

−4.5

− 2

−1.5

Nyquist Diagram

w = 0.2

0.3

0.5

6

1 2

20 10

Figure 7–129
Polar plot of G(jv)
given in Problem
A–7–13.