Modern Control Engineering

294 Chapter 6 / Control Systems Analysis and Design by the Root-Locus Method

Real Axis

–4 –3 –2 –1 0 1 2 3 4

Imag Axis

4

–1

3

–3

–4

2

0

1

–2

Root-Locus Plot of G(s)= K/[s(s+0.5)(s^2 +0.6s+10)]

Figure 6–18

Root-locus plot.

MATLAB Program 6–3

% --------- Root-locus plot ---------

num = [1];

den = [1 1.1 10.3 5 0];

K1 = 0:0.2:20;

K2 = 20:0.1:30;

K3 = 30:5:1000;

K = [K1 K2 K3];

r = rlocus(num,den,K);

plot(r, 'o')

v = [-4 4 -4 4]; axis(v)

grid

title('Root-Locus Plot of G(s) = K/[s(s + 0.5)(s^2 + 0.6s + 10)]')

xlabel('Real Axis')

ylabel('Imag Axis')

By a conventional trial-and-error approach or using the command rlocfindto be presented

later in this section, we find the particular region of interest to be 20K30. By entering

MATLAB Program 6–3, we obtain the root-locus plot shown in Figure 6–18. From this plot, it

is clear that the two branches that approach in the upper half-plane (or in the lower half-plane)

do not touch.

EXAMPLE 6–5 Consider the system shown in Figure 6–19. The system equations are

u =r-y

y =Cx+Du

x#=Ax+Bu

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