Modern Control Engineering

Section 6–3 / Plotting Root Loci with MATLAB 301

To obtain a root-locus plot with MATLAB, enter the numerator and denominator

as usual. For example, if T=1sec and enter the following numandden

in the program:

num = [-0.5 1]

den = [1 1 0]

MATLAB Program 6–8 gives the plot of the root loci shown in Figure 6–27.

Ta=0.5 sec,

MATLAB Program 6–8

num = [-0.5 1];

den = [1 1 0];

k1 = 0:0.01:30;

k2 = 30:1:100;

K3 = 100:5:500;

K = [k1 k2 k3];

rlocus(num,den,K)

v = [-2 6 -4 4]; axis(v); axis('square')

grid

title('Root-Locus Plot of G(s) = K(1 - 0.5s)/[s(s + 1)]')

% Place 'x' mark at each of 2 open-loop poles.

% Place 'o' mark at open-loop zero.

gtext('x')

gtext('x')

gtext('o')

Root-Locus Plot of G(s) = K(1− 0.5s)/[s(s + 1)]

Real Axis

Imag Axis

1

− 3

− (^4) − 2 − 10123456
2
− 1
− 2
0
3
4
Figure 6–27
Root-locus plot of
G(s)=
K(1-0.5s)
s(s+1)

.

Orthogonality of Root Loci and Constant-Gain Loci. Consider the negative

feedback system whose open-loop transfer function is G(s)H(s).In the G(s)H(s)plane,

the loci of constant are circles centered at the origin, and the loci corre-

sponding to /G(s)H(s)=; 180 °(2k+1)(k=0,1, 2,p) lie on the negative real axis

∑G(s)H(s)∑=