Modern Control Engineering

Section 6–3 / Plotting Root Loci with MATLAB 297

distance of the pole from the origin is determined by the undamped natural frequen-

cy vn. The constant vnloci are circles.

To draw constant zlines and constant vncircles on the root-locus diagram with

MATLAB, use the command sgrid.

Plotting Polar Grids in the Root-Locus Diagram. The command

sgrid

overlays lines of constant damping ratio (z=0 ~1 with 0.1 increment) and circles of

constantvnon the root-locus plot. See MATLAB Program 6–5 and the resulting diagram

shown in Figure 6–22.

MATLAB Program 6–5

sgrid

v = [-3 3 -3 3]; axis(v); axis('square')

title('Constant \zeta Lines and Constant \omega_n Circles')

xlabel('Real Axis')

ylabel('Imag Axis')

If only particular constant zlines (such as the z=0.5line and z=0.707line) and

particular constant vncircles (such as the vn=0.5circle,vn=1circle, and vn=2cir-

cle) are desired, use the following command:

sgrid([0.5, 0.707], [0.5, 1, 2])

If we wish to overlay lines of constant zand circles of constant vnas given above to a

root-locus plot of a negative feedback system with

num = [0 0 0 1]

den = [1 4 5 0]

− 3 − 2 − 1 0 1 2 3

0

1

3

2

− 1

− 3

− 2

Real Axis

Constantz Lines and Constant vn Circles

Imag Axis

2

1

2

1

0.64 0.5 0.340.16

0.64 0.5 0.340.16

0.76

0.86

0.94

0.985

0.76

0.86

0.94

0.985

Figure 6–22
Constantzlines and
constantvncircles.