Question

Given the transfer function 4. G(s)H(s) - (s + 8) (s +6s + 13) (a) Sketch the root locus plot using Matlab. (b) Estimate the system gain when the damping ratio is 7 0.707 (c) Add a simple pole, (s 2), to G (s)H (s) and examine the resulting root locus (d) Add a simple zero, (s +2), to G(s)H(s) and examine the resulting root locus

Answer 1

a)

Root Locus:

clear;clc;
G = tf(1,[1 8]);
H = tf(1,[1 6 13]);
GH = G*H;

rlocus(GH);

Root Locus 15 0.7 0.82 0.56 0.42 0.28 0.14 10 0.91 0.975 17.5 15 12.5 10 7.5 2.5 0.975 -5 0.91 10 0.82 0.7 0.56 0.42 0.28 15 0.14 15 -5 Real Axis (seconds

b)

The required value is between: 23.6 and 20.2.

By inspection it is close to 21.2

It can be verified.

Root Locus 0.8 0.7 0.6 0.5 0.38 3.5 System: GH Gain: 23.6 Pole: -2.67 2.76i Damping: 0.696 Overshoot (%): 4.77 System: GH Gain: 20.2 Pole: -2.712.67 Damping: 0.713 Overshoot(%) 4.12 0.88 requency (rad/s: 3.8equency (rad's): 3.84 2.5 2 0.95 D 1.5 10.985 0.5 5 2 -0.5 0.985 5.5 -4.5 3.5 .3 2.5 -2 Real Axis (seconds1 1.5

clear;clc; Gtf (1,[1 8]) Htf (1,[1 6 13]); GHGH; 1 3 5 6 rlocus (GH) damp (feedback (GH 21.2,1)) 10 12 14 Command Window Pole Time Constant Damping Frequency (rad/seconds) (seconds) 2.70e+00+ 2.70e+00i 2.70e+00-2.70e+00i 8.60e+00 7.08e-01 7.08e-01 1.00e+00 3.82e+00 3.82e+00 8.60e+00 3.70e-01 3.70e-01 1.16e-01

c)

Root Locus 15 0 0 10 10 15 10 15 Real Axis (seconds)

d)

Root Locus 15 10 5 10 15 -2 -6 -10 Real Axis (seconds]