Question

Matlab

Create a root locus plot to determine design a control system for the following system which has a standard negative unity feedback system. G(s) = K (s2 - 4s +20)/[(s+2)(s+4)]

needs to be done by matlab

Answer 1

Matlab script to plot root locus

n=[1 -4 20];
d=[1 6 8];
h=tf(n,d); % calculates the transfer function
rlocus(h) % plots the root locus

rootlocus matlab plot

Root Locus E 0 4 3 System: h Gain: 0.416 Pole: -1.53 +3.03i Damping: 0.451 Overshoot (%): 20.5 Frequency (rad/s): 3.39 System: h Gain: 1.51 Pole: 0.00736 + 3.91 Damping: -0.00189 Overshoot (%): 101 Frequency (rad/s): 3.9 Imaginary Axis (seconds) System: h Gain: 1.49 Pole:-0.00783 - 3.91 Damping: 0.00201 Overshoot (%): 99.4 Frequency (rad/s): 3.9 -5 -5 -4 -3 -2 -1 0 1 2 3 Real Axis (seconds)

from this root locus plot

question 1

the damping ratio is 0.45 when gain is 0.417

so answer is option 2

question 2

the system becomes unstable when plot(poles) moves to right side of jw axis

this happes at k =1.5

so answer is option2