Question

1:31 HW12.pdf 312 DYNAMIC SYSTEMS AND CONTROL University of Illinois, Chicago Department of Mechanical Engineering Spring 2019 PROBLEM SET# 12, DUE : 4/12/2019 (1) For the following system in a unity feedback loop, sketch the root locus, plot the asymptotes and find the gain such that they system remains stable. KG(s) = 긁(FF10)(s + 11) 1010 1)

Answer 1

The poles are 0,0,-11,-10

The zeros are -1.

Root locus exists when there is odd number of poles and zeros when seeing from a particular point.

Pole-Zero Map 1 0.6 0.4 0.2 NRI 0 RL li 0.6 -0.8 -1 -12 -10 .8 -6 1 Real Axis (seconds)

Hie pole at -11 reaches zero at infinity; the pole at -10 reaches zero at -1. The two poles at origin have no poles. So. they should reach zeros at infinity. So. definitely there is a breakaway point between the poles that is at the origin itself.
Since three branches will approach to infinity at the infinite gain, having an angle of asymptotes.

(2 x k+1)7T finitepoles

(2 x k+1)7T

(2 x k+1)T

for k=0

(2x 0+1) = ±( -)

$\theta_a =\pm \frac{\pi}{3}$

Root Locus:

Root Locus 40 30 20 E 10 60 -60 10 -20 -30 -40 -50 40 -30 -20 -10 10 20 Real Axis (seconds)

Calculating the Closed loop Transfer function

01:43 Using RH cnieatofindk 3 2 2.1 2

23lo-k 23lo >K 231o-k 21 2 System k shad b