Question

How can we know if the system is oscillating and if it is decaying with only a quadratic equation?
PS:
<1> Do not involve any calculus
<2> Haven't learnt damping.

Answer 1

@ash2326

Answer 2

First of all find the closed loop transfer function of the system

Answer 3

Suppose it is
\[Y= \frac{1}{1-kz^{-1} + bz^{-2}}\]where k is an unknown constant and b is a known constant.

Answer 4

I meant \[\frac{Y}{X}=...\]

Answer 5

Solving the denominator =0,
\[1-kz^{-1}+bz^{-2}=0\]\[z^2 - kz + b =0\]\[z = \frac{k \pm \sqrt{k^2 - 4b}}{2}\]

Answer 6

You'd need to find the poles of the system, sorry I was dealing in the s- domain

Answer 7

Poles are at \[\frac{k\pm\sqrt{k^2-4b}}{2}\]

Answer 8

There will be many cases,
\[ K=2\sqrt b, K>2\sqrt b\ and\ K<2\sqrt b\]
for first case,
\[z= \sqrt b\]
if b<1
then system will decay
if b=1 system will be constant
if b>1 then system will be unbounded.

Answer 9

@Callisto I found a good document on this, please see it. You'll understand better
http://www.eng.ox.ac.uk/~conmrc/dcs/dcs-lec4.pdf

Answer 10

Let me know if you have doubt anywhere

Answer 11

Now, I see why I can never get the answer. The reason why I have been stressing that no calculus is involved is because when we learnt this topic, our lecturer didn't not teach us using calculus, i.e. solving D.E., using expressions in exponential forms, nor mentioning those fancy terms like damping.
Anyway, thanks for trying to help!

Answer 12

so you undestood now ?

Answer 13

No.

Answer 14

oops, did you ask this doubt to your teacher?

Answer 15

Ha! I don't even have to ask as in the lecture, he has written "you will find out the reason if you take EEE"

Answer 16

*in the lecture notes

Answer 17

umm, where did calculus was used?

Answer 18

He has NEVER used calculus in this course.

Answer 19

But the solution requires just solving the quadratics. then depending on the poles, we classify the system

Answer 20

p = pole
p = 1 => remains (unchanged)
|p| >1 => diverge
|p| < 1 => converge

Answer 21

now you need to check which one lies in, out or on the unit circle. then you can classify the system

Answer 22

@Callisto are you here?

Answer 23

checking the magnitude? I did it.
But the problem is how I can identify if the system oscillates.
Sorry, I was on other page.

Answer 24

if the poles are on the unit circle, system will oscillate
if they are inside, it'll decay
if they are outside, oscillations will grow unboundedly

Answer 25

???

Answer 26

Hmm... I think the magnitude of the pole only tell us if the system is converging/diverging/remaining unchanged?!

Answer 27

I think I understand how to analyze the system now, thanks :)

Answer 28

welcome :P