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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.

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First of all find the closed loop transfer function of the system
Suppose it is \[Y= \frac{1}{1-kz^{-1} + bz^{-2}}\]where k is an unknown constant and b is a known constant.

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I meant \[\frac{Y}{X}=...\]
Solving the denominator =0, \[1-kz^{-1}+bz^{-2}=0\]\[z^2 - kz + b =0\]\[z = \frac{k \pm \sqrt{k^2 - 4b}}{2}\]
You'd need to find the poles of the system, sorry I was dealing in the s- domain
Poles are at \[\frac{k\pm\sqrt{k^2-4b}}{2}\]
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.
@Callisto I found a good document on this, please see it. You'll understand better
Let me know if you have doubt anywhere
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!
so you undestood now ?
oops, did you ask this doubt to your teacher?
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"
*in the lecture notes
umm, where did calculus was used?
He has NEVER used calculus in this course.
But the solution requires just solving the quadratics. then depending on the poles, we classify the system
p = pole p = 1 => remains (unchanged) |p| >1 => diverge |p| < 1 => converge
now you need to check which one lies in, out or on the unit circle. then you can classify the system
@Callisto are you here?
checking the magnitude? I did it. But the problem is how I can identify if the system oscillates. Sorry, I was on other page.
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
Hmm... I think the magnitude of the pole only tell us if the system is converging/diverging/remaining unchanged?!
I think I understand how to analyze the system now, thanks :)
welcome :P

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