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Callisto
 3 years ago
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.
Callisto
 3 years ago
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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ash2326
 3 years ago
Best ResponseYou've already chosen the best response.1First of all find the closed loop transfer function of the system

Callisto
 3 years ago
Best ResponseYou've already chosen the best response.0Suppose it is \[Y= \frac{1}{1kz^{1} + bz^{2}}\]where k is an unknown constant and b is a known constant.

Callisto
 3 years ago
Best ResponseYou've already chosen the best response.0I meant \[\frac{Y}{X}=...\]

Callisto
 3 years ago
Best ResponseYou've already chosen the best response.0Solving the denominator =0, \[1kz^{1}+bz^{2}=0\]\[z^2  kz + b =0\]\[z = \frac{k \pm \sqrt{k^2  4b}}{2}\]

ash2326
 3 years ago
Best ResponseYou've already chosen the best response.1You'd need to find the poles of the system, sorry I was dealing in the s domain

Callisto
 3 years ago
Best ResponseYou've already chosen the best response.0Poles are at \[\frac{k\pm\sqrt{k^24b}}{2}\]

ash2326
 3 years ago
Best ResponseYou've already chosen the best response.1There 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.

ash2326
 3 years ago
Best ResponseYou've already chosen the best response.1@Callisto I found a good document on this, please see it. You'll understand better http://www.eng.ox.ac.uk/~conmrc/dcs/dcslec4.pdf

ash2326
 3 years ago
Best ResponseYou've already chosen the best response.1Let me know if you have doubt anywhere

Callisto
 3 years ago
Best ResponseYou've already chosen the best response.0Now, 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!

ash2326
 3 years ago
Best ResponseYou've already chosen the best response.1oops, did you ask this doubt to your teacher?

Callisto
 3 years ago
Best ResponseYou've already chosen the best response.0Ha! I don't even have to ask as in the lecture, he has written "you will find out the reason if you take EEE"

ash2326
 3 years ago
Best ResponseYou've already chosen the best response.1umm, where did calculus was used?

Callisto
 3 years ago
Best ResponseYou've already chosen the best response.0He has NEVER used calculus in this course.

ash2326
 3 years ago
Best ResponseYou've already chosen the best response.1But the solution requires just solving the quadratics. then depending on the poles, we classify the system

Callisto
 3 years ago
Best ResponseYou've already chosen the best response.0p = pole p = 1 => remains (unchanged) p >1 => diverge p < 1 => converge

ash2326
 3 years ago
Best ResponseYou've already chosen the best response.1now you need to check which one lies in, out or on the unit circle. then you can classify the system

ash2326
 3 years ago
Best ResponseYou've already chosen the best response.1@Callisto are you here?

Callisto
 3 years ago
Best ResponseYou've already chosen the best response.0checking the magnitude? I did it. But the problem is how I can identify if the system oscillates. Sorry, I was on other page.

ash2326
 3 years ago
Best ResponseYou've already chosen the best response.1if 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

Callisto
 3 years ago
Best ResponseYou've already chosen the best response.0Hmm... I think the magnitude of the pole only tell us if the system is converging/diverging/remaining unchanged?!

Callisto
 3 years ago
Best ResponseYou've already chosen the best response.0I think I understand how to analyze the system now, thanks :)
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