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Callisto

  • 2 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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  1. Callisto
    • 2 years ago
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    @ash2326

  2. ash2326
    • 2 years ago
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    First of all find the closed loop transfer function of the system

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

  4. Callisto
    • 2 years ago
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    I meant \[\frac{Y}{X}=...\]

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

  6. ash2326
    • 2 years ago
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    You'd need to find the poles of the system, sorry I was dealing in the s- domain

  7. Callisto
    • 2 years ago
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    Poles are at \[\frac{k\pm\sqrt{k^2-4b}}{2}\]

  8. ash2326
    • 2 years ago
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    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.

  9. ash2326
    • 2 years ago
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    @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

  10. ash2326
    • 2 years ago
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    Let me know if you have doubt anywhere

  11. Callisto
    • 2 years ago
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    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!

  12. ash2326
    • 2 years ago
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    so you undestood now ?

  13. Callisto
    • 2 years ago
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    No.

  14. ash2326
    • 2 years ago
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    oops, did you ask this doubt to your teacher?

  15. Callisto
    • 2 years ago
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    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"

  16. Callisto
    • 2 years ago
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    *in the lecture notes

  17. ash2326
    • 2 years ago
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    umm, where did calculus was used?

  18. Callisto
    • 2 years ago
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    He has NEVER used calculus in this course.

  19. ash2326
    • 2 years ago
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    But the solution requires just solving the quadratics. then depending on the poles, we classify the system

  20. Callisto
    • 2 years ago
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    p = pole p = 1 => remains (unchanged) |p| >1 => diverge |p| < 1 => converge

  21. ash2326
    • 2 years ago
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    now you need to check which one lies in, out or on the unit circle. then you can classify the system

  22. ash2326
    • 2 years ago
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    @Callisto are you here?

  23. Callisto
    • 2 years ago
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    checking the magnitude? I did it. But the problem is how I can identify if the system oscillates. Sorry, I was on other page.

  24. ash2326
    • 2 years ago
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    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

  25. ash2326
    • 2 years ago
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    ???

  26. Callisto
    • 2 years ago
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    Hmm... I think the magnitude of the pole only tell us if the system is converging/diverging/remaining unchanged?!

  27. Callisto
    • 2 years ago
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    I think I understand how to analyze the system now, thanks :)

  28. ash2326
    • 2 years ago
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    welcome :P

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