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Callisto

  • one year 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
    • one year ago
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    @ash2326

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

  3. Callisto
    • one year 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
    • one year ago
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    I meant \[\frac{Y}{X}=...\]

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

  8. ash2326
    • one year 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
    • one year 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
    • one year ago
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    Let me know if you have doubt anywhere

  11. Callisto
    • one year 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
    • one year ago
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    so you undestood now ?

  13. Callisto
    • one year ago
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    No.

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

  15. Callisto
    • one year 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
    • one year ago
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    *in the lecture notes

  17. ash2326
    • one year ago
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    umm, where did calculus was used?

  18. Callisto
    • one year ago
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    He has NEVER used calculus in this course.

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

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

  21. ash2326
    • one year 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
    • one year ago
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    @Callisto are you here?

  23. Callisto
    • one year 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
    • one year 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
    • one year ago
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    ???

  26. Callisto
    • one year 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
    • one year ago
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    I think I understand how to analyze the system now, thanks :)

  28. ash2326
    • one year ago
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    welcome :P

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