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Callisto Group Title

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.

  • one year ago
  • one year ago

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  1. Callisto Group Title
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    @ash2326

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

    • one year ago
  3. Callisto Group Title
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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.

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

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

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

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

    • one year ago
  8. ash2326 Group Title
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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.

    • one year ago
  9. ash2326 Group Title
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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

    • one year ago
  10. ash2326 Group Title
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    Let me know if you have doubt anywhere

    • one year ago
  11. Callisto Group Title
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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!

    • one year ago
  12. ash2326 Group Title
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    so you undestood now ?

    • one year ago
  13. Callisto Group Title
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    No.

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

    • one year ago
  15. Callisto Group Title
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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"

    • one year ago
  16. Callisto Group Title
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    *in the lecture notes

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

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

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

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

    • one year ago
  21. ash2326 Group Title
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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

    • one year ago
  22. ash2326 Group Title
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    @Callisto are you here?

    • one year ago
  23. Callisto Group Title
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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.

    • one year ago
  24. ash2326 Group Title
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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

    • one year ago
  25. ash2326 Group Title
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    ???

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

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

    • one year ago
  28. ash2326 Group Title
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    welcome :P

    • one year ago
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