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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

@ash2326

2. ash2326 Group Title

First of all find the closed loop transfer function of the system

3. Callisto Group Title

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 Group Title

I meant $\frac{Y}{X}=...$

5. Callisto Group Title

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 Group Title

You'd need to find the poles of the system, sorry I was dealing in the s- domain

7. Callisto Group Title

Poles are at $\frac{k\pm\sqrt{k^2-4b}}{2}$

8. ash2326 Group Title

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 Group Title

@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 Group Title

Let me know if you have doubt anywhere

11. Callisto Group Title

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 Group Title

so you undestood now ?

13. Callisto Group Title

No.

14. ash2326 Group Title

oops, did you ask this doubt to your teacher?

15. Callisto Group Title

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 Group Title

*in the lecture notes

17. ash2326 Group Title

umm, where did calculus was used?

18. Callisto Group Title

He has NEVER used calculus in this course.

19. ash2326 Group Title

But the solution requires just solving the quadratics. then depending on the poles, we classify the system

20. Callisto Group Title

p = pole p = 1 => remains (unchanged) |p| >1 => diverge |p| < 1 => converge

21. ash2326 Group Title

now you need to check which one lies in, out or on the unit circle. then you can classify the system

22. ash2326 Group Title

@Callisto are you here?

23. Callisto Group Title

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 Group Title

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 Group Title

???

26. Callisto Group Title

Hmm... I think the magnitude of the pole only tell us if the system is converging/diverging/remaining unchanged?!

27. Callisto Group Title

I think I understand how to analyze the system now, thanks :)

28. ash2326 Group Title

welcome :P