## 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

@ash2326

2. ash2326

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

3. Callisto

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

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

5. Callisto

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

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

7. Callisto

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

8. ash2326

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

@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

Let me know if you have doubt anywhere

11. Callisto

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

so you undestood now ?

13. Callisto

No.

14. ash2326

15. Callisto

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

*in the lecture notes

17. ash2326

umm, where did calculus was used?

18. Callisto

He has NEVER used calculus in this course.

19. ash2326

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

20. Callisto

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

21. ash2326

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

22. ash2326

@Callisto are you here?

23. Callisto

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

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

???

26. Callisto

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

27. Callisto

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

28. ash2326

welcome :P