Question

I'm confused as to how I'm supposed to work out whether or not a system is BIBO stable.
If I have a differential equation in terms of my outputs, it seems easy enough. For example, I know how to determine BIBO stability if I have something like (D^2 _ 4D + 8)y(t) = (D-3)x(t).
My problem is if I just have something like y(t) = Dx(t), or some other system where y(t) doesn't have derivatives. How do I go about solving those?

Answer 1

You just have to worry about if the input and output are bounded or not

Just a quick examples would be \(\large y(t) = \frac{1}{x(t)}\)

If we let \(\large |x(t)| < R\) then we know \(\large |y(t)| > |\frac{1}{R}|\)

This system if NOT stable because of the fact if x(t) is bounded and small, y(t) can still be extremely large meaning unbounded

Basically meaning...for every bounded input, we get a bounded output

Answer 2

If you just take an arbitrary system \(\large y(t) = x(t)\) it's easy to see that \(\large |y(t)|\) will never be smaller or larger than \(\large |x(t)|\)

so when you bound x(t) as an input you get a bounded y(t) as an output

Hence stable

Answer 3

So if we have y(t) = Dx(t), and we say that |x(t)| < R (what does R stand for by the way? something Real? or a Region?) then we have that |y(t)| < D|x(t)| which means the system is stable?
Assuming this is correct, is this a systematic method I can always apply? I'm a little confused as to the role of derivatives in this (D denotes d/dt)

Answer 4

I found an explanation but it's not totally clear to me.
In that scenario, y(t) = Dx(t-1), which is just a time shift of the question I have. The system is apparently not stable and the solution uses a square wave input as an example - the derivative of a square wave is infinity at its bounds, which means y(t) is unbounded at those points, and thus the system is BIBO unstable.
Am I right to assume that the time shift has no significance? And do I always have to employ a sort of trial-and-error approach when working these out, by feeding example inputs and seeing if they work?

Thanks for your help so far!