BME 343 - BIBO Stability

Bibo stability

If the input signal x(t) is bounded, then the output signal y(t) is bounded. An unstable system would be the opposite: even if the you know the bound of the input, the output would be unbounded.

This might seem simple until we get to actual problems

Example 1

The question is always, if the input x(t) is bounded, is the response y(t) also bounded?

Let’s imagine you’re shooting a basketball in a fenced basketball court with assumed bounds for y(t). Your strength is the input x(t) which is bounded from -M to M. How do you prove that the y(t) is actually bounded?

I like to break up the system into its components, in this case the exponent and the integral.

Let’s first look at the exponent:

We know that the curve of something to the negative power is bounded between 1 and 0.

So, the exponent is bounded.

How to we tackle the integral? Pause to think…

We can change the integral to look like this:

• We know that X(t) is at most M, which makes it a constant

• We also know that X(0) is a constant in the bounds of x(t) because t > 0

• Then the difference of these constant is a constant, making it bounded

Since we know that the exponent is bounded and the integral is bounded, the response y(t) is bounded.

No matter how hard you throw the ball ( x(t) ), it will always be in bound ( y(t) ).

So, this is a BIBO stable system.

Example 2

Lets make x(t) bounded from -M to M again.

Let’s also separate the components again: 9 and x(t) and ramp(t)

• 9 is a constant so it is bounded

• x(t) is bounded bc we declared it as such

• HOWEVER, ramp(t) is not bounded because it infinitely grows

Even if x(t) is always between -M and M, ramp(t) will always grow so the response can not be bounded.

How would that look like?

Lets go back to the basketball example:

• The input would be your strength again x(t)

• The output would still be the ground distance y(t)

• Now you are playing in a hurricane, so your system is the problem 2 equation

• You thought that y(t) was bounded, but it turns out it was not! Even though your strength is bounded x(t), the hurricane wind picked up the ball (ramp(t)) over the misplaced bounded fence and hit your nice old grandma!!!

So, this is a BIBO unstable system.

Example 3

However, if the system was like this, then ramp(x(t)) would be bounded because the highest it can grow is M. Now your grandma is saved!

You can also use integration to prove if something is bounded, because the area under the curve of a bounded signal will always be less than infinity. However, I like this method because you can see where the bounded and unbounded aspect is.