# How to stabilize this system?

I'm trying to understand system stability basics. Let us say I have a system $$G(s) = \frac{500}{s^2 - 500}$$

Since it has two poles in the right half s-plane, it is unstable.

How can I stabilize it? I have read that a simple pole-zero cancellation would mostly be unreliable.

• What happens if you add unity gain negative feedback? After you analyze that, try negative feedback with a gain of 2 in the feedback path. Jan 14, 2021 at 19:19
• Start here ... This is the expression for the closed-loop response. You want the poles of that expression to be stable. You can work it out algebraically. The root locus method helps develop intuition for cases such as yours. (note the wiki explanation for this is pretty confusing, there are some good youtube vids that might be better) Jan 14, 2021 at 19:36
• Note: you will add some other function in front of "your" G(s), let's call it F(s). This is the "compensator". It might simply be a fixed gain, if so call that K. When the notation of the wiki page says G(s) , it will really be F(s)G(s), or KG(s) if it is just a fixed gain. H(s) can simply be 1 for now, for simplicity. Most of the time, the "design task" is to figure out what F(s) should be to make the whole loop perform as desired. Jan 14, 2021 at 19:47

You can use a Routh–Hurwitz matrix to stabilize the system.

A Routh–Hurwitz matrix is a method for checking wheather or not a system is stable. It can also be used to stabilize the system by giving you a new denominator function for G(s).

According to this method, a system will be stable if the first column of the Routh–Hurwitz matrix contains only positive values.

You can stabilize the system by introducing a gain k and re-calculate the Routh–Hurwitz matrix to determine this k value that makes the system stable. In other words you can stabilize the system by inteoducing feed-back into it to conpensate for the unstable components of G(s).

Check these additional resources if you're still unsure about how to do this:

http://control.asu.edu/Classes/MAE318/318Lecture10.pdf