# Feedback stabilizes unstable systems?

Can feedback stabilize a unstable system?I guess so since we may get rid of any poles in the positive complex plane by feedback but I am not sure.

## 2 Answers

I'll try a somewhat "sloppy" answer

1. Systems without feedback are always stable
2. Ergo: any unstable system already has a feedback path.
3. We can add a second feedback path. The resulting feedback is simply the sum of the two feedback paths.
4. The resulting feedback can be smaller or larger (and/or have a different phase) than the original feedback by itself.

Can feedback stabilize a unstable system?

Yes, if the additional feedback path reduces the feedback of the already existing feedback path below the stability threshold.

This, of course, can all be expressed formally in terms of poles and zeros and stability criteria but the general idea is: adding feedback to an unstable system can indeed make it stable, but only if the new feedback cancels "enough" of the feedback that's already there.

• I like the sloppy answer; it adds an interesting intuition beyond simply ensuring all poles are in the left half plane (inside the unit circle for discrete) in the new closed loop system. Apr 2, 2023 at 21:43

One thing you might wanna look into are concepts we learn in Control Theory about Observability and Controllability. A prerequisite to these concepts is the State-variable linear system.

I believe that an unstable system must be both "completely observable" and "completely controllable" in order for it to be stabilized with feedback. If there is an unstable state inside the system that cannot be observed at the output or cannot be controlled from the input, yer sorta screwed.