# Is there a correlation between conditionally stable systems and non-minimum phase systems?

I know bode plots cannot be used for non-minimum phase systems ( eg: delay, Boost converters).

1. Can we identify the closed loop conditional stability from open-loop bode plot response?
2. Conditional stability means stable for certain values of gain [Ref].

The word conditional comes from the fact that the gain has an upper/lower bounds to keep it this way, and crossing them makes the system unstable

Aren't all stable systems conditionally stable in that sense? If we change the gain, the bode plot changes and can become unstable.

I know bode plots cannot be used for non-minimum phase systems ( eg: delay, Boost converters).

Bode plots can, with a minimum of care, be used very profitably for designing control loops for non-minimum phase systems.

The easiest (but not the only) way is to start from a known-stable closed-loop configuration, and tune gains from there.

Can we identify the closed loop conditional stability from open-loop bode plot response?

As long as the transfer function is rational, and as long as the number of unstable zeros is known, yes. It's far far easier to do with a Nyquist plot, however -- but since the Nyquist plot uses the same input data as a Bode plot, this is not a big stretch.

The word conditional comes from the fact that the gain has an upper/lower bounds to keep it this way, and crossing them makes the system unstable.
Aren't all stable systems conditionally stable in that sense? If we change the gain, the bode plot changes and can become unstable.

In theory, there are continuous-time open-loop transfer functions for which any amount of gain can be chosen to result in a stable closed-loop transfer function. They are not physically realizable, though -- at some gain there will be an excess of poles that will drive the system to instability.

A system can be absolutely stable, i.e. all poles are in the left hand plane for all values of the system components. So no, it is not the case that all stable systems are conditionally stable. A given stable system can be a non-minimum phase system, or it can be a minimum phase system. This is a property of the stability of the inverse (i.e. locations of the zeros of the system transfer function) and is independent of the stability of the original system.

• My apologies, I misremembered the definition of open loop transfer function (assumed it meant just the plant transfer function). Edited Commented May 16, 2023 at 19:59