# unstable systems with positive phase/gain margin

Open loop Phase/Gain margins provide information on how much margin the closed loop system have until it becomes unstable. In other words, how much you can increase the effective loop gain and or effective phase lag until system becomes unstable.

Is it possible that a system shows positive phase/gain margins (at a range of frequency) but the overall system being unstable (potentially due to system behavior at other frequencies)?

• @Andyaka I removed the example request. Mar 15, 2021 at 14:30

As underlined by LvW, the Bode stability criterion applies to so-called minimum-phase functions. A minimum-phase function does not host pure delays and no right-half-plane zeroes (RHPZ) or right-half-plane poles (RHPP). If it does, it becomes a non-minimum-phase function and information obtained reading the Bode plot can be misleading. The only way is to resort to the Nyquist plot invoking the Cauchy's argument principle.

In the below example from my APEC 2019 seminar, I have purposely selected a wrong crossover frequency knowing the resonance of the $$\LC\$$ filter. If the crossover frequency is well below the resonance, you see that the Bode plot is perfect with good phase and gain margins. However, as you observe on the scope, there is some ringing in response to a step:

The problem is coming from the lack of gain at the resonance which is beyond crossover. Remember the sentence for a closed-loop system: no gain, no feedback. The below figure comes from my APEC 2009 seminar and shows that with the previous compensation, the loop is closed in dc up to 100 Hz. Beyond, it operates in ac open-loop conditions and cannot fight the $$\LC\$$ filter ringing:

As a conclusion, despite nice-looking values in a Bode plot, always apply engineering judgement by considering other approaches like Nyquist for the modulus margin for instance. A plot of the open-loop output impedance would have also probably revealed the abnormal peaking.

The stability of a linear systems depends solely on the position of all poles of the closed loop transfer function as a function of s.

If all poles of the closed loop transfer function are in the left part of the Gauss plane then the system is stable.

That means that you can feed the system with ANY sinus function at any frequency and all state or output variables will be stable.

To answer your question: If the system is linear and stable then there are no frequencies that might stimulate instability.

• I agree with "If the system is linear and stable then there are no frequencies that might stimulate instability." but my question is somewhat other way around: If all phase/gains margins are happy, could the overall system be unstable? Yes or No? Mar 15, 2021 at 13:57
• YES: if phase and gains margins are "happy", the linear system is stable at ANY frequency. Mar 15, 2021 at 14:49
• @Alborz, when a circuit is unstable, it will always and under all conditions show this instability (oscillations or saturation). As to your other question: When the loop gain crosses the negative-real axis more than once, the method of both gain and phase margin cannot be applied to check stability properties. In this case, you must apply the full Nyquist criterion. Such a circuit can have a loop gain showing a positive gain margin and the closed-loop can be unstable.
– LvW
Mar 15, 2021 at 15:41
• Thanks @LvW. Makes sense. Mar 15, 2021 at 18:46

It is possible -- and you directly indicate, although not quite elaborate on, the possible scenario in which instability emerges despite the system seemingly satisfies "open loop Phase/Gain margins" criterion. If the system features multiple gain crossover frequencies, and you make sure that the criterion holds at only one of the frequencies where the phase $$\\phi_{crossover} = -π\$$ (omitting other possibilities $$\-π - 2πn\$$), the gain margin at any other of these unattended frequencies may lead to instability of the overall system (the gain exceeds unity at those points of the frequency response diagram).

Another option may be that your understanding of the "open loop Phase/Gain margins" criterion embraces the multiple gain/phase crossover scenarios and yours is a "tricky" question. Indeed, a transfer function $$\G(s)\$$ in the loop feedback may lead to an infinite number of crossover points, with the $$\G(s)\$$ magnitude monotonically increasing and the phase angle monotonically decreasing with frequency and the notch frequency having a phase lag of less than $$\-\pi\$$ and the gain being less than unity. This scenario complicates application of your criterion and the criterion formulation must (and can) be revised to cover the "exceptional" scenarios.

What example would deserve your appreciation, depends on the depth of your understanding of stability criterion.

• Even the opposite is possible: The closed loop can be stable in spite of a loop gain function with a negative gain margin. This is because this simplified stabiliy criterion (using phase and gain margins) is applicable not for all functions,
– LvW
Mar 15, 2021 at 8:00