What's the interpretation of the margins when the curve of magnitude never crosses 0 dB or the phase curve never crosses the -180º? For example:

$$G(s) = \frac{s+20}{(s+1)(s+7)(s+50)}$$

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I know that this TF is stable looking it's pole/zero map. But how I can achieve this information just looking the bode plot, when the margins aren't explicity?


In theory, this means you can increase the gain of your controller by an infinite amount without losing stability. However, as one might notice, by increasing the gain the magnitude of the controller will at some point cross the 0dB line, which means that you will suddenly have a phase margin. See, gain margin and phase margin indicate how far you can tweak your controller until the feedback loop loses stability. The feedback loop (GC/(1+GC)) loses stability when GC = -1, because -1/(1-1) = -1/0. in terms of magnitude and phase, -1 has a magnitude of 1 (0db) and a phase of -180 (think of the unit circle). So, as you can see, there exist no frequency at which your open loop system will destabilize the closed loop system.

Checking for stability using only data (suppose you have done some frequency identification on a practical model) can be done in 3 ways:

  1. I just explained above using the gain and phase margin
  2. Unstable poles will have a different kind of phase compared to stable poles. this phase starts at -180 and moves towards -90 for high frequencies. So look for parts in the bode plot where the sloop of the magnitude decreases (for instance from -20db/dec to -40db/dec), but the phase increases. Those might indicate unstable poles.
  3. by checking the nyquist stability criterion using the nyquist plot. Assuming the open loop is stable (which is very likely as you have performed open loop measurements on it), if the nyquist plot encircles the -1 point, the closed loop system is unstable.

Also slightly unrelated note, even though having this system might look amazing as it appears to be very hard to destabilize, it must be noted the closed loop performance is terrible. try plotting the sensitivity function (1/(1+GC)). low gain means high noise / disturbance suppression.

Hope this clarifies your problem, if you have any questions please ask. if anyone spots a mistake or disagrees with my explanation, please let me know. I am doing most of this from my brain right now, but can get some references if desired.


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