Given a Bode plot of the open loop system, I can figure out its gain and phase margins. However, I'm not sure how those can be used to determine whether the closed-loop system with unity feedback is asymptotically stable, marginally stable, or unstable just by looking at the Bode plot. It would be great to hear some suggestions without using a converted Nyquist plot.
Without knowing the number of unstable poles in the plant you can't tell, at least not without doing higher-falutin math than I like to do.
Even with a Nyquist plot you still have to have some idea about the stability of the plant (i.e. how many unstable poles) before you can estimate the stability of the closed-loop system.
Thanks for your answer Tim. I'm studying for an exam by doing past exams. One of the questions asked the candidate to state whether the system is asymptotically stable, marginally stable, or unstable supplied only with the system's magnitude and phase bode plots. How would I determine the unstable poles from a bode plot (or is that even possible)?
I _think_ you _may_ be able to tell if the open-loop transfer function had unstable poles, because I _do_ know that for a transfer function that has stable poles and zeros the phase response is the Hilbert transform of the log magnitude (or something like that).
So it makes sense that you could go the other way and determine the extent of the divergence of the system from stability.
In fact, in a way when you're looking at a Bode plot of a system you're looking at the Fourier transform of the system's impulse response; in theory you could work it out backward for an answer. So, as I said I kinda know it can be done, but the 'falutin' level of the math is higher than the hifalutin math that I usually do...
But that's not exactly a question that you could ask sitting down in an exam looking at a piece of paper, at least not in general. I _think_ (there's that qualification again!) that if there are no phase crossings with gain higher than 0dB that the system will be unconditionally stable (but I _will_ be checking!). Any system that's marginally stable will have a coincident phase and 0dB crossing. A real system that has only one phase crossing at a gain greater than 0dB is _probably_ unstable -- but any three-integrator system will have a phase crossing with a gain greater than 0dB and at least one more with a gain less than 0dB, so you can't look at just one phase crossing point.
Generally when I do exams I assume that the examiner is asking me questions that can be answered, unless it has been made explicit that an explanation of the question's deficiencies is acceptable as an answer. So I would assume that a system that meets all my "thinks" and "probablys" is what it is. The only thing that would really make me hesitate is if it's an exam that marks you down for wrong answers more than non-answers.
Why not look at the gain at where the phase lag is at 180 degrees? If the gain is greater than 1 it isn't stable. If it is less than 1 it is stable. I don't know what other info you can get from a bode plot. As Tim pointed out a proof for all transfer functions would be very difficult.
Peter Nachtwey
Marginal stability occurs when both the gain and phase margin are zero; i.e. the phase is exactly (or whatever passes for exactly in your case) 0 around the loop when the gain is exactly one. Hard to do unless you design a control loop to do it:) You probably should read the article:
In my experience, if a teacher gives you a Bode Plot, then you can assume there are no poles in the RHP. The interpretation of the bode plot changes when poles are in the RHP.
After you make the assumption of no RHP poles, then the above is the direction to go.
This is pretty much the short version of what it took me several paragraphs to state.
In practice it's rare to find a system with RHP poles -- but they do happen, and both integrators and high-Q resonances can confound Bode plot design as the gain goes to infinity. By far and away the most sure way of using Bode plots is to start with a working system and treat your tuning exercises as incremental changes.
Bode plot stability analysis is only applicable for minimum-phase systems. For the complete picture in the general cases, you need the full Nyquist Stability Theorem (together with winding numbers etc).
Fred.
Yup. And where do you get the winding numbers from? I could see myself using Bode/Nyquist plots to tune up a non-minimum-phase system, but if I needed to stabilize it initially I'd probably be using some sort of pole locating or state-space control, rather than fumbling around with Nyquist and Bode plots.
But you never know; the occasion may come up.
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