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
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
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
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).
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.
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:
Polytechforum.com is a website by engineers for engineers. It is not affiliated with any of manufacturers or vendors discussed here.
All logos and trade names are the property of their respective owners.