I have been using both (Bode and Nyquist diagrams) to design simple digital feedback controllers for low-order SISO systems P(z), by analysing the loop transfer function L(z) = P(z)*C(z). But I am getting frustrated by the fact that both graphical representations, in some cases, may suggest that I have derived a nice controller C(z), for a good closed loop transfer function T(z) = P(z)*C(z)/{1+P(z)*C(z)}, as indicated by:
- a substantial gain margin,
- a phase margin of 60 deg,
- no encirclements of -1 in the Nyquist chart,
- good high frequency attenuation,
- large low-frequency gain,
- a near-unity peak sensitivity S(z) = 1/{1+L(z)}, etc
however, I may still end up with closed-loop poles in T(z) (slightly) outside the unit circle, which, as far as I can tell, are not apparent in these diagrams!
Fortunately, Matlab's margin() command picks up on this and issues a closed-loop instability warning, even though the returned gain and phase margins indicate stability.
Is there any point using Bode and Nyquist diagrams if closed loop instability can be concealed like this?
Or am I doing something wrong?
Could there be some evidence of instability in the Nyquist diagram, that I am missing?
... could this be interpreted as there being 'last minute' encirclements of -1, right down at the omega->0 limit, that are beyond machine precision and not rendered in graphical (Nyquist) representations?
If you know the number of unstable poles of loop gain (C(z)) and you have the polar diagram of the loop gain as well, you can always determine the stability of the closed loop system. There is no exception. But in the case that C(z) has some poles on the unit circle (at z=1 or anywhere else of that), you should pay attention that polar diagram would have some circles with infinite radius that Matlab does not plot them. If you consider these possible circles, Nyquist criterion always works.
... OK, I think the problem is that there was a plant pole cancelled by a controller zero outside the unit circle. So there should be a net of zero encirclements around -1, as I observed. But imperfect pole/zero cancellation (due to finite machine precision) is still sufficient to cause closed-loop instability.
Possible answer ...
Yes, they are very useful, if you already know that the closed loop system is stable.
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