Chaos In Control Loops

[rickman]

I was simulating a control loop and noticed that it was very regular in
the small perturbations that occur. I zoomed the scale out and the
pattern shrunk a bit but as it was zoomed out more another pattern
started to emerge, similar to the initial, but at a slower rate.
Continuing to zoom out to wider ranges of time more and more patterns
showed up, all somewhat familiar, but none quite the same.

I believe that is the sort of thing that is predicted by Chaos Theory. I
wonder what the fractal number of this data set is?

 

[Tim Williams]

It can be very difficult to simulate long-term values accurately (setting
smaller tolerances, usually RELTOL first, will cause it to make finer
timesteps -- note the simulation will slow down proportionally).

I once tried simulating a theremin (on the transistor level); it works,
but the lowest reasonable beat frequency I can seem to simulate is on the
order of 2kHz (out of ~1MHz); less and the phase shift slowly rolls around
without a well defined waveform (if it were phase locking due to
injection, the waveform would be humpy).

Some circuits are, by nature, chaotic. You can, of course, implement the
logistic function with samplers and multipliers; anything from the van der
Pol oscillator to Lorentz attractors and beyond can be built from analog
and calculus function blocks.

People don't always appreciate that peak-current-mode SMPS controllers are
inherently chaotic: the graph of duty cycle(s) vs. setpoint has precisely
the same form as varying the "r" parameter in the logistic equation. That
is, it exhibits a logistic map behavior, and limit cycles. This can be
improved, but cannot be eliminated, with slope compensation and feedback.
Such circuits whine and hiss when driven into overload.

Tim