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Chaos In Control Loops

Discussion in 'Electronic Design' started by rickman, Jan 31, 2013.

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  1. rickman

    rickman Guest

    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?
  2. Tim Williams

    Tim Williams Guest

    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.

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