This was a question someone posed on the Mathematics stack exchange site that didn't receive a satisfactory answer. I thought it mighy be of interest to the folks here.
It is straightforward to prove that in a 2nd order system with a linear damping term, that a damping ratio of 1 (critical damping) mimimizes the "time to rest" of the system. Now consider nonlinear damping: is there a specific nonlinear function, solely of the first derivative of the state variable that among all nonlinear functions minimizes the time to rest of the system?
Maybe I didn't explain very well. I mean the damping function is some nonlinear function solely of V, like v^2, v^3...etc. A delta function is not a function of V, so wouldn't qualify for this problem.
I expect delta(t) is one such function (scaled and time-shifted appropriately). Well... quasi function...
Presumably, you're going to want a few more constraints. Like, continuity, the magnitude (of the function and N derivatives) must be bounded for bounded t, and some other properties, perhaps one-to-one (over the whole range for an odd function, or for t > 0 for an even function), perhaps along the lines of some familiar polynomial or exponential functions (MOV, junction diode, etc.). And then from this somewhat more limited, but more practical space, you can try to solve, select or optimize functions that are minimal-time within given bounds and such.
A look at a physical model might be more illuminating. What kind of system do you have, is it excited (to use an electrical analogy) by voltage or current, in series or in parallel? Would it be better to, say, shunt an MOV across something, or a current-limiting diode in series with something? Do you need feed-forward from the excitation (e.g., damping the snubber in a power converter), or should that be damped as well (e.g., ESD/lightning arrestor)?
Tim
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"bitrex" wrote in message
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> This was a question someone posed on the Mathematics stack
> exchange site that didn't receive a satisfactory answer. I
> thought it mighy be of interest to the folks here.
>
> It is straightforward to prove that in a 2nd order system with a
> linear damping term, that a damping ratio of 1 (critical
> damping) mimimizes the "time to rest" of the system. Now
> consider nonlinear damping: is there a specific nonlinear
> function, solely of the first derivative of the state variable
> that among all nonlinear functions minimizes the time to rest of
> the system?
> --
>
>
>
>
> ----Android NewsGroup Reader----
> http://www.piaohong.tk/newsgroup
Ok, I see what you mean. If we define the point of rest as the point where the system has minimal potential energy, then a delta function of V offset so the damping is infinite at the point of maximum velocity will bring it to rest right there.
Looking here:
formatting link
it seems that if the damping control force is restricted to be no more than the weight of the pendulum, then the optimum damping is going to be bang-bang, with the switching points of the step function dependent upon initial conditions.
Perhaps to make this particular problem interesting would be to find the optimum nonlinear damping function among all continuous, C^infinity functions of V; i.e. excluding distributions.
I assume you mean damping linear in velocity. linear in distance would be friction. (well "ideal" friction) As Tim said a delta function works. I step in front of a pendulum, as long as it's not too big, I can grab it and stop it dead... and of course if it is too big...
I'm not sure that one could be had even for a fixed step size, but you could probably make some significant improvements if you had access to the error that you're trying to reduce.
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Tim Wescott
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http://www.wescottdesign.com
Sure; if you bungee-jump, instead of bouncing around, just wait 'till you hit zero velocity and hug onto the nearest tree. You're at rest, you'll stay at rest.
is minimum time to rest also minimal lest mean square error? which allows for overshoot(s)
In a paper moving system I used non-linear damping to get that paper stopped fast. Consider such a high speed system that the overshoot would [if the gain were allowed to be constant] ring like a bell. but in a sense you rip the carpet out from under it just as it starts to get near rest. Not sure, but I think you can make that trajectory arbitrary. Sorry, didn't analyze too much, just saw the effect and used it until systme met spec, done. Just realized I did NOT control damping, but loop gain. ...never mind.
Now that I understand your question, probably empirically derive an answer using Matlab or octave or such. Then go back and prove it,
A delta function of V could do exactly what George said. It would even be a continuously differentiable function of V, with the disadvantage that the derivatives would be infinite around zero velocity.
In reality, we can't generate infinite forces, so the "delta function" George H had in mind is imperfect, but serves the purpose (if the pendulum isn't too heavy).
Hi bitrex, Well my answer was a bit tongue in cheek. I'm not much of a math guy... but if you'll allow me to speculate wildly.. (and how ya gonna stop me? BWA Ha haa..
Try a hyperbolic sine function of velocity, say an ode like y'' + sinh(y') + y = 0. The hyperbolic sine has the property that it starts out undamped, and then goes to exponential damping.
Upon rereading original, I thought you meant an apples to apples comparison: put in step voltage and obtain a step response, with minimum settling time. period. no playing games with the voltage. A double order system use critically damped. A double order system WITH variable damping, WHAT is the best damping vs time function to gain minimum settling?
I'm going to guess: Go from highly undamped to criticaly damped [may need just overdamped here, where the poles lie on top each other, may have to split in order to get good braking] with an exponential time waveform for the critically damping term. time waveform of the critically damping term matches the 'original' critically damped exponential in some way, probably
1/2, or sqrt, or something. So, should get to rest at least twice as fast. That's my guess.
The idea is at first get free ringing which really accelerates towards rest, but then as approach rest simply start damping like crazy, like applying brakes. Should get to rest position faster that way. Assume that qualifying is done using the same minimum rms error over time that is used to justify critical damping, right? Isn't that a single overshoot and then slide to rest.
If you think in terms of warping time [or should that be 'stretching/shrinking' time?], might gain some intuitive insights.
Anybody going to calculate it? I plan on using octave to 'test' the above approach.
Yep. I like hyperbolic functions ;-) ...Jim Thompson
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