PSN-L Email List Message

Subject: Re: fine structure nonlinearity vs dithering
From: Brett Nordgren Brett3mr@.............
Date: Sat, 09 Feb 2008 14:04:05 -0500

At 11:29 PM 2/8/2008 -0500, you wrote:
>Hi Brett,
>     I would be quite happy to 'go public' if no one else objects?
>I'm uncomfortable with taking the approach that because there exist some 
>fairly small (I think) nonlinear effects, then no quantitative analysis 
>can be valid at all.  Although it's somewhat beyond
>my experience, I believe that feedback designers today routinely deal with 
>highly nonlinear, time-varying, and stochastic system variables and still 
>are able to obtain quite useful results.   If they couldn't there would be 
>a lot fewer airplanes out there and our cars wouldn't handle as well.
>     Read through the papers on Randall's Website?
>     Your car analogy misses the point. We are concerned mostly with 
> microscopic as opposed to macroscopic variations.
>     The mechanical properties of springs have a 'fine structure' of 
> discontinuous steps, a bit like ferro magnetic domains. This gives small 
> 'step function' variations and limits the ultimate performance of 
> seismometers, clocks, MEMS devices, etc. The macroscopic properties are 
> also not quiite linear and are time dependant. Hooke's Law is only an 
> approximation.

Certainly for the tiny MEMS stuff the world looks a lot different, and the 
effects may be an important consideration in a successful 
design.  Otherwise, I'm not yet sure I see why I should get out my 
microscope to look for the fine structure effects when there are plenty of 
other error terms which I think are quite a lot larger.  For example, in 
the STM-8 vertical the spring has a temperature sensitivity which amounts 
to about 200,000 nm / deg C.

>     How would you suggest incorporating step functions which are random 
> in time, sense and amplitude into the calculations / properties of a 
> feedback loop? The stochastic processes you mentioned?

I'm no nonlinear guru, but there are approaches out there that should be 
able to deal with it.  The easiest, is to prove that the effects are small 
enough to not affect the results and treat the system as linear.  Deep down 
that's what I really think is the situation, though am certainly not in a 
position to prove it.  It could be that the effects show up as some form of 
noise in the system, which is straight forward to analyze.

Many feedback systems today are digital, in which all the signals are 
quantized, so dealing with that sort of issue, in general, hasn't posed any 
insurmountable problems to the design community.  In fact they are doing 
things with digital feedback that could never have been considered 
otherwise, like making airplanes appear to be well behaved which without 
the feedback are inherently unstable and impossible to fly.

You could, on paper, start by treating the system as linear, then inject a 
signal of random step functions at the appropriate point in the feedback 
loop to simulate the situation and look at the effect at the output. That 
would probably be how I would first approach the analysis.



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