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? thermal feedback
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Cheng Cosine
science forum Guru Wannabe


Joined: 26 May 2005
Posts: 168

PostPosted: Sun Apr 16, 2006 4:21 am    Post subject: ? thermal feedback Reply with quote

In Luenberger's Intro to dynamics sys, there is an example describing how

a simple feedback system works. For a system governed by

dx/dt = u(t),

suppose u(t) is designed so that u(t) = a*( x0-x(t) ), then

the system becomes

dx/dt = a*( x0-x(t) ), which will approach to x0 as time approaches
infinite.

This somehow mimics a thermal feedback regulator at home. But in real work,

temperature decays after a constant heating, so a more realistic system eqn
should be

dx/dt = -k*x+u(t), k>0 and u(t) = function of x(t) to send feedback to
original system.

A simplest one is just: dx/dt = -k*x+b*( x0-x(t) )-k*x0 so that we have

dx/dt = ( b-k )*( x0-x(t) ) and then system approaches to x0 as time
approaches to inifnite.

But how does one have a feedback u(t) that moves the system to x0 in
shortest time, instead

of at inifite time?

Furthermore, how about a cheap thermal regulator works like this: heat
up for a fixed time

period (tH) and then wait for a fixed time (tC). The amount of heat is
adjusted according to

the deviation of x(t) to desired level x0, say, u(t) = b*( x0-x(t) ). Will
this appraoches to

desired x0 finally? How to estimate the time required for x(t) to reach x0
in, say, 99%?

Thanks,
by Cheng Cosine
Apr/16/2k6 NC
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