Cheng Cosine science forum Guru Wannabe
Joined: 26 May 2005
Posts: 168

Posted: Sun Apr 16, 2006 4:21 am Post subject:
? thermal feedback



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*( x0x(t) ), then
the system becomes
dx/dt = a*( x0x(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*( x0x(t) )k*x0 so that we have
dx/dt = ( bk )*( x0x(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*( x0x(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 
