Re: Optimization Halp!
From: BEC Charlie (beccharlie_at_krampus.phys.uconn.edu)
Date: 03/27/05
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Date: Sun, 27 Mar 2005 10:46:01 -0500
Hi Rishi,
was reading your question on sunday morning, and was wandering why do you
have as an optimization constraint the number of flippings of the switch.
If, say, the tanks are on left and right wing of an airplane, I would
probably want to minimize magnitude of oscillations of the center of
gravity of my airplane, or something to that effect.
Your intuition seems right to me as well. Given the local status of fuel
in the tanks, the solution which will drain the most fuel out of a tank
with the least number of switching is the one you described: let fuel be
taken out of one tank until the c-o-g moves to the furthest position allowed
in the direction of the opposite tank.
This said, let's go back to my first paragraph. What if you add the
constraint on switching so that the timings between the switching are
equal. This would produce either a wobble in the motion of the c-o-g with known
frequency, which could then be compensated for by the plane control
system, or would require a simple controlled switch (i.e., with a simple
timing device rather than with complex feedback structure as your original
solution seem to be).
Regards,
BEC Charlie
On Mon, 21 Mar 2005 10:22:26 -0800, Rishi Tandon wrote:
> This is a quite interesting (though somewhat tedious to explain)
> problem:
>
> The System:
>
> ...[Tank 1]............(-a)........(0).........(a)............[Tank
> 2]...
>
> We have two fuel tanks, initially each with point mass (m1) at a
> distance (x) on either side of the origin (0). A point mass (m0) is at
> the origin.
>
> Initially, the centre of gravity (c.g.) is at the origin (0).
>
> (-a) and (+a) are limits to the c.g. travel in the 1-D problem.
>
> I wish to consume the fuel in the two tanks, taking fuel out from only
> one tank at a time without violating the constraints on c.g. travel.
>
> The Problem:
>
> When I change to taking fuel from one tank to the other, i call it a
> "switch". When a switch occurs, the position of the c.g. is anywhere
> within the permitted 1-d travel region.
>
> Now to empty out the tanks, there will be a sequence of such switches
> at points lying in the constraint region. *I wish to find that sequence
> which requires the MINIMUM number of switches between tanks to EMPTY
> out the fuel in the tanks!*
>
> How do I go about finding this optimal sequence?
>
> Intuition tells us that that sequence which always switches at the
> extrema will yield the minimum number of switches needed. How do I
> prove or disprove this?
>
> Let me define this problem better:
>
> 1. R(n) is a value between 1 and -1.
> 2. a*R(n) gives us the Nth switching point on the 1-d constraint
> segment. (a) is defined above.
> 3. R(0)=0 the origin.
> 4. R(1),R(2),R(3),....R(k) is a sequence of (k) steps (say) that is
> required to remove empty the tanks.
> 5. What such sequence needs the minimum number of steps (k) to empty
> the tanks? And is the same as the 1,-1,1,-1,1,-1,...... sequence that
> would result if the extrema were taken?
>
> More definitions:
> 1. Dm(n) is the mass removed from the appropriate tank at when the c.g.
> moves from a*R(n-1) to a*R(n).
> 2. Thus successive masses removed are... Dm(1),Dm(2),Dm(3).....Dm(k).
> 3. Is the problem of minimizing (k) <the no. of steps> the same as
> maximizing the sum of these masses removed for (k) steps? If so, what
> sequence of R(1),R(2)...etc does this resilt in?
>
>
> *******************Appendix*********************
> More detail. A reduction formula was derived....
>
> R(n)-R(n-1) [x/a-R(1)] [x/a+R(2)] [x/a-R(3)]
> [x/a-R(n-2)]
> Dm(n)=-----------------------x----------x----------x----------x..------------
> [x/a +R(n)][x/a-R(n-1) [x/a+R(1)] [x/a-R(2)] [x/a+R(3)]
> [x/a+R(n-2)]
>
> Here, we assume that
> 1. (n) is an odd number. If you want it otherwise, reverse the signs of
> R(n,n-1,n-2) in equation above.
> 2. This formulation assumes that Fuel is initially taken from Tank 1
> such that c.g. moves towards (+a).
> ******************************************************
>
> Question: How should I *approach* this problem to prove/disprove that
> the extreme R(n) sequence is optimal in terms of no. of switches (k)?
>
> Thank you for persevering enough to read this. Any further quries may
> be addressed to tandon.rishi [AT] gmail DOT com.
>
> Any help or directions regarding this will be greatly appreciated!
>
> Rishi
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