I'm working on electronics test system, where a certain measurement take an awfully long time. Basically its find a balance point of a circuit and checks the point is within the test limits. The circuit is a bridge which has an unknown impedance attached, then changing know values to balance out the unknown load.
The current method uses a SAR(successive approximation) method, system take an educated guess on where the point should be and try's points around it to find the balance point of the circuit.
The test takes an age to run as it take 10-20 attempts to find the correct values.
My idea is to use a triangulation technique to find a point that point. Basically my idea is to use 3 point, measure the magnatude of the error signal(error signal will be zero at the balance point, and increase the further from the balance point), and plot the point where the arc's cross. Sounds simple enough in theory.
However the measurements taken are not perfect(as this is the real world) and do not alway give a trivial solutions, the magantudes do not always meet up neatly to define a single point.
The question is does any have any suggestions on algorthyms that could be used to define the approximate region/crossing point of the 3 vectors even if the vectors do not actually meet.
The setup is a RC bridge. The value measured(Error Voltage) is a scalar. Bridge works by switching in R C combinations untill the error voltage is at a minimum. The voltage applied is AC.See Pic 2
Pic 1 shows the a graph with 3 measurement points(arbitary, but near the to limit circle). I can equate the error voltage to give me a scalar value on the plot around.
However as this a real world measurement, the results are never quite perfect, and the scalar values do not always a intersect, and if they do not intersect you cannot calculate thier angle(so calculate a vector to the intersect point), and so work out where the balance point for the bridge is.(see pic 3).
What I need to know is thier a mathimatical method that could define the region shaded orange in pic 4.
Does that clarify things at all?
I don't know if this is of any value for you, but...
In terrestrial navigation you take the bearings of at least three objects which should meet in one point: your position. That's the theory.
Normally you get a triangle. The estimated position is the triangle centroid. This point can be calculated easily, especially if you use vectors.
(My post has nothing to do with your technical problem, of course. It's only one possibility to get an estimated result. Maybe this is of some help for you)
triangle (called the cocked hat) made from the lines, but for reasons I
don't recall the vertex with the largest angle is quite a good approximation
to the centroid of the posterior distribution of position (I think the analysis
is in one of my reference books on OR in World War 2, which are all at work).
In fact if I recall correctly there is in fact only a 12.5% chance that the
actual position is inside the cocked hat!