CrossCoupling Measurements
Before measurig the crosscoupling effects, we put the plate back to our setup. Since the plate has magnets attached to it, we expect that it will affect the magnetic field produced by the coils. Therefore, a more realistic measurement of the crosscoupling effect that will resemble the realtime feedback control should take place in the presence of the plate. Of course, we want the plate to be still during the measurements, so that our Halleffect sensors are not affected by their displacement (if the plate is still, it will only affect the reading of the sensors at 0 frequency, but we are looking at the transfer function over the whole frequency range). To fix the plate, we pressed all motors against it, ignoring any small fluctuations.
VECTOR FITTING
In this configuration, we used Diagnostic Tools software to measure the transfer function beween ACs coils' output and ACs sensors' reading. Then, we used Vector Fitting to obtain a fitting transfer function with zeros, poles, and gain that resembles the behavior of the crosscoupling. In few words, Vector Fitting uses a guess function σ(s)=, whose zeros are the poles of the desired function
. Finding the zeros of σ(s) equals to finding the poles of the function and vector fitting manages to find a linear leastsquares fit model by trying values for the poles and solving for the linear coefficients. More information can be found at http://www.sintef.no/Projectweb/VECTFIT/Algorithm/.
We developed a code for the vector fitting and found the best model with 20 pairs of poles (40 poles, since they are complex conjugates) and about 10 iterations. At the end, we applied compensation feedback given by the modelling transfer function so as to cancel the crosscoupling effects.Below, one can see the crosscoupling measurements before (left figure) and after (right figure) applying the modelled function. We modelled the coupling transfer functions only for ACi_Sensor  ACi_coil (i=1,2,3) pairs, because crosscoupling between e.g. AC1 coil and AC2 sensor was very small (as is evident in the figures). Using this technique, we minimized coupling effects from roughly 10dB to roughly 70dB.
