GPS Hold: Integrator limiter evaluation

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Here I provide a little analysis and discourse of what I think is incorrect 
about the way how the I-term is limited.

I flew with settings P:0.015, I:0.005 and D:0.010.  I noticed that the quad is 
very nervous and overshoots significantly. With I:0.005 I could not keep the 
GPS Hold on, because it was swaying 15 meters side to side. Dangerous. I:0.001 
significantly reduced this condition, but it was now hovering around the 
setpoint in a circle of a certain radius between 1-2 meters.

The I-term is formed by multiplying the accumulated error with the I-gain. The 
accumulated error is limited to some value X. In my experiments, I recorded the 
gps_lon_I and gps_lat_I on the DataFlash, together with the gps_lon_D terms.

MOD:GPS Hold mode
GPS:20.00,1.00,102.00,6.00
GPS:80.00,-40.00,40.00,-43.00
GPS:120.00,-88.00,-19.00,-6.00
GPS:230.00,-130.00,69.00,6.00
GPS:300.00,-221.00,-40.00,-50.00
GPS:360.00,-269.00,-9.00,42.00
GPS:350.00,-249.00,-69.00,67.00
GPS:360.00,-168.00,20.00,62.00
GPS:470.00,-13.00,101.00,75.00
GPS:690.00,159.00,109.00,18.00
GPS:980.00,333.00,69.00,0.00
GPS:1200.00,427.00,-49.00,-79.00
GPS:1200.00,385.00,-99.00,-134.00

The first two terms are the integrated values, the last two terms are the 
D-terms. The P and D settings here are not ideal, so that must be taken into 
account. Weather was relatively quiet, so more aggressive P and lower D would 
have been better.

In gusty wind conditions however it is not unlikely that the quad temporarily 
reaches 1 meter difference, especially at some higher altitudes. The way it 
returns (or fails to return) is I believe mostly attributable to the I-term 
development. The I-term @ 1m difference saturates in 10 iterations the way the 
limiter is currently set to 1200. An I-gain of 0.005 then results in a fixed 
tilt angle of 6 degrees, which only reduces in size as soon as the quad 
overshoots. 0.001 results in a tilt angle of 1.2 degrees. These angles are only 
reduced as soon as the quad overshoots (and it can be observed and calculated 
that it will overshoot). I think this is too aggressive a setting, so there are 
two issues I dealt with.

The I-term is supposed to resolve bias and to speed up the return to a 
particular setpoint. ( bias occurs when for example there is wind and a certain 
angle is needed to remain in position. This will be downwind when only a P-term 
is used, but the I-term will slowly bring the quad back to the real intended 
position. With P-term decreasing there, it shows how P and I work together to 
remain roughly at the right spot all the time).

I intend to resolve this (so have not yet tested) by setting the limiter max 
value to 12,000 and pre-multiplying the I-gain with 0.1. This should provide a 
better settings range for the I-term that can be set by the configurator, 
should reduce overshoot and should increase stability.

Original issue reported on code.google.com by [email protected] on 18 Jun 2011 at 9:13