Looking at the totality of the evidence from the four graphics presented in the BEA report it seems to me that the situation is as follows:
The HAP mode of the EFCS has no knowledge of what the engines are doing. Indeed, with thrust control switched to manual no part of the EFCS uses thrust information. Consequently any aircraft motions produced by thrust changes are treated in exactly the same way as perturbations coming from any other “outside” source such as atmospheric disturbances.
The HAP mode contains logic that in effect says: “Do not allow pitch to increase if the speed is falling”. Introduced to ensure trajectory stability in the phugoid mode, it has its own logic – if flying at or near alphamax with a falling airspeed the last thing one wants to do is allow changes that might take the aircraft over that limit and nearer to alphastall.
The “job” the EFCS was being asked to do has been stated as “Provide the commanded alphamax”, but although at first reading that seems a straightforward objective, it is in fact simplistic. A more accurate task description would be “Provide the commanded alphamax in a controlled manner so as to minimise the risk and potential magnitude of overshoots above alphamax”
The details vary a little depending on the timing of thrust and sidestick applications, but basically when starting from a condition of level flight in the HAP operative AOA range with idle thrust and a reducing airspeed and applying maximum thrust and full back stick the sequence of events would be as follows:
Because the speed is initially reducing, the HAP logic requires that a down elevator signal be added to cancel the pilot’s nose up command. At this point the thrust is low and increasing only slowly, so the pitch remains constant whilst the speed builds up slowly.
As soon as the thrust has increased enough to arrest the deceleration, the HAP logic changes to allow the aircraft to pitch up. By this time though,the thrust has increased enough to provide a noticeable nose up pitching moment so the pilot’s demand can be satisfied without much change to the elevator command. At the end of this phase the speed is still building up slowly and the aircraft has just started to climb. [Note**]
The engines are by now accelerating quite rapidly, being on the steep part of the “S” curve, and if left uncorrected the associated pitching moments would increase the pitch rate to a level where there would be a very real risk of overshooting alphamax. The HAP therefore applies corrective elevator (actually a reduction of ‘up’ elevator) as shown on the graphs.
The aircraft, at this point, is pitching up towards alphamax and starting to climb more steeply.
As the engines near TOGA power their acceleration tails off along with any increase in pitching moment. The HAP elevator command depends on the details of the pitch rate and the proximity to alphamax at this point. If it seems that the pitch rate could be safely increased then the HAP might apply a small nose up correction, but if the pitch rate means that the aircraft is approaching alphamax too rapidly it might need to do the reverse to avoid an overshoot.
It is quite difficult to see exactly what is driving the elevator at this time because it is trying to control the difference between pitch and flight path angle, but we don’t have direct information on either alpha or FPA from the simulator records, although it could in principle be calculated from the data available.
Under this line of reasoning the only substantive difference between any of the four “flights” would be that F-JFKC’s was truncated (literally!)
[Note**] It was at about this point in the sequence that F- GFKC entered the trees.