Settled comfortably into your seats, you wait impatiently for takeoff, though not without a few butterflies in your stomach. You marvel at the technological feat that will allow this machine, weighing several hundred tonnes, to stay aloft as if by magic and travel at over 800 km/h. But you probably have little idea of the immensely sophisticated calculations that the aircraft manufacturer, the airline and the pilot had to perform before boarding even began…
Several problems must be solved before takeoff. One of the crew's first concerns is, of course, to load exactly the right amount of fuel into the tanks to complete the mission. Too much makes the aircraft unnecessarily heavy, increasing its kerosene consumption. Too little—and the consequences are easy to imagine.
Another important consideration is ensuring that the aircraft can actually complete its mission: in particular, that the runway is long enough for takeoff; that any obstacles encountered along the way—the building at the end of the runway, the mountain that must somehow be cleared…—can be safely overflown; and that an accessible airport will always be available for diversion should a failure occur en route. Aeronautical regulations therefore lay down a whole range of requirements relating to aircraft performance, which is itself intimately linked to the trajectory.
Calculating the trajectory -----------------------
To answer all these questions, we must first calculate the trajectory, understood here in the broad sense of "the time evolution of parameters describing aircraft performance." These parameters include its position (latitude δ, longitude λ, altitude z), the aircraft's mass m (which decreases during the flight as fuel is consumed), its speed v and its thrust T. Some are set by the pilot, who keeps altitude and speed constant during cruise, whereas during the climb the pilot instead sets thrust and speed (during the descent, the flight-path angle and thrust are fixed). Parameters whose values are set directly by the pilot are called control variables or control inputs (denoted by u); the others are state variables (denoted by x).