Assessment and Future Directions of Nonlinear Model Predictive ...

NMPC for Complex Stochastic Systems 277A difference from usual control engineering is that the decision variables (controlinputs) do not follow arbitrary trajectories in time. Particularly in the terminalmanoeuvering area, there is a finite number of possible standard maneouvres,such as ‘descend’, ‘circle while descending’, ‘turn right through 90 degrees’, etc,and an expected sequence of these maneouvres. The available decisions are (realvalued)parameters of these manoeuvres, such as the initiation time, diameterof the circle, rate of descent, etc.In both the scenarios defined above, there are safety-related constraints (minimumhorizontal and vertical separation between aircraft), legislative constraints(eg not entering certain urban zones at night), and desirable objectives (minimumdisruption of scheduled flight-plan, landing aircraft X as soon as possible,etc). The objectives are liable to change with time — for example, an aircraftmight enter the TMA which has already been delayed en route, and should thereforebe given priority over aircraft which are already in the area. It is thereforeimportant to have a solution methodology that can accept a wide range of performancefunctions to be maximised. This is not to suggest that formulationsof such performance functions should be done in real time; a set of performancefunctions should be devised for a set of scenarios, and their suitability evaluatedcarefully before being put into use. Then the choice of a suitable function fromthe set could be made in real time.The time available for making a decision in the ATC context is of the order ofa few minutes. (Emergency manoeuvres required for avoiding imminent collisionrequire much faster decisions of course, but these are handled locally by theaffected aircraft, without involving ATC.) Our current implementation of theMCMC approach is much slower than this, even when only two decision variablesare involved. A speed-up of about one order of magnitude should result frommore efficient coding at an elementary level (much of the time is currently lostby very inefficient hand-overs between the Java-based simulator and the MatlabbasedMCMC algorithm), but more sophisticated algorithm development will berequired to obtain some further speed-up.A typical scenario involves two aircraft, one of which is following a fixed setof instructions, while the other will fly a straight course to a waypoint whichneeds to be selected, after which it will fly to a fixed waypoint. The performanceobjective is that they should arrive at a final waypoint (glide-slope capture)separated in time by 300 sec; this is represented by the objective functionexp{−a|(|T 1 − T 2 |) − 300|}, whereT 1 and T 2 are the arrival times of the twoaircraft at the final waypoint, and a>0. The constraint is that their minimumseparation should be 5 nautical miles horizontally and 1000 feet verticallyat all times, and the probability of violating this constraint should be smallerthan ɛ =0.1. As formulated, this simple scenario is a finite-time problem, whichmay be re-solved as time proceeds, but it is a ‘shrinking-horizon’ rather thana receding-horizon scenario. This problem was solved by initially choosing theinstrumental distribution g(ω) uniform over the possible parameter space (arectangle in R 2 ), and J = 10. This gave two rather large ‘clouds’ of possiblesolutions, but all of them safe ones. Another instrumental distribution g(ω) was