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with ATC in terminal and approach sectors and Section 4.2 provides a brief overview of the simulator used to carry out the experiments. Sections 4.3 and 4.4 present results on benchmark problems in terminal and approach sectors respectively. Conclusions and future objectives are discussed in Section 5. For the readers convenience the acronyms used in the paper are summarized in an appendix.
2 Conflict resolution with an expected value criterion
We formulate conflict resolution as a constrained optimization problem. Given a set of aircraft involved in a conflict, the conflict resolution maneuver is determined by a parameter ω which defines the nominal paths of the aircraft. From the point of view of the ATC, the execution of the maneuver is affected by uncertainty, due to wind, imprecise knowledge of aircraft parameters (e.g. mass) and Flight Management System (FMS) settings, etc. Therefore, the sequence of actual positions of the aircraft (for example, the sequence of positions observed by ATC every 6 seconds, which is a typical time interval between two successive radar sweeps) during the resolution maneuver is, a-priori of its execution, a random variable, denoted by X. A conflict is defined as the event that two aircraft get too close during the execution of the maneuver. The goal is to select ω to maximize the expected value of some measure of performance associated to the execution of the resolution maneuver, while ensuring a small probability of conflict. In this section we introduce the formulation of this problem in a general framework.
Let X be a random variable whose distribution depends on some parameter ω. The distribution of X is denoted by pd (x) with x ∀ X. The set of all possible values of ω is denoted by n. We assume that a constraint on the random variable X is given in terms of a feasible set Xf ç X. We say that a realization x, of random variable X, violates the constraint if x ∼∀ Xf . The probability of satisfying the constraint for a given ω is denoted by P(ω)
P(ω) =
pd(x)dx .
xιXf
¯
The probability of violating the constraint is denoted by P(ω)=1 − P(ω).
For a realization x ∀ Xf we assume that we are given some definition of performance of x. In general performance can depend also on the value of ω, therefore performance is measured by a function perf(·, ·): n × Xf ∈ [0, 1]. The expected performance for a given ω ∀ n is denoted by Perf(ω), where
ι
Perf(ω) =perf(ωd (x)dx .
,x)p
xXf
Ideally one would like to select ω to maximize the performance, subject to a bound ¯
on the probability of constraint satisfaction. Given a bound P ∀ [0, 1], this corresponds to solving the constrained optimization problem
Perfmax | ¯= sup Perf(ω) (1)
P
dιo
¯¯
subject to P(ω) < P. (2)
　
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