3.2 STATE SPACE
The state of the world is represented using .ve variables:
.
h: altitude of the intruder relative to the own aircraft,

.
h˙0: vertical rate of the own aircraft,

.
h˙1: vertical rate of the intruder aircraft,

.
τ: time to closest approach (horizontally), and

.
sRA: the state of the resolution advisory.

The variable τ is the time to horizontal closest approach, which is the same as the time to zero horizontal separation in this model because the encounter is assumed head-on with no turning. Instead of τ, the horizontal range could have been used, but τ allows the MDP to be formulated independently of the actual closure rate, assumed to be constant.
The variable sRA is discrete. It allows the system to track which advisory is currently active (if
any)
and
the
time
remaining
until
the
pilot
responds.
For
the
advisories
in
Section
3.1,
sRA can assume one of 23 di.erent values. The state COC indicates that there is no resolution advisory currently active. If DES1500 is issued, for example, the state transitions to DES1500-4, indicating that there are four more seconds until the pilot will execute the descend. If the descend advisory is continued,
the
state
will
countdown
deterministically,
as
in
Fig.
2.
The
pilot
begins
executing
the
descent at DES1500-0. The transition model for CL1500 is similar.

Figure 2. Advisory transition model when DES1500 is issued.
Fig.
3
shows
the
transition
model
when
a
subsequent
advisory,
SDES1500,
is
issued.
The
SDES1500 advisory can be issued following any climb advisory or SDES2500. It cannot be issued following COC or DES1500. When SDES1500 is issued for the .rst time, the state transitions to SDES1500-2, indicating two seconds until execution. If SDES1500 is continued, the state counts

Figure 3. Advisory transition model when SDES1500 is issued.
down until reaching SDES1500-0, at which point the advisory is followed. The transition models for SCL1500, SDES2500, and SCL2500 are similar. When the advisory is terminated, the state transitions to COC immediately, discontinuing any response of the pilot to advisories.
The state space is discretized using a multidimensional grid. This report uses the discretiza-tion
shown
in
Table
2,
which
results
in
8.7
million
grid
vertices
that
correspond
to
discrete
states.
The discretization can be made .ner to improve the quality of the discrete model approximation, but it would be at the expense of additional computation and storage to calculate and store the expected cost table.
TABLE 2
Discretization scheme
Variable Grid Edges

.1000, .900,..., 1000ft .2500, .2250,..., 2500ft/min .2500, .2250,..., 2500ft/min 0, 1,..., 40s
sRA N/A (already discrete)

3.3 DYNAMIC MODEL

The dynamics of the aircraft involved in the encounter are governed by sequences of accelerations. These accelerations are used to update the vertical rates of the aircraft and, consequently, their positions. The maximum vertical rate of both aircraft is assumed to be ±2500 ft/min, although this is easily changed at the expense of a larger state space. In a real system, the limits can be adjusted to meet the performance constraints for the particular aircraft.
When pilots are not following a resolution advisory due to the response delay or the absence of an advisory, the aircraft follow a white-noise acceleration model. At each step, the aircraft
　
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