Figure 9. Parametric robustness with varying environment and constant logic optimized at σ¨ = 8ft/s2.
h
Inthesecondset ofexperiments, themodelednoisewasvariedfrom0ft/s2 to 15ft/s2, but the environment noise was kept .xed at 8ft/s2.
The
performance
is
summarized
in
Fig.
10.
Increasing
the
noise
modeled
by
the
logic
resulted
in
more
alerts
and
fewer
NMACs.
As
in
Fig.
9,
the
number
of NMACs is nearly minimized when the modeled noise matches the environment. Divergence between the environment dynamics and the modeled dynamics slightly degrades performance. The performance of TCAS is constant because it is invariant to the modeled noise parameter.

4.4 ROBUSTNESS TO MODEL STRUCTURE VARIATION
Di.erences between the environment and the model used by the logic may be more signi.cant than simply variations in the process noise parameter. To determine the robustness of the system to structural inaccuracies in the modeled dynamics, both TCAS and the DP logic were evaluated on one
million
encounters
generated
by
the
correlated
encounter
model.
Figure
11
shows
the
DP
logic
performance for two di.erent costs of alerting.
The logic derived with an alert cost of 0.01 resulted in fewer NMACs than TCAS but a comparable number of alerts. When the alert cost was increased to 0.1, the DP logic was still more successful than TCAS in preventing NMACs while alerting much less frequently. The two DP logic curves show how choosing the alert cost can trade o. safety for alert rate. These results indicate that even if the environment model is very di.erent from the model used by DP to optimize the logic, the DP logic can still perform better than TCAS.

4.5 ROBUST DYNAMIC PROGRAMMING
Robust dynamic programming (RDP) has been proposed as a technique for making the policies obtained through dynamic programming less sensitive to the particular choice of transition model
×10.3

1
0.8
Pr(Alert)

Pr(NMAC)
0.6
0.4
1
0.2
0
0
0 5 1015 0 5 1015
Modeled noise (ft/s2) Modeled noise (ft/s2)
(a) Safety performance. (b) Alert performance.
Figure 10. Parametric robustness with varying logic and constant environment using σ¨ = 8ft/s2.
h
×10.2

1
0.8
1.5

Pr(NMAC)
0.6
1
0.4
0.5
0.2
0
0
0 5 1015 0 5 1015
Modeled noise (ft/s2) Modeled noise (ft/s2)
(a) Safety performance. (b) Alert performance.
Figure 11. Model structure robustness. DP alert cost is in parentheses.
×10.3

1
0.8
6
Pr(Alert)

Pr(NMAC)
4
0.6
0.4
2
0.2
00  5  10  15  00  5  10  15 
Modeled noise (ft/s2)  Modeled noise (ft/s2) 
(a) Safety performance.  (b) Alert performance. 

Figure 12. Robust logic performance on white-noise encounter model. DP alert cost is in parentheses.
[42].
If
the
set
of
transition
models
is
given
by
T1,...,TN , then the value iteration algorithm is modi.ed so that
J .(s) = minmax C(s, a)+ Ti(s, a, s )Jk..1(s ) . (17)
k ai s
Similarly,
Jk .(s, a) = maxC(s, a)+ Ti(s, a, s )Jk..1(s ) . (18)
i s
To determine the e.ect of robust dynamic programming on the performance of the system, the logic was optimized using models with process noise parameters of 4ft/s2, 6ft/s2, and 8ft/s2. Figure
　
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