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Why use fuzzy sets as a basis for flight segment identification? After all, the flight segment decision is a crisp decision. The control of the HITS display requires a specific selection of the next flight segment. While the flight segment decision must be a crisp all-or-nothing decision, the flight segments themselves do overlap in the aircraft state space and fuzzy sets are an excellent model of the ambiguity in defining some flight segments. However, there is another motivation for using fuzzy models of the flight segments. The degree of memberships in the fuzzy flight segments can be interpreted as a measure of certainty and used to derive confidences of the flight segment decisions. These certainties and confidence factors allow for filtering the mode decision, a feature we have found valuable in practice.
A. One Dimensional Fuzzy Sets
The membership function indicates the degree of membership of a crisp value in a fuzzy set. A variety of basic shapes can be used to design the fuzzy membership functions. The trapezoidal fuzzy membership function is the most popular. Trapezoids are easily specified and calculated. By far, fuzzy sets are usually defined in one domain at a time. That is, the fuzzy sets are one-dimensional, based on a single variable. Multiple domains are combined using rules, which have fuzzy sets as their antecedents.
The first implementation of flight segment identification logic in ASTRA was performed using a rule-base of one-dimensional fuzzy sets.11 The rule base is shown in Fig. 10. Each row lists the one-dimensional fuzzy sets for a state-variable. Each column corresponds to a flight segment. The aircraft is operating in a given flight segment to the degree that the current values for power, angle of attack, roll, etc. match the one-dimensional membership function in the flight segment’s column.
taxi takeoff climbout cruise initapp finalapp landing
power
[%]

alpha
[degrees]

roll
[degrees]

landing
gear

flaps
[degrees]

airspeed
[knots]
altitude
[feet]

rate of
climb
[fpm]

The one-dimensional FSI provided a glimpse of the usefulness of an avionics system which maintains a qualitative assessment of the current flight procedure. However, it also revealed the challenges of the flight segment identification problem. Tuning the fuzzy set definitions and rule base is a time-consuming, trial-and-error process. And, more importantly, one-dimensional fuzzy sets have a fundamental shortcoming – they do not model correlation between variables in defining the flight segments.

While one-dimensional fuzzy reasoning is still largely the state-of-the-art for fuzzy systems, correlation between input variables of a fuzzy system can lead to complications.  By “correlation” is meant the condition that a fuzzy set describing a system state is represented by an
x
irregular, smoothly connected
Two-variable composition region in a multivariable state
max(mX, mY)> 0 space.  The “footprint” of such a
mode on the x-y plane could look something like the solid ellipse in Fig. 11. One-dimensional membership functions cannot by themselves represent such a relationship. The current practice approximates a smooth representation by composition of two or more single-variable regions.  Such a composition is shown in dashed lines in Fig. 11.
The use of fuzzy inference for flight mode interpretation has revealed that this standard fuzzy logic approach is insufficient for application in complex systems. To address some fundamental shortcomings in the current state-of-the-art, the authors developed the hypertrapezoidal fuzzy membership function (HFMF).
B. Bayesian Isomorphism
Before introducing a method for specifying multi-dimensional fuzzy sets, it is useful to consider a somewhat theoretical question of the relationship between fuzzy set theory and Bayesian decision theory. There is an isomorphic relationship between the two. That relationship has been shown in Refs. Error! Reference source not found. and 13. The parallel between the two approaches can be developed by assuming certain constraints on the design of the fuzzy sets and on the logical connectives used to operate on fuzzy sets.
　
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