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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