The above example shows that a fuzzy controller is a nonlinear
controller. The form of the characteristic depends only on the
rule base and the membership functions of 
and 
. In the following
discussions about the influence of membership functions the following
assumptions for the fuzzy controller with the input signal and
the output signal will be used:
and for the OR
connectives the 
operator will be used.
,
and at first, only the three linguistic
terms NS (negative small), AZ (approximate zero) and PS (positive
small) are considered. The rule base is that of
a proportional fuzzy controller
have two overlaps in the intervals
and 
that correspond precisely with the ranges with the positive slope
of the curve. The reason for this is just that two rules in these
ranges are simultaneously active. On the other hand, in the
non-overlapping ranges only one rule is active. The membership function of the
output depends in this case only on the degree of relevance and thus the centre
of gravity of the membership function remains constant.
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If the number of linguistic terms for the input and output is increased, the characteristic is similar, but with more sections. The number of sections depends only on the number of linguistic terms and the width of the sections depends on the degree of overlapping. In the special case without overlapping in the input one obtains the characteristic of a three-level controller, as shown in Figure 17.5. In this case only one rule is active such that only the three crisp values -1, 0 and 1 are generated. Now, consider varying the degree of overlap of the output membership functions. Figure 17.6 shows the case with full overlap on input and output, where the result is approximately a linear behaviour.
A modification of the output membership functions so that they do not overlap will cause the characteristic to become close to that of Figure 17.6, compare Figure 17.7. Therefore one can establish the fact that the degree of overlap in the input membership functions has a strong influence on the static characteristic of a fuzzy controller. While small overlaps in the input membership functions generate step characteristics, with a higher degree of overlap the curves become smoother. The influence of overlap in the output membership functions has less effect on the characteristic. For a reduction of the support of the output membership functions the characteristic of Figure 17.8 is obtained which does not differ significantly from that of Figure 17.6.
The size of the individual output membership function has a strong influence on the characteristic. Figure 17.9 shows the case for a very small support of the output membership function AZ, which generates an S-type characteristic with a high gain at the origin. Widening the support of the membership function AZ inverts the S-curve with a small gain at the origin, as shown in Figure 17.10. Thus the form of the characteristic depends strongly on the support of the individual output membership function.
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The effects of a modified rule base will be demonstrated by an example. The same full overlapping membership functions are used as in Figure 17.6a and b. A modified rule base of the form
