Elektronika 2010-11.pdf - Instytut Systemów Elektronicznych ...

tion about a set. That is why we can define an interval type-2
fuzzy set with upper and lower bounds of function defined on
(x,u) plane (a shaded area in Fig. 1). They are called respectively:
upper and lower membership functions and both are
type-1 membership functions. An example is shown in Fig. 2.
All the fuzzy operations on interval fuzzy sets are defined
by using well known classical fuzzy logic [9,13]. Type-2 fuzzy
sets bring an additional degree of freedom comparing to type-
1 fuzzy sets. In the case of interval fuzzy sets, designers don’t
loose that degree of freedom. Because of that, and the fact
that general type-2 fuzzy sets require complex computations,
interval type-2 fuzzy sets are widely used in practical applications.
Type-2 fuzzy logic controller
The general structure of a type-2 fuzzy logic controller, depicted
in Fig. 3, consists of four basic modules: fuzzification, fuzzy inference
with the rule base, type reduction and defuzzification.
In the fuzzifier block crisp values from inputs are mapped
into type-2 fuzzy sets. As a result, for each input signal two
membership levels are obtained, one called lower membership
level and the other called upper membership level, which
are computed from the lower and upper membership functions
respectively.
In the inference block type-2 fuzzy antecedents sets are
combined and computed to the type-2 fuzzy consequent sets.
These operations are based on a set of fuzzy rules which are
stored in the rule base. In this paper the following form of the
rules was used:
R if
x
is
X
and
x
is
X
...or
x
is
X
...
then
y
is
Y
,
y
is
Y
... (4)
k
: 1 1
j
2
2
l
i
ij
1
1
j
2
2
l
where x i
and y i
are input and output linguistic variables (eg.
speed, height, temperature), X k
and Y l
represents labels of
interval type-2 fuzzy sets (eg. slow, fast, low, medium, high).
The connectives and, or represent operations on fuzzy sets:
intersection and union, respectively, defined for type-1 fuzzy
sets as stated in the previous section. Several assumptions
have been made on fuzzy reasoning process:
• Mamdani method of fuzzy reasoning is employed,
• “and” connective is implemented as min,
Fig. 3. General block diagram of a type-2 fuzzy logic controller
Rys. 3. Schemat blokowy sterownika rozmytego 2-go rzędu
• “or” connective is implemented as max,
• rules connective is implemented as max.
The type reduction module represents mapping from interval
type-2 fuzzy sets into classical interval fuzzy sets. Since
we operate on interval sets, it computes an interval [y L
, y R
].
To date two algorithms have been proposed for interval type-
2 fuzzy sets: the Karnick-Mendel iterative procedure [5] and
the Wu-Mendel closed forms [7]. The former method provides
an exact computation, the latter provides an approximation.
Although the KM algorithm is computationally more costly, we
decided to implement this solution. As mentioned earlier, the
aim of this algorithm is to find an interval. In order to do so,
centroids of certain type-1 membership functions are computed.
The algorithm looks as follows.
Algorithm 1. The Karnick-Mendel procedure of type reduction.
Step 1. Initialization: set µ i
and calculate y’ for each sample of
the output type-2 fuzzy set (i indicates consecutive samples):
s
−
1
∑
x
i *
µ
i
µ
+
= 1
i
µ
i
i
y
' =
,
µ
= , i = 0,1,..., s −1 s−1
i
, (5)
2
µ
∑
i
=
1
i
Step 2. Find index k (1 ≤ k ≤ s-1) such that x k-1
≤ y’ ≤ x k+1
.
Fig. 4. An example of type-2 fuzzy reasoning
Rys. 4. Przykład wnioskowania rozmytego opartego na zbiorach rozmytych 2-go rzędu
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