Fuzzy relations

In the introduction to the fuzzy control methodology, section 14.3, rules have been introduced, which in mathematical notation are connective operations over fuzzy sets. For example, the operations on premises in Eq.(14.1) can be handled for each rule already by the elementary standard operators introduced in section 15.3. The means are now available to handle steps 1 and 2 of the methodology. But to cope with step 3 something more is needed to complete the modelling of rules. That is now added in this section.

First, relations are explained by a simple example from daily life using discrete fuzzy sets. Let us describe the relationship between the colour of a fruit and the grade of maturity and characterise the linguistic variable colour by a crisp set with three linguistic terms as

and similarly the grade of maturity as

One knows that a crisp formulation of a relation between the two crisp sets would look like this in tabular form:

	verdant	half-mature	mature
green	1	0	0
yellow	0	1	0
red	0	0	1

The zeros and ones describe the grade of membership to this relation. This relation is now a new kind of crisp set that is built from the two crisp base sets and . This new set is now called and can be expressed also by the rules:

(1) IF the colour is green THEN the fruit is verdant

(2) IF the colour is yellow THEN the fruit is half-mature

(3) IF the colour is red THEN the fruit is mature

As can be seen from this example, a relation, which is called a rule or rule base, can be used to provide a model.

Demonstration Example 15.5   Test this with your own tomatoes

This crisp relation represents the presence or absence of association, interaction or interconnection between the elements of these two sets. This can be generalised to allow for various degrees of strength of association or interaction between elements. Degrees of association can be represented by membership grades in a fuzzy relation in the same way as degrees of the set membership are represented in a fuzzy set. Applying this to the fruit example, the table can be modified to

	verdant	half-mature	mature
green	1	0.5	0
yellow	0.3	1	0.4
red	0	0.2	1

where there are now real numbers in . This table represents a fuzzy relation and models the connectives in a fuzzy rule base. It is a two-dimensional fuzzy set and the question now is, how can this set be determined from its elements.

Demonstration Example 15.6   Test this with your own strawberries

In order to do this, the elements are generalised. In the above example, the linguistic terms where treated as crisp terms. For example, when one represents the colours on a colour spectrum scale, the colours would be described by their spectral distribution curves that can be interpreted as membership functions and then a particular colour is a fuzzy term. Treating also the grades of maturity as fuzzy terms, the above relation is a two-dimensional fuzzy set over two fuzzy sets. For example, taking from the fruit example the relation between the linguistic terms red and mature, and represent them by the membership functions as shown in Figure 15.5a, a fruit can be characterised by the

Figure 15.5: Relation between two fuzzy sets: (a) membership functions, (b) 3-D view of the membership functions, (c) membership function of the relation after applying the operation to (b)

property red AND mature. This expression can be re-written in mathematical form using elementary connective operators (see Eqs. (15.15) or (15.18)) for the membership functions by

or

Figure 15.5b shows a 3-dimensional view of these two fuzzy terms and Figure 15.5c the result of the connective operation according to Eq.(15.20). This result combines the two fuzzy sets by an operation that is a Cartesian product

From this example it is obvious that the connective operation in a rule for the operation is simply performed by a fuzzy intersection in two dimensions. For this, both intersection operators from Eqs. (15.15) or (15.18) can be used.

Combining rules into a rule base the example from above may help when it is rewritten as

(1) IF the colour is green THEN the fruit is verdant

OR

(2) IF the colour is yellow THEN the fruit is half-mature

OR

(3) IF the colour is red THEN the fruit is mature

which describes in a linguistic way a union of three rules. For the complete rule base one can combine the relations formed for each individual rule with a fuzzy union operator, which is the fuzzy OR according to Eqs. (15.16) or (15.20).

Now, step 4 of the methodology introduced in section 14.3 can be specified by taking the rule base from Eq. (14.1) and applying the union operator by writing the rule base with max/min operators as follows:

where is the premise of the th rule. This representation is the standard max/min representation of a rule base that will be later used for fuzzy controllers. Instead of the max/min representation a so called max-prod representation is also usual, where the algebraic product

is used to build the relation between the premise and the conclusion.