**6.1 Membership functions**

Let us now quantify the meaning of the linguistic variables using the membership functions. Depending on the specific application and the designer (expert), we may select from various membership functions.

The fuzzy partitions for both the input variables (error and error-variation) and for the output variable (increase-energy-supplied) will consist of seven diffuse groups uniformly distributed in a normalized universe of discourse with range [-1,1]. Figure 7 shows the partition for the input variables, and Figure 8 that corresponding to the output variable.

The membership functions for the controller's input variables, at the edge of the universe of discourse, are saturated. This means that at a given point, the expert regards all values above a given value as capable of being grouped under the same linguistic description of "large-positive" or "large-negative". The membership function of the controller's output variable, however, cannot be saturated at the edge if the controller is to function properly. The basic reason is that the controller cannot tell the actuator that any value above a given value is valid; instead, a specific value must always be specified. Moreover, from a practical standpoint, we could not carry out a defuzzification process that considers the area of conclusion of the rule if, as an output, we have membership functions with an infinite area.

A convenient way of representing the set of rules when the number of inputs to the fuzzy controller is low (three or fewer) is by using a table. Each square represents the linguistic value of the consequent of a rule, with the left column and the top row containing the linguistic values of the antecedent's variables. A temperature control example is shown in Table 1. Note the symmetry exhibited by the table. This is not coincidental, and corresponds

error/error-variation LN MN SN ZE SP MP LP LP LN LN LN LP LP LP LP MP LN LN LN MP LP LP LP SP LN LN LN SP SP LP LP ZE LN LN LN ZE MP MP LP SN LN LN LN SN ZE SP MP MN LN LN LN MN SN ZE SP LN LN LN LN LN MN SN ZE

Until now we have only quantified the expert's knowledge of how to control a system in an abstract manner. Next, we shall see how, using fuzzy logic, we can quantify the meaning of the linguistic descriptions so as to automate the control rules specified by the expert in a

Let us now quantify the meaning of the linguistic variables using the membership functions. Depending on the specific application and the designer (expert), we may select from various

The fuzzy partitions for both the input variables (error and error-variation) and for the output variable (increase-energy-supplied) will consist of seven diffuse groups uniformly distributed in a normalized universe of discourse with range [-1,1]. Figure 7 shows the partition for the input variables, and Figure 8 that corresponding to the output variable.

The membership functions for the controller's input variables, at the edge of the universe of discourse, are saturated. This means that at a given point, the expert regards all values above a given value as capable of being grouped under the same linguistic description of "large-positive" or "large-negative". The membership function of the controller's output variable, however, cannot be saturated at the edge if the controller is to function properly. The basic reason is that the controller cannot tell the actuator that any value above a given value is valid; instead, a specific value must always be specified. Moreover, from a practical standpoint, we could not carry out a defuzzification process that considers the area of conclusion of the rule if, as an output, we have membership functions with an infinite area.

to the symmetrical behavior of the system to be controlled.

Table 1. Rule base for controlling temperature.

**6. Fuzzy quantification of knowledge** 

fuzzy controller.

**6.1 Membership functions** 

membership functions.

Fig. 7. Fuzzy partition of controller input variables.

Fig. 8. Fuzzy partition of controller output variable.
