# Spatial Decision Support Systems/Fuzzy Controller

Fuzzy Logic Examples
Spatial Fuzzy Logic

A fuzzy control system is a control system based on fuzzy logic—a mathematical system that analyzes analog input values in terms of logical variables that take on continuous values between 0 and 1, in contrast to classical or digital logic, which operates on discrete values of either 1 or 0 (true or false, respectively).[1][2].

For a spatial application of fuzzy controllers continuous values between 0 and 1 are assigned to all geolocations. E.g. “temperature is optimal for mosquito X”) will take the current temperature a the geolocation as input variable and will assign a value 0.1 for the temperature ${\displaystyle -5^{o}C}$ in a cold area and 0.9 in a region where the temperature in ${\displaystyle 19^{o}C}$

## Overview

Fuzzy logic is widely used in machine control. The term "fuzzy" refers to the fact that the logic involved can deal with concepts that cannot be expressed as the "true" or "false" but rather as "partially true". Although alternative approaches such as genetic algorithms and neural networks can perform just as well as fuzzy logic in many cases, fuzzy logic has the advantage that the solution to the problem can be cast in terms that human operators can understand, so that their experience can be used in the design of the controller. This makes it easier to mechanize tasks that are already successfully performed by humans.[1]

## Logical interpretation of fuzzy control

In spite of the appearance there are several difficulties to give a rigorous logical interpretation of the IF-THEN rules. As an example, interpret a rule as IF (temperature is "cold") THEN (heater is "high") by the first order formula Cold(x)→High(y) and assume that r is an input such that Cold(r) is false. Then the formula Cold(r)→High(t) is true for any t and therefore any t gives a correct control given r. A rigorous logical justification of fuzzy control is given in Hájek's book (see Chapter 7) where fuzzy control is represented as a theory of Hájek's basic logic.[2] Also in Gerla 2005 [3] another logical approach to fuzzy control is proposed based on fuzzy logic programming.Indeed, denote by f the fuzzy function arising of an IF-THEN systems of rules. Then we can translate this system into a fuzzy program P containing a series of rules whose head is "Good(x,y)". The interpretation of this predicate in the least fuzzy Herbrand model of P coincides with f. This gives further useful tools to fuzzy control.

## References

1. Pedrycz, Witold (1993). Fuzzy control and fuzzy systems (2 ed.). Research Studies Press Ltd..
2. Hájek, Petr (1998). Metamathematics of fuzzy logic (4 ed.). Springer Science & Business Media.
3. Gerla, Giangiacomo (2005). "Fuzzy logic programming and fuzzy control". Studia Logica 79 (2): 231–254. doi:10.1007/s11225-005-2977-0.