Fuzzy Washing Machine: Comparison with Ordinary Washing MachinesFUZZY SYSTEMS

Fuzzy Washing Machine: Comparison with ordinary systems How could we have approached this problem with ordinary expert systems?

The nine rules would have been the same, but the values of Dirtiness and Soil would have been crisp sets (fixed intervals)

If Dirtiness = x and Soil = xx, then Time = xxxFUZZY SYSTEMS

If Dirtiness = x and Soil = xx, then Time = xxxFUZZY SYSTEMS

Fuzzy Washing Machine: Comparison with ordinary systems

For the specific measured values of D and S, we check the rulesFUZZY SYSTEMS

Fuzzy Washing Machine: Comparison with ordinary systems The only rule fired would have been:If Dirt = Low & Soil = Medium then Time = ShortFUZZY SYSTEMS

Fuzzy Washing Machine: Comparison with ordinary systemsFUZZY SYSTEMS

Fuzzy Washing Machine: Comparison with ordinary systems In non-fuzzy washing machines only five fixed timer settings would have been possible

In the fuzzy case we have the whole range from 0 to tmaxFUZZY SYSTEMS

Fuzzy ControllersFUZZY SYSTEMS

Controllers

A controller is used to control some system, or plant

The system has a desired response that must be maintained under whatever inputs are received

The task of the controller is then to take corrective action by providing a set of inputs that ensure the desired responseFUZZY SYSTEMS

Main Components

Fuzzy Rule Base

Condition Interface (fuzzifier)

Action Interface (defuzzifier)

Inference Engine (fuzzy controller)FUZZY SYSTEMS

Main Components Fuzzy Rule Base:

The rule base, or knowledge base, contains the fuzzy rules that represent the knowledge and experience of a human expert of the system

Condition Interface (fuzzifier):

The fuzzifier receives the actual outputs of the system, and transforms these non-fuzzy values into membership degrees to the corresponding fuzzy sets. The fuzzification of the input values also occurs via this interfaceFUZZY SYSTEMS

Main Components Action Interface (defuzzifier) :

It defuzzifies the outcome of the inference engine to produce a non-fuzzy value

Inference Engine:

The inference engine performs inferencing upon fuzzified inputs to produce a fuzzified outputFUZZY SYSTEMS

Mamdani Fuzzy Controller Developed by Mamdani & Assilian in 1975. Mamdani-type controllers follow the following simple steps: FUZZY SYSTEMS

Alternate Type of Inference: Max-Product InferenceFUZZY SYSTEMS

Max-Product Inference

Let A = [0/100, 0.5/125, 1/150, 0.5/175, 0/200] B = [0/10, 0.6/20, 1/30, 0.6/40, 0/50]

thenM = 0 0 0 0 00 0.3 0.5 0.3 0by taking the product0 0.6 1 0.6 0of each pair0 0.3 0.5 0.3 00 0 0 0 0FUZZY SYSTEMS

Max-Product Inference Let Temperature = 125 oF = AcurrentThen it may be thought of as a new fuzzy set defined over the universe of discourse of variable TemperatureAcurrent = [0/100, 0.5/125, 0/150, 0/175, 0/200]

We can find the FAM by taking the product

A = Acurrent x A = 0 x [0 0.5 1 0.5 0]0.50 = [0 0.25 0.5 0.25 0]0(if we eliminate all 0 rows) 0FUZZY SYSTEMS

Max-Product Inference Now the composition of A and M will produce a new relationship, which we call Bmax product composition: bj= maxi {(ai.mij)}

A M = B0 0 0 0 0 0 0.3 0.5 0.3 0[0 0.25 0.5 0.25 0] 0 0.6 1.0 0.6 0 0 0.3 0.5 0.3 0 0 0 0 0 0 = [0 0.3 0.5 0.3 0]FUZZY SYSTEMS

Max-Product InferenceMax-product inferenceFUZZY SYSTEMS

Multiple Fuzzy RulesFUZZY SYSTEMS

Multiple Fuzzy RulesFUZZY SYSTEMS

References

Engelbrecht Chapter 18, 19 & 20

Scan of Fuzzy Washing Machine on FolderFUZZY SYSTEMS