Literature Review
SIMPLE FL CONTROL SYSTEM
4.2.2 Fuzzification and membership functions
To apply the rules the system uses the membership function that is a graphical
representation o f the magnitude o f participation of each input. It associates a
weighting with each of the inputs that are processed, defines functional overlap
There is a unique membership function associated with each input parameter. The
membership functions associate a weighting factor with values of each input and the
effective rules. These weighting factors determine the degree of influence or degree of
membership (DOM) each active rule has. By computing the logical product of the
membership weights for each active rule, a set o f fuzzy output response magnitudes
are produced.
The shape of membership functions can be different. In general, the triangular is
common (see Figure 4.3), but bell, trapezoidal, haversine and, exponential have been
used. More complex functions are possible but require greater computing overhead to
implement4. MEMBERSHIP FUNCTIONS
I t
Q ) 2 (B v - o O Q . <D ,>s Shoddered Centers/
Negative Zero Positive Z & P[4
Width » | EngheeringUnits(Typically lbs, deg F, or deg/m, ftfsec, etc.)
Fig. 4.3 - The features of a m em bership function
(SOURCE: Mitsuishi T., Endou T., Shidama Y., (2003) - The concept o f fuzzy set
and membership function and basic properties o f fuzzy set operation - Journal
of formalized mathematics, Vol. 12)
The degree of membership (DOM) is determined by plugging the selected input
parameter into the horizontal axis and projecting vertically to the upper boundary of
the membership function(s).
4.2.3 Defuzzification
Once the functions are inferred, scaled, and combined, they are defuzzified into a
crisp output which drives the system.
Before undertaking the defuzzification process for crisp output generation, the logical
products for each rule must be combined or inferred. There are several inference
methods:
The MAX-MIN method tests the magnitudes o f each rule and selects the
highest one. The horizontal coordinate o f the "fuzzy centroid" of the area
under that function is taken as the output. This method does not combine the
effects of all applicable rules but does produce a continuous output function
and is easy to implement.
The MAX-DOT or MAX-PRODUCT method scales each member function to
fit under its respective peak value and takes the horizontal coordinate of the
"fuzzy" centroid of the composite area under the fimction(s) as the output.
Essentially, the member fimction(s) are shrunk so that their peak equals the
magnitude o f their respective function ("negative", "zero", and "positive").
This method combines the influence of all active rules and produces a smooth,
continuous output.
example, if three "negative" rules fire, but only one "zero" rule does,
averaging will not reflect this difference since both averages will equal 0.5.
Each function is clipped at the average and the "fuzzy" centroid of the
composite area is computed.
The ROOT-SUM-SQUARE (RSS) method combines the effects of all
applicable rules, scales the functions at their respective magnitudes, and
computes the "fuzzy" centroid o f the composite area. This method is more
complicated mathematically than other methods, but was selected for this
example since it seemed to give the best weighted influence to all firing rules.
The defuzzification of the data into a crisp output is accomplished by
combining the results of the inference process and then computing the "fuzzy
centroid" of the area. The weighted strengths o f each output member function
are multiplied by their respective output membership function center points
and summed.
Finally, this area is divided by the sum o f the weighted member function
strengths and the result is taken as the crisp output as shown in Figure 4.4.
OUTPUT M E M B E R SH IP FUNCTION
P o sitiv e
-1 0 0 t - 5 0 0 +50 +100
(-63.5%)
P e rc e n t O utput - (-100 to 0 = C o o lin g , 0 to + 1 0 0 = H e ^ in g )
Fig. 4.4 - The horizontal coordinate of the centroid is taken as the crisp output
(SOURCE: Mitsuishi T., Endou T., Shidama Y., (2003) - The concept o f fuzzy set
and membership function and basic properties o f fuzzy set operation - Journal
of formalized mathematics, Vol. 12)
The logical product of each rule is inferred to arrive at a combined magnitude for each
output membership function. Once inferred, the magnitudes are mapped into then-
respective output membership functions, delineating all or part of them.
4.3 Artificial Neural Networks
Artificial Neural Networks (ANNs) are relatively crude electronic models based on
the neural structure of the brain5. The brain basically learns from experience.
Neural networks attempt to produce the human way o f processing data and learning.
They exhibit certain analogies to the way in which arrays o f neurons function in
biological learning and memory. The fundamental blocks are processing units, called
nodes, which can be likened to neurons, and weighted connections, which can be
The basic unit o f neural networks, the artificial neurons, simulate the four basic
functions of natural neurons as shown in Figure 4.5.
Sun T ra n s fe r
Wei a r ts
Fig. 4.5 - A basic Artificial Neuron
(SOURCE: Mehrotra K., Mohan C.K., and Ranka S., (1996) - Elements o f Artificial
Neural Networks — Cloth)
The various inputs to the network are represented by the mathematical symbol, x(n).
Each of these inputs are multiplied by a connection weight. These weights are
represented by w(n). In the simplest case, these products are simply summed, fed
through a transfer function to generate a result, and then output. This process lends
itself to physical implementation on a large scale in a small package.
In currently available software packages these artificial neurons are called
"processing elements" and have many more capabilities than the simple artificial
neuron described above.
Simmadcn F in c tio n Transfer Function Inputs Learning and Recall Schedule Learning C ycle Hyperbolic Tangent Linear Sgmoid etc. And Min etc. Max Sum Outputs
Fig. 4.6 - A model of a “Processing Element”
(SOURCE: Mehrotra K., Mohan C.K., and Ranka S., (1996) - Elements o f Artificial
Neural Networks — Cloth)
In Figure 4.6, inputs enter into the processing element from the upper left. The first
step is for each of these inputs to be multiplied by their respective weighting factor
(w(n)). Then these modified inputs are fed into the summing function, which usually
just sums these products. Yet, many different types o f operations can be selected.
These operations could produce a number o f different values which are then
propagated forward; values such as the average, the largest, the smallest, etc.
Sometimes the summing function is further complicated by the addition of an
activation function which enables the summing function to operate in a time sensitive
way. The output of the summing function is then sent into a transfer function. This
algorithm that takes the input and turns it into a zero or a one, a minus one or a one, or
some other number. The transfer functions that are commonly supported are sigmoid,
sine, hyperbolic tangent, etc. This transfer function also can scale the output or
control its value via thresholds. The result o f the transfer function is usually the direct
output of the processing element.
Finally, the processing element is ready to output the result of its transfer function.
This output is then input into other processing elements, or to an outside connection,
as dictated by the structure of the network.
A typical neural network consists of many inter -connected nodes that are organized
into a sequence of layers. The first layer is called the input layer, and it is responsible
for converting the input data in order to be processed it in the next layer. The next
layer, may be more than one layer, is called hidden layer in which nodes perform the
operations on the data received from the first layer. The output layer produces the
estimation by performing a differently defined mathematical function. Evidently, the
calculated output will not be similar or close to the existing output data in one cycle.
Therefore, by repeating and memorizing with unique weight and performing error
calculation of each cycle, the best estimation result is gained.
INPUT LAYER
HIDDEN LAYER
(there may be several hidden layers)
OUTPUT LAYER
Fig. 4.7 - A Simple Neural Network Diagram
(SOURCE: Hassoun M.H., (1995) - Fundamentals o f Artificial Neural Networks
(Hardcover) - The MIT Press)
Once a network has been structured for a particular application, that network is ready
to be trained.
n
The Operation of a Neural Network is controlled by three properties :
1. The pattern of its interconnections, architecture.
2. Method of determining and updating the weights on the interconnections, training.
3. The function that determines the output of each individual neuron, activation or