Fuzzy logic

AN INTRODUCTION TO FUZZY LOGIC
(Seattle Robotics)
AN INTRODUCTION TO FUZZY LOGIC..................................................................1
FUZZY LOGIC - AN INTRODUCTION - PART 1.....................................................2
INTRODUCTION......................................................................................................2
WHERE DID FUZZY LOGIC COME FROM?........................................................2
WHAT IS FUZZY LOGIC?.......................................................................................2
HOW IS FL DIFFERENT FROM CONVENTIONAL CONTROL METHODS?...2
HOW DOES FL WORK?..........................................................................................3
SUMMARY...............................................................................................................3
REFERENCES.......................................................................................................3
INTRODUCTION......................................................................................................4
WHY USE FL?..........................................................................................................4
HOW IS FL USED?...................................................................................................5
LINGUISTIC VARIABLES......................................................................................5
SUMMARY...............................................................................................................5
REFERENCES.......................................................................................................6
INTRODUCTION......................................................................................................7
THE RULE MATRIX................................................................................................7
STARTING THE PROCESS.....................................................................................7
WHAT IS BEING CONTROLLED AND HOW:.....................................................9
SUMMARY.............................................................................................................10
REFERENCES.....................................................................................................10
FUZZY LOGIC - AN INTRODUCTION - PART 4...................................................12
INTRODUCTION....................................................................................................12
MEMBERSHIP FUNCTIONS................................................................................12
ERROR & ERROR-DOT FUNCTION MEMBERSHIP........................................14
SUMMARY.............................................................................................................15
REFERENCES.....................................................................................................15
INTRODUCTION....................................................................................................16
PUTTING IT ALL TOGETHER.............................................................................16
REFERENCES.....................................................................................................18
INTRODUCTION....................................................................................................19
INFERENCING.......................................................................................................19
A "FUZZY CENTROID" ALGORITHM ...............................................................20
TUNING AND SYSTEM ENHANCEMENT.........................................................21
SUMMARY.............................................................................................................21
CONCLUSION........................................................................................................21
REFERENCES.....................................................................................................22

FUZZY LOGIC - AN INTRODUCTION - PART 1
by Steven D. Kaehler
INTRODUCTION
This is the first in a series of six articles intended to share information and experience
in the realm of fuzzy logic (FL) and its application. This article will introduce FL.
Through the course of this article series, a simple implementation will be explained in
detail. Each article will include additional outside resource references for interested
readers.
WHERE DID FUZZY LOGIC COME FROM?
The concept of Fuzzy Logic (FL) was conceived by Lotfi Zadeh, a professor at the
University of California at Berkley, and presented not as a control methodology, but
as a way of processing data by allowing partial set membership rather than crisp set
membership or non-membership. This approach to set theory was not applied to
control systems until the 70's due to insufficient small-computer capability prior to
that time. Professor Zadeh reasoned that people do not require precise, numerical
information input, and yet they are capable of highly adaptive control. If feedback
controllers could be programmed to accept noisy, imprecise input, they would be
much more effective and perhaps easier to implement. Unfortunately, U.S.
manufacturers have not been so quick to embrace this technology while the Europeans
and Japanese have been aggressively building real products around it.
WHAT IS FUZZY LOGIC?
In this context, FL is a problem-solving control system methodology that lends itself
to implementation in systems ranging from simple, small, embedded micro-
controllers to large, networked, multi-channel PC or workstation-based data
acquisition and control systems. It can be implemented in hardware, software, or a
combination of both. FL provides a simple way to arrive at a definite conclusion
based upon vague, ambiguous, imprecise, noisy, or missing input information. FL's
approach to control problems mimics how a person would make decisions, only much
faster.
HOW IS FL DIFFERENT FROM CONVENTIONAL CONTROL
METHODS?
FL incorporates a simple, rule-based IF X AND Y THEN Z approach to a solving
control problem rather than attempting to model a system mathematically. The FL
model is empirically-based, relying on an operator's experience rather than their
technical understanding of the system. For example, rather than dealing with
temperature control in terms such as "SP =500F", "T <1000F", or "210C <TEMP
<220C", terms like "IF (process is too cool) AND (process is getting colder) THEN
(add heat to the process)" or "IF (process is too hot) AND (process is heating rapidly)
THEN (cool the process quickly)" are used. These terms are imprecise and yet very

descriptive of what must actually happen. Consider what you do in the shower if the
temperature is too cold: you will make the water comfortable very quickly with little
trouble. FL is capable of mimicking this type of behavior but at very high rate.
HOW DOES FL WORK?
FL requires some numerical parameters in order to operate such as what is considered
significant error and significant rate-of-change-of-error, but exact values of these
numbers are usually not critical unless very responsive performance is required in
which case empirical tuning would determine them. For example, a simple
temperature control system could use a single temperature feedback sensor whose
data is subtracted from the command signal to compute "error" and then time-
differentiated to yield the error slope or rate-of-change-of-error, hereafter called
"error-dot". Error might have units of degs F and a small error considered to be 2F
while a large error is 5F. The "error-dot" might then have units of degs/min with a
small error-dot being 5F/min and a large one being 15F/min. These values don't have
to be symmetrical and can be "tweaked" once the system is operating in order to
optimize performance. Generally, FL is so forgiving that the system will probably
work the first time without any tweaking.
SUMMARY
FL was conceived as a better method for sorting and handling data but has proven to
be a excellent choice for many control system applications since it mimics human
control logic. It can be built into anything from small, hand-held products to large
computerized process control systems. It uses an imprecise but very descriptive
language to deal with input data more like a human operator. It is very robust and
forgiving of operator and data input and often works when first implemented with
little or no tuning.
REFERENCES
[1] "Europe Gets into Fuzzy Logic" (Electronics Engineering Times, Nov. 11, 1991).
[2] "Fuzzy Sets and Applications: Selected Papers by L.A. Zadeh", ed. R.R. Yager et
al. (John Wiley, New York, 1987).
[3] "U.S. Loses Focus on Fuzzy Logic" (Machine Design, June 21, 1990).
[4] "Why the Japanese are Going in for this 'Fuzzy Logic'" by Emily T. Smith
(Business Week, Feb. 20, 1993, pp. 39).

INTRODUCTION
This is the second in a series of six articles intended to share information and
experience in the realm of fuzzy logic (FL) and its application. This article will
continue the introduction with a more detailed look at how one might use FL. A
simple implementation will be explained in detail beginning in the next article.
Accompanying outside references are included for interested readers.
In the last article, FL was introduced and the thrust of this article series presented. The
origin of FL was shared and an introduction to some of the basic concepts of FL was
presented. We will now look a little deeper.
WHY USE FL?
FL offers several unique features that make it a particularly good choice for many
control problems.
1) It is inherently robust since it does not require precise, noise-free inputs and can be
programmed to fail safely if a feedback sensor quits or is destroyed. The output
control is a smooth control function despite a wide range of input variations.
2) Since the FL controller processes user-defined rules governing the target control
system, it can be modified and tweaked easily to improve or drastically alter system
performance. New sensors can easily be incorporated into the system simply by
generating appropriate governing rules.
3) FL is not limited to a few feedback inputs and one or two control outputs, nor is it
necessary to measure or compute rate-of-change parameters in order for it to be
implemented. Any sensor data that provides some indication of a system's actions and
reactions is sufficient. This allows the sensors to be inexpensive and imprecise thus
keeping the overall system cost and complexity low.
4) Because of the rule-based operation, any reasonable number of inputs can be
processed (1-8 or more) and numerous outputs (1-4 or more) generated, although
defining the rulebase quickly becomes complex if too many inputs and outputs are
chosen for a single implementation since rules defining their interrelations must also
be defined. It would be better to break the control system into smaller chunks and use
several smaller FL controllers distributed on the system, each with more limited
responsibilities.
5) FL can control nonlinear systems that would be difficult or impossible to model
mathematically. This opens doors for control systems that would normally be deemed
unfeasible for automation.

HOW IS FL USED?
1) Define the control objectives and criteria: What am I trying to control? What do I
have to do to control the system? What kind of response do I need? What are the
possible (probable) system failure modes?
2) Determine the input and output relationships and choose a minimum number of
variables for input to the FL engine (typically error and rate-of-change-of-error).
3) Using the rule-based structure of FL, break the control problem down into a series
of IF X AND Y THEN Z rules that define the desired system output response for
given system input conditions. The number and complexity of rules depends on the
number of input parameters that are to be processed and the number fuzzy variables
associated with each parameter. If possible, use at least one variable and its time
derivative. Although it is possible to use a single, instantaneous error parameter
without knowing its rate of change, this cripples the system's ability to minimize
overshoot for a step inputs.
4) Create FL membership functions that define the meaning (values) of Input/Output
terms used in the rules.
5) Create the necessary pre- and post-processing FL routines if implementing in S/W,
otherwise program the rules into the FL H/W engine.
6) Test the system, evaluate the results, tune the rules and membership functions, and
retest until satisfactory results are obtained.
LINGUISTIC VARIABLES
In 1973, Professor Lotfi Zadeh proposed the concept of linguistic or "fuzzy"
variables. Think of them as linguistic objects or words, rather than numbers. The
sensor input is a noun, e.g. "temperature", "displacement", "velocity", "flow",
"pressure", etc. Since error is just the difference, it can be thought of the same way.
The fuzzy variables themselves are adjectives that modify the variable (e.g. "large
positive" error, "small positive" error ,"zero" error, "small negative" error, and "large
negative" error). As a minimum, one could simply have "positive", "zero", and
"negative" variables for each of the parameters. Additional ranges such as "very
large" and "very small" could also be added to extend the responsiveness to
exceptional or very nonlinear conditions, but aren't necessary in a basic system.
SUMMARY
FL does not require precise inputs, is inherently robust, and can process any
reasonable number of inputs but system complexity increases rapidly with more
inputs and outputs. Distributed processors would probably be easier to implement.
Simple, plain-language IF X AND Y THEN Z rules are used to describe the desired
system response in terms of linguistic variables rather than mathematical formulas.
The number of these is dependent on the number of inputs, outputs, and the designer's
control response goals.

REFERENCES
[5] "Clear Thinking on Fuzzy Logic" by L.A. Bernardinis (Machine Design, April 23,
1993).
[6] "Fuzzy Fundamentals" by E. Cox (IEEE Spectrum, October 1992, pp. 58-61).
[7] "Fuzzy Logic in Control Systems" by C.C. Lee (IEEE Trans. on Systems, Man,
and Cybernetics, SMC, Vol. 20, No. 2, 1990, pp. 404-35).
[8] "Fuzzy Sets" by Ivars Peterson (Science News, Vol. 144, July 24, 1993, pp. 55).

INTRODUCTION
This is the third in a series of six articles intended to share information and experience
in the realm of fuzzy logic (FL) and its application. This article and the three to follow
will take a more detailed look at how FL works by walking through a simple example.
Informational references are included at the end of this article for interested readers.
THE RULE MATRIX
In the last article the concept of linguistic variables was presented. The fuzzy
parameters of error (command-feedback) and error-dot (rate-of-change-of-error) were
modified by the adjectives "negative", "zero", and "positive". To picture this, imagine
the simplest practical implementation, a 3-by-3 matrix. The columns represent
"negative error", "zero error", and "positive error" inputs from left to right. The rows
represent "negative", "zero", and "positive" "error-dot" input from top to bottom. This
planar construct is called a rule matrix. It has two input conditions, "error" and "error-
dot", and one output response conclusion (at the intersection of each row and
column). In this case there are nine possible logical product (AND) output response
conclusions.
Although not absolutely necessary, rule matrices usually have an odd number of rows
and columns to accommodate a "zero" center row and column region. This may not
be needed as long as the functions on either side of the center overlap somewhat and
continuous dithering of the output is acceptable since the "zero" regions correspond to
"no change" output responses the lack of this region will cause the system to
continually hunt for "zero". It is also possible to have a different number of rows than
columns. This occurs when numerous degrees of inputs are needed. The maximum
number of possible rules is simply the product of the number of rows and columns,
but definition of all of these rules may not be necessary since some input conditions
may never occur in practical operation. The primary objective of this construct is to
map out the universe of possible inputs while keeping the system sufficiently under
control.
STARTING THE PROCESS
The first step in implementing FL is to decide exactly what is to be controlled and
how. For example, suppose we want to design a simple proportional temperature
controller with an electric heating element and a variable-speed cooling fan. A
positive signal output calls for 0-100 percent heat while a negative signal output calls
for 0-100 percent cooling. Control is achieved through proper balance and control of
these two active devices.

Figure 1 - A simple block diagram of the control system.
It is necessary to establish a meaningful system for representing the linguistic
variables in the matrix. For this example, the following will be used:
"N" = "negative" error or error-dot input level
"Z" = "zero" error or error-dot input level
"P" = "positive" error or error-dot input level
"H" = "Heat" output response
"-" = "No Change" to current output
"C" = "Cool" output response
Define the minimum number of possible input product combinations and
corresponding output response conclusions using these terms. For a three-by-
three matrix with heating and cooling output responses, all nine rules will need
to be defined. The conclusions to the rules with the linguistic variables
associated with the output response for each rule are transferred to the matrix.

WHAT IS BEING CONTROLLED AND HOW:
Figure 2 - Typical control system response
Figure 2 shows what command and error look like in a typical control system
relative to the command setpoint as the system hunts for stability. Definitions
are also shown for this example.
DEFINITIONS:
INPUT#1: ("Error", positive (P), zero (Z), negative (N))
INPUT#2: ("Error-dot", positive (P), zero (Z), negative (N))
CONCLUSION: ("Output", Heat (H), No Change (-), Cool (C))
INPUT#1 System Status
Error = Command-Feedback
P=Too cold, Z=Just right, N=Too hot
INPUT#2 System Status
Error-dot = d(Error)/dt
P=Getting hotter Z=Not changing N=Getting colder
OUTPUT Conclusion & System Response
Output H = Call for heating - = Don't change anything C = Call for cooling
SYSTEM OPERATING RULES
Linguistic rules describing the control system consist of two parts; an
antecedent block (between the IF and THEN) and a consequent block
(following THEN). Depending on the system, it may not be necessary to
evaluate every possible input combination (for 5-by-5 & up matrices) since
some may rarely or never occur. By making this type of evaluation, usually
done by an experienced operator, fewer rules can be evaluated, thus
simplifying the processing logic and perhaps even improving the FL system
performance.

Figures 3 & 4 - The rule structure.
After transferring the conclusions from the nine rules to the matrix there is a
noticeable symmetry to the matrix. This suggests (but doesn't guarantee) a
reasonably well-behaved (linear) system. This implementation may prove to
be too simplistic for some control problems, however it does illustrate the
process. Additional degrees of error and error-dot may be included if the
desired system response calls for this. This will increase the rulebase size and
complexity but may also increase the quality of the control. Figure 4 shows the
rule matrix derived from the previous rules.
SUMMARY
Linguistic variables are used to represent an FL system's operating parameters.
The rule matrix is a simple graphical tool for mapping the FL control system
rules. It accommodates two input variables and expresses their logical product
(AND) as one output response variable. To use, define the system using plain-
English rules based upon the inputs, decide appropriate output response
conclusions, and load these into the rule matrix.
REFERENCES
[9] "Fundamentals of Fuzzy Logic: Parts 1,2,3" by G. Anderson (SENSORS,
March-May 1993).

[10] "Fuzzy Logic Flowers in Japan" by D.G. Schartz & G.J. Klir (IEEE
Spectrum, July 1992, pp. 32-35).
[11] "Fuzzy Logic Makes Guesswork of Computer Control" by Gail M.
Robinson (Design News, Vol. 47, Nov. 28, 1991, pp. 21).
[12] "Fuzzy Logic Outperforms PID Controller" by P. Basehore (PCIM,
March 1993).

INTRODUCTION
This is the fourth in a series of six articles intended to share information and
continue the example from Part 3 by introducing membership functions and
explaining how they work. The next two articles will examine FL inference and
defuzzification processes and how they work. For further information, several general
references are included at the end of this article.
MEMBERSHIP FUNCTIONS
In the last article, the rule matrix was introduced and used. The next logical question
is how to apply the rules. This leads into the next concept, the membership function.
The membership function is a graphical representation of the magnitude of
participation of each input. It associates a weighting with each of the inputs that are
processed, define functional overlap between inputs, and ultimately determines an
output response. The rules use the input membership values as weighting factors to
determine their influence on the fuzzy output sets of the final output conclusion. Once
the functions are inferred, scaled, and combined, they are defuzzified into a crisp
output which drives the system. There are different membership functions associated
with each input and output response. Some features to note are:
SHAPE - triangular is common, but bell, trapezoidal, haversine and, exponential have
been used. More complex functions are possible but require greater computing
overhead to implement.. HEIGHT or magnitude (usually normalized to 1) WIDTH (of
the base of function), SHOULDERING (locks height at maximum if an outer
function. Shouldered functions evaluate as 1.0 past their center) CENTER points
(center of the member function shape) OVERLAP (N&Z, Z&P, typically about 50%
of width but can be less).

Figure 5 - The features of a membership function
Figure 5 illustrates the features of the triangular membership function which is
used in this example because of its mathematical simplicity. Other shapes can
be used but the triangular shape lends itself to this illustration.
The degree of membership (DOM) is determined by plugging the selected
input parameter (error or error-dot) into the horizontal axis and projecting
vertically to the upper boundary of the membership function(s).

Figure 6 - A sample case
In Figure 6, consider an "error" of -1.0 and an "error-dot" of +2.5. These
particular input conditions indicate that the feedback has exceeded the
command and is still increasing.
ERROR & ERROR-DOT FUNCTION MEMBERSHIP
The degree of membership for an "error" of -1.0 projects up to the middle of
the overlapping part of the "negative" and "zero" function so the result is
"negative" membership = 0.5 and "zero" membership = 0.5. Only rules
associated with "negative" & "zero" error will actually apply to the output
response. This selects only the left and middle columns of the rule matrix.
For an "error-dot" of +2.5, a "zero" and "positive" membership of 0.5 is
indicated. This selects the middle and bottom rows of the rule matrix. By
overlaying the two regions of the rule matrix, it can be seen that only the rules
in the 2-by-2 square in the lower left corner (rules 4,5,7,8) of the rules matrix
will generate non-zero output conclusions. The others have a zero weighting
due to the logical AND in the rules.

SUMMARY
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 of
fuzzy output response magnitudes are produced. All that remains is to
combine and defuzzify these output responses.
REFERENCES
[13] "Fuzzy but Steady" (1991 Discover Awards) (Discover, Vol. 12, Dec.
1991, pp. 73).
[14] "Neural Networks and Fuzzy Systems--A Dynamic Systems Approach to
Machine Intelligence" by B. Kosko (Prentice-Hall, Englewood Cliffs, N.J.,
1992).
[15] "Putting Fuzzy Logic into Focus" by Janet J. Barron (Byte, Vol. 18, Apr.
1993, pp. 11).
[16] "Putting Fuzzy Logic in Motion" by Dr. P. Miller (Motion Control, April
1993, pp. 42-44).

INTRODUCTION
This is the fifth in a series of six articles intended to share information and experience
in the realm of fuzzy logic (FL) and its application. This article will continue the
tutorial discussion on FL by looking at the output membership function and several
inference processes. The next article will wrap up the discussion of the ongoing
example. To further explore the topic of FL, references are included for interested
readers.
In the last article, we left off with the inference engine producing fuzzy output
response magnitudes for each of the effective rules. These must be processed and
combined in some manner to produce a single, crisp (defuzzified) output.
PUTTING IT ALL TOGETHER
As inputs are received by the system, the rulebase is evaluated. The antecedent (IF X
AND Y) blocks test the inputs and produce conclusions. The consequent (THEN Z)
blocks of some rules are satisfied while others are not. The conclusions are combined
to form logical sums. These conclusions feed into the inference process where each
response output member function's firing strength (0 to 1) is determined.

Figure 7 - Degree of membership for the error and error-dot functions in the current
example
Data summary from previous illustrations:
INPUT DEGREE OF MEMBERSHIP
"error" = -1.0: "negative" = 0.5 and "zero" = 0.5
"error-dot" = +2.5: "zero" = 0.5 and "positive" = 0.5
ANTECEDENT & CONSEQUENT BLOCKS (e = error, er = error-dot or
error-rate)
Now referring back to the rules, plug in the membership function weights
from above. "Error" selects rules 1,2,4,5,7,8 while "error-dot" selects rules 4
through 9. "Error" and "error-dot" for all rules are combined to a logical
product (LP or AND, that is the minimum of either term). Of the nine rules
selected, only four (rules 4,5,7,8) fire or have non-zero results. This leaves
fuzzy output response magnitudes for only "Cooling" and "No_Change" which
must be inferred, combined, and defuzzified to return the actual crisp output.

In the rule list below, the following ddefinitions apply: (e)=error, (er)=error-
dot.
1. If (e < 0) AND (er < 0) then Cool 0.5 & 0.0 = 0.0
2. If (e = 0) AND (er < 0) then Heat 0.5 & 0.0 = 0.0
3. If (e > 0) AND (er < 0) then Heat 0.0 & 0.0 = 0.0
4. If (e < 0) AND (er = 0) then Cool 0.5 & 0.5 = 0.5
5. If (e = 0) AND (er = 0) then No_Chng 0.5 & 0.5 = 0.5
6. If (e > 0) AND (er = 0) then Heat 0.0 & 0.5 = 0.0
7. If (e < 0) AND (er > 0) then Cool 0.5 & 0.5 = 0.5
8. If (e = 0) AND (er > 0) then Cool 0.5 & 0.5 = 0.5
9. If (e > 0) AND (er > 0) then Heat 0.0 & 0.5 = 0.0
SUMMARY
The inputs are combined logically using the AND operator to produce output
response values for all expected inputs. The active conclusions are then
combined into a logical sum for each membership function. A firing strength
for each output membership function is computed. All that remains is to
combine these logical sums in a defuzzification process to produce the crisp
output.
REFERENCES
[17] "Estimation of Fuzzy Membership from Histograms, Information
Sciences" by B.B. Devi et al (Vol. 35, 1985, pp. 43-59).
[18] "Fuzzy Logic" by Bart Kosko and Satoru Isaka (Scientific American,
Vol. 269, July 1993, pp. 76).
[19] "Fuzzy Sets, Uncertainty, and Information", by G.J. Klir and T.A. Folger
(Prentice-Hall, Englewood Cliffs, N.J., 1988).
[20] "Industrial Applications of Fuzzy Control" ed. M. Sugeno (North-
Holland, New York, 1985).

INTRODUCTION
This is the sixth and final article in a series intended to share information and
conclude the tutorial discussion of the ongoing FL example. For the interested reader,
informational references are included.
INFERENCING
The last step completed in the example in the last article was to determine the firing
strength of each rule. It turned out that rules 4, 5, 7, and 8 each fired at 50% or 0.5
while rules 1, 2, 3, 6, and 9 did not fire at all (0% or 0.0). The logical products for
each rule must be combined or inferred (max-min'd, max-dot'd, averaged, root-sum-
squared, etc.) before being passed on to the defuzzification process for crisp output
generation. Several inference methods exist.
The MAX-MIN method tests the magnitudes of each rule and selects the highest one.
The horizontal coordinate of 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 function(s) as the output. Essentially, the
member function(s) are shrunk so that their peak equals the magnitude of their
respective function ("negative", "zero", and "positive"). This method combines the
influence of all active rules and produces a smooth, continuous output.
The AVERAGING method is another approach that works but fails to give increased
weighting to more rule votes per output member function. For 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 of 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.
DEFUZZIFICATION - GETTING BACK TO CRISP NUMBERS
The RSS method was chosen to include all contributing rules since there are so few
member functions associated with the inputs and outputs. For the ongoing example,

an error of -1.0 and an error-dot of +2.5 selects regions of the "negative" and "zero"
output membership functions. The respective output membership function strengths
(range: 0-1) from the possible rules (R1-R9) are:
"negative" = (R1^2 + R4^2 + R7^2 + R8^2) (Cooling) = (0.00^2 + 0.50^2 +
0.50^2 + 0.50^2)^.5 = 0.866
"zero" = (R5^2)^.5 = (0.50^2)^.5 (No Change) = 0.500
"positive" = (R2^2 + R3^2 + R6^2 + R9^2) (Heating) = (0.00^2 + 0.00^2 +
0.00^2 + 0.00^2)^.5 = 0.000
A "FUZZY CENTROID" ALGORITHM
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 of each output member function
are multiplied by their respective output membership function center points
and summed. Finally, this area is divided by the sum of the weighted member
function strengths and the result is taken as the crisp output. One feature to
note is that since the zero center is at zero, any zero strength will automatically
compute to zero. If the center of the zero function happened to be offset from
zero (which is likely in a real system where heating and cooling effects are not
perfectly equal), then this factor would have an influence.
(neg_center * neg_strength + zero_center * zero_strength + pos_center *
pos_strength) = OUTPUT
(neg_strength + zero_strength + pos_strength)
(-100 * 0.866 + 0 * 0.500 + 100 * 0.000) = 63.4%
(0.866 + 0.500 + 0.000)
Figure 8 - The horizontal coordinate of the centeriod is taken as the crisp
output

The horizontal coordinate of the centroid of the area marked in Figure 8 is
taken as the normalized, crisp output. This value of -63.4% (63.4% Cooling)
seems logical since the particular input conditions (Error=-1, Error-dot=+2.5)
indicate that the feedback has exceeded the command and is still increasing
therefore cooling is the expected and required system response.
TUNING AND SYSTEM ENHANCEMENT
Tuning the system can be done by changing the rule antecedents or
conclusions, changing the centers of the input and/or output membership
functions, or adding additional degrees to the input and/or output functions
such as "low", "medium", and "high" levels of "error", "error-dot", and output
response. These new levels would generate additional rules and membership
functions which would overlap with adjacent functions forming longer
"mountain ranges" of functions and responses. The techniques for doing this
systematically are a subject unto itself.
SUMMARY
The logical product of each rule is inferred to arrive at a combined magnitude
for each output membership function. This can be done by max-min, max-dot,
averaging, RSS, or other methods. Once inferred, the magnitudes are mapped
into their respective output membership functions, delineating all or part of
them. The "fuzzy centroid" of the composite area of the member functions is
computed and the final result taken as the crisp output. Tuning the system
amounts to "tweaking" the rules and membership function definition
parameters to achieve acceptable system response.
CONCLUSION
This completes this article series on FL control and one way it can be done.
The author has applied something close to this particular approach to a PC-
based temperature controller which could be the topic of a future article series
if there is interest. The PC solution has been implemented in Qbasic 1.0 and
Borland's Turbo C running on the PC using iotech hardware ADC's and
DAC's. This functionality has also been implemented using PIC's and 68HC11
processors.
Fuzzy Logic provides a completely different, unorthodox way to approach a
control problem. This method focuses on what the system should do rather
than trying to understand how it works. One can concentrate on solving the
problem rather than trying to model the system mathematically, if that is even
possible. This almost invariably leads to quicker, cheaper solutions. Once
understood, this technology is not difficult to apply and the results are usually
quite surprising and pleasing.

REFERENCES
[21] "The Coming Age of Fuzzy Logic" Proceedings of the 1989 IFSA
Congress, J.C. Bezdek, ed. (University of Washington, Seattle, WA 1989).
[22] "The Current Mode Fuzzy Logic Integrated Circuits Fabricated by
Standard CMOS Process" (IEEE Trans. on Computers, Vol. C-35, No. 2, pp.
161-7, February 1986).
[23] "Fuzzy Logic - From Concept to Implementation", (Application Note
EDU01V10-0193. ((c) 1993 by Aptronix, Inc, (408) 428-1888).
[24] "Fuzzy Motor Controller" Huntington Technical Brief, D. Brubaker ed.
(April 1992, No. 25, Menlo Park, CA 1992).

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