O'Hara,
Steve Fuzzy Systems, 10/29/07
Fuzzy Logic and Fuzzy Expert Systems
Steve O'Hara UTSA CS 7123
Steve O'Hara, Fuzzy Systems, 10/29/07
Preview
Fuzzy sets and logic must be viewed as a formal mathematical theory for the representation of uncertainty. Uncertainty is crucial for the management of real systems: if you had to park your car PRECISELY in one place, it would not be possible. Fuzzy logic is a mathematical formalism, and a membership function is a precise number. What's crucial to realize is that fuzzy logic is a logic OF fuzziness, not a logic which is ITSELF fuzzy. But that's OK: just as the laws of probability are not random, so the laws of fuzziness are not vague.
Source: http://www.cs.cmu.edu/Groups/AI/html/faqs/ai/fuzzy/part1/faqdoc8.html
Steve O'Hara, Fuzzy Systems, 10/29/07
Sources
Compendium of many sources: Main source for today:
http://www.austinlinks.com/Fuzzy/index.html
http://www.esru.strath.ac.uk/Reference/concepts/fuzz y/fuzzy.htm
"The" seminal paper:
Zadeh, L.A., "Fuzzy Sets," Information and Control, Vol. 8, pp. 338353, 1965.
Steve O'Hara, Fuzzy Systems, 10/29/07
What is Fuzzy Logic?
Fuzzy Logic. Logic that allows values other than "true" and "false". Every statement has a numerical range from 0.0 (false) to 1.0 (true). Fuzzy Set Membership. A given element can be member of a set with a numerical range of 0.0 (not at all a member) to 1.0 (completely a member)
Steve O'Hara, Fuzzy Systems, 10/29/07
Example
What is a "big" house?
What is an "expensive" house? What is an "old" house?
"small" might be under 1800 sf "medium" might be 1500 to 2200 sf "large" might be 2000 to 3000 sf "huge" might be over 2800 sf
What do you call a 2100 sf house?
"Mediumlargeish"? partially "medium" AND partially "large"
Steve O'Hara, Fuzzy Systems, 10/29/07
Is it Just Probability?
If a house is partially "medium" and partially "large", can it be quantified?
Does this mean there is a 60% chance it is medium and a 30% chance is large?
Can it be 60% of a member in the "medium" category, and 30% a member in the "large" category at the same time?
Steve O'Hara, Fuzzy Systems, 10/29/07
Membership Functions #1
House Sizes
"piecewise linear" extends easily to 2d (planes) and larger dimensions
100%
small
0%
medium
large
huge
1500 sf
2100 sf
2700 sf
3300 sf
Steve O'Hara, Fuzzy Systems, 10/29/07
Membership Functions #2
Radial Basis (Gaussian) Sigmoid (Logistic)
Steve O'Hara, Fuzzy Systems, 10/29/07
Fuzzy "Logic"
For the fuzzy variables A and B:
A
not A = 1 A
B
A and B = min (A, B)
A or B = max (A, B)
Steve O'Hara, Fuzzy Systems, 10/29/07
Fuzzy Qualifiers
Hedges for extreme values
VERY <x> = more extreme SOMEWHAT <x> = less extreme
Powering
Steve O'Hara, Fuzzy Systems, 10/29/07
Shifting
Fuzzy Expert Systems
Fuzzy Logic and Expert Systems Rulebased System Requires Domain Expertise Tuning is Difficult Lowdimensional Problems Many Rules fire at the same time
Steve O'Hara, Fuzzy Systems, 10/29/07
Inverted Pendulum
http://cnx.org/content/m12977/latest/
O'Hara, Steve Fuzzy Systems, 10/29/07
Inverted Pendulum Control
Two Inputs:
One Output:
Steve O'Hara, Fuzzy Systems, 10/29/07
Inverted Pendulum Rules
SPEED = fn (ANGLE, ANGULAR VELOCITY) ANGLE
NH NL Z NH NL NH NL Z Z PL PH
Steve O'Hara, Fuzzy Systems, 10/29/07
PL
PH
ANGULAR VELOCITY
NH NL Z PL PH
Z PL PH
Sample Inputs for Inv. Pend.
Steve O'Hara, Fuzzy Systems, 10/29/07
Sample Rule "Firing"
Steve O'Hara, Fuzzy Systems, 10/29/07
Four Rules "Fire"
if angle is zero and angular velocity is zero then speed is zero
if angle is positive low and angular velocity is zero then speed is PL
}
if angle is zero and angular velocity is NL then speed is NL
if angle is positive low and angular velocity is NL then speed is zero
Steve O'Hara, Fuzzy Systems, 10/29/07
A Few Fuzzy Applications
Automatic control of dam gates for hydroelectricpowerplants (Tokyo Electric Power) Simplified control of robots (Hirota, Fuji Electric, Toshiba, Omron) Camera aiming for the telecast of sporting events (Omron) Substitution of an expert for the assessment of stock exchange activities (Yamaichi, Hitachi) Preventing unwanted temperature fluctuations in airconditioning systems (Mitsubishi, Sharp) Efficient and stable control of carengines (Nissan) Cruisecontrol for automobiles (Nissan, Subaru) Improved efficiency and optimized function of industrial control applications (Aptronix, Omron, Meiden, Sha, Micom, Mitsubishi, NisshinDenki, OkuElectronics) Positioning of wafersteppers in the production of semiconductors (Canon) Optimized planning of bus timetables (Toshiba, NipponSystem, KeihanExpress) Archiving system for documents (Mitsubishi Elec.) Prediction system for early recognition of earthquakes (Inst. of Seismology Bureau of Metrology, Japan) Medicine technology: cancer diagnosis (Kawasaki Medical School) Combination of Fuzzy Logic and Neural Nets (Matsushita) Recognition of handwritten symbols with ...