Collaborative Learning using Fuzzy Inferencing
(CLIFF) Part 1
Sponsored by: The National Science Foundation
Grant ID No: DUE-0756921
University of Cincinnati AY-REU Program
2012 - 2013
Sophia M. Mitchell
Junior Aerospace Engineering ACCEND Student, College of Engineering and Applied Science,
University of Cincinnati, Cincinnati, Ohio, 45219
Dr. Kelly Cohen
Mentor and Professor, Department of Aerospace Engineering, College of Engineering and
Applied Science
University of Cincinnati, Cincinnati, Ohio, 45219
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I. Goals and Objectives
The overall research endeavor is to use type-2 fuzzy logic to create a team of robots that
learn from their environment to work effectively and collaborate within an uncertain spatiotemporal environment. This overall objective will be met through completing three major tasks:
1. Developing type-2 fuzzy logic software in the MATLAB environment using theoretical
literature as a guide. This software will then be verified through comparison and
improvement of a determined benchmark problem.
2. Implementation of this new type-2 fuzzy logic software into the Collaborative Robotic
PONG environment developed in FLIP.
3. Create a robotic coach that uses the new type-2 fuzzy software to learn from its opponent
and use this intelligence to alter the logic of the players, allowing them to develop over
time into a team that can beat its opponent. Both ideas of using fuzzy logic to alter fuzzy
logic as well as having it “learn” over time from its environment will be two interesting,
novel and useful developments.
In the current research endeavor, the first task is being attempted.
II. Research Methods
A. Development of the fuzzy system
Type-2 fuzzy logic is unique in that its membership functions not only come in different
shapes but also in different sizes depending on if the mean or standard deviation is being varied
and if so by how much. Therefore before performing any actual coding it was necessary to learn
about the different types of type-2 fuzzy logic that exists, and from that understanding decide on
what type of type-2 membership function would be used in the simulation.
First, it was decided that a Gaussian primary membership function would make the most
sense to use, as its gradual nature would allow for very smooth membership function outputs;
something that is important for an effective and efficient decision making system. Research
conducted by Baklouti, Robert and Adel [5] suggested that a type-2 system that varies in
standard deviation is more effective in controlling a system than one which varies by mean, so it
was decided to use a Gaussian function that varies along its standard deviation as can be seen in
equation 1 below.
(1)
Finally, through studying a benchmark problem provided by Dongrui and Woei [2] it was seen
that using a singleton type-2 system would be best for comparison with a type-1 system as the
outputs are more similar. Finally it was decided that the type-2 system would be an interval
system, which means that along the third dimension the type-2 membership functions would be
constant which allows for a more simplistic function.
B. Creation of MATLAB Code
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Once the type of type-2 logic to use was
decided upon, it was then necessary to create a
toolbox in MATLAB for the type of type-2 logic
that would be used. Several matlab functions had to
be created including the functions that determine
the shape and effects of each type-2 membership
function, as well as the function to communicate
between the system being controlled and the type-2
system.
The MATLAB code works as follows: an input
Figure 1: Schematic diagram of the dynamic system
is taken into the fuzzy system where it is then
being controlled for the benchmark problem. Note:
fuzzified by the type-2 fuzzy membership
probes do not restrict or effect water flow.
functions. These membership functions are more
complex than type-1 functions since a crisp input point results in a fuzzified set of numbers.
Therefore a special MATLAB function was necessary to determine (1) overlapping membership
function results and (2) what rules would be fired as both of these steps are extremely different
and much more complex than a similar situation in a type-1 system. The firing of the rules
creates an array which is then type reduced; a step that is unique to type-2 fuzzy logic. Type
reduction takes the array from the rules and turns it back into a set of numbers that can then be
defuzzified to produce a crisp output from the system. The system was then tested manually with
a few points to determine if the results it produced at least made sense before attempting to have
it solve an entire system iteratively in the benchmark problem. These results are discussed in the
results and discussion of this paper.
Creating the entire toolbox for just this one type of type-2 fuzzy logic was extremely time
consuming, but very rewarding as it provided the researcher with a much better understanding of
the inter-workings of type-2 fuzzy logic. Developing the toolbox to be able to compute more
types of type-2 systems would be a very useful future endeavor.
C. Verification with the Benchmark Problem
In order to fully verify that the created type-2 system and toolbox were in true working order,
it was necessary to re-create a simulation environment for testing that had results for comparison.
For this, the problem presented by Dongrei and Woei [2] was used, as it had a straightforward
problem scenario, included the dynamic equations that control the dynamic system and had
interesting results for comparison.
To start, the dynamic scenario had to be re-created in a MATLAB environment. The
presented scenario was that the type-2 system was to control the flow rate of water being pumped
into a set of drums which are divided by a wall. There are holes in the bottom of both drums as
well as a space in the wall for water to flow into the second drum, as can be seen in Figure 6. In
order to simulate this dynamic system, some equation manipulation was necessary in order to use
the equations given in the text with the MATLAB ode45 solver. The ode45 solver uses a RungeKutta method in order to bring a set of differential equations to their solution iteratively. The
following differential equations (Equations 2 and 3) were given to describe the system, where H1
is the height of the water in drum one, and H2 is the height of the water in drum two.
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𝑑𝐻1
𝐴1
𝐴2
𝑑𝑡
𝑑𝐻2
𝑑𝑡
= 𝑄1 −∝1 √𝐻1 +∝3 √𝐻1 − 𝐻2
(2)
= −∝2 √𝐻2 +∝3 √𝐻1 − 𝐻2
(3)
A was the area of each drum, while α was a constant for the amount of water lost through holes
in the bottom of the drums. These constants could be defined as follows: A1=A2= 36.52 cm2, α1=
α2=5.6186 and α3=10.
Each of these equations was then used to derive the following differential equations
which define the height of the water in drum one and the change in that height in terms of the
height of the water in drum two. This derivation was necessary as the optimization of the
problem was to fill drum two to a certain height; not drum one.
𝑑𝐻1
2
∝
3
(4)
= 2 [𝐴2 𝐻̇2 +∝2 √𝐻2 ] ∙ (𝐴2 𝐻̈ + (𝐻2 ) ⁄2 )
𝑑𝑡
2
∝3
+ 𝐻̇2
1
𝐻1 =
[𝐴 𝐻̇ +∝2 √𝐻2 ]2
(5)
∝3 2 2
+ 𝐻2
Equations 4 and 5 were then combined and inserted
into equation two to create equation 6.
2
𝑑𝐻
∝
𝐹 = [ 𝑑𝑡1 − 𝐻̇2 ] − 23 ∙ (𝐴
1
̇
𝐻
+∝
2 2
2 √𝐻2)
(6)
From this equation the following solution for Ḧ was then derived.
𝐻̈ =
−∝2
2𝐴2
(𝐻2 )
3⁄
2
+𝐹
(7)
The MATLAB model could then use equation 7 in the ODE45 solver to simulate the solution for
the benchmark problem.
This whole system was tested with a type-1 system to ensure its reliability, as can be
found in the results.
III. Results
The results produced by this study are as follows. First the software was tested to see if it
created useful Gaussian singleton interval type-2 fuzzy membership functions. The system
created both input membership functions which can be found in Figures 3 and 4 below.
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Figure 2: Produced type-2 membership functions for input variable E
Figure 3: Produced type-2 membership functions for input variable Edot
The creation of the membership functions was considered a success, and when tested
manually with inputs the system produced results from these membership functions that made
sense for the filling and control of water level in a drum, meaning that a low amount of water
resulted in a higher filling rate, and a situation in which the drum was overflowing resulted in no
fill rate (so the water would drain out).
Before comparing this type-2 system to the benchmark problem, it was first necessary to
solve the problem using the more familiar type-1 system to ensure that the differential equation
solver (ode45) was being implemented correctly in MATLAB to produce correct results. This
verification would then ensure that any changes or discrepancies between data would be a result
of a change in the type-2 system and not because of an error in the simulation. Figures 5 and 6
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are plots of the benchmark problem results obtained by the previous researcher, and results
obtained by the system that was created for this project.
Figure 4: Results obtained by the benchmark problem paper
Figure 5: Results obtained by this study
One can see that both systems fill the drum to the target height of 15cm; however, the
results for this project do not overshoot the limit. It can also be seen that both results fill the
drum in roughly the same amount of time.
IV. Computer Software and Training
This project utilized a University of Cincinnati licesnsed version of MATLAB 2012. The
student was already quite familiar with this software coming into the research project and
therefore did not need any formal training on its basic use.
However, a type-2 fuzzy logic toolbox was needed in order to complete the research. As
this toolbox did not exist at the time of the research, the student took it upon them to create this
toolbox, which is generally outlined in the methods section of this paper. In order to create this
toolbox the student had to self-teach type-2 fuzzy logic with the help of their mentor as no
classes and few textbooks exist on the subject.
V. Discussion and Conclusion
The data suggests two things: the type-2 MATLAB fuzzy logic toolbox that has been
created during this research gives results which make sense, and that the simulation created for
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the benchmark problem to verify this type-2 system works. Sadly, due to time constraints the
type-2 system could not yet be tested for results. Nevertheless, some interesting ideas did come
from the results that were produced.
First, it is interesting to see that with a simple type-1 system, this research project was
able to produce results with a type-1 system that were almost better than the results produced by
the type-2 system in the benchmark problem. This was a very interesting note because the whole
argument for using type-2 fuzzy logic is that it is supposed to perform better than a type-1
system. If the type-1 system produced by this project could be improved and then compared, this
whole reasoning for using type-2 falls apart, unless that simply means a much better type-2
system could also be created. Therefore future work will include not only improving the type-1
system to see what results could be obtained, but also improving the type-2 system to see how
much better it really is.
In conclusion the creation of a type-2 fuzzy logic toolbox for a Gaussian singleton
interval type-2 fuzzy Inferencing system was considered a success, as well as the creation of the
simulation environment for the benchmark problem. Future work will include several goals:
finding the best possible solution to the problem using both type-1 and type-2 fuzzy logic
systems, as well as implementing the type-2 logic into the aforementioned fuzzy PONG game if
said logic does prove to be a better solution than type-1.
VI. Acknowledgements
I would like to thank my mentor, Dr. Kelly Cohen, for all of his wisdom and support
while I perused this endeavor. I would also like to thank the University of Cincinnati AY-REU
program as well as NSF for funding me and providing me with a framework with which I could
conduct this research. Finally I would like to thank my family and friends for the moral support
as I worked through teaching myself type-2 fuzzy logic.
VII. References
[1] Baklouti, Nesrine, Robert John, and Adel Alimi. "Interval Type-2 Fuzzy Logic Control of
Mobile Robots."Journal of Intelligent Learning Systems and Applications. 4.November 2012
(2012): 291-302. Web. 18 Feb. 2 013.
[2] Dongrui Wu, Woei Wan Tan, “Genetic learning and performance evaluation of interval type2 fuzzy logic controllers, Engineering Applications of Artificial Intelligence”, Volume 19,
Issue 8, December 2006, Pages 829-841, ISSN 0952-1976, 10.1016/j.engappai.2005.12.011.
(http://www.sciencedirect.com/science/article/pii/S0952197606000388)
[3] Mendel, Jerry. Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New
Directions. Upper Saddle River, NJ: Prentice Hall PTR, 2001. Print.
[4] Castillo, Oscar, and Patricia Melin. Type-2 Fuzzy Logic: Theory and Applications. 1.
Heidelberg: Springer, 2008. Print.
[5] Castillo, Oscar. Type-2 Fuzzy Logic in Intelligent Control Applications. 1. Heidelberg:
Springer, 2012. eBook.
VIII. Appendix A - Pictures
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Figure 6: Brainstorming area
Figure 7: Sophia, working on her research
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Figure 8: Sophia, working on her research
Figure 9: A screen shot of a sample output from the research project