ORIGINAL_ARTICLE
Cover Special Issue vol. 13, no. 7, Decemberr 2016
http://ijfs.usb.ac.ir/article_2949_c2a2d61733497224fc24c7f242f74085.pdf
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10.22111/ijfs.2016.2949
ORIGINAL_ARTICLE
SYSTEM MODELING WITH FUZZY MODELS: FUNDAMENTAL DEVELOPMENTS AND PERSPECTIVES
In this study, we offer a general view at the area of fuzzy modeling and fuzzymodels, identify the visible development phases and elaborate on a new and promisingdirections of system modeling by introducing a concept of granular models. Granularmodels, especially granular fuzzy models constitute an important generalization of existingfuzzy models and, in contrast to the existing models, generate results in the form ofinformation granules (such as intervals, fuzzy sets, rough sets and others). We present arationale and deliver some key motivating arguments behind the emergence of granularmodels and discuss their underlying design process. Central to the development of granularmodels are granular spaces, namely a granular space of parameters of the models and agranular input space. The development of the granular model is completed through anoptimal allocation of information granularity, which optimizes criteria of coverage andspecificity of granular information. The emergence of granular models of type-2 and type-n,in general, is discussed along with an elaboration on their formation. It is shown thatachieving a sound coverage-specificity tradeoff (compromise) is of paramount relevance inthe realization of the granular models.
http://ijfs.usb.ac.ir/article_2940_16d7b0c0bed5ba69ff2b3b46e7f4336c.pdf
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14
10.22111/ijfs.2016.2940
Fuzzy models
Granular computing
information granules of higher type
Granular spaces
WITOLD
PEDRYCZ
pedrycz@ee.ualberta.ca
true
1
DEPARTMENT OF ELECTRICAL & COMPUTER ENGINEERING
UNIVERSITY OF ALBERTA EDMONTON T6R 2V4 AB CANADA, DEPARTMENT OF ELECTRICAL & COMPUTER ENGINEERING FACULTY OF ENGINEERING KING ABDULAZIZ
UNIVERSITY JEDDAH, 21589 SAUDI ARABIA AND SYSTEMS RESEARCH INSTITUTE POLISH
ACADEMY OF SCIENCES, WARSAW POLAND.
DEPARTMENT OF ELECTRICAL & COMPUTER ENGINEERING
UNIVERSITY OF ALBERTA EDMONTON T6R 2V4 AB CANADA, DEPARTMENT OF ELECTRICAL & COMPUTER ENGINEERING FACULTY OF ENGINEERING KING ABDULAZIZ
UNIVERSITY JEDDAH, 21589 SAUDI ARABIA AND SYSTEMS RESEARCH INSTITUTE POLISH
ACADEMY OF SCIENCES, WARSAW POLAND.
DEPARTMENT OF ELECTRICAL & COMPUTER ENGINEERING
UNIVERSITY OF ALBERTA EDMONTON T6R 2V4 AB CANADA, DEPARTMENT OF ELECTRICAL & COMPUTER ENGINEERING FACULTY OF ENGINEERING KING ABDULAZIZ
UNIVERSITY JEDDAH, 21589 SAUDI ARABIA AND SYSTEMS RESEARCH INSTITUTE POLISH
ACADEMY OF SCIENCES, WARSAW POLAND.
LEAD_AUTHOR
[1] R. Alcala, M. J. Gacto and F. Herrera, A fast and scalable multiobjective genetic fuzzy system for
1
linguistic fuzzy modeling in high-dimensional regression problems, IEEE Trans. Fuzzy Systems 19 (2011),
2
666–681.
3
[2] J. M. Alonso, L. Magdalena and S. Guillaume, Linguistic knowledge base simplification regarding
4
accuracy and interpretability, Mathware Soft Comput., 13 (2006), 203–216.
5
[3] J. C. Bezdek, Pattern recognition with fuzzy objective function algorithms plenum press, N. York, 1981.
6
[4] C. Hwang and F. C. H Rhee, Uncertain fuzzy clustering: Interval Type-2 fuzzy approach to C-Means,
7
IEEE Trans. on Fuzzy Systems, 15 (12) (2007), 107-120.
8
[5] Y. Jin, Fuzzy modeling of high-dimensional systems: complexity reduction and interpretability
9
improvement, IEEE Trans. Fuzzy Systems, 8 (2000), 212–221.
10
[6] T. A. Johansen and R. Babuska, Multiobjective identification of Takagi-Sugeno fuzzy models, IEEE
11
Trans. Fuzzy Systems, 11 (2003), 847–860.
12
[7] R. Mikut, J. Jäkel and L. Gröll, Interpretability issues in data-based learning of fuzzy systems, Fuzzy
13
Sets & Systems, 150 (2005), 179–197.
14
[8] W. Pedrycz, Granular computing - The emerging paradigm, Journal of Uncertain Systems 1(1) (2007),
15
[9] W. Pedrycz, Granular computing: analysis and design of intelligent systems CRC press/francis taylor,
16
Boca Raton, 2013.
17
[10] W. Pedrycz and A. Bargiela, An optimization of allocation of information granularity in the
18
interpretation of data structures: toward granular fuzzy clustering, IEEE Trans on Systems, Man, and
19
Cybernetics, Part B, 42 (2012), 582-590.
20
[11] W. Pedrycz and W. Homenda, Building the fundamentals of granular computing: A principle of
21
justifiable granularity, Applied Soft Computing, 13 (2013), 4209-4218.
22
[12] W. Pedrycz, Knowledge-Based Fuzzy Clustering John Wiley, N. York, 2005.
23
[13] R. R. Yager, Ordinal measures of specificity, Int. J. of General Systems, 17 (1990), 57-72.
24
[14] J. T. Yao, A. V. Vasilakos and W. Pedrycz, Granular computing: perspectives and challenges, IEEE
25
Transactions on Cybernetics, 43(6) (2013), 1977 – 1989.
26
[15] L. A. Zadeh, Towards a theory of fuzzy information granulation and its centrality in human reasoning
27
and fuzzy logic, Fuzzy Sets and Systems, 90 (1997), 111-117.
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29
172 (2005), 1- 40.
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[18] S. M. Zhou and J. Q. Gan, Low-level interpretability and high-level interpretability: a unified view of
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data-driven interpretable fuzzy system modelling, Fuzzy Sets and Systems, 159(23) (2008), 3091–3131.
33
[19] B. Zhu, C. Z. He, P. Liatsis and X. Y. Li, A GMDH-based fuzzy modeling approach for constructing TS
34
model, Fuzzy Sets and Systems, 189 (2012), 19–29.
35
ORIGINAL_ARTICLE
ON THE COMPATIBILITY OF A CRISP RELATION WITH A FUZZY EQUIVALENCE RELATION
In a recent paper, De Baets et al. have characterized the fuzzytolerance and fuzzy equivalence relations that a given strict order relation iscompatible with. In this paper, we generalize this characterization by consideringan arbitrary (crisp) relation instead of a strict order relation, while payingattention to the particular cases of a reflexive or irreflexive relation. The reasoninglargely draws upon the notion of the clone relation of a (crisp) relation,introduced recently by Bouremel et al., and the partition of this clone relationin terms of three different types of pairs of clones. More specifically, reflexive related clones and irreflexive unrelated clones turn out to play a key role in thecharacterization of the fuzzy tolerance and fuzzy equivalence relations that agiven (crisp) relation is compatible with.
http://ijfs.usb.ac.ir/article_2941_e66348aeee3c2b2b3fb70d708b5956cd.pdf
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10.22111/ijfs.2016.2941
Crisp relation
Fuzzy relation
Clone relation
Compatibility
Tolerance relation
Equivalence relation
B. De
Baets
true
1
KERMIT, Department of Mathematical Modelling, Statistics and
Bioinformatics, Ghent University, Coupure links 653, B-9000, Gent, Belgium
KERMIT, Department of Mathematical Modelling, Statistics and
Bioinformatics, Ghent University, Coupure links 653, B-9000, Gent, Belgium
KERMIT, Department of Mathematical Modelling, Statistics and
Bioinformatics, Ghent University, Coupure links 653, B-9000, Gent, Belgium
LEAD_AUTHOR
H.
Bouremel
true
2
Department of Mathematics, Faculty of Mathematics and Informatics,
Med Boudiaf University of Msila, P.O. Box 166 Ichbilia, Msila 28000, Algeria
Department of Mathematics, Faculty of Mathematics and Informatics,
Med Boudiaf University of Msila, P.O. Box 166 Ichbilia, Msila 28000, Algeria
Department of Mathematics, Faculty of Mathematics and Informatics,
Med Boudiaf University of Msila, P.O. Box 166 Ichbilia, Msila 28000, Algeria
AUTHOR
L.
Zedam
l.zedam@yahoo.fr
true
3
Department of Mathematics, Faculty of Mathematics and Informatics,
Med Boudiaf University of Msila, P.O. Box 166 Ichbilia, Msila 28000, Algeria
Department of Mathematics, Faculty of Mathematics and Informatics,
Med Boudiaf University of Msila, P.O. Box 166 Ichbilia, Msila 28000, Algeria
Department of Mathematics, Faculty of Mathematics and Informatics,
Med Boudiaf University of Msila, P.O. Box 166 Ichbilia, Msila 28000, Algeria
AUTHOR
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1
Publishers/Plenum Publishers, New York, 2002.
2
[2] R. Belohlavek, Concept lattices and order in fuzzy logic, Ann. Pure Appl. Logic, 128(1-3)
3
(2004), 277-298.
4
[3] U. Bodenhofer, A new approach to fuzzy orderings, Tatra Mt Math Publ, 16(1) (1999), 1-9.
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[4] U. Bodenhofer, Representations and constructions of similarity-based fuzzy orderings, Fuzzy
6
Sets and Systems, 137(1) (2003), 113-136.
7
[5] U. Bodenhofer and M. Demirci, Strict fuzzy orderings in a similarity-based setting, Proc. of
8
EUSFLAT-LFA 2005, Barcelona, Spain, (2005), 297-302.
9
[6] U. Bodenhofer, B. De Baets and J. Fodor, A compendium of fuzzy weak orders: Representa-
10
tions and constructions, Fuzzy Sets and Systems, 158(8) (2007), 811-829.
11
[7] H. Bouremel, R. Perez-Fernandez, L. Zedam and B. De Baets, The clone relation of a binary
12
relation, Information Sciences, doi: 10.1016/j.ins.2016.12.008, accepted.
13
[8] A. Burusco and R. Fuentes-Gonzales, The study of the L-fuzzy concept lattice, Mathware and
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Soft Computing, 3 (1994), 209-218.
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[9] B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, Second ed., Cambridge
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University Press, Cambridge, 2002.
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104(1) (1999), 61-75.
19
[11] B. De Baets, L. Zedam and A. Kheniche, A clone-based representation of the fuzzy tolerance
20
or equivalence relations a strict order relation is compatible with, Fuzzy Sets and Systems,
21
296 (2016), 35-50.
22
[12] M. Demirci, Foundations of fuzzy functions and vague algebra based on many-valued equiva-
23
lence relations, Part I: fuzzy functions and their applications, Internat. J. General Systems,
24
32(2) (2003), 123-155.
25
[13] M. Demirci, A theory of vague lattices based on many-valued equivalence relations|I: general
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representation results, Fuzzy Sets and Systems, 151(3) (2005), 437-472.
27
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35(2) (1985), 133-144.
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[16] A. Kheniche, B. De Baets and L. Zedam, Compatibility of fuzzy relations, International
31
Journal of Intelligent Systems, 31(3) (2015), 240-256.
32
[17] P. Martinek, Completely lattice L-ordered sets with and without L-equality, Fuzzy Sets and
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Systems, 166(1) (2011), 44-55.
34
[18] I. Perlieva, Normal forms in BL-algebra and their contribution to universal approximation
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of functions, Fuzzy Sets and Systems, 143(1) (2004), 111-127.
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[19] I. Perlieva, Fuzzy function: theoretical and practical point of view, Proc. EUSFLAT 2011,
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Aix-les-Bains, France, (2011), 480-486.
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[20] I. Perlieva, D. Dubois, H. Prade, F. Esteva, L. Godo and P. Hoddakova, Interpolation of
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fuzzy data: Analytical approach and overview, Fuzzy Sets and Systems, 192 (2012), 134-158.
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[23] K. Wang and B. Zhao, Join-completions of L-ordered sets, Fuzzy Sets and Systems, 199
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(2012), 92-107.
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(2009), 2275-2291.
47
ORIGINAL_ARTICLE
DC-DC CONVERTER WITH FUZZY CONTROLLER FOR SOLAR CELL APPLICATIONS ON MOBILE ROBOTS
Emerging technologies increase the needs on self efficient mobile robotic applications that bring a new concern of sustainable and continuous power supply for the robotic platforms. This paper covers the various techniques and technologies used to design a solar powered robot, exploring the currently available products, software and limitations to this application. The main aim is to integrate a fuzzy logic based charging system which allows the batteries to be charged from solar panels, wall outlet, and a deploy-able solar charging station. The goal of this paper is to summarize the tested methods and results to expedite future researchers in the correct direction. This paper will cover only up to the design of the DC-DC converter and simulation, as further work is still pending implementation on actual hardware.Simulations results are provided to evaluate the feasibility of the paper for future implementations.
http://ijfs.usb.ac.ir/article_2942_93977f91069127aa27550c83972e06e9.pdf
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10.22111/ijfs.2016.2942
Solar
Renewable
LiPo
Lithium Polymer
MPPT
Robotics
Fuzzy controller
Energy
J.
Cruz-Lambert
true
1
Electrical and Computer Engineering Department, The University
of Texas at San Antonio, San Antonio, Texas, USA
Electrical and Computer Engineering Department, The University
of Texas at San Antonio, San Antonio, Texas, USA
Electrical and Computer Engineering Department, The University
of Texas at San Antonio, San Antonio, Texas, USA
AUTHOR
P.
Benavidez
true
2
Electrical and Computer Engineering Department, The University
of Texas at San Antonio, San Antonio, Texas, USA
Electrical and Computer Engineering Department, The University
of Texas at San Antonio, San Antonio, Texas, USA
Electrical and Computer Engineering Department, The University
of Texas at San Antonio, San Antonio, Texas, USA
LEAD_AUTHOR
J.
Ortiz
true
3
Electrical and Computer Engineering Department, The University of Texas
at San Antonio, San Antonio, Texas, USA
Electrical and Computer Engineering Department, The University of Texas
at San Antonio, San Antonio, Texas, USA
Electrical and Computer Engineering Department, The University of Texas
at San Antonio, San Antonio, Texas, USA
AUTHOR
N.
Gallardo
true
4
Electrical and Computer Engineering Department, The University of
Texas at San Antonio, San Antonio, Texas, USA
Electrical and Computer Engineering Department, The University of
Texas at San Antonio, San Antonio, Texas, USA
Electrical and Computer Engineering Department, The University of
Texas at San Antonio, San Antonio, Texas, USA
AUTHOR
B. A.
Erol
true
5
Electrical and Computer Engineering Department, The University of
Texas at San Antonio, San Antonio, Texas, USA
Electrical and Computer Engineering Department, The University of
Texas at San Antonio, San Antonio, Texas, USA
Electrical and Computer Engineering Department, The University of
Texas at San Antonio, San Antonio, Texas, USA
AUTHOR
J.
Richey
true
6
Electrical and Computer Engineering Department, The University of
Texas at San Antonio, San Antonio, Texas, USA
Electrical and Computer Engineering Department, The University of
Texas at San Antonio, San Antonio, Texas, USA
Electrical and Computer Engineering Department, The University of
Texas at San Antonio, San Antonio, Texas, USA
AUTHOR
S.
Morris
true
7
Electrical and Computer Engineering Department, The University of
Texas at San Antonio, San Antonio, Texas, USA
Electrical and Computer Engineering Department, The University of
Texas at San Antonio, San Antonio, Texas, USA
Electrical and Computer Engineering Department, The University of
Texas at San Antonio, San Antonio, Texas, USA
AUTHOR
M.
Jamshidi
true
8
Electrical and Computer Engineering Department, The University of
Texas at San Antonio, San Antonio, Texas, USA
Electrical and Computer Engineering Department, The University of
Texas at San Antonio, San Antonio, Texas, USA
Electrical and Computer Engineering Department, The University of
Texas at San Antonio, San Antonio, Texas, USA
AUTHOR
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images/doc8088.pdf, 2016.
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(2013), 89{98.
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mer battery, In Technological Advances in Electrical, Electronics and Computer Engineering
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(TAEECE), 2015 Third International Conference on, IEEE, (2015), 33{38.
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power management for solar-powered unmanned ground vehicles, 2015 IEEE International
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[9] J. Leitner and W. Chamberlain and D. G. Dansereau and M. Dunbabin and M. Eich and
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T. Peynot and J. Roberts and R. Russell and N. Snderhauf, LunaRoo: Designing a hopping
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lunar science payload, 2016 IEEE Aerospace Conference, (2016), 1-12.
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[10] J. H. Lever, A. Streeter and LR. Ray, Performance of a solar-powered robot for polar in-
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strument networks, Proceedings of the 2006 IEEE International Conference on Robotics and
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Automation, 2006, ICRA 2006, (2006), 4252{4257.
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[11] Ned Mohan, Power Electronics: A First Course, Wiley, 2012.
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[12] Projects/avr bc100.git., Available: http://git.kpe.io/?p=avr bc100.git, 2016.
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Autonomous rover for polar science support and remote sensing, 2014 IEEE Geoscience and
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31
ORIGINAL_ARTICLE
A NOTE TO INTERPRETABLE FUZZY MODELS AND THEIR LEARNING
In this paper we turn the attention to a well developed theory of fuzzy/lin\-guis\-tic models that are interpretable and, moreover, can be learned from the data.We present four different situations demonstrating both interpretability as well as learning abilities of these models.
http://ijfs.usb.ac.ir/article_2943_96ab638fc5e0fd03be6c8ba7e35c5e6f.pdf
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10.22111/ijfs.2016.2943
Fuzzy Natural Logic
Perception-based logical deduction
Learning. } newlineindent{footnotesize {The paper has been supported by the project IT4I XS (LQ1602)
Vilem
Novak
true
1
University of Ostrava, Institute for Research and Applications of
Fuzzy Modeling, NSC IT4Innovations, 30. dubna 22, 701 03 Ostrava 1, Czech Republic
University of Ostrava, Institute for Research and Applications of
Fuzzy Modeling, NSC IT4Innovations, 30. dubna 22, 701 03 Ostrava 1, Czech Republic
University of Ostrava, Institute for Research and Applications of
Fuzzy Modeling, NSC IT4Innovations, 30. dubna 22, 701 03 Ostrava 1, Czech Republic
LEAD_AUTHOR
[1] A. Dvorak, H. Habiballa, V. Novak and V. Pavliska, The software package LFLC 2000 | its
1
specicity, recent and perspective applications, Computers in Industry, 51 (2003), 269{280.
2
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3
Fuzzy/Linguistic IF-THEN Rules under Perception-based Logical Deduction Inference, Fuzzy
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Sets and Systems.
5
[3] E. Hullermeier, Does machine learning need fuzzy logic?, Fuzzy Sets and Systems, 281 (2015),
6
[4] V. Novak, Linguistically oriented fuzzy logic controller, in: Proc. of the 2nd Int. Conf. On
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Fuzzy Logic and Neural Networks IIZUKA'92, Fuzzy Logic Systems Institute, Iizuka, (1992),
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[5] V. Novak, Fuzzy relation equations with words, in: M. Nikravesh, L. Zadeh, V. Korotkikh
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(Eds.), Fuzzy Partial Dierential Equations and Relational Equations, Springer, Berlin,
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(2004), 167{185.
11
[6] V. Novak, Perception-based logical deduction, in: B. Reusch (Ed.), Computational Intelligence,
12
Theory and Applications, Springer, Berlin, (2005), 237{250.
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[7] V. Novak, Mathematical fuzzy logic in modeling of natural language semantics, in: P. Wang,
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D. Ruan, E. Kerre (Eds.), Fuzzy Logic { A Spectrum of Theoretical & Practical Issues,
15
Elsevier, Berlin, (2007), 145{182.
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[8] V. Novak, A comprehensive theory of trichotomous evaluative linguistic expressions, Fuzzy
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Sets and Systems, 159 (22) (2008), 2939{2969.
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281 (2015), 81{87.
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E. Seising, R.and Trillas, J. Kacprzyk (Eds.), Fuzzy Logic: Towards the Future, Springer,
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(2015), 137{165.
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[12] V. Novak, Linguistic characterization of time series, Fuzzy Sets and Systems, 285 (2016),
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[13] V. Novak and J. Kova, Linguistic IF-THEN rules in large scale application of fuzzy control,
26
in: R. Da, E. Kerre (Eds.), Fuzzy If-Then Rules in Computational Intelligence: Theory and
27
Applications, Kluwer Academic Publishers, Boston, (2000), 223{241.
28
[14] V. Novak and S. Lehmke, Logical structure of fuzzy IF-THEN rules, Fuzzy Sets and Systems,
29
157 (2006), 2003{2029.
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[15] V. Novak, V. Pavliska and Valasek, Specialized software for fuzzy natural logic and fuzzy
31
transform applications, in: Proc. Int. Conference FUZZ-IEEE'2014, Beijing, China, (2014),
32
2337{2344.
33
[16] V. Novak, V. Pavliska, M. Stepnicka and L. Stepnickova, Time series trend extraction and
34
its linguistic evaluation using F-transform and fuzzy natural logic, in: L. Zadeh, A. Abbasov,
35
R. Yager, S. Shahbazova (Eds.), Recent Developments and New Directions in Soft Computing,
36
Springer, Berlin, (2014), 429{442.
37
[17] V. Novak and I. Perlieva, Smooth fuzzy logic deduction with words, in: Proc. Int. Conf.
38
Fuzzy Information Processing: Theories and Applications, Vol. II, Tsinghua University
39
Press/Springer, Beijing, (2003), 599{604.
40
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41
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42
[19] V. Novak, I. Perlieva and A. Dvorak, Insight into Fuzzy Modeling, Wiley & Sons, Hoboken,
43
New Jersey, 2016.
44
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making using intelligent techniques, in: E. Rakus-Anderson, R. Yager, N. Ichalkaranje, L. Jain
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(Eds.), Recent Advances in Fuzzy Decision-Making, Springer, Heidelberg, (2009), 103{120.
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67
ORIGINAL_ARTICLE
MINING FUZZY TEMPORAL ITEMSETS WITHIN VARIOUS TIME INTERVALS IN QUANTITATIVE DATASETS
This research aims at proposing a new method for discovering frequent temporal itemsets in continuous subsets of a dataset with quantitative transactions. It is important to note that although these temporal itemsets may have relatively high \textit{support} or occurrence within particular time intervals, they do not necessarily get similar \textit{support} across the whole dataset, which makes it almost impossible to extract them using the existing traditional algorithms. This paper directly addresses this problem and introduces a new algorithm called Fuzzy Solid Linguistic Itemset Mining (FSLIM) to discover Solid Linguistic Itemsets (SLIs) within a quantitative dataset. SLI is a new concept introduced here as an essential part of the solution presented in this paper. The proposed method consists of two phases. In the first phase, fuzzy set theory is used to transform each quantitative value to a linguistic item; and in the second phase, all SLIs are extracted. Finally, the efficiency of FSLIM is compared in terms of execution time, scalability and the number of frequent patterns with those of two classic approaches on synthetic datasets. The proposed approach is also applied to an actual Mashhad Urban Traffic dataset in order to illustrate FSLIM's ability in discovering the hidden knowledge that could not be extracted by traditional methods.
http://ijfs.usb.ac.ir/article_2944_d38c9bdaf4139b353082432c484adc12.pdf
2016-12-30T11:23:20
2018-02-25T11:23:20
67
89
10.22111/ijfs.2016.2944
Fuzzy data mining
Temporal data mining
Frequent itemset
Temporal quantitative dataset
Mahnaz
Kadkhoda
true
1
Department of Computer Engineering, Center of Excellence
on Soft Computing and Intelligent Information Processing, Ferdowsi University of
Mashhad, Mashhad, Iran
Department of Computer Engineering, Center of Excellence
on Soft Computing and Intelligent Information Processing, Ferdowsi University of
Mashhad, Mashhad, Iran
Department of Computer Engineering, Center of Excellence
on Soft Computing and Intelligent Information Processing, Ferdowsi University of
Mashhad, Mashhad, Iran
AUTHOR
Mohammad-R.
Akbarzadeh-T
true
2
Department of Computer Engineering, Center of Excellence on Soft Computing and Intelligent Information Processing, Ferdowsi University of Mashhad, Mashhad, Iran
Department of Computer Engineering, Center of Excellence on Soft Computing and Intelligent Information Processing, Ferdowsi University of Mashhad, Mashhad, Iran
Department of Computer Engineering, Center of Excellence on Soft Computing and Intelligent Information Processing, Ferdowsi University of Mashhad, Mashhad, Iran
LEAD_AUTHOR
S. Mahmoud
Taheri
taher@cc.iut.ac.ir;sm_taheri@ut.ac.ir
true
3
Faculty of Engineering Science, College of Engineering, University of Tehran, Tehran, Iran
Faculty of Engineering Science, College of Engineering, University of Tehran, Tehran, Iran
Faculty of Engineering Science, College of Engineering, University of Tehran, Tehran, Iran
AUTHOR
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large databases, In: ACM SIGMOD Rec., ACM, (1993), 207{216.
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2000 ACM Symp. Appl. Comput.-Vol. 1, ACM, (2000), 294{300.
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[3] R. Agrawal and R. Srikant, Others, Fast algorithms for mining association rules, In: Proc
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20th Int Conf Very Large Data Bases VLDB, (1994), 487{499.
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[4] X. Chen and I. Petrounias, Discovering temporal association rules: Algorithms, language
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and system, In: 16th Int. Conf. Data Eng. ICDE, IEEE Computer Society, San Diego, CA,
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(2000), 306-306.
9
[5] C. Y. Chang, M. S. Chen and C. H. Lee, Mining general temporal association rules for items
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with dierent exhibition periods, In: Data Min. 2002 ICDM 2003 Proc. 2002 IEEE Int. Conf.
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On, IEEE, (2002), 59{66.
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[6] C. H. Chen, T. P. Hong and S. B. Lin, Mining fuzzy temporal knowledge from quantitative
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transactions, In: Syst. Sci. Eng. ICSSE 2011 Int. Conf. On, IEEE, (2011), 405{409.
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[7] J. Han, J. Pei, Y. Yin and R. Mao, Mining frequent patterns without candidate generation:
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A frequent-pattern tree approach, Data Min. Knowl. Discov., 8 (2004), 53{87.
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[8] T. P. Hong, Y. Y. Wu and S. L. Wang,An eective mining approach for up-to-date patterns,
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Expert Syst. Appl., 36 (2009), 9747{9752.
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[9] T. P. Hong, C. S. Kuo and S. C. Chi, Trade-o between computation time and number of rules
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for fuzzy mining from quantitative data, Int. J. Uncertain. Fuzziness Knowl.-Based Syst., 9
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(2001), 587{604.
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[10] J. W. Huang, B. R. Dai and M. S. Chen, Twain: Two-end association miner with precise
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frequent exhibition periods, ACM Trans. Knowl. Discov. Data TKDD., 1 (2007), 8.
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systems, Fuzzy Syst. IEEE Trans. On., 13 (2005), 428{435.
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temporal association rules in a publication database, In: Hybrid Intell. Syst. HIS 2011 11th
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Int. Conf. On, IEEE, (2011), 611{615.
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mining general temporal association rules, Knowl. Data Eng. IEEE Trans. On., 15 (2003),
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1004{1017.
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Part B Cybern. IEEE Trans. On., 34 (2004), 2330{2342.
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Eng., 65 (2008), 442{462.
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rules, Data Knowl. Eng., 44 (2003), 193{218.
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Appl., 38 (2011), 15143{15150.
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[18] S. G. Matthews, M. A. Gongora and A. A. Hopgood, Evolving temporal association rules
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with genetic algorithms, In: Res. Dev. Intell. Syst. XXVII, Springer, (2011), 107{120.
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[19] S. G. Matthews, M. A. Gongora, A. A. Hopgood and S. Ahmadi, Web usage mining with
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evolutionary extraction of temporal fuzzy association rules, Knowl.-Based Syst., 54 (2013),
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[20] S. G. Matthews, M. A. Gongora and A. A. Hopgood, Evolutionary algorithms and fuzzy sets
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for discovering temporal rules, Int. J. Appl. Math. Comput. Sci., 23 (2013), 855{868.
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[21] S. G. Matthews, M. A. Gongora and A. A. Hopgood, Evolving temporal fuzzy itemsets from
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quantitative data with a multi-objective evolutionary algorithm, In: Genet. Evol. Fuzzy Syst.
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GEFS 2011 IEEE 5th Int. Workshop On, IEEE, (2011), 9{16.
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for mining association rules, Knowl. Data Eng. IEEE Trans. On., 9 (1997), 813{825.
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frequent patterns in large databases, In: Data Min. 2001 ICDM 2001 Proc. IEEE Int. Conf.
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On, IEEE, (2001), 441{448.
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the context, Knowl. Inf. Syst., 28 (2011), 311{331.
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of frequent itemsets, Int. J. Intell. Syst. Technol. Appl., 13 (2014), 222{243.
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windows on real-time transaction database, Int. J. Innov. Comput. Inf. Control., 7 (2011),
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3239{3253.
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rules mining in a temporal database using a variable neighbourhood search, Comput. Oper.
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Res., 52 (2014), 241{250.
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IEEE Trans. On., 21 (2009), 1147{1161.
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[29] L. A. Zadeh, Fuzzy sets, Inf. Control., 8 (1965), 338{353.
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[30] C. Zhuo, L. Jiahui and L. Chen, A fuzzy calendar-based algorithm for mining temporal as-
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sociation rules and its application, In: Fuzzy Syst. Knowl. Discov. 2009 FSKD09 Sixth Int.
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Conf. On, IEEE, (2009), 28{33.
69
ORIGINAL_ARTICLE
SOLUTION-SET INVARIANT MATRICES AND VECTORS IN FUZZY RELATION INEQUALITIES BASED ON MAX-AGGREGATION FUNCTION COMPOSITION
Fuzzy relation inequalities based on max-F composition are discussed, where F is a binary aggregation on [0,1]. For a fixed fuzzy relation inequalities system $ A \circ^{F}\textbf{x}\leq\textbf{b}$, we characterize all matrices $ A^{'} $ For which the solution set of the system $ A^{' } \circ^{F}\textbf{x}\leq\textbf{b}$ is the same as the original solution set. Similarly, for a fixed matrix $ A $, the possible perturbations $ b^{'} $ of the right-hand side vector $ b $ not modifying the original solution set are determined. Several illustrative examples are included to clarify the results of the paper.
http://ijfs.usb.ac.ir/article_2945_c340fe680c20557fc27f32ba5cc9cf8f.pdf
2016-12-30T11:23:20
2018-02-25T11:23:20
91
100
10.22111/ijfs.2016.2945
Aggregation function
Max-aggregation function composition
Solution-set invariant matrices
Solution-set invariant vectors
System of fuzzy relation inequalities
F.
Kouchakinejad
true
1
Department of Mathematics, Graduate University of Advanced
Technology, Kerman, Iran
Department of Mathematics, Graduate University of Advanced
Technology, Kerman, Iran
Department of Mathematics, Graduate University of Advanced
Technology, Kerman, Iran
LEAD_AUTHOR
M.
Mashinchi
ijfs-editor@usb.ac.ir
true
2
Department of Statistics, Faculty of Mathematics and Computer Sciences, Shahid Bahonar University of Kerman, Kerman, Iran
Department of Statistics, Faculty of Mathematics and Computer Sciences, Shahid Bahonar University of Kerman, Kerman, Iran
Department of Statistics, Faculty of Mathematics and Computer Sciences, Shahid Bahonar University of Kerman, Kerman, Iran
AUTHOR
R.
Mesiar
true
3
Slovak University of Technology in Bratislava, Faculty of Civil Engineering, Radlinskeho 11, 810 05 Bratislava, Slovak Republic
Slovak University of Technology in Bratislava, Faculty of Civil Engineering, Radlinskeho 11, 810 05 Bratislava, Slovak Republic
Slovak University of Technology in Bratislava, Faculty of Civil Engineering, Radlinskeho 11, 810 05 Bratislava, Slovak Republic
AUTHOR
[1] M. Baczynski and B. Jayaram, Fuzzy implications, Studies in Fuzziness and Soft Computing,
1
[2] G. Beliakov, A. Pradera and T. Calvo, Aggregation functions: A guide for practitioners,
2
Springer, Berlin: Heidelberg, 2007.
3
[3] J. Drewniak and Z. Matusiewicz, Properties of max-fuzzy relation equations, Soft Computing,
4
14 (2010), 1037-1041.
5
[4] F. Durante, J. J. Quesada-Molina and C. Sempi, Semicopulas: characterizations and appli-
6
cability, Kybernetika, 42 (2006), 287-302.
7
[5] M. J. Fernandez and P. Gil, Some specific types of fuzzy relation equations, Inform. Science,
8
164 (2004), 189-195.
9
[6] M. Grabisch, J. L. Marichal, R. Mesiar and E. Pap, Aggregation functions, Cambridge: Cam-
10
bridge University press, 2009.
11
[7] E. P. Klement, R. Mesiar and E. Pap, Triangular norms, Kluwer, Dordrecht, 2000.
12
[8] K. Peeva and Y. Kyosev, Fuzzy relational calculus - theory, applications and software, Ad-
13
vances in Fuzzy Systems-Applications and Theory, World Scientific, 2005.
14
[9] E. Sanchez, Resolution of composite fuzzy relation equations, Inform. and Control, 30 (1976),
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[11] G. B. Stamou and S. G. Tzafestas, Resolution of composite fuzzy relation equations based on
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archimedean triangular norms, Fuzzy Set. Syst., 120 (2001), 395-407.
19
[12] H. F. Wang and H. M. Hsu, Sensitivity analysis of fuzzy relation equations, Int. J. Gen. Syst.,
20
19 (1991), 155-169.
21
ORIGINAL_ARTICLE
AN OBSERVER-BASED INTELLIGENT DECENTRALIZED VARIABLE STRUCTURE CONTROLLER FOR NONLINEAR NON-CANONICAL NON-AFFINE LARGE SCALE SYSTEMS
In this paper, an observer based fuzzy adaptive controller (FAC) is designed fora class of large scale systems with non-canonical non-affine nonlinear subsystems. It isassumed that functions of the subsystems and the interactions among subsystems areunknown. By constructing a new class of state observer for each follower, the proposedconsensus control method solves the problem of unmeasured states of nonlinear noncanonicalnon-affine subsystems. The main characteristics of the proposed observer-basedintelligent controller are: 1) on-line adaptation of the controller and the observer parameters,2) ultimate boundedness of both the output and the observer errors, 3) boundedness of allsignals involved, 4) employing experts’ knowledge in the controller design procedure and 5)chattering avoidance. The simulation results are further carried out to demonstrate better theeffectiveness of the proposed fuzzy based consensus controller method.
http://ijfs.usb.ac.ir/article_2946_548f6d22a30e721cee9755ec84977197.pdf
2016-12-30T11:23:20
2018-02-25T11:23:20
101
130
10.22111/ijfs.2016.2946
Lyapunov stability
Adaptive control
Non-affine nonlinear system
large scale systems
Fuzzy systems
Nonlinear observer
REZA
GHASEMI
true
1
DEPARTMENT OF ELECTRICAL ENGINEERING, UNIVERSITY OF QOM, QOM, IRAN
DEPARTMENT OF ELECTRICAL ENGINEERING, UNIVERSITY OF QOM, QOM, IRAN
DEPARTMENT OF ELECTRICAL ENGINEERING, UNIVERSITY OF QOM, QOM, IRAN
LEAD_AUTHOR
MOHAMMAD BAGHER
MENHAJ
true
2
DEPARTMENT OF ELECTRICAL ENGINEERING, AMIRKABIR UNIVERSITY
OF TECHNOLOGY, TEHRAN, IRAN, AND QIAU’S INCUBATOR CENTER OF TECHNOLOGY UNITS (CENTER
OF COGNITIVE SYSTEMS), QAZVIN, IRAN
DEPARTMENT OF ELECTRICAL ENGINEERING, AMIRKABIR UNIVERSITY
OF TECHNOLOGY, TEHRAN, IRAN, AND QIAU’S INCUBATOR CENTER OF TECHNOLOGY UNITS (CENTER
OF COGNITIVE SYSTEMS), QAZVIN, IRAN
DEPARTMENT OF ELECTRICAL ENGINEERING, AMIRKABIR UNIVERSITY
OF TECHNOLOGY, TEHRAN, IRAN, AND QIAU’S INCUBATOR CENTER OF TECHNOLOGY UNITS (CENTER
OF COGNITIVE SYSTEMS), QAZVIN, IRAN
AUTHOR
[1] C.C. Cheng, S.H. Chien, Adaptive Sliding Mode Controller Design Based On T–S Fuzzy System Models,
1
Elsevier Science, Automatica, 42 (2006), 1005-1010.
2
[2] L. Chen, G. Chen, Y.W. Lee, Fuzzy Modeling And Adaptive Control Of Uncertain Chaotic Systems,
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Elsevier Science, Information Sciences, 121 (1999), 27-37.
4
[3] C.C. Chiang, Adaptive Fuzzy Sliding Mode Control For Time-Delay Uncertain Large-Scale Systems,
5
Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control
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Conference, pp. 4077-4082,Seville, Spain, December, (2005), 12-15.
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[4] C. L. P. Chen, G. X. Wen, Y. J. Liu, Adaptive Consensus Control for a Class of Nonlinear Multiagent
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Time-Delay Systems Using Neural Networks, IEEE Transaction of neural network learning systems, 25
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(2014), 1217-1226.
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[5] C. L. P. Chen, Y. J. Liu, G. X. Wen, Fuzzy neural network-based adaptive control for a class of
11
uncertain nonlinear stochastic systems, IEEE Trans. Cybernetics, 44 (2014), 583-593.
12
[6] G. Feng, S.G. Cao, N.W. Rees, Stable Adaptive Control Of Fuzzy Dynamic Systems, Elsevier Science,
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Fuzzy Sets and Systems, 131 (2002), 217 – 224.
14
[7] G. Feng, An Approach To Adaptive Control Of Fuzzy Dynamic Systems, IEEE TRANSACTIONS ON
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FUZZY SYSTEMS, 10 (2002), 268-275.
16
[8] R. Ghasemi, M.B. Menhaj, A. Afshar, A decentralized stable fuzzy adaptive controller for large scale
17
nonlinear systems, Journal of Applied Science, 9 (2009), 892-900.
18
[9] R. Ghasemi, M.B. Menhaj and A. Afshar, A New Decentralized Fuzzy Model Reference Adaptive
19
Controller for a Class of Large-scale Non-affine Nonlinear Systems, European Journal of Control, 5
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(2009), 1–11.
21
[10] R Ghasemi, MB Menhaj, A variable structure observer based control design for a class of large scale
22
MIMO nonlinear systems, Amirkabir International Journal of Modeling, Identification, Simulation &
23
Control, 48 (2016), 47-56.
24
[11] N. Golea, A. Golea, K. Benmahammed, Stable Indirect Fuzzy Adaptive Control, Elsevier Science,
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Fuzzy Sets and Systems, 137 (2003), 353-366.
26
[12] A. Hamzaoui, N. Essounbouli, K. Benmahammed, and J. Zaytoon, State Observer Based Robust
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Adaptive Fuzzy Controller for Nonlinear Uncertain and Perturbed Systems, IEEE TRANSACTIONS
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ON SYSTEMS, MAN, AND CYBERNETICS—PART B, 34 (2004), 23-28.
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[13] H.F. Ho, Y.K. Wong, A.B. Rad, W.L. Lo, State Observer Based Indirect Adaptive Fuzzy Tracking
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Control, Simulation Modeling Practice and Theory, 13 (2005), 646–663.
31
[14] Y.C. Hsu, G. Chen, S. Tong, H.X. Li, Integrated Fuzzy Modeling And Adaptive Control For Nonlinear
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Systems, Elsevier Science, Information Sciences, 153 (2003), 217-236.
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[15] J. Hu, Y. Hong, Leader-following coordination of multiagent systems with coupling time delays,
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Physica A, 374 (2007), 853-863.
35
[16] S. Jagannathan, Adaptive Fuzzy logic control of feedback linearization discrete time dynamical systems
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under persistence of excitation, Automatica, 34 (1998), 1295-1310.
37
[17] X. Jiang,W. Xu, Q.L. Han, Observer-based fuzzy control design with adaptation to delay parameter for
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time-delay systems, Elsevier Science, Fuzzy Sets and Systems, 152 (2005), 637–649.
39
[18] S. Labiod, M. S. Boucherit, T. M. Guerra, Adaptive fuzzy control of a class of MIMO nonlinear
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systems, Elsevier Science, Fuzzy Sets and Systems, 151 (2005), 59–77.
41
[19] S. Labiod, T. M. Guerra, Adaptive fuzzy control of a class of SISO non-affine nonlinear systems,
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Elsevier Science, Fuzzy Sets and Systems, 158 (2007), 1126 –1137.
43
[20] Z. Li, X. Liu, P. Lin, W. Ren, Consensus of linear multi-agent systems with reduced-order observerbased
44
protocols, Systems & Control Letters, 60- 7 (2011), 510-516.
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Elsevier Science, Information Sciences, 4 (2007), 1-17.
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[22] Y. J. Liu, S. C. Tong, C. L. P. Chen, Adaptive fuzzy control via observer design for uncertain
48
nonlinear systems with unmodeled dynamics, IEEE Trans. Fuzzy Syst., 21 (2013), 275-288.
49
[23] Y. J. Liu, S. C. Tong, D. Wang, T. S. Li, C. L. P. Chen, Adaptive neural output feedback controller
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design with reduced-order observer for a class of uncertain nonlinear SISO systems, IEEE Trans.
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Neural Netw., 22 (2011), 1328-1334.
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and timedelays, IEEE Trans. Automatic Control, 49 (2004), 1520-1533.
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To indirect adaptive fuzzy control design, Elsevier science, Information Sciences, 159 (2004), 125-139.
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delays, Physica A, 388 (2009), 193-208.
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Proc. ICNSC, 129 (2006), 457 - 462.
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[28] T. Shaocheng, C. Bin, W. Yongfu, fuzzy adaptive output feedback control for MIMO nonlinear
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systems, Elsevier Science, Fuzzy Sets and Systems, 156 (2005), 285–299.
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Science, Fuzzy Sets and Systems, 104 (1999), 279-288.
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[31] S. Tong, H.X. Li, W. Wang, Observer-Based Adaptive Fuzzy Control For SISO Nonlinear Systems,
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Elsevier Science, Fuzzy Sets and Systems, 148 (2004), 355–376.
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nonlinear systems, IEEE Transactions on Systems, Man, and Cybernetics—Part B, 34 (2004), 24-27.
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[33] S. C. Tong, Y. Li, Y. M. Li, and Y. J. Liu, Observerbased adaptive fuzzy backstepping control for a
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class of stochastic nonlinear strict-feedback systems, IEEE Trans. Syst., Man, Cybern. Part B, 41
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(2011), 1693-1704.
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[34] S. C. Tong, Y. M. Li, G. Feng, and T. S. Li, Observer based adaptive fuzzy backstepping dynamic
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surface control for a class of MIMO nonlinear systems, IEEE Trans. Syst., Man, Cybern. Part B, 41
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(2011), 1124- 1135.
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Elsevier Science, Fuzzy Sets and Systems, 111 (2000), 153-167.
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Systems, Man, And Cybernetics—Part B: Cybernetics, 34 (2004), 34-39.
80
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inverted-pendulum servomechanism, IEEE Transactions On Systems, Man, and Cybernetics—Part B:
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Cybernetics, 38 (2008), 67-77.
83
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84
for a class of uncertain strict-feedback discrete-time nonlinear systems, Neurocomputing, 138 (2014),
85
[39] H. Wu, Decentralized adaptive robust control for a class of large-scale systems including delayed
86
state perturbations in the interconnections, IEEE Transactions On Automatic Control, 47 (2002), 1745-
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[40] P. Ying-guo, Z. Hua-guang, Design of fuzzy direct adaptive controller and stability analysis for a
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class of nonlinear system, Proceedings of the American Control conference, Philadelphia, Pennsylvania,
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(1998), 2274-2275.
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[41] T. Yiqian, W. Jianhui, G. Shusheng, Q. Fengying, Fuzzy Adaptive Output Feedback Control For
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Nonlinear MIMO Systems Based On Observer, Proceedings of the 5th World Congress on Intelligent
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93
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IEEE International Conference on Systems, Man and Cybernetics, 5 (2004), 5246-5251.
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103
ORIGINAL_ARTICLE
MINIMAL AND STATEWISE MINIMAL INTUITIONISTIC GENERAL L-FUZZY AUTOMATA
In this note, by considering the notions of the intuitionistic general L-fuzzy automaton and $(\alpha, \beta)$-language, we show that for any $(\alpha, \beta)$-language $\mathcal{L}$, there exists a minimal intuitionistic general L-fuzzy automaton recognizing $\mathcal{L}$.We prove that the minimal intuitionistic general L-fuzzy automaton is isomorphic with threshold $(\alpha,\beta)$ to any $(\alpha, \beta)$-reduced max-min intuitionistic general L-fuzzy automaton.%Also, we prove that the minimal intuitionistic general L-fuzzy automaton is an $(\alpha, \beta)$-reduced.Also, we show that for any strong deterministic max-min intuitionistic general L-fuzzy automaton there exists a statewise $(\alpha, \beta)$-minimal intuitionistic general L-fuzzy automaton.In particular, a connection between the minimal and statewise $(\alpha, \beta)$-minimal intuitionistic general L-fuzzy automaton is presented.%We show if $\tilde{F}^*$ is an $(\alpha, \beta)$-complete $(\alpha, \beta)$-accessible deterministic max-min intuitionistic general L-fuzzy automaton and it is recognizing $(\alpha, \beta)$-language $\mathcal{L}$, then the minimal $\tilde{F}^*_{\mathcal{L}}$ is homomorphism with threshold $(\alpha, \beta)$ to statewise $(\alpha, \beta)$-minimal $\tilde{F}_{m}^*$, where $\tilde{F}_{m}^*$ is statewise $(\alpha, \beta)$-equivalent to $\tilde{F}^*$.Also, for a given intuitionistic general L-fuzzy automaton, we present two algorithms, which determinesstates of the minimal intuitionistic general L-fuzzy automaton and the statewise $(\alpha, \beta)$-minimal intuitionistic general L-fuzzy automaton.Finally, by giving some examples, we comparison minimal intuitionistic general L-fuzzy automaton and statewise $(\alpha, \beta)$-minimal intuitionistic general L-fuzzy automaton.
http://ijfs.usb.ac.ir/article_2947_e25316858488812f742b91e5709605c4.pdf
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152
10.22111/ijfs.2016.2947
Minimal automata
$(alpha
beta)$-language
Statewise minimal automata
Homomorphism with threshold $(alpha
beta)$
M.
Shamsizadeh
true
1
Department of Mathematics, Graduate University of Advanced
Technology, Kerman, Iran
Department of Mathematics, Graduate University of Advanced
Technology, Kerman, Iran
Department of Mathematics, Graduate University of Advanced
Technology, Kerman, Iran
LEAD_AUTHOR
M. M.
Zahedi
zahedi_mm@ mail.uk.ac.ir
true
2
Department of Mathematics, Graduate University of Advanced Tech-
nology, Kerman, Iran
Department of Mathematics, Graduate University of Advanced Tech-
nology, Kerman, Iran
Department of Mathematics, Graduate University of Advanced Tech-
nology, Kerman, Iran
AUTHOR
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of Approximate Reasoning, 38 (2005), 175-214.
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nite state machines", Journal of Applied Mathematics and Computing, 51 (2016), 413-423.
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ORIGINAL_ARTICLE
SOFT TOPOLOGY AND SOFT PROXIMITY AS FUZZY PREDICATES BY FORMULAE OF LUKASIEWICZ LOGIC
In this paper, based in the \L ukasiewicz logic, the definition offuzzifying soft neighborhood structure and fuzzifying soft continuity areintroduced. Also, the fuzzifying soft proximity spaces which are ageneralizations of the classical soft proximity spaces are given. Severaltheorems on classical soft proximities are special cases of the theorems weprove in this paper.
http://ijfs.usb.ac.ir/article_2948_ecf1fe1f138a39e8d7a8dd747ecdb98f.pdf
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168
10.22111/ijfs.2016.2948
Soft set
Soft topology
Fuzzifying soft topology
Fuzzifying soft proximity
O. R.
Sayed
true
1
Department of Mathematics, Faculty of Science, Assiut University,
Assiut, Egypt
Department of Mathematics, Faculty of Science, Assiut University,
Assiut, Egypt
Department of Mathematics, Faculty of Science, Assiut University,
Assiut, Egypt
AUTHOR
R. A.
Borzooei
borzooei@sbu.ac.ir
true
2
Department of Mathematics, Shahid Beheshti University, Tehran,
Iran
Department of Mathematics, Shahid Beheshti University, Tehran,
Iran
Department of Mathematics, Shahid Beheshti University, Tehran,
Iran
LEAD_AUTHOR
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2
[3] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96.
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[4] K. Atanassov, Operators over interval valued intuitionistic fuzzy sets, Fuzzy Sets and Systems,
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64 (1994), 159-174.
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[5] F. Feng, Y. B. Jun and X. Z. Zhao, Soft semirings, Comput. Math. Appl., 56(10) (2008),
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2621-2628.
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fuzzy sets, Fuzzy Sets and Systems, 21 (1987), 1{17.
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problem, Comput. Math. Appl., 44(8-9) (2002), 1077-1083.
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[15] M. Shabir and M. Naz, On soft topological spaces, Comput. Math. Appl., 61 (2011), 1786-
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61(10) (2011), 2952-2957.
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Stat., 41(5) (2012), 731{741.
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doi:10.2991/eus
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at. (2011). 52 883-890.
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[20] X. B. Yang, T. Y. Lin, J. Y. Yang, Y. Li and D. J. Yu, Combination of interval-valued fuzzy
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36
ORIGINAL_ARTICLE
Persian-translation vol. 13, no. 7, Decemberr 2016
http://ijfs.usb.ac.ir/article_2950_655a94a8efc837e6b4e0c64d101dd333.pdf
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171
179
10.22111/ijfs.2016.2950