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List of Abstracts :

1. A Novel Unsupervised Neuro-Fuzzy System Applied to Circuit Analysis

2. A Numerical Approach Based on Neuro-Fuzzy Systems for Obtaining Functional Inverse

3. Improved Genetic Algorithm-Based Optimization of Fuzzy Logic Controllers

4. FUZZY REAL NUMBERS AND THEIR RELATION TO THE TOPOLOGICAL VECTOR SPACE

5. Fuzzy Seminormed Linear Space

6. A Novel Algorithm for Tuning of the Type-2 Fuzzy System

7. Fuzzy Um-set in Sostak sense

8. Velocity Control of an Electro Hydraulic Servosystem by Sliding Mamdani

9. An On-line Fuzzy Backstepping Controller for Rotary Inverted Pendulum System

10. Optimal Design of Type_1 TSK Fuzzy Controller Using GRLA

11. System of linear fuzzy differential equations

12. A method for fully fuzzy linear system of equations

13. Full fuzzy linear systems of the form Ax+b=Cx+d

14. Solving fuzzy polynomial equation by ranking method

15. Stabilization of Autonomous Bicycle by an Intelligent Controller

16. Solving fuzzy linear programming problems with piecewise linear membership functions by the determination of a crisp maximizing decision

17. Fuzzy Convex Subalgebras of Residuated Lattice

18. Algebraic Fuzzy Subsets of Non-commutative Join Spaces

19. An Ant Based Algorithm Approach to Vehicle Navigation

20. Novel ranking method of fuzzy numbers

21. A Trapezoidal Membership Function Block for Fuzzy Applications

22. Ranking Fuzzy Numbers by Sign Length

23. Ranking of fuzzy numbers by left and right ranking Functions

24. An Application of Possibility Goal Programming to the Time-Cost Trade off Problem

25. A Multi Hybrid Genetic Algorithm for the Quadratic Assignment Problem

26. Implementation of Evolutionary Algorithm and Fuzzy Sets for Reliability Optimization of Engineering Systems

27. One-sided Process Capability Indices and Fuzzy lower Confidence Bounds for them

28. Fuzzy Confidence regions for the Taguchi Index in Fuzzy Process

29. A Method of Generating a Random Sample From a Fuzzy Distribution Function

30. Fuzzy linear regression analysis with trapezoidal coefficients

31. Probability on balanced lattices

32. Dpq-DISTANCE AND THE STRONG LAWOF LARGE NUMBER FOR NEGATIVELY DEPENDENT FUZZY RANDOM VARIABLES

33. Reducible Fuzzy Markov Chain and Fuzzy Absorption Probability

34. Fuzzy Traffic Rate Controller (FTRC) for Mobile Ad hoc Network Routers

35. A Fuzzy Routing Algorithm for Low Earth Satellite Networks

36. A metaheuristic Algorithm for New Models of Fuzzy Bus Terminal Location Problem with a Certain Ranking Function

37. Finding the Inversion of a Square Matrix and Pseudo-inverse of a Non-square Matrix by Hebbian Learning Rule

38. Solving Fuzzy Integral equations by Differential Transformation Method

39. FUZZY TOPOLOGICAL SPACES

40. COMMON FIXED POINT THEOREM IN COMPLETE FUZZY METRIC SPACES

41. Extension Principle for Vauge Sets

42. Signed Decomposition of Fully Fuzzy Linear Systems

43. Iteration approach of solving interval and fuzzy linear system of equations

44. FUZZY BANACH ALGEBRA

45. Some Properties of Zariski Topology on the Spectrum of Prime Fuzzy Submodules

46. A NEURAL NETWORK MODEL FOR SOLVING STOCHASTIC FUZZY MULTIOBJECTIVE LINEAR FRACTIONAL PROGRAMS

47. Application of a hybrid GA-BP optimized neural network for springback estimation in sheet metal forming process

48. Vehicle Type Recognition Using Probabilistic Constraint Support Vector Machine

49. On-line Identification and Prediction of Lorenz's Chaotic System Using Chebyshev Neural Networks

50. A Novel Hybrid Structure and Criteria in Modeling and Identification

51. Design of a VLSI Hamming Neural Network For arrhythmia classification

52. Designing an optimal PID controller using Imperialist Competitive Algorithm

53. Fractional PID Controller Design based on Evolutionary Algorithms for Robust two-inertia Speed Control

54. Neural Networks for Fault Detection and Isolation of

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- 5. First Joint Congress on Fuzzy and Intelligent Systems Ferdowsi University of Mashhad, Iran 29-31 Aug 2007 Intelligent Systems Scientific Society of Iran A Novel Unsupervised Neuro-Fuzzy System Applied to Circuit Analysis Hadi Sadoghi Yazdi Seyed Ebrahim Hosseini Engineering Department, Tarbiat Moallem University of Sabzevar, Sabzevar, Iran E-mail: {sadoghi, ehosseini}@sttu.ac.ir Abstract: In this paper, for the first time, unsupervised neuro-fuzzy system is presented and is applied in circuit analysis. Usually Neuro-fuzzy systems have a learning phase in which the system is trained with input data. But if the training set is unavailable, conventional procedures encounter serious problem. Due to unsupervised character, no learning data is needed. To investigate the method, linear circuit is analyzed. Results are compared with the exact solution. Keywords: Unsupervised neuro-fuzzy system, Circuit analysis. Where ψ (x ) denotes the solution, G is the functional 1. Introduction defining the structure of the differential equation, and Artificial neural networks (ANN), as intelligent ∇ is some differential operator. The basic idea, called computational tools have been used for modeling and collocation method, is to discretize the domain D over optimization of analog circuit design [1]. For example a finite set of points D . Thus (1) becomes a system of optimization of high-speed very large scale integration equations. Suppose an approximation of the solution (VLSI) interconnects [2], global modeling [3, 4] and ψ(x) is given by the trail solution ψt(x). As a measure circuit synthesis [5]. for the degree of fulfillment of the original differential Numerous problems in science and engineering can be equation (1), an error function similar to the mean converted to a set of differential equations. Basic squared error is defined: numerical methods can be used to solve differential E= 1 [( )] ∑ G xi ,ψ t , ∇ψ t , ∇ 2ψ t ,... . D xt ∈D 2 (2) equations such as the finite difference method, the finite element method, the finite volume method and the boundary element method. Therefore, finding an approximation of the solution Neural networks (NNs) are used to approximate of (1) is equal to finding a function that minimizes the stochastically unknown functions and relations. This error E . It is well known that a multilayer feed forward approximation is a kind of implicit model of the neural network is a universal approximator [7], unknown dependencies. In contrast, differential therefore the trail solution ψt(x) can be represented by equations are used to model explicitly all relations. such an artificial neural network. In case of a given Solving ordinary and partial differential equations can network architecture the problem is reduced to finding be learned to a good degree by an artificial neural a configuration of weights that minimizes (2). As E is network. The basic ideas were presented by Lagaris, differentiable with respect to the weights for most Likas, and Fotiadis [6]. Let the differential equation to differential equations, efficient gradient-based learning be solved be given by: G (x,ψ ( x ), ∇ψ (x ), ∇ 2ψ ( x ),...) = 0 x ∈ D ⊆ R n , algorithms for artificial neural networks can be (1) employed for minimizing (2). 1 ﻫﻔﺘﻤﻴﻦ ﻛﻨﻔﺮاﻧﺲ ﺳﻴﺴﺘﻤﻬﺎي ﻓﺎزي ، 9-7 ﺷﻬﺮﻳﻮر 6831 ، داﻧﺸﮕﺎه ﻓﺮدوﺳﻲ ﻣﺸﻬﺪ
- 6. First Joint Congress on Fuzzy and Intelligent Systems Ferdowsi University of Mashhad, Iran 29-31 Aug 2007 Intelligent Systems Scientific Society of Iran A Numerical Approach Based on Neuro-Fuzzy Systems for Obtaining Functional Inverse Hadi Sadoghi Yazdi1, Sohrab Effati2 1-Engineering Department, Tarbiat Moallem University of Sabzevar, Sabzevar, Iran 2-Department of Mathematics, Tarbiat Moallem University of Sabzevar, Sabzevar, Iran Abstract: In this paper, a new and easy available numerical method is presented in calculation of functional inverse. As for ANFIS (Adaptive Network based Fuzzy Inference System) model is available without great difficulty in MATLAB software also due to important of obtaining functional inverse in wide range applications, we present ANFIS-based approach for the first time for obtaining inverse of mathematical functions. The proposed approach includes two main stages, in the first step; some limited points are sampled from desired function using Monte Carlo simulation and second step contains training of ANFIS system that input system is output points of function and ANFIS desired values are input values to function. One of notes in the proposed approach is calculation of inverse of time varying functions only with tuning of generated fuzzy model with the least square and the back propagation (BP) algorithms to form of hybrid learning algorithm which able into tracking of time varying functions. Experimental results over many functions show superiority of the proposed method relative MATLAB software. Keywords: Functional inverse; Adaptive Network based Fuzzy Inference System; Monte Carlo method; Hybrid learning; MATLAB software. 1. Introduction resolution image which is as close as possible to the Functional inverse not only is used in mathematics original high resolution image subject to certain but also have many applications in engineering and constraints [4]. sciences. Finding sources from measured data is a common problem in a real world. From the Inverse learning in signal processing: mathematic point of view, solving the equation obtain The developments of inverse learning was initiated inverse of function or an iterative approach give us from adaptive signal processing and control of linear functional inverse. If f is a continuous function systems approach [2, 3] and due to its potential [ ] applicability, until now several modifications and defined in interval a, b , g is inverse of f if applications are still being developed in the literature. ( f o g )(x ) = x or f (g (x )) = x ; so −1 g is f or is a In data communications, receiver equalization has become an essential building block to mitigate the functional inverse. Functional inverse can study from inter-symbol interference problem, which is due to different views: limited bandwidth of low cost channel materials. Several equalization methods have been developed Finding source of generated data: which are types of functional inverse [5]. The inverse problem of finding the source from the data is other form of functional inverse. In practice a Constructing inverse signal: process has two elements, the physical process and Creating of inverse signal for reducing of the measurement. In the measurement procedure undesirable signal is another view of functional noise is inserted to data. So it is needed a smoother is inverse. Active noise control systems are based on the selected for noise reduction then backward mapping principle of superposition: an unwanted noise is is required for finding source [1]. cancelled out by the action of a secondary noise that In the field of image processing, finding main generates a sound wave of equal amplitude and image from degraded image is based on dealing with opposite phase [6]. This idea has evolved and new the problem as an inverse problem. Based on the applications have been developed in which the observation model, our objective is to obtain a high 15 ﻫﻔﺘﻤﻴﻦ ﻛﻨﻔﺮاﻧﺲ ﺳﻴﺴﺘﻤﻬﺎي ﻓﺎزي ، 9-7 ﺷﻬﺮﻳﻮر 6831 ، داﻧﺸﮕﺎه ﻓﺮدوﺳﻲ ﻣﺸﻬﺪ
- 7. First Joint Congress on Fuzzy and Intelligent Systems Ferdowsi University of Mashhad, Iran 29-31 Aug 2007 Intelligent Systems Scientific Society of Iran Improved Genetic Algorithm-Based Optimization of Fuzzy Logic Controllers Amin Chegeni, Abdollah Khoei, Khayrollah Hadidi Microelectronics Research Laboratory Urmia University st_a.chegeni@urmia.ac.ir a.khoei@urmia.ac.ir kh.hadidi@urmia.ac.ir Abstract: This paper presents a new method to degree of freedom and degrades the performance. improve genetic algorithm-based optimization of fuzzy Second, since the plant is nonlinear, as we will logic controllers considering both membership explain in this paper, if the FLC is optimized for a functions and rules. We indicate that eliminating the special input signal, the system may be unstable limitation on symmetric membership functions and for other inputs. Thus, optimizing the FLC for rules results better optimization. Furthermore, we only one special input is a faulty optimization. optimize fuzzy logic controller using genetic algorithm In this paper, we indicate that eliminating the considering whole of system states, which cause the reliability of the controller to improve significantly. mentioned limitation on symmetric MFs and rule table results in more degree of freedom and consequently better optimization. Furthermore, we Keywords: Fuzzy logic controller, genetic analyze the effect of optimizing FLC for different algorithm, optimization. states of the system, and show that the reliability of the controller improves significantly when 1 Introduction involving more states of the system in the optimization. In the last decade, many researches have focused This paper is organized as follows. In section II, a on the learning and tuning algorithms for both brief description of GA is presented. Optimization model based and model-free design of fuzzy logic mechanism will be presented in section III, and in controllers (FLCs). The literature report successful section IV, two well-known examples along with applications of methods based on fuzzy clustering, their simulation results will be presented to verify neural networks (NNs), reinforcement learning, the feasibility and advantage of this method. and genetic algorithms (Gas) for the most widely Section V concludes this work. adopted techniques. In many researches, only membership functions (MFs) are used in the 2 Genetic Algorithms (GA’s) optimization process, and in some of them, both MFs and rules are optimized. However, two GA’s are search algorithms modeled after the important points usually are not considered in the mechanics of natural genetics [8]. They are useful practical optimization. First, in almost all articles, approaches to problems requiring effective and the resulted MFs and/or rule-table are constrained efficient searching, and their use is widespread in to be symmetrical [1, 3, 4, 5]. Considering that the applications to business, scientific, and plant controlled by FLC is generally nonlinear, the engineering fields. In an optimally designed optimal MFs and rule table are not necessarily application, GA’s can be used to obtain an symmetrical; therefore, restricting the approximate solution for single variable or optimization to symmetric results decreases the multivariable optimal problems. Before a GA is 25 ﻫﻔﺘﻤﻴﻦ ﻛﻨﻔﺮاﻧﺲ ﺳﻴﺴﺘﻤﻬﺎي ﻓﺎزي ، 9-7 ﺷﻬﺮﻳﻮر 6831 ، داﻧﺸﮕﺎه ﻓﺮدوﺳﻲ ﻣﺸﻬﺪ
- 8. First Joint Congress on Fuzzy and Intelligent Systems Ferdowsi University of Mashhad, Iran 29-31 Aug 2007 Intelligent Systems Scientific Society of Iran FUZZY REAL NUMBERS AND THEIR RELATION TO THE TOPOLOGICAL VECTOR SPACE I. SADEQI AND M. SALEHI esadeqi@sut.ac.ir Abstract: : In this paper first we show that the fuzzy normed spaces (FNS) constructed by fuzzy real numbers are topological vector space, and then we prove that the definition of fuzzy continuity and topological continuity are equivalent. By using that we can easily establish all results in the topological vector spaces for the fuzzy normed spaces.. Keywords: Fuzzy real number, Fuzzy norm, Fuzzy bounded operator, topological vector space. 1 Introduction Felbin [1] put forward the concept of a fuzzy We denote the origin of a linear space by θ; and the normed linear space (briefly , FNS) by applying set of all fuzzy real number by F. If h∈ F and h(t) the notion of fuzzy distance to the linear space. It = 0 whenever t < 0, then h is called an non- is more natural and practical that the norm of a negative fuzzy real number and by F+ we mean the point is a fuzzy number rather than a real number. set of all non-negative fuzzy real number. The Xiao and Zhu [2] investigated the linear number 0 stands for the fuzzy number satisfying topological structure and some basis properties of FNS under the condition of weaker right norm and 0 (t) = 1 and 0 (t) = 1 if t ≠0 clearly, 0 ∈ F+. The left norm. We will show that a fuzzy normed space set of all real numbers can be embedded in F is also a topological vector space and we will because if r ∈ (-∞ , ∞), then r ∈ F satisfies r (t) = prove that the fuzzy continuity and topological continuity (fuzzy boundedness and topological 0 (t - r). For h ∈ F , r ∈ (0,∞) and α∈ (0,1], r • h boundedness) are equivalent. So all results and is defined as (r • h)(t) = h(t/r) and 0 • h is defined theorems in the topological vector spaces hold for the fuzzy normed spaces in general. There are to be 0 ; the α-level set [h]α = {t : h(t) ≥ α} is a − + many important known theorems in the topological closed interval and we denote it by [h]α= [h α ,h α ]. vector spaces; among them are Uniform Definition 2.1. Let X be a linear space over real boundedness, Open mapping theorem, Closed number field; L and R (respectively, left norm and Graph theorem, Han-Banach theorem,... which right norm) be symmetric and non decreasing remain true for fuzzy normed spaces. Therefore as mapping from [0, 1] × [0, 1] into [0,1] satisfying it is cited above already proven in classical analysis, for fuzzy normed spaces. The only L(0,0) = 0 , R(1,1) = 1. Then ∥ .∥ is called a fuzzy problem remain is that we investigate the norm and (X, ∥ .∥ , L , R ) a fuzzy normed space properties in fuzzy normed spaces which does not hold in classical analysis. (abbreviated to FNS). If the mapping ∥ .∥ from X into F+ satisfies the following axioms where − + 2 preliminaries [∥ x∥ ]α = [∥ x∥ α ,∥ x∥ α ] for x∈ X and Our notation and definition follow [2]. α∈ (0,1] ) 45 ﻫﻔﺘﻤﻴﻦ ﻛﻨﻔﺮاﻧﺲ ﺳﻴﺴﺘﻤﻬﺎي ﻓﺎزي ، 9-7 ﺷﻬﺮﻳﻮر 6831 ، داﻧﺸﮕﺎه ﻓﺮدوﺳﻲ ﻣﺸﻬﺪ
- 9. First Joint Congress on Fuzzy and Intelligent Systems Ferdowsi University of Mashhad, Iran 29-31 Aug 2007 Intelligent Systems Scientific Society of Iran Fuzzy Seminormed Linear Space I . Sadeqi and F. Solaty kia esadeqi@sut.ac.ir Abstract: In this note we introduce and study the notion of fuzzy seminorm and we get some new results. Finally we give an example of a fuzzy seminorm space which is fuzzy normable but is not classical normable. Keywords: fuzzy normed space, fuzzy seminorm, Minkowski functional, fuzzy metric. (N5) N( x , . ) is a non -decreasing function on 1 Introduction ℜ , and lim N(x, t)=1 as t → ∞ . The pair (X, N) is said to be a fuzzy normed In 1984, Kastaras [3] first introduced the idea of space. fuzzy norm on a linear space. In 1992, Felbin [4] introduced an idea of a fuzzy norm on a linear 2 Fuzzy seminorm space whose associated metric is Kaleva [5] type. In 1994, S.C.Chang and J.N.Mordeson [6] Definition: A fuzzy seminorm on a vector space introduced an idea of a fuzzy norm on a linear X is a function on X × ℜ such that space whose associated metric is Kramosil and P1) ∀ t ∈ ℜ with t ≤ 0, ρ (x, t) =0; Michalek type [7]. Following Chang and P2) ∀ t ∈ ℜ with t > 0, ρ (cx, t) = ρ (x, t/|c|) Mordeson, In 2003, T.Bag and S.K.Samanta introduced in [1] a definition of fuzzy norm and if c ≠ 0 proved the decomposition theorem of fuzzy norm P3) ∀ t, s ∈ ℜ , x, u ∈ X , ρ (x+u, t+s) ≥ min to a family of crisp norms. We introduce the fuzzy { ρ (x, t), ρ (u, s) } seminorm and study some of it's properties, then P4) ρ ( x , . ) is a non-decreasing function of ℜ by an example we will show that a family of fuzzy and lim ρ (x, t)=1 as t → ∞ seminorm implies a fuzzy norm on C ( Ω ) space but it is not true in classical case. Note : A fuzzy seminorm ρ is a fuzzy norm N if Preliminary it satisfies ; ρ (x, t)= 1 ∀ t ∈ ℜ with t > 0 then x = 0. Definition: [1] Let X be a vector space. A fuzzy subset N of X × ℜ is called a fuzzy norm on Definition: A family P of fuzzy seminorms on X X if the following condition, are satisfied for all is said to be separating if to each x ≠ 0 x, y ∈ X and c ∈ ℜ : corresponds at least one ρ ∈ P and t ∈ ℜ such (N1) N(x, t) = 0 ; ∀ t ∈ ℜ with t ≤ 0 that ρ (x, t) ≠ 1. (N2) N(x, t) = 1 ; ∀ t ∈ ℜ , t > 0 iff x=0 (N3) N(c x, t) = N(x, t/|c|), ∀ t ∈ ℜ , t >0 and Theorem: Let ρ is a fuzzy seminorm on a vector c ≠ 0 space X. Define ρ α (x) = inf { t : ρ (x ,t) ≥ α }, (N4) N(x + y, t +s) ≥ min {N(x, s), N( y, t)}, for all x ,y ∈ X , s ,t ∈ ℜ α ∈ (0,1). Then {ρ α : α ∈ (0,1)} an ascending 51 ﻫﻔﺘﻤﻴﻦ ﻛﻨﻔﺮاﻧﺲ ﺳﻴﺴﺘﻤﻬﺎي ﻓﺎزي ، 9-7 ﺷﻬﺮﻳﻮر 6831 ، داﻧﺸﮕﺎه ﻓﺮدوﺳﻲ ﻣﺸﻬﺪ
- 10. First Joint Congress on Fuzzy and Intelligent Systems Ferdowsi University of Mashhad, Iran 29-31 Aug 2007 Intelligent Systems Scientific Society of Iran A Novel Algorithm for Tuning of the Type-2 Fuzzy System Sayed Mohammad Ali Mohammadi1 Ali Akbar gharaveisi1 Mashaalah Mashinchi2 s_m_ali_mohammadi@yahoo.com a_gharaveisi@yahoo.com mmachinci@yahoo.com 1. Department of Electrical Engineering, Shahi Bahonar University of Kerman-Iran 2. Department of Mathematic, Shahi Bahonar University of Kerman –Iran Abstract- Type-2 fuzzy logic systems (FLSs) let uncertainties that occur in rule-based FLSs be modeled using the new third dimension of type-2 fuzzy sets. Although a complete theory of type-2 FLSs exists for general type-2 fuzzy sets, it is only for interval type-2 fuzzy sets that type-2 FLSs are practical. Type 2 fuzzy sets allow for linguistic grades of membership. A type-2 fuzzy inferencing systems uses type-2 fuzzy sets to represent uncertainty in both the representation and inferencing. However; as with type-l fuzzy systems there is till an issue with regard to the design of the appropriate membership functions. One of the best performance the fuzzy inference system is optimized by the least square and numerical method .The key advantages of the least square method are the efficient use of samples and the simplicity of the implementation but it can be take long time for convergence. This paper presents a novel type-2 adaptive system for learning the membership grades of type-2 fuzzy sets which can be important. The results from the application problems lead us to believe that this approach offers the capability to allow linguistic descriptors to be learnt by an adaptive network and we can use some new algorithm same as Reinforcement Learning Methods for adaptation. Key word: Fuzzy Decision, Type 2 Fuzzy Sets, Adaptive Fuzzy Systems, FOU . I. Introduction 1- The meanings of the words that are used in One possible approach to deal with the vague the antecedents and consequents of rules concepts is fuzzy logic systems, which are can be uncertain (words mean different based on the fuzzy set theory, formulated and things to different people). developed by Zadeh [1]. Fuzzy set theory is a 2-Consequents rnay have a histogram of generalization of classical set theory that values associated with them, especially provides a way to absorb the uncertainty when knowledge is extracted from a group inherent to phenomena whose information is of experts who do not all agree. vague and supplies a strict mathematical 3-Measurements that activate a FLS may be framework, which allows its study with some noisy and therefore uncertain. precision and accuracy. A fuzzy logic system 4-The data that are used to tune the (FLS) can deal with the vagueness and parameters of a FLS may also be noisy. uncertainty residing in the knowledge All of’ these uncertainties translate into possessed by human beings or implicated in uncertainties about fuzzy set membership the numerical data, and it allows us to functions. Traditional type-I fuzzy sets are not represent the system parameters with linguistic able to directly model such uncertainties terms[2]. Since the introduction of the basic because their membership functions are totally conceptions of the fuzzy set theory, FLS have crisp. On the other hand, type-2 fuzzy sets are been studied for more than 30 years. The able to model such uncertainties because their success of their applications is in various membership functions are themselves fuzzy. fields. They can be very helpful to achieve classification tasks, offline process simulation and diagnosis, decision support tools, and process control [3]. Fuzzy rules and membership functions have been used as a key tool to express knowledge. There are (at least) four sources of uncertainties associated with fuzzy logic systems (FLSs) which can be listed as follows: Figure.1-Structure of a Type-2 FLS 63 ﻫﻔﺘﻤﻴﻦ ﻛﻨﻔﺮاﻧﺲ ﺳﻴﺴﺘﻤﻬﺎي ﻓﺎزي ، 9-7 ﺷﻬﺮﻳﻮر 6831 ، داﻧﺸﮕﺎه ﻓﺮدوﺳﻲ ﻣﺸﻬﺪ
- 11. First Joint Congress on Fuzzy and Intelligent Systems Ferdowsi University of Mashhad, Iran 29-31 Aug 2007 Fuzzy Um-set in Sostak sense Intelligent Systems Scientific Society of Iran Mohsen Alimohammady and Mehdi Roohi Department of Mathematics, Faculty of Basic Sciences University of Mazandaran, Babolsar 47416–1468, Iran, amohsen@umz.ac.ir, mehdi.roohi@gmail.com Abstract. This paper is devoted to extending the notion of fuzzy minimal space in Sostak sense. The concepts of fuzzy Um -set, (U, m)-open set and (U, m)-closed set will be introduced and some char- acterizations of them are achieved. Finally, some brand of fuzzy minimal continuous functions and their relations with fuzzy Um -set, (U, m)-open set and (U, m)-closed set are given. 1 Introduction set where 1X is the characteristic function on X. A family τ of fuzzy sets in X is called a fuzzy topology After the discovery of the fuzzy sets, many at- for X if tempts have been made to extend various branches (a) α1X ∈ τ for each α ∈ I, of mathematics to the fuzzy setting. Fuzzy topo- (b) A ∧ B ∈ τ , where A, B ∈ τ and logical spaces as a very natural generalization of W (c) α∈A Aα ∈ τ whenever, Aα ∈ τ for all α topological spaces were ﬁrst put forward in the lit- in A. The pair (X, τ ) is called a fuzzy topological erature by Chang [11] in 1968. He studied a number space [17]. Every member of τ is called fuzzy open of the basic concepts including interior and closure set and its complement is called fuzzy closed sets of a fuzzy set, fuzzy continuous mapping and fuzzy [17]. In a fuzzy topological space X the interior and compactness. Several authors used Chang’s deﬁni- the closure of a fuzzy set A (denoted by Int(A) and tion in many direction to obtain some results which Cl(A) respectively) are deﬁned by are compatible with results in general topology. In 1976 Lowen [17] suggested an alternative and _ Int(A) = {U : U ≤ A, U is fuzzy open set} and more natural deﬁnition for achieving more results ^ which are compatible to the general case in topol- Cl(A) = {F : A ≤ F, F is fuzzy closed set}. ogy. For example with Chang’s deﬁnition, constant functions between fuzzy topological spaces are not Let f be a function from X to Y . It is a fuzzy necessarily fuzzy continuous but in Lowen’s sense function deﬁned by all of the constant functions are fuzzy continuous. 8 W In 1985 Sostak [25] introduced the smooth fuzzy < A(x) f −1 ({y}) = ∅ topology as an extension of Chang’s fuzzy topol- f (A)(y) = x∈f −1 ({y}) : 0 f −1 ({y}) = ∅, ogy. It has been developed in many directions [12], [14] and [15]. for all y in Y , where A is an arbitrary fuzzy set in In [20] authors introduced minimal structures X [27]. and minimal spaces. Some result about minimal spaces can be found in [2], [7], [13], [18], [19], [22] Deﬁnition 2.1. Let (X, τ ) be a fuzzy topological and [24]. The concept of fuzzy minimal structure space. A fuzzy set A in X is said to be was introduced and studied in [1], [3], [4], [5], [6] (a) fuzzy semi open if A ≤ Cl(Int(A)) [9], and [8] which it extended fuzzy topology deﬁned by Lowen [17]. In this paper we redeﬁne fuzzy (b) fuzzy preopen if A ≤ Int(Cl(A)) [10], minimal structure as a function which is a general- (c) fuzzy α-open if A ≤ Int(Cl(Int(A))) [10], ization of fuzzy topology introduced by Sostak [25]. (d) fuzzy β-open if A ≤ Cl(Int(Cl(A))) [21]. The family of all fuzzy semi open, fuzzy pre- open, fuzzy α-open and fuzzy β-open sets is 2 Preliminaries denoted by F SO(X), F P O(X), F αO(X) and F βO(X) respectively. The complement of a fuzzy For easy understanding of the material incorpo- semi open, fuzzy preopen, fuzzy α-open and fuzzy rated in this paper we recall some basic deﬁnitions β-open set is called fuzzy semi closed, fuzzy pre- and results. For details on the following notions closed, fuzzy α-closed and fuzzy β-closed respec- we refer to [23], [25] and [26]. A fuzzy set in(on) tively. The union of all fuzzy semi open, fuzzy pre- a universe set X is a function with domain X and open, fuzzy α-open and fuzzy β-open sets of X con- values in I = [0, 1]. The class of all fuzzy sets on tained in A is called fuzzy semi interior, fuzzy prein- X will be denoted by I X and symbols A,B,... is terior, fuzzy α-interior and fuzzy β-interior of A used for fuzzy sets on X. 01X is called empty fuzzy and is denoted by sInt(A), pInt(A), αInt(A) and 1 71 ﻫﻔﺘﻤﻴﻦ ﻛﻨﻔﺮاﻧﺲ ﺳﻴﺴﺘﻤﻬﺎي ﻓﺎزي ، 9-7 ﺷﻬﺮﻳﻮر 6831 ، داﻧﺸﮕﺎه ﻓﺮدوﺳﻲ ﻣﺸﻬﺪ
- 12. First Joint Congress on Fuzzy and Intelligent Systems Ferdowsi University of Mashhad, Iran 29-31 Aug 2007 Intelligent Systems Scientific Society of Iran Velocity Control of an Electro Hydraulic Servosystem by Sliding Mamdani M. Aliyari Shoorehdeli1, M. Teshnehlab2 1. Computer department of Science and Research Branch, Islamic Azad University of Tehran, Iran. 1, 2. K. N. Toosi University of Tech Tehran, Iran. Aliyari@gmail.com, {m_aliyari, teshnehlab}@eetd.kntu.ac.ir Abstract- This paper addresses new hybrid approaches for velocity control of an electro hydraulic servosystem (EHSS) in presence of flow nonlinearities and internal friction. In our new approaches, we combined classical method based-on sliding mode control and Mamdani networks. The control by using adaptive networks need plant’s Jacobean, but here this problem solved by sliding surface. It is demonstrated that this new technique have good ability control performance. It is shown that this technique can be successfully used to stabilize any chosen operating point of the system. All derived results are validated by computer simulation of a nonlinear mathematical model of the system. The controllers which introduced have big range for control the system. Keywords: Electro Hydraulic, Mamdani networks, Strict Feedback, Sliding mode, Sliding Surface. 1 Introduction in some detail. In section 4, system properties and The EHSS is used in many industrial applications, the relation of new method and simulation results because of its ability to handle large inertia and describe and section 5 is the conclusion part. torque loads and, at the same time, achieve fast responses and a high degree of both accuracy and 2 System Description performance [1, 2]. Depending on the desired control objective, an A scheme of an electrohydraulic velocity EHSS can be classified as either a position, servosystem is shown in Figure 1. velocity or force/torque EHSS. Therefore, control The basic parts of this system are: 1. hydraulic techniques for electro-hydraulic servo system have power supply, 2. accumulator, 3. charge valve, 4. been widely studied over the past decade; the pressure gauge device, 5. filter, 6. two-stage details of these systems are given in the reference electrohydraulic servovalve, 7. hydraulic motor, 8. [3]. In [8] an intelligent CMAC neural network measurement device, 9. personal computer, and 10. controller using feedback error learning approach Voltage- to- current converter. introduced which is very complex, and [3] presents methods based on feedback linearization and backstepping approaches, that have good performances but the controller designing is not very simple procedure, in [9] authors proposed a simpler method than other methods based on Lyapunov stability approach but the results in this new hybrid approaches are very better and faster than other mentioned methods. The paper is organized like this: In section 2, the Figure 1: Electrohydraulic velocity servosystem. EHSS and its nonlinear mathematical model are described. In section 3, issues related to the sliding A mathematical representation of the system is mode design and Mamdani network are discussed derived using Newton’s Second Law for the 103 ﻫﻔﺘﻤﻴﻦ ﻛﻨﻔﺮاﻧﺲ ﺳﻴﺴﺘﻤﻬﺎي ﻓﺎزي ، 9-7 ﺷﻬﺮﻳﻮر 6831 ، داﻧﺸﮕﺎه ﻓﺮدوﺳﻲ ﻣﺸﻬﺪ
- 13. First Joint Congress on Fuzzy and Intelligent Systems Ferdowsi University of Mashhad, Iran 29-31 Aug 2007 Intelligent Systems Scientific Society of Iran An On-line Fuzzy Backstepping Controller for Rotary Inverted Pendulum System Mahsa Rahmanian1, Mahdi Aliyari Shoorehdeli2, Mohammad Teshnehlab3 1, 2 Computer Department, Science & Research Branch, Islamic Azad University 3 Electrical Department, K. N. Toosi University of Tech 1 Mahsa_r_360@yahoo.com, 2Aliyari@gmail.com, 3Teshnehlab@eetd.kntu.ac.ir Abstract: In this study a new combination of This paper introduces a new combination of nonlinear backstepping scheme with on-line fuzzy backstepping scheme with an on-line fuzzy system system is presented for the rotary inverted pendulum to obtain a nonlinear controller to stabilize the system to achieve better performance in nonlinear pendulum at the upright equilibrium point. controller. The inverted pendulum, a popular mechatronic application, exists in many different 2 System Model and Dynamics forms. The common thread among these systems is Here, in this part the description of model and their goal: to balance a link on end using feedback dynamics are given. Fig (1-a) depicts the rotary control. The purpose of this study is to design a inverted pendulum in motion. Fig (1-b) depicts the stabilizing controller that balances the inverted pendulum as a lump mass at half the length of the pendulum in the upright position. pendulum. By applying the Euler-Lagrange equations, we can obtain the equations of motion as Keywords: Nonlinear backstepping, Fuzzy follows: approximator, Rotary inverted pendulum. (mr 2 ) + J eq θ& + mrLSin (α ) α 2 − mrLCos (α ) α = T − Beqθ & & && & 1 Introduction mL α − mrLCos (α )θ& − mgLSin(α ) = 0 4 2 & && (1) Inverted pendulum has been widely used in both 3 linear and nonlinear control education with applications to other under actuated mechanical where L is the length to pendulum's center of mass, systems, involving nonlinear dynamics, robotics and m is the mass of pendulum arm, r is the rotating arm aerospace vehicles testing [1]. length, θ is the servo load gear angle, α is the Recently, the pioneering work on fuzzy control via pendulum arm deflection, J eq is the equivalent backstepping has been done in [2-4]. In which, a fuzzy system is used to approximate the unknown moment of inertia at the load, Beq is the equivalent nonlinear function in each design step, and an viscous damping coefficient, g is the gravitational adaptive fuzzy controller was developed by meaning acceleration and T is the control torque. of backstepping technique for a class of SISO The Eq.(1) can be rewritten as the following state uncertain nonlinear systems [5]. equations: The rotary motion inverted pendulum, which is shown in Fig (1-a), is driven by a rotary servo motor x1 = x 3 & system [6]. The zero position for α and θ are x2 = x4 & defined as the pendulum being vertical ‘up’. 107 ﻫﻔﺘﻤﻴﻦ ﻛﻨﻔﺮاﻧﺲ ﺳﻴﺴﺘﻤﻬﺎي ﻓﺎزي ، 9-7 ﺷﻬﺮﻳﻮر 6831 ، داﻧﺸﮕﺎه ﻓﺮدوﺳﻲ ﻣﺸﻬﺪ
- 14. First Joint Congress on Fuzzy and Intelligent Systems Ferdowsi University of Mashhad, Iran 29-31 Aug 2007 Intelligent Systems Scientific Society of Iran Optimal Design of Type_1 TSK Fuzzy Controller Using GRLA F. Naderi, A. A. Gharaveisi and M. Rashidinejad Electrical Engineering Dept. of Shahid Bahonar University of Kerman Kerman, Iran a_gharaveisi@yahoo.com ABSTRACT A new methodology for designing optimal systematic GA-based fuzzy controller is presented in this paper. Our design is based on Genetic Reinforcement Learning Algorithm (GRLA), unlike conventional GA that is based on the competition between chromosomes only to survive, this method is based on competition and cooperation between chromosomes, GA tries to find good chromosomes and good combination for them to form an optimal fuzzy controller. The proposed GRLA design method has been applied to the cart-pole balancing system. The controller was capable of balancing ° the pole for initial conditions up to 80 . As a comparison we applied a Mamdani controller which is designed through normal GA and uses five membership functions for inputs and output variables to the same problem. the results show the efficiency of the proposed method. Keywords : Genetic Reinforcement Learning Algorithm (GRLA), TSK Fuzzy Controller I. INTRODUCTION if-then rules. So this fact has forced researchers to find a method that Fuzzy theory has been developed by automatically determines the parameters. L.A.Zadeh in 1965 [1]. Since conventional Some papers propose automatic methods control schemes are limited in their range of using neural networks (NN) [7,8] ,Fuzzy practical applications, fuzzy logic clustering [6] ,genetic algorithms(GA) controllers are receiving increased attention [3,9,10,11,12] , gradient methods[13,14], or for intelligent control applications [2]. Evolutionary Algorithm (EA) [4,22,23]Karr Fuzzy control systems employ a mode of used a GA to generate membership function approximate reasoning that resembles the for a fuzzy controller [15] in Karr’s work , decision-making process of humans. The the user has to define a method to set the behavior of a fuzzy controller is easily rules at first or hand-design this exhaustive understood by a human expert as knowledge task, then use the GA to design the mf only. is expressed by means of intuitive, linguistic Since the mf and rule set are highly rules. In the design of a fuzzy controller the dependent, hand-design a one, and GA- definition of membership functions and the design of the other, does not use the GA to establishment of control rules (if-then rules) its full advantage. So in most automatic are very important. designs For an FCS the optimization In [5], Procyk and Mamdani show that a of both the Membership function (mf) change in the membership function (mf) and the control rules of the FCS are may alter the fuzzy control system (FCS) required. performance significantly. Berenji [16, 17] introduced a method that Unfortunately the human experts are not learns to adjust the fuzzy mf of the linguistic sometimes able to express their knowledge labels used in different control rules through in the form of fuzzy 113 ﻫﻔﺘﻤﻴﻦ ﻛﻨﻔﺮاﻧﺲ ﺳﻴﺴﺘﻤﻬﺎي ﻓﺎزي ، 9-7 ﺷﻬﺮﻳﻮر 6831 ، داﻧﺸﮕﺎه ﻓﺮدوﺳﻲ ﻣﺸﻬﺪ
- 15. First Joint Congress on Fuzzy and Intelligent Systems Ferdowsi University of Mashhad, Iran 29-31 Aug 2007 Intelligent Systems Scientific Society of Iran System of linear fuzzy differential equations M. Otadi a , S. Abbasbandy b and M. Mosleh a a Department of Mathematics, Islamic Azad University, Firuozkooh Branch, Firuozkooh, Iran b Department of Mathematics, Faculty of Science, Imam Khomeini International University, Ghazvin, 34149-16818, Iran Mahmood_Otadi@yahoo.com , Abbasbandy@yahoo.com, Maryam_mosleh79@yahoo.com Abstract: In this paper, we discuss the solution of a system of linear fuzzy differential equations, where A and B are real n × n matrix and the initial condition x 0 is described by a vector made up of n fuzzy numbers. In this paper, we investigated a necessary and sufficient condition for the existence . derivative x ( t ) of a fuzzy process x(t). Keywords: Fuzzy number, Linear fuzzy differential equations, Fuzzy system 1. Introduction The concept of fuzzy numbers and fuzzy rithmetic exactly. One method of treating this uncertainty is operations were first introduced by Zadeh [19], to use a fuzzy set theory formulation of problem Dubois and Prade[8]. We refer the reader to [9] for [18]. The topic of Fuzzy Differential Equations more information on fuzzy numbers and fuzzy (FDEs) has been rapidly growing in recent years. arithmetic. Fuzzy systems are used to study a The concept of fuzzy derivative was first variety of problems ranging from fuzzy topological introduced by Chang and Zadeh [5] , it was spaces [6] to control chaotic systems [11], fuzzy followed up by Dobois and Prade [7] who used the metric spaces [17], fuzzy differential equations [3], extension principle in their approach. Other ethods fuzzy linear systems [1,2]. have been discussed by Puri and Ralescu [10]. One of the major applications of fuzzy number fuzzy differential equations were first formulated arithmetic is treating system of linear fuzzy by Kaleva [9] and Seikkala [18] in time dependent differential equations [20]. In modelling real form. Kaleval had formulated fuzzy differential systems one can be frequently confronted with a equations, in term of derivative [9]. Buckley and Feuring cite{bf} have give a very general differential equation formulation of a fuzzy first-order initial value problem. They first find the crisp solution, fuzzify where the structure of the equation is known, it and then check to see if it satisfies the FDE. represented by the vector field f, but the model Pearson cite{pear} has a property of linear fuzzy parameters and the initial value x0 are not known differential equations, 123 ﻫﻔﺘﻤﻴﻦ ﻛﻨﻔﺮاﻧﺲ ﺳﻴﺴﺘﻤﻬﺎي ﻓﺎزي ، 9-7 ﺷﻬﺮﻳﻮر 6831 ، داﻧﺸﮕﺎه ﻓﺮدوﺳﻲ ﻣﺸﻬﺪ
- 16. First Joint Congress on Fuzzy and Intelligent Systems Ferdowsi University of Mashhad, Iran 29-31 Aug 2007 Intelligent Systems Scientific Society of Iran A method for fully fuzzy linear system of equations H. Rouhparvar ∗, T. Allahviranloo † ∗ Department of Mathematics, Saveh Branch, Islamic Azad University, Saveh, Iran † Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran Abstract:Fuzzy linear systems of equations play a aspects and the works on them are too limited. Some major role in various applications such as ﬁnancial, of the most interesting works on these matrices can be economics, engineering and physics. We employ new seen in [5, 9]. In [8] a computational method repre- product-type operation which are introduced and studied sented for solving a FFLSEs based on Zadehs exten- in [4], as e.g. the cross product of fuzzy numbers that sion principle and in [18] provided a method to solve from the theoretical and practical point of view in fuzzy this problem based on their cuts. Now we provide a arithmetic, is eﬃciently of the multiplication based on method based on r-cuts and new deﬁnition of product Zadeh’s extension principle for ﬁnding a fuzzy vector x fuzzy numbers, i.e., the cross product to solve FFLSEs which satisﬁes Ax = b, where An×n and b are a fuzzy is computational and practical. In this paper we use matrix and a fuzzy vector, respectively. We transform triangular fuzzy numbers. fully fuzzy linear system of equations (FFLSEs) to The structure of this paper is organized as follows: extended crisp linear system in [12]. This systems can In Section 2, we recall some concepts from fuzzy arith- be solve as numerically, see [1, 2, 3, 6]. metic. We illustrate summary of the cross product in Section 3. In Section 4, we solve A ⊗ x = b by pro- Keywords: Fully fuzzy linear system of equations, posed method. The proposed method are illustrated Cross product, Fuzzy number, Fuzzy matrix. by solving some examples in section 5 and conclusion are drawn in Section 6. 1 Introduction Fuzzy linear system, Ax = b, where the coeﬃcient ma- 2 Preliminaries trix A is crisp, while b is a fuzzy number vector, is In this section we represent deﬁnitions needful for next solved in [12, 13, 14]. Friedman et al. [12] use the em- sections. We denote by E 1 the set of all fuzzy numbers. bedding method given in [19]. Allahviranloo [1, 2, 3] uses the iterative Jacobi and Gauss Siedel method, the Adomian method and SOR method, respectively. Of 2.1 Fuzzy numbers course, the a lot of papers with diﬀerent methods ex- ist in this content but in continuation to these works, Deﬁnition 2.1. A fuzzy subset u of the real line R people worked the case in which all parameters in a with membership function u(t) : R → [0, 1] is called a fuzzy linear system are fuzzy numbers, which we call fuzzy number if: it a Fully Fuzzy Linear System of Equations(FFLSEs), for example see [8, 16, 18]. (a) u is normal, i.e., there exist an element t0 such Here we will solve A ⊗ x = b, where A is a fuzzy ma- that u(t0 ) = 1, trix and x and b are fuzzy vectors. We will use fuzzy (b) u is fuzzy convex, i.e., u(λt1 + (1 − λ)t2 ) ≥ matrix deﬁned in [10]. This class of fuzzy matrices con- min{u(t1 ), u(t2 )} ∀t1 , t2 ∈ R, ∀λ ∈ [0, 1], sist of applicable matrices, which can model uncertain ∗ Email address: rouhparvar59@gmail.com. (c) u(t) is upper semi continuous, 1 129 ﻫﻔﺘﻤﻴﻦ ﻛﻨﻔﺮاﻧﺲ ﺳﻴﺴﺘﻤﻬﺎي ﻓﺎزي ، 9-7 ﺷﻬﺮﻳﻮر 6831 ، داﻧﺸﮕﺎه ﻓﺮدوﺳﻲ ﻣﺸﻬﺪ
- 17. First Joint Congress on Fuzzy and Intelligent Systems Ferdowsi University of Mashhad, Iran 29-31 Aug 2007 Intelligent Systems Scientific Society of Iran Full fuzzy linear systems of the form Ax+b=Cx+d M. Mosleh a , S. Abbasbandy b and M. Otadi a a Department of Mathematics, Islamic Azad University, Firuozkooh Branch, Firuozkooh, Iran b Department of Mathematics, Faculty of Science, Imam Khomeini International University, Ghazvin, 34149-16818, Iran Maryam_mosleh79@yahoo.com, Abbasbandy@yahoo.com, Mahmood_Otadi@yahoo.com Abstract: This paper mainly intends to discuss the solution of the full fuzzy linear systems (FFLS) Ax+b=Cx+d, where A and C are fuzzy matrices, b and d are fuzzy vectors. Ming Ma et al. introduced a new fuzzy arithmetic based on parametric form of fuzzy numbers, which we apply it for our purpose. Keywords: Fuzzy number, Full fuzzy linear systems, Fuzzy arithmetic, Parametric epresentation 1. Introduction column is an arbitrary fuzzy number vector. They The concept of fuzzy numbers and fuzzy used the parametric form of fuzzy numbers and arithmetic operations were first introduced by replaced the original fuzzy n × n linear system by Zadeh [16], Dubois and Prade [7]. Fuzzy systems a crisp 2n × 2n linear system and studied duality are used to study a variety of problems ranging in fuzzy linear systems Ax=Bx+y where A , B from fuzzy topological spaces [5] to control haotic are real n × n matrices, the unknown vector x is systems [8,11], fuzzy metric spaces [14], fuzzy vector consisting of n fuzzy numbers and the differential equations [3], fuzzy linear systems constant y is vector consisting of n fuzzy numbers, [1,2]. in [10]. In [1,2] the authors presented conjugate One of the major applications of fuzzy number gradient, LU decomposition method for solving arithmetic is treating fuzzy linear systems and fully general fuzzy linear systems or symmetric fuzzy fuzzy linear systems, several problems in various linear systems. Also, Wang et al. [15] presented an areas such as economics, engineering and physics iterative algorithm for solving dual linear system boil down to the solution of a linear system of of the form x=Ax+u, where A is real n × n matrix, equations. In many applications, at least some of the unknown vector x and the constant u are all the parameters of the system should be represented vectors consisting of fuzzy numbers and by fuzzy rather than crisp numbers. Thus, it is abbasbandy [4] investigated the existence of a immensely important to develop numerical minimal solution of general dual fuzzy linear procedures that would appropriately treat fuzzy equation system of the form Ax+f=Bx+c, where A, linear systems and solve them. B are real m × n matrices, the unknown vector x is vector consisting of n fuzzy numbers and the Friedman et al. [9] introduced a general model onstant f, c are vectors consisting of $m$ fuzzy for solving a fuzzy n × n linear system whose numbers. Recently, Muzziloi et al. [13] considered coefficient matrix is crisp and the right-hand side fully fuzzy linear systems of the form 135 ﻫﻔﺘﻤﻴﻦ ﻛﻨﻔﺮاﻧﺲ ﺳﻴﺴﺘﻤﻬﺎي ﻓﺎزي ، 9-7 ﺷﻬﺮﻳﻮر 6831 ، داﻧﺸﮕﺎه ﻓﺮدوﺳﻲ ﻣﺸﻬﺪ
- 18. First Joint Congress on Fuzzy and Intelligent Systems Ferdowsi University of Mashhad, Iran 29-31 Aug 2007 Intelligent Systems Scientific Society of Iran Solving fuzzy polynomial equation by ranking method H. Rouhparvar Department of Mathematics, Saveh Branch, Islamic Azad University, Saveh, Iran rouhparvar59@gmail.com Abstract: In this paper, we find real roots for fuzzy polynomial equations (if exists) by ranking fuzzy numbers. We transform fuzzy polynomial equation to system of crisp polynomial equations, this transformation perform with ranking method based on three parameters Value, Ambiguity and Fuzziness. Provided system of crisp polynomial equations can be solved numerically. Finally, we illustrate our approach by numerical examples. Keywords: Fuzzy number, Fuzzy polynomial, Value, Ambiguity, Fuzziness. 1 Introduction In Section 2, we review basic definition and the Polynomial equations play a major role in various notions of fuzzy numbers, Value, Ambiguity and areas such as mathematics, engineering and social Fuzziness. In Section 3, we represent fuzzy sciences. We interested in finding real roots of polynomial equation and proposed method for polynomial equation like solving it and provide two total examples. Some numerical examples are considered in Section 4 , and Conclusion comes in Section 5. that x ∈ ℜ (if exists) and C 0 , C1 , K , C n are fuzzy numbers, by a ranking method of fuzzy numbers. 2 Preliminaries This ranking method is provided for canonical representation of the solution of fuzzy linear In this section, we review some preliminaries system in [10]. The applications of fuzzy which are needed in the next section. For more polynomial equations are considered by [2]. Also, details, see [7,8,9]. numerical solution of fuzzy polynomial equations by fuzzy neural network is solved in [3] and linear Definition 2.1. A fuzzy subset C of the real line and nonlinear fuzzy equations are shown by ℜ with membership function C ( x) : ℜ → [0,1] is [1,4,5,6]. called a fuzzy number if In this paper, we propose a new method for solving 1. C is normal, i.e., there exist an element x0 an fuzzy polynomial equation based on ranking such that C ( x 0 ) = 1 , method which is introduced by Delgado et. al. 2. C is fuzzy convex, i.e., C (λx1 + (1 − λ ) x 2 ) [7,8]. They introduced three real indices called Value, Ambiguity and Fuzziness to obtain ≥ min{C ( x1 ), C ( x 2 )} ∀x1 , x 2 ∈ ℜ, ∀λ ∈ [0,1] , "simple", fuzzy numbers, that could be used to 3. C (x) is upper semi continuous, represent more arbitrary fuzzy numbers. Therefore, 4. support C is bounded, where support C = cl{x ∈ ℜ : C ( x) > 0},and cl is we use this parameters and transform fuzzy polynomial equation to system of crisp polynomial equations then with solving this system we can the closure operator. find real roots of fuzzy polynomial equation. 141 ﻫﻔﺘﻤﻴﻦ ﻛﻨﻔﺮاﻧﺲ ﺳﻴﺴﺘﻤﻬﺎي ﻓﺎزي ، 9-7 ﺷﻬﺮﻳﻮر 6831 ، داﻧﺸﮕﺎه ﻓﺮدوﺳﻲ ﻣﺸﻬﺪ
- 19. First Joint Congress on Fuzzy and Intelligent Systems Ferdowsi University of Mashhad, Iran 29-31 Aug 2007 Intelligent Systems Scientific Society of Iran Stabilization of Autonomous Bicycle by an Intelligent Controller N. Noroozi and F. Shabani Nia Department of Electrical Engineering, Shiraz University, Shiraz, Iran Abstract: Nowadays, urban traffic jams cause waste of time and air pollution. Bicycle is a convenient alternative as a mean of transportation for being confronted with these problems. However, bicycle is unstable in itself and it will fall down without human assistance like steering handle or moving upper body. It needs practice to ride a bicycle [1]. In this paper, an intelligent controller is proposed, which can stabilize a bicycle with suitable performance for the whole of considered range of disturbance. The original part of controller is consists mainly of a fuzzy controller. Also it is used genetic algorithms and neural networks for optimization and adaptation of scaling factors respectively. Keywords: Bicycle, Fuzzy Controller, Stability, Genetic Algorithms, Neural Networks genetic algorithms for optimization of response. I. Introduction Neural networks are proposed the controller to Nowadays, urban traffic jams cause waste of conform. time and air pollution. Bicycle will help to solve This paper is composed of five sections. In these problems. In addition, the energy conversion section II, dynamic modeling is shown. Section III efficiency of bicycle is critically higher than other shows the proposed control strategy. In section IV vehicles [2]. simulation results are implemented to verify the Soft computing approaches in decision proposed control strategy. The conclusions are making have become increasingly popular in summarized in section V. many disciplines. This evident from the vast number of technical papers appearing in journals II. The Model and conference proceeding in all areas of Assuming that the rider doesn't move upper engineering, manufacturing, sciences, medicine, body, the dynamic model of bicycle is represented and business. Soft computing is a rapidly evolving as follows [1] field that combines knowledge, techniques from neural networks, fuzzy set theory, and Iθ& = TGR + FCF . h & (1) approximate reasoning, and using optimization methods such as genetic algorithms. The where I, TGR, FCF, h, and θ mean moment of integration of these and other methodologies inertia, gravitational force, centrifugal force, forms the core of soft computing [6]. height of COG, and camber angle of bicycle This paper presents an intelligent controller respectively as shown Fig.1. for stabilization of bicycle by controlling its steering. The original part of controller is consists mainly of a fuzzy controller. Also it is used 153 ﻫﻔﺘﻤﻴﻦ ﻛﻨﻔﺮاﻧﺲ ﺳﻴﺴﺘﻤﻬﺎي ﻓﺎزي ، 9-7 ﺷﻬﺮﻳﻮر 6831 ، داﻧﺸﮕﺎه ﻓﺮدوﺳﻲ ﻣﺸﻬﺪ
- 20. First Joint Congress on Fuzzy and Intelligent Systems Ferdowsi University of Mashhad, Iran 29-31 Aug 2007 Intelligent Systems Scientific Society of Iran Solving fuzzy linear programming problems with piecewise linear membership functions by the determination of a crisp maximizing decision S. Effati and H. Abbasiyan e-mail:effati911@yahoo.com e-mail:abbasiyan58@yahoo.com Abstract: In this paper, we concentrate on linear programming problems in which both the right-hand side and the technological coefficients are fuzzy numbers.We consider here only the case of fuzzy numbers with linear membership function. The determination of a crisp maximizing decision [2] is used for a defuzzification of these problems. The crisp problems obtained after the defuzzification are non-linear and non-convex in general. We propose here the ”augmented lagrangian penalty function method” and use it for solving these problems. We also compare the new proposed method with well known ”fuzzy decisive set method”. Finally, we give illustrative example and this solve by the new proposed method and compare the numerical solution with the solution obtained from fuzzy decisive set method. Keywords: Fuzzy linear programming, fuzzy number, augmented lagrangian penalty function method, fuzzy decisive set method. 1 Introduction A model in which the objective function is crisp– constraints. For solving these problems we use and that is, has to be maximized or minimized– and in compare two methods. One of them called the which the constraints are or partially fuzzy is no fuzzy decisive set method, as introduced by longer symmetrical. Fuzzy linear programming Sakawa and Yana [7]. In this methoda combination problem with fuzzy coefficients was formulated by with the bisection method and phase one of the Negoita [3] and called robust programming. simplex method of linear programming is used to Dubois and Prade [4] investigated linear fuzzy obtained a feasible solution. The second method constraints. Tanaka and Asai [5] also proposed a we use, is the ”augmented lagrangian penalty formulation of fuzzy linear programming with method”. In this method a combines the fuzzy constraints and gave a method for its algorithmic aspects of both Lagrangian duality solution which bases on inequality relation methods and penalty function methods. For this between fuzzy numbers. kind of problems we consider, this method is We consider linear programming problems in applied to solve concrete examples. which both technological coefficients and right- The paper is outlined as follows. In section 2, we hand-side numbers are fuzzy numbers. Each study the linear programming problem in which problem is first converted into an equivalent crisp both technological coefficients and right-hand-side problem. This is a problem of finding a point are fuzzy numbers. The general principles of the which satisfies the constraints and the goal with the augmented lagrangian penalty method is presented maximum degree. The crisp problems, obtained by in section 3. In section 4, we examine the such a manner, can be non-linear (even non- application of this method and fuzzy decisive set convex), where the non-linearity arises in method to concrete example. 205 ﻫﻔﺘﻤﻴﻦ ﻛﻨﻔﺮاﻧﺲ ﺳﻴﺴﺘﻤﻬﺎي ﻓﺎزي ، 9-7 ﺷﻬﺮﻳﻮر 6831 ، داﻧﺸﮕﺎه ﻓﺮدوﺳﻲ ﻣﺸﻬﺪ
- 21. First Joint Congress on Fuzzy and Intelligent Systems Ferdowsi University of Mashhad, Iran 29-31 Aug 2007 Intelligent Systems Scientific Society of Iran Fuzzy Convex Subalgebras of Residuated Lattice Shokoofeh Ghorbani and Abbas Hasankhani Department of Mathematics Shahid Bahonar University of Kerman Kerman, Iran sh.ghorbani@graduate.uk.ac.ir abhasan@mail.uk.ac.ir Abstract: In this paper we define the concept of fuzzy congruence relations, fuzzy subalgebras, fuzzy convex subalgebras on a residuated lattice and we obtain some related results. In particular, we show that there is a one to one corresponding between the set of all fuzzy convex subalgebras and the set of all fuzzy congruence relations in the residuated lattice. Keywords: Residuated lattice, Congruence relation, Convex subalgebra, Fuzzy set, Fuzzy relation. 2 Preliminaries 1 Introduction Definition 2.1[2, 9] A residuated lattice is an The concept of residuated lattice was introduced algebra ( L,∧,∨,∗, →,0,1) where and studied by Krull [7], Dilworth [4], Ward and LR1) ( L,∧,∨,0,1) is a bounded lattice, Dilworth [11], Ward [10], Bables and Dwinger [1] LR2) ( L,∗,1) is a commutative monoid, and Pavelka [8]. The origins of residuated theory lie in the study of ideal lattices of rings. Zadeh in LR3) c ≤ a → b if and only if c ∗ a ≤ b for all [12] introduced the notion of fuzzy set μ in a a, b, c ∈ L . nonempty set X , as a function from X into [0,1]. Theorem 2.2 [6, 8, 9] Let ( L,∧,∨,∗, →,0,1) be a In the next section we review the basic residuated lattice. Then we have the following: definitions and some theorems from [2], [3], [5], 1 1 → x = x , x → x = 1, x → 1 = 1 , [9] and [12]. In section 3 we introduce the concept 2 x∗ y ≤ x ∧ y, of fuzzy congruence relations on a residuated 3 x ≤ y if and only if x → y = 1 , lattice and we see that the set of all fuzzy 4 x ∗ ( x → y ) ≤ x, y , congruence relations on a residuated lattice is a complete lattice. In section 4, we define the 5 x → y ≤ x∗z → y∗z , concept of fuzzy subalgebras; fuzzy convex 6 x ≤ y if and only if x ∗ z ≤ y ∗ z , subalgebras and we obtain some related results. 7 x → y ≤ ( z → x) → ( z → y ) , Finally, we show that there is a bijection between 8 x → y ≤ ( y → z) → ( x → z) , the set of all fuzzy convex subalgebras and the set of all fuzzy congruence relations on a residuated 9 x → ( y → z) = x ∗ y → z , lattice. Hence the set of all fuzzy convex 10 x ∗ ( y ∨ z ) = x ∗ y ∨ x ∗ z , subalgebras of residuated lattice is a complete for all x, y, z ∈ L . lattice. 211 ﻫﻔﺘﻤﻴﻦ ﻛﻨﻔﺮاﻧﺲ ﺳﻴﺴﺘﻤﻬﺎي ﻓﺎزي ، 9-7 ﺷﻬﺮﻳﻮر 6831 ، داﻧﺸﮕﺎه ﻓﺮدوﺳﻲ ﻣﺸﻬﺪ
- 22. First Joint Congress on Fuzzy and Intelligent Systems Ferdowsi University of Mashhad, Iran 29-31 Aug 2007 Intelligent Systems Scientific Society of Iran Algebraic Fuzzy Subsets of Non-commutative Join Spaces H. Hedayatia , R. Amerib a Department of Basic Sciences, Babol University of Technology, Babol, Iran b Department of Mathematics, Faculty of Basic Sciences, University of Mazandaran, Babolsar, Iran Email: {h.hedayati, ameri}@umz.ac.ir Abstract liminaries of hypergroups and fuzzy sets which will be used in the next sections. In section 3 we study fuzzy The aim of this note is the study of important alge- closed and fuzzy normal subhypergroups. In section 4 braic fuzzy subsets of transposition hypergroups (non- we introduce the notions of fuzzy invertible and fuzzy commutative join spaces). In this regards, we ﬁrst intro- reﬂexive subset of a hypergroup, and then we investi- duce the notions of fuzzy closed, fuzzy normal, fuzzy gate the relationships between fuzzy closed, fuzzy nor- reﬂexive and fuzzy invertible subsets of transposition mal, fuzzy invertible and fuzzy reﬂexive subsets. hypergroups and, then we investigate their basic prop- erties. Keywords: Fuzzy Hypergroup, Fuzzy Closed set, 2 Preliminaries Fuzzy Normal, Fuzzy Invertible, Fuzzy Reﬂexive. In this section we gather all deﬁnitions and simple prop- erties we require of hypergroups and fuzzy subsets and 1 Introduction set the notions. The notion of hypergroup introduced by F. Marty [24]. Let H be a nonempty set and P∗ (H) the family of all Since then many researchers have studied this ﬁeld and nonempty subsets of H. developed it, for example see [2, 4, 8, 9, 10, 11, 12, 19, A map . : H × H −→ P∗ (H) is called hyperoperation 26]. Transposition hypergroups are large class of multi or join operation. valued systems. Many well-known hypergroups such as The join operation is extended to subsets of H in nat- hypergroups, weak cogroups, join spaces, polygroups, ural way , so that A.B or AB is given by canonical hypergroups, as well as ordinary groups are all transposition hypergroups [19]. AB = {ab|a ∈ A and b ∈ B }. The notion of a fuzzy subset introduced by L. Zadeh in 1965 [27] as a function from a nonempty set X to unit The relational notation A ≈ B (read A meets B) is used real interval I = [0, 1]. Rosenfeld deﬁned the concept of to asserts that A and B have an element in common, that a fuzzy subgroup of a given group G [25] and then many / is, A ∩ B = 0. The notations aA and Aa are used for researchers has developed it in all subject of algebra. {a}A and A{a} respectively. Generally, the singleton Also fuzzy sets has been a good developed in hyper- {a} is identiﬁed by its element a. alebraic structures theory(for example see [1],[3],[4], In H right extension / and left extension are deﬁned [5], [6], [7], [13], [14], [15], [16], [17], [21], [22], by [23],[29]). a/b = { x ∈ H | a ∈ xb}, The second author in [1] introduced some basic re- sults of fuzzy transposition hypergroups. Now in this ba = { x ∈ H | a ∈ bx }. note we follow [1] and introduce some another impor- tant fuzzy subsets of transposition hypergroups, such A hypergroup is a structure (H, .) that satisﬁes two as fuzzy normal, fuzzy reﬂexive and fuzzy invertible axioms, subsets and then we obtain some related basic results (Reproductivity) aH = H = Ha for all a ∈ H; of these notions. In section 2 we gather all the pre- (Associativity) a(bc) = (ab)c for all a, b, c ∈ H. 1 217 ﻫﻔﺘﻤﻴﻦ ﻛﻨﻔﺮاﻧﺲ ﺳﻴﺴﺘﻤﻬﺎي ﻓﺎزي ، 9-7 ﺷﻬﺮﻳﻮر 6831 ، داﻧﺸﮕﺎه ﻓﺮدوﺳﻲ ﻣﺸﻬﺪ
- 23. First Joint Congress on Fuzzy and Intelligent Systems Ferdowsi University of Mashhad, Iran 29-31 Aug 2007 Intelligent Systems Scientific Society of Iran An Ant Based Algorithm Approach to Vehicle Navigation Hojjat Salehi-nezhad and Fereydoun Farrahi-Moghaddam Department of Electrical Engineering, S.B University of Kerman, Kerman, Iran h.salehi@mail.uk.ac.ir ffarrahi @mail.uk.ac.ir Abstract This paper presents an algorithm to search for the best direction between two desired origin and destination intersections in cities using ant algorithms, which is known as Ant-based Vehicle Navigation (AVN) algorithm. As travelers have their own parameters to select a direction, adjustable parameters were considered in this algorithm in order to satisfy the most important passengers’ needs. A practical model of this algorithm has applied on a part of Kerman city and its validity was examined. By using this AVN algorithm the traffic problem could be decreased efficiently without employing geographical positioning system (GPS). Keywords: Ant algorithms; Vehicle navigation; Vehicle traffic; Travel time 1. Introduction ACO algorithm has plenty of applications in release on the route transiting in. By means of this, network problems such as water distribution they do their complex daily activities such as networks [18], computer networks [7] and finding and sorting foods. transportation networks [17]. For example, in The idea of employing the foraging behavior of ants computer network problems, routing of packets as a method of stochastic combinatorial and finding paths from source to destination nodes optimization was initially introduced by Dorigo in has a closed relation with the cost and type of his PhD thesis [1]. This is very important for utilization of that network. ACO with its specific travelers to get the best direction relevant to their type of agents is a perfect tool for optimization of preferences, even though they be familiar or such networks, which results in reducing cost in unfamiliar with a city. We want to use ants for path finding and packet routing [7]. Transportation finding the best path between two specific locations networks are similar to computer networks by according to different local and statistical considering vehicles as network packets and routes parameters important for city travelers supporting as network connections, therefore same solution them. could be used for both types of networks. This paper is organized as follows. The next section In this paper an algorithm is proposed to find the and section 3 reviews the basic principles of the best direction satisfying city travelers using ant ACS algorithm and AVN algorithm respectively. algorithms. Although real ants are blind, they are Sections 4 renders the details of the proposed capable of finding the shortest path from food method and experimental results are presented in source to their nest by exploiting information of a section 5. Finally, the paper in concluded in section liquid substance called pheromone, which they 6. 237 ﻫﻔﺘﻤﻴﻦ ﻛﻨﻔﺮاﻧﺲ ﺳﻴﺴﺘﻤﻬﺎي ﻓﺎزي ، 9-7 ﺷﻬﺮﻳﻮر 6831 ، داﻧﺸﮕﺎه ﻓﺮدوﺳﻲ ﻣﺸﻬﺪ
- 24. First Joint Congress on Fuzzy and Intelligent Systems Ferdowsi University of Mashhad, Iran 29-31 Aug 2007 Intelligent Systems Scientific Society of Iran Novel ranking method of fuzzy numbers M. Barkhordary a T. Allahviranloo b T. Hajjari c a Department of Mathematics, Bandar Abbas Branch, Islamic Azad University, Bandar Abbas, Iran b Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, 14515, Iran c Department of Mathematics, Firuz Kuh Branch, Islamic Azad University, Firuz Kuh, 132, Iran Mahnaz_barkhordari@yahoo.com, tofigh@allahviranloo.com, tayebehajjari@yahoo.com Abstract: In this paper, a novel method for ranking of fuzzy numbers is proposed. This method is based on the center of mass at some α-cuts of a fuzzy number. Our method can rank more than two fuzzy numbers simultaneously. Also some properties of method are described. At last, we present some numerical examples to illustrate our proposed method, and compare them with other ranking methods. Keywords: Ranking fuzzy numbers; Center of mass. human intuition, indiscrimination, and difficulty of 1 Introduction interpretation. What seems to be clear is that there exists no uniquely best method for comparing For ranking of fuzzy numbers, a fuzzy number fuzzy numbers, and different methods may satisfy needs to be evaluated and compared with the different desirable criteria. In the existing fuzzy others, but this may not be easy. Since fuzzy number ranking methods, many of them are based numbers are represented by possibility on the area measurement with the integral value distributions, they can overlap with each other and about the membership function of fuzzy numbers it is difficult to determine clearly whether one [5, 20, 17, 21, 7, 11, 13, 9, 14, 18, 16, 15]. Yager fuzzy number is larger or smaller than another. [8] proposed centroid index ranking method with Fuzzy set ranking has been studied by many weighting function. Cheng [4] proposed a centroid researchers. Some of these ranking methods have index ranking method that calculates the distance been compared and reviewed by Bortolan and of the centroid point of each fuzzy number and Degain [2]. More recently by Chen and Hwang [3], original point to improve the ranking method [23]. and it still receives much attention in recent years They also proposed a coefficient of variation (CV [5, 10, 17, 18]. Many methods for ranking Fuzzy index) to improve Lee and Lis method [16]. numbers have been proposed, such as representing Recently, Tsu and Tsao [6] pointed out the them with real numbers or using fuzzy relations. inconsistent and counter intuition of these two Wang and Kerre [18, 12] proposed some axioms as indices and proposed ranking fuzzy numbers with reasonable properties to determine the rationality the area between the centriod point and original of a fuzzy ranking method and systematically point. The rest of our work is organized as follows. compared a wide array of existing fuzzy ranking Section 2, contains the basic definitions and methods. Almost each method, however, has notations used in the remaining parts of the paper. pitfalls in some aspect, such as inconsistency with In Section 3, introduces the ranking method and 287 ﻫﻔﺘﻤﻴﻦ ﻛﻨﻔﺮاﻧﺲ ﺳﻴﺴﺘﻤﻬﺎي ﻓﺎزي ، 9-7 ﺷﻬﺮﻳﻮر 6831 ، داﻧﺸﮕﺎه ﻓﺮدوﺳﻲ ﻣﺸﻬﺪ

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