Search results for: type 2 fuzzy system

Authors: Erion Luga, Aksel Seitllari, Kemal Pervanqe

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This study investigates the relation between physical and mechanical properties of concrete paving stones using neuro-fuzzy approach. For this purpose 200 samples of concrete paving stones were selected randomly from different sources. The first phase included the determination of physical properties of the samples such as water absorption capacity, porosity and unit weight. After that the indirect tensile strength test and compressive strength test of the samples were performed. İn the second phase, adaptive neuro-fuzzy approach was employed to simulate nonlinear mapping between the above mentioned physical properties and mechanical properties of paving stones. The neuro-fuzzy models uses Sugeno type fuzzy inference system. The models parameters were adapted using hybrid learning algorithm and input space was fuzzyfied by considering grid partitioning. It is concluded based on the observed data and the estimated data through ANFIS models that neuro-fuzzy system exhibits a satisfactory performance.

Keywords: paving stones, physical properties, mechanical properties, ANFIS

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Authors: Muhammad M. A. S. Mahmoud

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Natural gas sweetening process is a controlled process that must be done at maximum efficiency and with the highest quality. In this work, due to complexity and non-linearity of the process, the H₂S gas separation and the intelligent fuzzy controller, which is used to enhance the process, are simulated in MATLAB – Simulink. The new design of fuzzy control for Gas Separator is discussed in this paper. The design is based on the utilization of linear state-estimation to generate the internal knowledge-base that stores input-output pairs. The obtained input/output pairs are then used to design a feedback fuzzy controller. The proposed closed-loop fuzzy control system maintains the system asymptotically-stability while it enhances the system time response to achieve better control of the concentration of the output gas from the tower. Simulation studies are carried out to illustrate the Gas Separator system performance.

Keywords: gas separator, gas sweetening, intelligent controller, fuzzy control

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Authors: B. Güney, Ç. Teke

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In recent years, rapid and correct decision making is crucial for both people and enterprises. However, uncertainty makes decision-making difficult. Fuzzy logic is used for coping with this situation. Thus, fuzzy linear programming models are developed in order to handle uncertainty in objective function and the constraints. In this study, a problem of a factory in food industry is investigated, required data is obtained and the problem is figured out as a fuzzy linear programming model. The model is solved using Zimmerman approach which is one of the approaches for fuzzy linear programming. As a result, the solution gives the amount of production for each product type in order to gain maximum profit.

Keywords: food industry, fuzzy linear programming, fuzzy logic, linear programming

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Authors: Tomáš Vyčítal

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In a simulation model of a railway system it is important, besides other crucial algorithms, to have correct behaviour of train overtaking in stochastic conditions. This problem is being addressed in many simulation tools focused on railway traffic, however these are not very human-realistic. The goal of this paper is to create a more human-realistic overtaking decision support system for the use in railway traffic simulations. A fuzzy system has been chosen for this task as fuzzy systems are well-suited for human-like decision making. The fuzzy system designed takes into account timetables, train positions, delays and buffer times as inputs and provides an instruction to overtake or not overtake.

Keywords: decision-making support, fuzzy systems, simulation, railway, transport

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Authors: S. H. Nasseri, A. Ebrahimnejad

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Fuzzy set theory has been applied to many fields, such as operations research, control theory, and management sciences. In this paper, we consider two classes of fuzzy linear programming (FLP) problems: Fuzzy number linear programming and linear programming with trapezoidal fuzzy variables problems. We state our recently established results and develop fuzzy primal simplex algorithms for solving these problems. Finally, we give illustrative examples.

Keywords: fuzzy linear programming, fuzzy numbers, duality, sensitivity analysis

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Authors: Luiz F. J. Maia

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This paper presents an approach to fuzzy control, with the use of new pertinence functions, applied in the case of an inverted pendulum. Appropriate definitions of pertinence functions to fuzzy sets make possible the implementation of the controller with only one control rule, resulting in a smooth control surface. The fuzzy control system can be implemented with analog devices, affording a true real-time performance.

Keywords: control surface, fuzzy control, Inverted pendulum, pertinence functions

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Authors: Firas M. Tuaimah, Huda M. Abdul Abbas

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Accurate Short Term Load Forecasting (STLF) is essential for a variety of decision making processes. However, forecasting accuracy can drop due to the presence of uncertainty in the operation of energy systems or unexpected behavior of exogenous variables. Interval Type 2 Fuzzy Logic System (IT2 FLS), with additional degrees of freedom, gives an excellent tool for handling uncertainties and it improved the prediction accuracy. The training data used in this study covers the period from January 1, 2012 to February 1, 2012 for winter season and the period from July 1, 2012 to August 1, 2012 for summer season. The actual load forecasting period starts from January 22, till 28, 2012 for winter model and from July 22 till 28, 2012 for summer model. The real data for Iraqi power system which belongs to the Ministry of Electricity.

Keywords: short term load forecasting, prediction interval, type 2 fuzzy logic systems, electric, computer systems engineering

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Authors: G. S. Thakur

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In this paper, four new operators (O1, O2, O3, O4) are proposed, defined and considered to study the new properties and identities on hesitant fuzzy sets. These operators are useful for different operation on hesitant fuzzy sets. The various theorems are proved using the new operators. The study of the proposed new operators has opened a new area of research and applications.

Keywords: vague sets, hesitant fuzzy sets, intuitionistic fuzzy set, fuzzy sets, fuzzy multisets

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Authors: Yas Barzegar, Atrin Barzegar

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Drinking water quality assessment is a major issue today; technology and practices are continuously improving; Artificial Intelligence (AI) methods prove their efficiency in this domain. The current research seeks a hierarchical fuzzy model for predicting drinking water quality in Rome (Italy). The Mamdani fuzzy inference system (FIS) is applied with different defuzzification methods. The Proposed Model includes three fuzzy intermediate models and one fuzzy final model. Each fuzzy model consists of three input parameters and 27 fuzzy rules. The model is developed for water quality assessment with a dataset considering nine parameters (Alkalinity, Hardness, pH, Ca, Mg, Fluoride, Sulphate, Nitrates, and Iron). Fuzzy-logic-based methods have been demonstrated to be appropriate to address uncertainty and subjectivity in drinking water quality assessment; it is an effective method for managing complicated, uncertain water systems and predicting drinking water quality. The FIS method can provide an effective solution to complex systems; this method can be modified easily to improve performance.

Keywords: water quality, fuzzy logic, smart cities, water attribute, fuzzy inference system, membership function

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Authors: Seungwoo Kim, Yeongcheol Cho

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In this paper, a holonomic mobile robot is designed in omnidirectional wheels and an adaptive fuzzy controller is presented for its precise trajectories. A kind of adaptive controller based on fuzzy multi-layered algorithm is used to solve the big parametric uncertainty of motor-controlled dynamic system of 3-wheels omnidirectional mobile robot. The system parameters such as a tracking force are so time-varying due to the kinematic structure of omnidirectional wheels. The fuzzy adaptive control method is able to solve the problems of classical adaptive controller and conventional fuzzy adaptive controllers. The basic idea of new adaptive control scheme is that an adaptive controller can be constructed with parallel combination of robust controllers. This new adaptive controller uses a fuzzy multi-layered architecture which has several independent fuzzy controllers in parallel, each with different robust stability area. Out of several independent fuzzy controllers, the most suited one is selected by a system identifier which observes variations in the controlled system parameter. This paper proposes a design procedure which can be carried out mathematically and systematically from the model of a controlled system. Finally, the good performance of a holonomic mobile robot is confirmed through live tests of the tracking control task.

Keywords: fuzzy adaptive control, fuzzy multi-layered controller, holonomic mobile robot, omnidirectional wheels, robustness and stability.

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Authors: Azam Salimi, Majid Delshad

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This paper presents a modified design of a sliding mode controller based on fuzzy logic for a New ZVThigh step up DC-DC Converter . Here a proportional - integral (PI)-type current mode control is employed and a sliding mode controller is designed utilizing fuzzy algorithm. Sliding mode controller guarantees robustness against all variations and fuzzy logic helps to reduce chattering phenomenon due to sliding controller, in that way efficiency increases and error, voltage and current ripples decreases. The proposed system is simulated using MATLAB / SIMULINK. This model is tested under variations of input and reference voltages and it was found that in comparison with conventional sliding mode controllers they perform better.

Keywords: switching mode power supplies, DC-DC converters, sliding mode control, robustness, fuzzy control, current mode control, non-linear behavior

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Authors: Soroor K. H. Al-Khafaji, Manal Mohammad Abed

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The research aim of this paper is to evaluate the reliability of a complex engineering system and to design a fuzzy model for the reliability estimation. The designed model has been applied on Vegetable Oil Purification System (neutralization system) to help the specialist user based on the concept of FMEA (Failure Mode and Effect Analysis) to estimate the reliability of the repairable system at the vegetable oil industry. The fuzzy model has been used to predict the system reliability for a future time period, depending on a historical database for the two past years. The model can help to specify the system malfunctions and to predict its reliability during a future period in more accurate and reasonable results compared with the results obtained by the traditional method of reliability estimation.

Keywords: fuzzy logic, reliability, repairable systems, FMEA

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Authors: František Včelař, Zuzana Pátíková

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Recently a new type of very general relational structures, the so called (L-)complete propelattices, was introduced. These significantly generalize complete lattices and completely lattice L-ordered sets, because they do not assume the technically very strong property of transitivity. For these structures also the main part of the original Tarski’s fixed point theorem holds for (L-fuzzy) isotone maps, i.e., the part which concerns the existence of fixed points and the structure of their set. In this paper, fundamental properties of (L-)complete propelattices are recalled and the so called L-fuzzy relatively isotone maps are introduced. For these maps it is proved that they also have fixed points in L-complete propelattices, even if their set does not have to be of an awaited analogous structure of a complete propelattice.

Keywords: fixed point, L-complete propelattice, L-fuzzy (relatively) isotone map, residuated lattice, transitivity

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Authors: Mounir Bekaik, Messaoud Ramdani

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We present in this work a new technique of stabilization for fault nonlinear systems. The approach we adopt focus on a fuzzy Luenverger observer. The T-S approximation of the nonlinear observer is based on fuzzy C-Means clustering algorithm to find local linear subsystems. The MOESP identification approach was applied to design an empirical model describing the subsystems state variables. The gain of the observer is given by the minimization of the estimation error through Lyapunov-krasovskii functional and LMI approach. We consider a three tank hydraulic system for an illustrative example.

Keywords: nonlinear system, fuzzy, faults, TS, Lyapunov-Krasovskii, observer

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Authors: Hafida Bouarfa, Mohamed Abed

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The main object of our work is the development and the validation of a system indicated Fuzzy Vulnerability. Fuzzy Vulnerability uses a fuzzy representation in order to tolerate the imprecision during the description of construction. At the the second phase, we evaluated the similarity between the vulnerability of a new construction and those of the whole of the historical cases. This similarity is evaluated on two levels: 1) individual similarity: bases on the fuzzy techniques of aggregation; 2) Global similarity: uses the increasing monotonous linguistic quantifiers (RIM) to combine the various individual similarities between two constructions. The third phase of the process of Fuzzy Vulnerability consists in using vulnerabilities of historical constructions narrowly similar to current construction to deduce its estimate vulnerability. We validated our system by using 50 cases. We evaluated the performances of Fuzzy Vulnerability on the basis of two basic criteria, the precision of the estimates and the tolerance of the imprecision along the process of estimation. The comparison was done with estimates made by tiresome and long models. The results are satisfactory.

Keywords: case based reasoning, fuzzy logic, fuzzy case based reasoning, seismic vulnerability

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Authors: Bader Alkandari, Jamal Y. Madouh, Ahmad M. Alkandari, Anwar A. Alnaqi

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A new formulation of fuzzy linear estimation problem is presented. It is formulated as a linear programming problem. The objective is to minimize the spread of the data points, taking into consideration the type of the membership function of the fuzzy parameters to satisfy the constraints on each measurement point and to insure that the original membership is included in the estimated membership. Different models are developed for a fuzzy triangular membership. The proposed models are applied to different examples from the area of fuzzy linear regression and finally to different examples for estimating the electrical load on a busbar. It had been found that the proposed technique is more suited for electrical load estimation, since the nature of the load is characterized by the uncertainty and vagueness.

Keywords: fuzzy regression, load estimation, fuzzy linear parameters, electrical load estimation

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Authors: N. Kaewpraek, W. Assawinchaichote

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This paper considers an H_∞ TS fuzzy state-derivative feedback controller for a class of nonlinear dynamical systems. A Takagi-Sugeno (TS) fuzzy model is used to approximate a class of nonlinear dynamical systems. Then, based on a linear matrix inequality (LMI) approach, we design an H_∞ TS fuzzy state-derivative feedback control law which guarantees L₂-gain of the mapping from the exogenous input noise to the regulated output to be less or equal to a prescribed value. We derive a sufficient condition such that the system with the fuzzy controller is asymptotically stable and H_∞ performance is satisfied. Finally, we provide and simulate a numerical example is provided to illustrate the stability and the effectiveness of the proposed controller.

Keywords: h-infinity fuzzy control, an LMI approach, Takagi-Sugano (TS) fuzzy system, the photovoltaic systems

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Authors: A. Mansouri, F. Krim

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This paper discusses the implementation and design of both linear PI and fuzzy controllers for DC-DC boost converters. Design of PI controllers is based on temporal response of closed-loop converters, while fuzzy controllers design is based on heuristic knowledge of boost converters. Linear controller implementation is quite straightforward relying on mathematical models, while fuzzy controller implementation employs one or more artificial intelligences techniques. Comparison between these boost controllers is made in design aspect. Experimental results show that the proposed fuzzy controller system is robust against input voltage and load resistance changing and in respect of start-up transient. Results indicate that fuzzy controller can achieve best control performance concerning faster transient response, steady-state response good stability and accuracy under different operating conditions. Fuzzy controller is more suitable to control boost converters.

Keywords: boost DC-DC converter, fuzzy, PI controllers, power electronics and control system

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Authors: Nazanin Pilevari, Kamyar Mahmoodi

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The purpose of this research is to provide a model based on fuzzy inference system for evaluating empowerment of Knowledge management. The first prototype of the research was developed based on the study of literature. In the next step, experts were provided with these models and after implementing consensus-based reform, the views of Fuzzy Delphi experts and techniques, components and Index research model were finalized. Culture, structure, IT and leadership were considered as dimensions of empowerment. Then, In order to collect and extract data for fuzzy inference system based on knowledge and Experience, the experts were interviewed. The values obtained from designed fuzzy inference system, made review and assessment of the organization's empowerment of Knowledge management possible. After the design and validation of systems to measure indexes ,empowerment of Knowledge management and inputs into fuzzy inference) in the AYANDEH Bank, a questionnaire was used. In the case of this bank, the system output indicates that the status of empowerment of Knowledge management, culture, organizational structure and leadership are at the moderate level and information technology empowerment are relatively high. Based on these results, the status of knowledge management empowerment in AYANDE Bank, was moderate. Eventually, some suggestions for improving the current situation of banks were provided. According to studies of research history, the use of powerful tools in Fuzzy Inference System for assessment of Knowledge management and knowledge management empowerment such an assessment in the field of banking, are the innovation of this Research.

Keywords: knowledge management, knowledge management empowerment, fuzzy inference system, fuzzy Delphi

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Authors: Ikbal Eski, Turan Gürgenç

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After nearly more than a century of research and development, internal combustion engines have become almost perfect. Along with such improvement in internal combustion engines, automotive manufacturers are conducting research on design of alternative fuel vehicles. Nevertheless an ideal interim solution is to increase overall efficiency of internal combustion vehicles. A potential solution to achieve that is using continuously variable transmission system which, despite being an old idea, has recently become a hope for automotive manufacturers. CVT system, by continuously varying speed ratio, raises vehicle efficiency. In this study, fuzzy logic controller is used in speed ratio control of pulley based CVT system.

Keywords: continuously variable transmission system, variator, speed ratio, fuzzy logic

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Authors: P. Bharathiraja

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Computer Aided decision making system is used to enhance diagnosis and prognosis of diseases and also to assist clinicians and junior doctors in clinical decision making. Medical Data used for decision making should be definite and consistent. Data Mining and soft computing techniques are used for cleaning the data and for incorporating human reasoning in decision making systems. Fuzzy rule based inference technique can be used for classification in order to incorporate human reasoning in the decision making process. In this work, missing values are imputed using the mean or mode of the attribute. The data are normalized using min-ma normalization to improve the design and efficiency of the fuzzy inference system. The fuzzy inference system is used to handle the uncertainties that exist in the medical data. Equal-width-partitioning is used to partition the attribute values into appropriate fuzzy intervals. Fuzzy rules are generated using Class Based Associative rule mining algorithm. The system is trained and tested using heart disease data set from the University of California at Irvine (UCI) Machine Learning Repository. The data was split using a hold out approach into training and testing data. From the experimental results it can be inferred that classification using fuzzy inference system performs better than trivial IF-THEN rule based classification approaches. Furthermore it is observed that the use of fuzzy logic and fuzzy inference mechanism handles uncertainty and also resembles human decision making. The system can be used in the absence of a clinical expert to assist junior doctors and clinicians in clinical decision making.

Keywords: decision making, data mining, normalization, fuzzy rule, classification

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Authors: R. H. Fattepur, Sameer R. Fattepur, D. K. Sreekantha

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Dairy Farming is one of the key industries in India. India is the leading producer and also the consumer of milk, milk-based products in the world. In this paper, we have attempted to the replace the human expert system and to develop an artificial expert system prototype to increase the speed and accuracy of decision making dairy farming credit risk evaluation. Fuzzy logic is used for dealing with uncertainty, vague and acquired knowledge, fuzzy rule base method is used for representing this knowledge for building an effective expert system.

Keywords: expert system, fuzzy logic, knowledge base, dairy farming, credit risk

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Authors: Rishabh Sharma

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The main design parameter of WSN (wireless sensor network) is the energy consumption. To compensate this parameter, hierarchical clustering is a technique that assists in extending duration of the networks life by efficiently consuming the energy. This paper focuses on dealing with the WSNs and the FIS (fuzzy interface system) which are deployed to enhance the BEENISH protocol. The node energy, mobility, pause time and density are considered for the selection of CH (cluster head). The simulation outcomes exhibited that the projected system outperforms the traditional system with regard to the energy utilization and number of packets transmitted to sink.

Keywords: wireless sensor network, sink, sensor node, routing protocol, fuzzy rule, fuzzy inference system

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Authors: Yousfi Abdelkader, Chaker Abdelkader, Bot Youcef

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This article proposes a five-level shunt active power filter for power quality improvement using a interval type-2 fuzzy logic controller (IT2 FLC). The reference compensating current is extracted using the P-Q theory. The majority of works previously reported are based on two-level inverters with a conventional Proportional integral (PI) controller, which requires rigorous mathematical modeling of the system. In this paper, a IT2 FLC controlled five-level active power filter is proposed to overcome the problem associated with PI controller. The IT2 FLC algorithm is applied for controlling the DC-side capacitor voltage as well as the harmonic currents of the five-level active power filter. The active power filter with a IT2 FLC is simulated in MATLAB Simulink environment. The simulated response shows that the proposed shunt active power filter controller has produced a sinusoidal supply current with low harmonic distortion and in phase with the source voltage.

Keywords: power quality, shunt active power filter, interval type-2 fuzzy logic controller (T2FL), multilevel inverter

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Authors: T. Pathinathan, M. Peter

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Fuzzy graphs incorporate concepts from graph theory with fuzzy principles. In this paper, we make a study on the properties of fuzzy graphs which are non-cyclic and are of two-dimensional in structure. In particular, this paper presents 2D structure or the structure of double layer for a non-cyclic fuzzy graph whose underlying crisp graph is non-cyclic. In any graph structure, introducing 2D structure may lead to an inherent cycle. We propose relevant conditions for 2D structured non-cyclic fuzzy graphs. These conditions are extended even to fuzzy graphs of the 3D structure. General theoretical properties that are studied for any fuzzy graph are verified to 2D structured or double layered fuzzy graphs. Concepts like Order, Degree, Strong and Size for a fuzzy graph are studied for 2D structured or double layered non-cyclic fuzzy graphs. Using different types of fuzzy graphs, the proposed concepts relating to 2D structured fuzzy graphs are verified.

Keywords: double layered fuzzy graph, double layered non–cyclic fuzzy graph, order, degree and size

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Authors: M. Doumi, A. Miloudi, A. G. Aissaoui, K. Tahir, C. Belfedal, S. Tahir

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The studies on the photovoltaic system are extensively increasing because of a large, secure, essentially exhaustible and broadly available resource as a future energy supply. However, the output power induced in the photovoltaic modules is influenced by an intensity of solar cell radiation, temperature of the solar cells and so on. Therefore, to maximize the efficiency of the photovoltaic system, it is necessary to track the maximum power point of the PV array, for this Maximum Power Point Tracking (MPPT) technique is used. Some MPPT techniques are available in that perturbation and observation (P&O) and Fuzzy logic controller (FLC). The fuzzy control method has been compared with perturb and observe (P&O) method as one of the most widely conventional method used in this area. Both techniques have been analyzed and simulated. MPPT using fuzzy logic shows superior performance and more reliable control with respect to the P&O technique for this application.

Keywords: photovoltaic system, MPPT, perturb and observe, fuzzy logic

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Authors: H. Mahdjoub, B. Hamaidi, B. Zerouali, S. Rouabhia

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The issue of fault detection and diagnosis (FDD) has gained widespread industrial interest in process condition monitoring applications. Accordingly, the use of neuro-fuzzy technic seems very promising. This paper treats a diagnosis modeling a strategic equipment of an industrial installation. We propose a diagnostic tool based on adaptive neuro-fuzzy inference system (ANFIS). The neuro-fuzzy network provides an abductive diagnosis. Moreover, it takes into account the uncertainties on the maintenance knowledge by giving a fuzzy characterization of each cause. This work was carried out with real data of a lubrication circuit from the gas turbine. The machine of interest is a gas turbine placed in a gas compressor station at South Industrial Centre (SIC Hassi Messaoud Ouargla, Algeria). We have defined the zones of good and bad functioning, and the results are presented to demonstrate the advantages of the proposed method.

Keywords: fault detection and diagnosis, lubrication system, turbine, ANFIS, training, pattern recognition

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Authors: Seyed Habib A. Rahmati, Mohsen Sadegh Amalnick

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Different terms of the statistical process control (SPC) has sketch in the fuzzy environment. However, measurement system analysis (MSA), as a main branch of the SPC, is rarely investigated in fuzzy area. This procedure assesses the suitability of the data to be used in later stages or decisions of the SPC. Therefore, this research focuses on some important measures of MSA and through a new method introduces the measures in fuzzy environment. In this method, which works based on Buckley approach, imprecision and vagueness nature of the real world measurement are considered simultaneously. To do so, fuzzy version of the gauge capability (Cg and Cgk) are introduced. The method is also explained through example clearly.

Keywords: measurement, SPC, MSA, gauge capability (Cg and Cgk)

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Authors: Saad M. Darwish, Magda M. Madbouly, Murad B. Khorsheed

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Sign Languages (SL) are the most accomplished forms of gestural communication. Therefore, their automatic analysis is a real challenge, which is interestingly implied to their lexical and syntactic organization levels. Hidden Markov models (HMM’s) have been used prominently and successfully in speech recognition and, more recently, in handwriting recognition. Consequently, they seem ideal for visual recognition of complex, structured hand gestures such as are found in sign language. In this paper, several results concerning static hand gesture recognition using an algorithm based on Type-2 Fuzzy HMM (T2FHMM) are presented. The features used as observables in the training as well as in the recognition phases are based on Singular Value Decomposition (SVD). SVD is an extension of Eigen decomposition to suit non-square matrices to reduce multi attribute hand gesture data to feature vectors. SVD optimally exposes the geometric structure of a matrix. In our approach, we replace the basic HMM arithmetic operators by some adequate Type-2 fuzzy operators that permits us to relax the additive constraint of probability measures. Therefore, T2FHMMs are able to handle both random and fuzzy uncertainties existing universally in the sequential data. Experimental results show that T2FHMMs can effectively handle noise and dialect uncertainties in hand signals besides a better classification performance than the classical HMMs. The recognition rate of the proposed system is 100% for uniform hand images and 86.21% for cluttered hand images.

Keywords: hand gesture recognition, hand detection, type-2 fuzzy logic, hidden Markov Model

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Authors: Peng Li, Chuanri Li, Tao Li

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Reliability allocation is quite important during early design and development stages for a system to apportion its specified reliability goal to subsystems. This paper improves the reliability fuzzy allocation method and gives concrete processes on determining the factor set, the factor weight set, judgment set, and multi-grade fuzzy comprehensive evaluation. To determine the weight of factor set, the modified trapezoidal numbers are proposed to reduce errors caused by subjective factors. To decrease the fuzziness in the fuzzy division, an approximation method based on linear programming is employed. To compute the explicit values of fuzzy numbers, centroid method of defuzzification is considered. An example is provided to illustrate the application of the proposed reliability allocation method based on fuzzy arithmetic.

Keywords: reliability allocation, fuzzy arithmetic, allocation weight, linear programming

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