Search results for: triangular fuzzy number.

Authors: A. M. Pashayev, C. Ardil, D. D. Askerov, R. A. Sadiqov, P. S. Abdullayev

Abstract:

In this paper is shown that the probability-statistic methods application, especially at the early stage of the aviation gas turbine engine (GTE) technical condition diagnosing, when the flight information has property of the fuzzy, limitation and uncertainty is unfounded. Hence is considered the efficiency of application of new technology Soft Computing at these diagnosing stages with the using of the Fuzzy Logic and Neural Networks methods. Training with high accuracy of fuzzy multiple linear and non-linear models (fuzzy regression equations) which received on the statistical fuzzy data basis is made. Thus for GTE technical condition more adequate model making are analysed dynamics of skewness and kurtosis coefficients' changes. Researches of skewness and kurtosis coefficients values- changes show that, distributions of GTE work parameters have fuzzy character. Hence consideration of fuzzy skewness and kurtosis coefficients is expedient. Investigation of the basic characteristics changes- dynamics of GTE work parameters allows to draw conclusion on necessity of the Fuzzy Statistical Analysis at preliminary identification of the engines' technical condition. Researches of correlation coefficients values- changes shows also on their fuzzy character. Therefore for models choice the application of the Fuzzy Correlation Analysis results is offered. For checking of models adequacy is considered the Fuzzy Multiple Correlation Coefficient of Fuzzy Multiple Regression. At the information sufficiency is offered to use recurrent algorithm of aviation GTE technical condition identification (Hard Computing technology is used) on measurements of input and output parameters of the multiple linear and non-linear generalised models at presence of noise measured (the new recursive Least Squares Method (LSM)). The developed GTE condition monitoring system provides stage-bystage estimation of engine technical conditions. As application of the given technique the estimation of the new operating aviation engine temperature condition was made.

Keywords: Gas turbine engines, neural networks, fuzzy logic, fuzzy statistics.

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Authors: N. A. Muhamad, S.A.M. Ali

Abstract:

Condition monitoring of electrical power equipment has attracted considerable attention for many years. The aim of this paper is to use Labview with Fuzzy Logic controller to build a simulation system to diagnose transformer faults and monitor its condition. The front panel of the system was designed using LabVIEW to enable computer to act as customer-designed instrument. The dissolved gas-in-oil analysis (DGA) method was used as technique for oil type transformer diagnosis; meanwhile terminal voltages and currents analysis method was used for dry type transformer. Fuzzy Logic was used as expert system that assesses all information keyed in at the front panel to diagnose and predict the condition of the transformer. The outcome of the Fuzzy Logic interpretation will be displayed at front panel of LabVIEW to show the user the conditions of the transformer at any time.

Keywords: LabVIEW, Fuzzy Logic, condition monitoring, oiltransformer, dry transformer, DGA, terminal values.

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Authors: Petr Morávek, Miloš Šeda

Abstract:

In practice, we often come across situations where it is necessary to make decisions based on incomplete or uncertain data. In control systems it may be due to the unknown exact mathematical model, or its excessive complexity (e.g. nonlinearity) when it is necessary to simplify it, respectively, to solve it using a rule base. In the case of databases, searching data we compare a similarity measure with of the requirements of the selection with stored data, where both the select query and the data itself may contain vague terms, for example in the form of linguistic qualifiers. In this paper, we focus on the processing of uncertain data in databases and demonstrate it on the example multi-criteria decision making in the selection of variants, specified by higher number of technical parameters.

Keywords: fuzzy logic, linguistic variable, multicriteria decision

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Authors: Reza Ghasemi, Negin Amiri Hazaveh

Abstract:

In this paper, control methodology based on the selection of the type of traffic light and the period of the green phase to accomplish an optimum balance at intersections is proposed. This balance should be flexible to the static behavior of time, and randomness in a traffic situation; the goal of the proposed method is to reduce traffic volume in transportation, the average delay for each vehicle, and control over the crash of cars. The proposed method was specifically investigated at the intersection through an appropriate timing of traffic lights by sampling a multi-agent system. It consists of a large number of intersections, each of which is considered as an independent agent that exchanges information with each other, and the stability of each agent is provided separately. The robustness against uncertainties, scalability, and stability of the closed-loop overall system are the main merits of the proposed methodology. The simulation results show that the fuzzy intelligent controller in this multi-factor system which is a Takagi-Sugeno (TS) fuzzy is more useful than scheduling in the fixed-time method and it reduces the lengths of vehicles queuing.

Keywords: Fuzzy intelligent controller, traffic-light control, multi-agent systems, state space equations, stability.

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Authors: Xinhua Zhang, Kelin Li

Abstract:

In this paper, a class of impulsive BAM fuzzy cellular neural networks with distributed delays and reaction-diffusion terms is formulated and investigated. By employing the delay differential inequality and inequality technique developed by Xu et al., some sufficient conditions ensuring the existence, uniqueness and global exponential stability of equilibrium point for impulsive BAM fuzzy cellular neural networks with distributed delays and reaction-diffusion terms are obtained. In particular, the estimate of the exponential convergence rate is also provided, which depends on system parameters, diffusion effect and impulsive disturbed intention. It is believed that these results are significant and useful for the design and applications of BAM fuzzy cellular neural networks. An example is given to show the effectiveness of the results obtained here.

Keywords: Bi-directional associative memory, fuzzy cellular neuralnetworks, reaction-diffusion, delays, impulses, global exponentialstability.

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Authors: E. Ardil, Erdem Uçar, Parvinder S. Sandhu

Abstract:

As the majority of faults are found in a few of its modules so there is a need to investigate the modules that are affected severely as compared to other modules and proper maintenance need to be done on time especially for the critical applications. In this paper, we have explored the different predictor models to NASA-s public domain defect dataset coded in Perl programming language. Different machine learning algorithms belonging to the different learner categories of the WEKA project including Mamdani Based Fuzzy Inference System and Neuro-fuzzy based system have been evaluated for the modeling of maintenance severity or impact of fault severity. The results are recorded in terms of Accuracy, Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE). The results show that Neuro-fuzzy based model provides relatively better prediction accuracy as compared to other models and hence, can be used for the maintenance severity prediction of the software.

Keywords: Software Metrics, Fuzzy, Neuro-Fuzzy, SoftwareFaults, Accuracy, MAE, RMSE.

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Authors: Yong-can Cao, Wei-dong Zhang

Abstract:

To deal with random delays in Networked Control System (NCS), Modified Fuzzy PID Controller is introduced in this paper to implement real-time control adaptively. Via adjusting the control signal dynamically, the system performance is improved. In this paper, the design process and the ultimate simulation results are represented. Finally, examples and corresponding comparisons prove the significance of this method.

Keywords: Fuzzy Control, Networked Control System, PID, Random Delays

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Authors: Şaban Çetin, Özgür Demir

Abstract:

In this study, Fuzzy PID Control scheme is designed for an active suspension system. The main goal of an active suspension system for using in a vehicle model is reducing body deflections and handling high comfort for a passenger car. The present system was modelled as a two-degree-of-freedom (2-DOF) nonlinear vehicle model.

Keywords: Active suspension system, Fuzzy PID controller, a nonlinear quarter car model.

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Authors: A. Faro, D. Giordano, M. Pennisi, G. Scarciofalo, C. Spampinato, F. Tramontana

Abstract:

The paper proposes a methodology to process the signals coming from the Transcranial Magnetic Stimulation (TMS) in order to identify the pathology and evaluate the therapy to treat the patients affected by demency diseases. In particular, a fuzzy model is developed to identify the demency of the patients affected by Subcortical Ischemic Vascular Dementia and to measure the positive effect, if any, of a repetitive TMS on their motor performances. A tool is also presented to support the mentioned analysis.

Keywords: TMS, SIVD, Electromiography , Fuzzy Logic.

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Authors: Sandeep Chandana, Rene V. Mayorga, Christine W. Chan

Abstract:

This article outlines conceptualization and implementation of an intelligent system capable of extracting knowledge from databases. Use of hybridized features of both the Rough and Fuzzy Set theory render the developed system flexibility in dealing with discreet as well as continuous datasets. A raw data set provided to the system, is initially transformed in a computer legible format followed by pruning of the data set. The refined data set is then processed through various Rough Set operators which enable discovery of parameter relationships and interdependencies. The discovered knowledge is automatically transformed into a rule base expressed in Fuzzy terms. Two exemplary cancer repository datasets (for Breast and Lung Cancer) have been used to test and implement the proposed framework.

Keywords: Knowledge Extraction, Fuzzy Sets, Rough Sets, Neuro–Fuzzy Systems, Databases

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Authors: Chokri Slim

Abstract:

The neural network's performance can be measured by efficiency and accuracy. The major disadvantages of neural network approach are that the generalization capability of neural networks is often significantly low, and it may take a very long time to tune the weights in the net to generate an accurate model for a highly complex and nonlinear systems. This paper presents a novel Neuro-fuzzy architecture based on Extended Kalman filter. To test the performance and applicability of the proposed neuro-fuzzy model, simulation study of nonlinear complex dynamic system is carried out. The proposed method can be applied to an on-line incremental adaptive learning for the prediction of financial time series. A benchmark case studie is used to demonstrate that the proposed model is a superior neuro-fuzzy modeling technique.

Keywords: Neuro-fuzzy, Extended Kalman filter, nonlinear systems, financial time series.

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Authors: O. Belalia Douma, B. Boukhatem, M. Ghrici

Abstract:

Self-compacting concrete (SCC) developed in Japan in the late 80s has enabled the construction industry to reduce demand on the resources, improve the work condition and also reduce the impact of environment by elimination of the need for compaction. Fuzzy logic (FL) approaches has recently been used to model some of the human activities in many areas of civil engineering applications. Especially from these systems in the model experimental studies, very good results have been obtained. In the present study, a model for predicting compressive strength of SCC containing various proportions of fly ash, as partial replacement of cement has been developed by using Fuzzy Inference System (FIS). For the purpose of building this model, a database of experimental data were gathered from the literature and used for training and testing the model. The used data as the inputs of fuzzy logic models are arranged in a format of five parameters that cover the total binder content, fly ash replacement percentage, water content, superplasticizer and age of specimens. The training and testing results in the fuzzy logic model have shown a strong potential for predicting the compressive strength of SCC containing fly ash in the considered range.

Keywords: Self-compacting concrete, fly ash, strength prediction, fuzzy logic.

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Authors: Ahmad Aljaafreh

Abstract:

This paper proposes a hierarchical hidden Markov model (HHMM) to model the detection of M vehicles in a wireless sensor network (WSN). The HHMM model contains an extra level of hidden Markov model to model the temporal transitions of each state of the first HMM. By modeling the temporal transitions, only those hypothesis with nonzero transition probabilities needs to be tested. Thus, this method efficiently reduces the computation load, which is preferable in WSN applications.This paper integrates several techniques to optimize the detection performance. The output of the states of the first HMM is modeled as Gaussian Mixture Model (GMM), where the number of states and the number of Gaussians are experimentally determined, while the other parameters are estimated using Expectation Maximization (EM). HHMM is used to model the sequence of the local decisions which are based on multiple hypothesis testing with maximum likelihood approach. The states in the HHMM represent various combinations of vehicles of different types. Due to the statistical advantages of multisensor data fusion, we propose a heuristic based on fuzzy weighted majority voting to enhance cooperative classification of moving vehicles within a region that is monitored by a wireless sensor network. A fuzzy inference system weighs each local decision based on the signal to noise ratio of the acoustic signal for target detection and the signal to noise ratio of the radio signal for sensor communication. The spatial correlation among the observations of neighboring sensor nodes is efficiently utilized as well as the temporal correlation. Simulation results demonstrate the efficiency of this scheme.

Keywords: Classification, decision fusion, fuzzy logic, hidden Markov model

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Authors: C. Ardil

Abstract:

Aircraft selection decisions can be challenging due to their multidimensional and interdisciplinary nature. They involve multiple stakeholders with conflicting objectives and numerous alternative options with uncertain outcomes. This study focuses on the analysis of aerial firefighting aircraft that can be chosen for the Air Fire Service to extinguish forest fires. To make such a selection, the characteristics of the fire zones must be considered, and the capability to manage the logistics involved in such operations, as well as the purchase and maintenance of the aircraft, must be determined. The selection of firefighting aircraft is particularly complex because they have longer fleet lives and require more demanding operation and maintenance than scheduled passenger air service. This paper aims to use the fuzzy proximity measure method to select the most appropriate aerial firefighting aircraft based on decision criteria using multiple attribute decision making analysis. Following fuzzy decision analysis, the most suitable aerial firefighting aircraft is ranked and determined for the Air Fire Service.

Keywords: Aerial firefighting aircraft selection, multiple criteria decision making, fuzzy sets, standard fuzzy sets, determinate fuzzy sets, indeterminate fuzzy sets, proximity measure method, Minkowski distance family function, Hausdorff distance function, MCDM, PMM, PMM-F

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Authors: Kuo-Lung Wu

Abstract:

The weighting exponent m is called the fuzzifier that can have influence on the clustering performance of fuzzy c-means (FCM) and mÎ[1.5,2.5] is suggested by Pal and Bezdek [13]. In this paper, we will discuss the robust properties of FCM and show that the parameter m will have influence on the robustness of FCM. According to our analysis, we find that a large m value will make FCM more robust to noise and outliers. However, if m is larger than the theoretical upper bound proposed by Yu et al. [14], the sample mean will become the unique optimizer. Here, we suggest to implement the FCM algorithm with mÎ[1.5,4] under the restriction when m is smaller than the theoretical upper bound.

Keywords: Fuzzy c-means, robust, fuzzifier.

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Authors: T. Juhana Hashim, A. Mohamed

Abstract:

This paper discusses the implementation of a fuzzy logic based coordinated voltage control for a distribution system connected with distributed generations (DGs). The connection of DGs has created a challenge for the distribution network operators to keep the voltage in the system within its acceptable limits. Intelligent centralized or coordinated voltage control schemes have proven to be more reliable due to its ability to provide more control and coordination with the communication with other network devices. In this work, voltage control using fuzzy logic by coordinating three methods of control, power factor control, on load tap changer and generation curtailment is implemented on a distribution network test system. The results show that the fuzzy logic based coordination is able to keep the voltage within its allowable limits. 

Keywords: Coordinated control, Distributed generation, Fuzzy logic, Voltage control.

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Authors: C.Kia, M.R.Arshad, A.H.Adom, P.A.Wilson

Abstract:

This paper presents recent work on the improvement of the robotics vision based control strategy for underwater pipeline tracking system. The study focuses on developing image processing algorithms and a fuzzy inference system for the analysis of the terrain. The main goal is to implement the supervisory fuzzy learning control technique to reduce the errors on navigation decision due to the pipeline occlusion problem. The system developed is capable of interpreting underwater images containing occluded pipeline, seabed and other unwanted noise. The algorithm proposed in previous work does not explore the cooperation between fuzzy controllers, knowledge and learnt data to improve the outputs for underwater pipeline tracking. Computer simulations and prototype simulations demonstrate the effectiveness of this approach. The system accuracy level has also been discussed.

Keywords: Fuzzy logic, Underwater target tracking, Autonomous underwater vehicles, Artificial intelligence, Simulations, Robot navigation, Vision system.

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Authors: E. A. Tkachenko, E. M. Rogova, V. V. Klimov

Abstract:

One of the basic issues of development management is connected with performance measurement as a prerequisite for identifying the achievement of development objectives. The aim of our research is to develop an improved model of assessing a company’s development results. The model should take into account the cyclical nature of development and the high degree of uncertainty in dealing with numerous management tasks. Our hypotheses may be formulated as follows: Hypothesis 1. The cycle of a company’s development may be studied from the standpoint of a project cycle. To do that, methods and tools of project analysis are to be used. Hypothesis 2. The problem of the uncertainty when justifying managerial decisions within the framework of a company’s development cycle can be solved through the use of the mathematical apparatus of fuzzy logic. The reasoned justification of the validity of the hypotheses made is given in the suggested article. The fuzzy logic toolkit applies to the case of technology shift within an enterprise. It is proven that some restrictions in performance measurement that are incurred to conventional methods could be eliminated by implementation of the fuzzy logic apparatus in performance measurement models.

Keywords: Fuzzy logic, fuzzy sets, performance measurement, project analysis.

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Authors: Sana Bouzaida, Anis Sakly, Faouzi M'Sahli

Abstract:

In this paper, a TSK-type Neuro-fuzzy Inference System that combines the features of fuzzy sets and neural networks has been applied for the identification of MIMO systems. The procedure of adapting parameters in TSK model employs a Shuffled Frog Leaping Algorithm (SFLA) which is inspired from the memetic evolution of a group of frogs when seeking for food. To demonstrate the accuracy and effectiveness of the proposed controller, two nonlinear systems have been considered as the MIMO plant, and results have been compared with other learning methods based on Particle Swarm Optimization algorithm (PSO) and Genetic Algorithm (GA).

Keywords: Identification, Shuffled frog Leaping Algorithm (SFLA), TSK-type neuro-fuzzy model.

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Authors: Kamalpreet Kaur, Harjit Pal Singh, Vikas Khullar

Abstract:

In Wireless Sensor Network (WSN), the main limitation is generally inimitable energy consumption during processing of the sensor nodes. Cluster head (CH) election is one of the main issues that can reduce the energy consumption. Therefore, discovering energy saving routing protocol is the focused area for research. In this paper, fuzzy-based energy aware routing protocol is presented, which enhances the stability and network lifetime of the network. Fuzzy logic ensures the well-organized selection of CH by taking four linguistic variables that are concentration, energy, centrality, and distance to base station (BS). The results show that the proposed protocol shows better results in requisites of stability and throughput of the network.

Keywords: ERA, fuzzy logic, network model, WSN.

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Authors: Ashok Kumar, Vinod Kumar

Abstract:

Wireless Sensor Networks (WSNs) are used to monitor/observe vast inaccessible regions through deployment of large number of sensor nodes in the sensing area. For majority of WSN applications, the collected data needs to be combined with geographic information of its origin to make it useful for the user; information received from remote Sensor Nodes (SNs) that are several hops away from base station/sink is meaningless without knowledge of its source. In addition to this, location information of SNs can also be used to propose/develop new network protocols for WSNs to improve their energy efficiency and lifetime. In this paper, range free localization protocols for WSNs have been proposed. The proposed protocols are based on weighted centroid localization technique, where the edge weights of SNs are decided by utilizing fuzzy logic inference for received signal strength and link quality between the nodes. The fuzzification is carried out using (i) Mamdani, (ii) Sugeno, and (iii) Combined Mamdani Sugeno fuzzy logic inference. Simulation results demonstrate that proposed protocols provide better accuracy in node localization compared to conventional centroid based localization protocols despite presence of unintentional radio frequency interference from radio frequency (RF) sources operating in same frequency band.

Keywords: localization, range free, received signal strength, link quality indicator, Mamdani fuzzy logic inference, Sugeno fuzzy logic inference.

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Authors: Jayanth Jacob, C. V. Hariharakrishnan, Suganthi L.

Abstract:

The rapid urbanization of cities has a bane in the form road accidents that cause extensive damage to life and limbs. A number of location based factors are enablers of road accidents in the city. The speed of travel of vehicles is non-uniform among locations within a city. In this study, the perception of vehicle users is captured on a 10-point rating scale regarding the degree of variation in speed of travel at chosen locations in the city. The average rating is used to cluster locations using fuzzy c-means clustering and classify them as low, moderate and high speed of travel locations. The high speed of travel locations can be classified proactively to ensure that accidents do not occur due to the speeding of vehicles at such locations. The advantage of fuzzy c-means clustering is that a location may be a part of more than one cluster to a varying degree and this gives a better picture about the location with respect to the characteristic (speed of travel) being studied.

Keywords: C-means clustering, Location Specific, Road Accidents.

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Authors: Shikha Maheshwari, Amit Srivastava

Abstract:

In medical investigations, uncertainty is a major challenging problem in making decision for doctors/experts to identify the diseases with a common set of symptoms and also has been extensively increasing in medical diagnosis problems. The theory of cross entropy for intuitionistic fuzzy sets (IFS) is an effective approach in coping uncertainty in decision making for medical diagnosis problem. The main focus of this paper is to propose a new intuitionistic fuzzy cross entropy measure (IFCEM), which aid in reducing the uncertainty and doctors/experts will take their decision easily in context of patient’s disease. It is shown that the proposed measure has some elegant properties, which demonstrates its potency. Further, it is also exemplified in detail the efficiency and utility of the proposed measure by using a real life case study of diagnosis the disease in medical science.

Keywords: Intuitionistic fuzzy cross entropy (IFCEM), intuitionistic fuzzy set (IFS), medical diagnosis, uncertainty.

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Authors: A. M. Pashayev, D. D. Askerov, C. Ardil, R. A. Sadiqov, P. S. Abdullayev

Abstract:

Researches show that probability-statistical methods application, especially at the early stage of the aviation Gas Turbine Engine (GTE) technical condition diagnosing, when the flight information has property of the fuzzy, limitation and uncertainty is unfounded. Hence the efficiency of application of new technology Soft Computing at these diagnosing stages with the using of the Fuzzy Logic and Neural Networks methods is considered. According to the purpose of this problem training with high accuracy of fuzzy multiple linear and non-linear models (fuzzy regression equations) which received on the statistical fuzzy data basis is made. For GTE technical condition more adequate model making dynamics of skewness and kurtosis coefficients- changes are analysed. Researches of skewness and kurtosis coefficients values- changes show that, distributions of GTE workand output parameters of the multiple linear and non-linear generalised models at presence of noise measured (the new recursive Least Squares Method (LSM)). The developed GTE condition monitoring system provides stage-by-stage estimation of engine technical conditions. As application of the given technique the estimation of the new operating aviation engine technical condition was made.

Keywords: aviation gas turbine engine, technical condition, fuzzy logic, neural networks, fuzzy statistics

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Authors: Mansoureh Naderipour, Susan Bastani, Mohammad Fazel Zarandi

Abstract:

Predicting links that may occur in the future and missing links in social networks is an attractive problem in social network analysis. Granular computing can help us to model the relationships between human-based system and social sciences in this field. In this paper, we present a model based on granular computing approach and Type-2 fuzzy logic to predict links regarding nodes’ activity and the relationship between two nodes. Our model is tested on collaboration networks. It is found that the accuracy of prediction is significantly higher than the Type-1 fuzzy and crisp approach.

Keywords: Social Network, link prediction, granular computing, Type-2 fuzzy sets.

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Authors: Sanjay K.Srivastava, Bhanu Gupta

Abstract:

In this paper some results on strict stability heve beeb extended for fuzzy differential equations with impulse effect using Lyapunov functions and Razumikhin technique.

Keywords: Fuzzy differential equations, Impulsive differential equations, Strict stability, Lyapunov function, Razumikhin technique.

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Authors: Liang Zhao, Jili Zhang, Kai Li

Abstract:

A microcontroller-based fan coil unit (FCU) fuzzy controller is designed and implemented in this paper. The controller employs the concept of duty ratio on the electric valve control, which could make full use of the cooling and dehumidifying capacity of the FCU when the valve is off. The traditional control method and its limitations are analyzed. The hardware and software design processes are introduced in detail. The experimental results show that the proposed method is more energy efficient compared to the traditional controlling strategy. Furthermore, a more comfortable room condition could be achieved by the proposed method. The proposed low-cost FCU fuzzy controller deserves to be widely used in engineering applications.

Keywords: Fan coil unit, duty ratio, fuzzy controller, experiment.

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Authors: Hamed Rafiei, Masoud Rabbani

Abstract:

This paper addresses one of the most important issues have been considered in hybrid MTS/MTO production environments. To cope with the problem, a mathematical programming model is applied from a tactical point of view. The model is converted to a fuzzy goal programming model, because a degree of uncertainty is involved in hybrid MTS/MTO context. Finally, application of the proposed model in an industrial center is reported and the results prove the validity of the model.

Keywords: Fuzzy sets theory, Hybrid MTS/MTO, Order penetration point, Quadratic programming.

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Authors: İres İskender, Ahmet Karaarslan

Abstract:

In this study, the performance of a high-frequency arc welding machine including a two-switch inverter is analyzed. The control of the system is achieved using two different control techniques i- fuzzy logic control (FLC) ii- state space averaging based sliding control. Fuzzy logic control does not need accurate mathematical model of a plant and can be used in nonlinear applications. The second method needs the mathematical model of the system. In this method the state space equations of the system are derived for two different “on" and “off" states of the switches. The derived state equations are combined with the sliding control rule considering the duty-cycle of the converter. The performance of the system is analyzed by simulating the system using SIMULINK tool box of MATLAB. The simulation results show that fuzzy logic controller is more robust and less sensitive to parameter variations.

Keywords: Fuzzy logic, arc welding, sliding state space control, PWM, current control.

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Authors: Zhu Jiaming, Yu Mengna

Abstract:

In this article, models based on quantitative analysis, physical geometry and regression analysis are established, by using analytic hierarchy process analysis, fuzzy cluster analysis, fuzzy photographic and data fitting. The reasons of various leaf shapes among different species and the differences between the leaf shapes on same tree have been solved by using software, such as Eviews, VB and Matlab. We also successfully estimate the leaf mass of a tree and the correlation with the tree profile.

Keywords: Leaf shape; Mass; Fuzzy cluster; Regression analysis; Eviews; Matlab

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