Shape Of Membership Functions Research Articles

Soft-Computing Control of Ball and Beam System

This article derives a mathematical model and compares different soft-computing techniques for control of a highly dynamic ball and beam system. The techniques which were incorporated for control of proposed system were fuzzy logic, proportional-integral-derivative (PID), adaptive neuro fuzzy inference system (ANFIS) and neural networks. Initially, a fuzzy controller has been developed using seven gaussian shape membership functions. The article illustrates briefly both learning ability and parameter estimation properties of ANFIS and neural controllers. The results of PID controller were collected and used for training of ANFIS and Neural controllers. A Matlab simulink model of a ball and beam system has been derived for simulating and comparing different controllers. The performances of controllers were measured and compared in terms of settling time and steady state error. Simulation results proved the superiority of ANFIS over other control techniques.

Analysing of exchange rate and gross domestic product (GDP) by adaptive neuro fuzzy inference system (ANFIS)

In this paper was investigated the effect of exchange rate pass-through (ERPT) into aggregate import prices. The results shown in the long run pass-through was not observed. The results of our research confirm the results of previous investigations according to which pass-through is somewhat higher in transition countries than in developed ones and the pass-through degree is declining. Afterwards exchange rate effect on gross domestic product (GDP) was investigated by adaptive neuro fuzzy inference system (ANFIS). Dataset was collected from EUROSTAT database for 37 countries. The dataset was divided into two sets, for training and for testing of the ANFIS model. The main goal was to predict GDP in relation to exchange rate. ANFIS model was created by three Bell shaped membership functions since it was the best model according to the obtained results. Linear membership functions was used for output. Results shown that the ANFIS model can be used for GDP prediction based on exchange rate values.

FPGA BASED P/PI/PD/PID-LIKE INTERVAL TYPE -2 FLC DESIGN FOR CONTROL SYSTEMS

Type-2iFuzzyilogicicontrol contains footprint of uncertainty (FOUi) that is able to handle the numerical uncertainties, nonlinearities, and linguistic associated with the inputs and outputs. This paper presents the structure of P/PI/PD/PID-like Interval Type-2 Fuzzy Logic Controller (IT2-FLC) that is designed and simulated using MATLAB SIMULINK. This controller is designed by combining the advantages of fuzzy inference and different structures of controller. The type of the controller is selected using two bits control inputs. The inputs to the controller are: input-output gains, set-point, and the actual output. Karnick-Mendel (KM) algorithm is used to implement the IT2-FLC. IT2FLC has two triangular shaped membership functions. The range of the universe of discourse (-1.5 to 1.5) is selected for each input and output. The defuzzification technique is selected as Centroid method. The sampling time is (0.01), and the choice of rules is by using trial and error. The programs are written in MATLAB SIMULINK and converted into VHDL and Verilog languages. The program generated must achieve small FPGA size and as a result less execution time must achieve vs. acceptable accuracy. Linear and nonlinear models were used to test the Controller. Trial and error method was used to tune the gain factors to obtain the best response with minimum overshoot, minimum settling time, minimum Root Mean Squared Error (RMSE), and minimum structure design.

Development of Genetically tuned Fuzzy dynamic model for nonlinear dynamical systems: Application on reaction section of Tennessee Eastman process

This work presents a new GA-Fuzzy method to model dynamic behavior of a process, based on Recurrent Fuzzy modeling through Mamdani approach whose inference system is optimized by Genetic Algorithms. By using the Mamdani approach, the proposed method surmounts the need to solve various types of mathematical equations governing the dynamic behavior of the process. The proposed method consists of two steps; i) constructing a startup version of the model, ii) optimizing the shape of membership functions of the fuzzy sets corresponding to the variables exist in the fuzzy model, along with the production rules constituting the inference such that the obtained fuzzy model can predict the dynamic behavior of the process fairly accurately. The proposed method is used to predict the dynamic behavior of the reaction section of the Tennessee Eastman (TE) benchmark. The overall accuracy of the obtained results compared to their corresponding counterparts in TE benchmark. The mean absolute percentage error (MAPE) of the key process variables which are temperature, pressure, and level of the reactor, and the reactor cooling water outlet temperature were calculated as 1.17%, 0.38%, 1.5%, and 1.57%, respectively, showing high prediction capability of the proposed method.

Selection of Fuzzy Controller Membership Functions Based on Adaptive Gaussian Cloud Transform

Aiming at the boundedness of existing methods of selecting membership functions, an adaptive Gaussian cloud transform algorithm which is guided by the threshold values of hybridization degree is proposed to construct concept hierarchy from original sample data, and then the number, shape and coverage area of membership functions can be derived from the distribution of Gaussian cloud. To test and verify the effectiveness of membership function that is extracted based on adaptive Gaussian cloud transform algorithm, a six-degree-of freedom model of unmanned aerial vehicles(UAV) is constructed, and a fuzzy controller of pitching angle is established with the platform of Simulink. The simulation results show that the fuzzy controller which includes membership functions derived from the distribution of Gaussian cloud transform can achieve perfect control performance of pitching angle and meanwhile obtain good dynamic response characteristics.

An Optimized Multiple Fuzzy Membership Functions based Image Contrast Enhancement Technique

Image enhancement is an emerging method for analyzing the images clearer for interpretation and analysis in the spatial domain. The goal of image enhancement is to serve an input image so that the resultant image is more suited to the particular application. In this paper, a novel method is proposed based on Mamdani fuzzy inference system (FIS) using multiple fuzzy membership functions. It is observed that the shape of membership function while converting the input image into the fuzzy domain is the essential important selection. Then, a set of fuzzy If-Then rule base in fuzzy domain gives the best result in image contrast enhancement. Based on a different combination of membership function shapes, a best predictive solution can be determined which can be suitable for different types of the input image as per application requirements. Our result analysis shows that the quality attributes such as PSNR, Index of Fuzziness (IOF) parameters give different performances with a selection of numbers and different sized membership function in the fuzzy domain. To get more insight, an optimization algorithm is proposed to identify the best combination of the fuzzy membership function for best image contrast enhancement.

Multi-objective optimisation of electrical discharge machining for Inconel 825 using Taguchi-fuzzy approach

This paper employs Taguchi-fuzzy approach for parametric optimisation of electric discharge machining with multiple performance measures. In this work, seven input parameters (one of two levels and six of three levels) and two performance measures have been considered and the experiments are designed using Taguchi’s L36 orthogonal array. A fuzzy model is formed using mamdani inference system and optimal combination of process parameters has been obtained on basis of multi-performance fuzzy index (MPFI) value calculated using different shapes of membership functions (MF) viz. triangular, trapezoidal and gaussian. Gaussian MF is found to provide better results as compared to triangular and trapezoidal MF. ANOVA analysis has also been carried out on MPFI to find out percentage contribution. It is shown with the help of confirmatory experiments that MRR and SR are improved by 103.25 and 32.11% respectively by employing the proposed approach.

Online Estimation of Wind Turbine Tip Speed Ratio by Adaptive Neuro-Fuzzy Algorithm

The efficiency of a wind turbine highly depends on the value of tip speed ratio during its operation. The power coefficient of a wind turbine varies with tip speed ratio. For maximum power extraction, it is very important to hold the tip speed ratio at optimum value and operate the variable-speed wind turbine at its maximum power coefficient. In this paper, an intelligent learning based adaptive neuro-fuzzy inference system (ANFIS) is proposed for online estimation of tip speed ratio (TSR) as a function of wind speed and rotor speed. The system is developed by assigning fuzzy membership functions (MFs) to the input-output variables and artificial neural network (ANN) is applied to train the system using back propagation gradient descent algorithm and least square method. During the training process, the ANN adjusts the shape of MFs by analyzing training data set and automatically generates the decision making fuzzy rules. The simulations are done in MATLAB for standard offshore 5 MW baseline wind turbine developed by national renewable energy laboratory (NREL). The performance of proposed neuro-fuzzy algorithm is compared with conventional multilayer perceptron feed-forward neural network (MLPFFNN). The results show the effectiveness of proposed model. The proposed system is more reliable for accurate estimation of tip speed ratio.

Оптимізація процесу одержання целюлози із лляного волокна із формалізацією критеріїв оптимальності

Flax fibre is an environmentally friendly, pure vegetable material with high durability. During the long flax fibres production, the short fibres, which have limited using, are formed. One of the ways of short flax fibre using is the cellulose with various purposes production because the short fibre is a valuable raw material. The problem of vegetable raw materials rational using is relevant. The previous results showed the possibility of obtaining cellulose from flax materials. Flax raw materials, in comparison with wood, has a higher content of cellulose and lower content of lignin and in its properties is approaching to the cotton. The soda boiling of flax cellulose from short flax fibre with preliminary aqueous hydrolysis is carried out in the research. The alkali consumption for boiling is 20% of the mass of dry raw materials with anthraquinonoid catalyst. An attempt to solve such a problem was carried out with the participation of the authors. The quality of the boiling process was controlled by the indexes: cellulose output ( Y 1 , %), lignin content ( Y 2 , %), α-cellulose content ( Y 3 , %), celluloses viscosity ( Y 4 , mPas/s). The second order regression models that describe the dependence of the process quality indexes was built. The aim of the research is solution the multi-criteria optimization problem of the flax celluloses soda boiling with the previous aqueous hydrolysis process. Optimality criteria are given in the linguistic expressions form: cellulose output – “as high as possible”; α-cellulose content – “above 90%”; the content of lignin – “less than 2.5%”; the viscosity – “about 20 mPas/s”. Therefore, for optimization problem solution the linguistically defined optimality criteria must be formalized. For the formalization of the optimality criteria, we use the methods of the fuzzy set theory. The optimality criteria are considered as fuzzy sets and the membership functions are built for these sets. We suggest using S -shaped membership functions for the indexes Y 1 and Y 2 , Z -shaped functions for index Y 3 and U-shaped function for index Y 4 . In order to facilitate the optimization problem solution, we consider expedient to use sigmoidal type of membership functions for indexes form Y 1 to Y 3 , and bell-shared type for index Y 4 . The upper and lower limits of the membership functions’ universe are the worst values of quality indexes obtained in the experiment and the best theoretically possible values of these indexes. Because the both sigmoidal and bell-shared membership functions generate not strictly normal, but subnormal fuzzy sets, we suggest the maximum deviation of 0.01. Based on the above considerations, we established the numerical values for the membership functions’ parameters. The Bellman-Zade optimality principle is used to find the optimal solution. According to this principle, the solution is considered the optimal one, for which the lowest of the membership functions has the maximum value. The optimum values of technological parameters for obtaining linseed cellulose are established, namely: the duration of aqueous pre-hydrolysis is 2 hours 12 minutes, the temperature of preliminary hydrolysis is 154 O C, the duration of sodium hydroxide is 3 hours, and the boiling temperature is 166 O C with the hydro-module is 6:1. As can be seen from the results, the content of lignin and the content of α-cellulose limit for the found optimum value. Our optimal results are in good agreement with solutions obtained by other methods. Thus, the proposed approach allows obtaining the high quality cellulose. Linseed cellulose can be used after bleaching for the cellulose nitrates or carboxymethyl-cellulose production.

Stabilization and Control of Elastic Inverted Pendulum System (EIPS) Using Adaptive Fuzzy Inference Controllers

Elastic Inverted Pendulum system (EIP) are very popular objects of theoretical investigation and experimentation in field of control engineering. The system becomes highly nonlinear and complex due to transverse displacement of elastic pole or pendulum. This paper presents a comparison study for control of EIP using fuzzy and hybrid adaptive neuro fuzzy inference system (ANFIS) controllers. Initially a fuzzy controller was designed, which was used for training and tuning of ANFIS controller using gbell shape membership functions (MFs). The performance of complete system was evaluated through output responses of settling time, steady state error and maximum overshoot. The study also highlights effect of varying number of MFs on training error of ANFIS. The results showed better performance of ANFIS controller compared to fuzzy controller.

Precision tension control technology of composite fiber tape winding molding

During the composite fiber tape winding process, tension fluctuation has a serious impact on the performance parameters of winding products, such as strength, compactness, resin content, fatigue resistance, and stress uniformity. Because of the influence of dynamic performance of tension control system caused by the torque motor, external disturbance, sensitivity of the parameter variation, inverter dead zone, ripple torque, cogging torque, nonlinear friction, and measurement noise, a fuzzy proportion integral derivative (PID) controller based on robust extended Kalman filter (REKF) is proposed, which considers membership function optimization of nonlinear dynamic fuzzy control system as systematic identification through designating activated membership function of input and output variable in each control step, and then, online adjustment of the activated membership functions is used to reduce the control error by REKF. By adjusting the shape and position of membership function, the designed controller can be adapted to the dynamic conditions. Simulation and experimental results show that overall performance of the control system has been significantly improved. Compared with traditional fuzzy PID, the fuzzy PID controller based on REKF has strong anti-interference performance and robustness, and the winding tension control precision has been increased by 40–42%.

An investigation of membership functions on performance of ANFIS for solving classification problems

Adaptive neuro-fuzzy inference system (ANFIS) is one of the efficient machine learning techniques, which has been successfully employed in wide variety of applications. The performance of ANFIS depends on the selection of the number and shape of membership functions as these two factors influence the most on computational complexity and accuracy of the designed ANFIS-based model. Mostly, an expert knowledge is required in this regard. However, there is an immense need of an investigative study for helping researchers make better decision on the number and shape of membership functions for thier ANFIS models. Hence, this study examines the role of four popular shapes of membership functions on the performance of ANFIS while solving various classification problems. According to experiments, Gaussian membership function demonstrated higher degree of accuracy with lesser computational complexity as compared to the counterparts.

Soft Computing Based Adaptive Error Optimisation for Control of Nonlinear System

This paper elaborates a novel hybrid learning approach for training error optimisation and control of highly dynamic triple-link inverted pendulum on cart. The study demonstrates a relationship between shape and number of membership functions (MFs) of both linear and constant type to determine training error tolerance of ANFIS controller. The results are plotted which clearly highlighted supremacy of constant type three triangular shape MFs. Mathematical model and simulink of proposed system has also been analysed. The learning ability and designing methodology of adaptive networks and robustness of PID controllers are briefly described. Finally, the study illustrates an offline mode comparison of PID based ANFIS and Neural controllers in terms of settling time, steady state error and overshoot.

Automated Control and Optimisation of Overhead Cranes

This study considers a fuzzy based computing technique for control and optimising performance of overhead gantry crane. The objective is to minimise load swing and stabilise crane position in the least possible time. The fuzzy controllers were designed using nine gaussian and triangular shape membership functions. The results clearly confirmed the effect of shape of memberships on performance of fuzzy controllers. Performance of overhead crane was measured in terms of settling time and overshoot. The study also demonstrates the influence of varying mass of the load, mass of crane and length of crane bar on stability of the crane. A mathematical model of the crane system has been derived to develop a simulink model of proposed system and performing simulations.

Type-1 OWA Unbalanced Fuzzy Linguistic Aggregation Methodology: Application to Eurobonds Credit Risk Evaluation

In decision making, a widely used methodology to manage unbalanced fuzzy linguistic information is the linguistic hierarchy (LH), which relies on a linguistic symbolic computational model based on ordinal 2-tuple linguistic representation. However, the ordinal 2-tuple linguistic approach does not exploit all advantages of Zadeh's fuzzy linguistic approach to model uncertainty because the membership function shapes are ignored. Furthermore, the LH methodology is an indirect approach that relies on the uniform distribution of symmetric linguistic assessments. These drawbacks are overcome by applying a fuzzy methodology based on the implementation of the type-1 ordered weighted average (T1OWA) operator. The T1OWA operator is not a symbolic operator and it allows to directly aggregate membership functions, which in practice means that the T1OWA methodology is suitable for both balanced and unbalanced linguistic contexts and with heterogeneous membership functions. Furthermore, the final output of the T1OWA methodology is always fuzzy and defined in the same domain of the original unbalanced fuzzy linguistic labels, which facilitates its interpretation via a visual joint representation. A case study is presented where the T1OWA operator methodology is used to assess the creditworthiness of European bonds based on real credit risk ratings of individual Eurozone member states modeled as unbalanced fuzzy linguistic labels.

Modeling of the Effects of Cyclic Voltammetry (CV) Using Fuzzy Logic with Different Membership Functions for Proton Exchange Membrane Fuel Cell (PEM) with Polyvinyl Alcohol/Nano Silver (PVA/Ag)

In this study, the effects of cyclic voltammetry (CV) has been modeled with Rule-based mamdani-type fuzzy (RBMTF), by using experimental data for proton exchange membrane fuel cell with PVA/AG. In the system developed, RBMTF apply input parameters are CV, scan rate and time, output parameters are current density and voltage. 12300 values for experimental study also obtained with RBMTF. Membership functions (MFs) are the building blocks of fuzzy set theory, i.e., fuzziness in a fuzzy set is determined by its MF. Accordingly, the shapes of MFs are important for a particular problem since they effect on a fuzzy inference system. They may have different shapes like triangular, trapezoidal, Gaussian, etc. When the results obtained from RBMTF and statistical analyses of experimental data have been compared, it has been determined that the two groups of data are coherent, and that there is not a significant difference between them. As a result, this study indicates that RBMTF with different membership functions can be safely used for CV.

Relaxed stability conditions for polynomial‐fuzzy‐model‐based control system with membership function information

This study presents a new approach for stability analysis of polynomial-fuzzy-model-based (PFMB) control system using membership function information. For the purpose of extracting the regional information of the membership functions, the operating domain is partitioned into several sub-domains. In each sub-domain, the boundaries of every single membership function overlap term and the numerical relation among all the membership function overlap terms are represented as a group of inequalities. Through the S-procedure, the regional membership function information is taken into account in the stability analysis to relax the stability conditions. The operating domain partition scheme naturally arises the motivation of constructing the PFMB control system with the sub-domain fuzzy controllers. Each polynomial fuzzy controller works in its corresponding sub-domain such that the compensation capability of controller is enhanced. The sum-of-squares (SOS) approach is proposed to obtain the stability conditions of the PFMB control system using the Lyapunov stability theory. The PFMB control system studied in this study has the feature that the number of fuzzy rules and the membership function shapes of the polynomial fuzzy controller can be designed independently from the polynomial fuzzy model. To verify the stability analysis result, a numerical example is given to demonstrate the validity of the proposed method.

Control Dynamics and Simulation of Inclined Cart and Pendulum System

This paper applies various soft-computing control strategies for offline mode control of highly non-linear cart and pendulum system moving on an inclined surface. The surface is considered at inclination of 12° from horizontal. The study compares performance of four different control techniques namely Proportional-integral-derivative (PID), Fuzzy logic, Adaptive neuro fuzzy inference system (ANFIS) and Neural networks for control of proposed system. A Matlab-Simulink model of system has been developed from mathematical equations derived using Newton's second law. The cart and pendulum system has been initially controlled using PID controllers and results were further used to train ANFIS and neural controllers. The ANFIS and fuzzy controllers were designed using three and nine gbell shape membership functions (MFs) respectively. The controllers were further compared in terms of settling time, overshoot and undershoot.

Semi-elliptic membership function: Representation, generation, operations, defuzzification, ranking and its application to the real-time task scheduling problem

A potentially important, yet under-stated membership function (MF) shape viz. semi-elliptic membership function (SEMF) has been elaborated in this paper so that researchers are able to see its true potential for realistic modelling of the decision variables in real-life applications. The concept of semi-elliptic membership function has the practical significance since it can nicely remove the drawbacks associated with the Gaussian membership function shape regarding its long tail with non-zero membership values. A semi-elliptic membership function can be mathematically expressed in terms of only two parameters. It is shown that, with the representation considered here, prominent arithmetic operations such as addition, subtraction, multiplication etc. can be applied on the semi-elliptic fuzzy numbers. As an extension, a novel membership function generation technique for interval type-2 semi-elliptic (IT2SE) MF has also been proposed. Then the applicability of the arithmetic operations on the IT2SE fuzzy numbers is shown. It is illustrated that semi-elliptic fuzzy numbers can suitably be ranked by existing distance measures. We have shown that the well-known INT (Improved Nie-Tan’s) defuzzification technique can be applied to the interval type-2 semi-elliptic fuzzy sets for producing defuzzified outputs. Finally, with application point of view, semi-elliptic fuzzy numbers has been applied to the real-time task scheduling problem. Results are compared with triangular fuzzy numbers. Suitable numerical examples are considered for demonstration purposes.

Using genetic algorithms to evolve type-2 fuzzy logic systems for predicting bankruptcy

PurposeType-2 fuzzy sets became attractive in practice because of their footprint of uncertainty that gives them more degrees of freedom. This paper aims to use genetic algorithms (GAs) to design an interval Type-2 fuzzy logic system (IT2FLS) for the purpose of predicting bankruptcy.Design/methodology/approachThe shape of type-2 membership functions, the parameters giving their spread and location in the fuzzy partitions and the set of fuzzy rules are evolved at the same time by encoding all together into the chromosome representation. The enhanced Karnik–Mendel algorithms are used for the centroid type-reduction and defuzzification stage. The performance in predicting bankruptcy is evaluated by benchmarking IT2FLSs against type-1 FLSs. The experimental setup consists of evolving 100 configurations for both the T1FLS and IT2FLS and comparing their in-sample and out-of-sample average accuracy.FindingsThe experiments confirm that representing and capturing uncertainty with more degrees of freedom is an important advantage. It is this extra potential of IT2FLSs that allows them to outperform T1FLS, especially in terms of generalization capability.Originality/valueThe strategy followed in this paper is to train an IT2FLS from scratch rather than tuning the parameters of an existing T1FLS. Because this leads to solving a mixed integer optimization problem, the GA-based approach is specifically designed and uses genetic operators that are most suited for such a case: tournament selection, extended Laplace crossover and power mutation. Finally, the trained IT2FLS is applied to bankruptcy prediction, and its generalization capability is compared with related techniques.