Suitable Membership Functions Research Articles

Intelligent decision support system for pulmonary tuberculosis detection using bipolar fuzzy utility matrix and bipolar Mamdani fuzzy inference system

Tuberculosis (TB) stands as the second leading global infectious cause of death, following closely behind the impact of COVID-19. The standard approach to diagnose TB involves skin tests, but these tests can yield inaccurate results due to limited access to healthcare and insufficient diagnostic resources. To enhance diagnostic accuracy, this study introduces a novel approach employing a Bipolar Fuzzy Utility Matrix Inference System (BFUMIS) and a Bipolar Mamdani Fuzzy Inference System (BMFIS) to assess TB disease levels. By considering factors associated with the causation of TB, the study devises suitable membership functions for bipolar fuzzy sets (BFS) using both triangular and trapezoidal fuzzy numbers. Using a point factor scale, the study clusters the rules systematically and assesses the level of uncertainty within these grouped rules by utilizing bipolar triangular fuzzy numbers (BTFN). To handle the BTFN, this study proposes converting bipolar triangular fuzzy into bipolar crisp score (CBTFBCS) algorithm as a defuzzification method. The optimal bipolar fuzzy utility sets (BFUS) are determined from the bipolar fuzzy utility matrix to identify patients’ TB disease levels. These sets play a pivotal role in characterizing the severity of TB disease levels in patients. Additionally, rigorous validation of the utility framework is accomplished through measures of bipolar fuzzy satisfactory factors and sensitivity analyses. Furthermore, the study introduces the BMFIS, which presents a novel perspective on the conventional fuzzy inference system. This innovative system integrates the Mamdani fuzzy inference system (MFIS) into a bipolar fuzzy context, enriching the diagnostic process with enhanced insights. To demonstrate the efficacy of the proposed methods, extensive validation is carried out using actual clinical data. The performance metrics used in this validation effectively demonstrate the superiority of the proposed approach.

Adaptive robust control for fuzzy underactuated mechanical systems: A Stackelberg game-theoretic optimization approach

This article proposes a Stackelberg game-theoretic optimization approach for adaptive robust controller design of fuzzy underactuated mechanical systems (UMSs). As a commonly encountered challenge, several factors render it difficult to further improve the control performance of these UMSs, such as the coupling effects and nonlinearities resulting from the underactuated structure. Meanwhile, various sources of unexpected time-varying uncertainties can further add to the difficulty in the controller design procedure pipeline. To overcome the above-mentioned issues and further enhance the system performance, a dual-parameter optimization framework is presented here for adaptive robust controller design, leveraging fuzzy set theory and Stackelberg game. In this development, the time-varying uncertainty is characterized via suitable membership functions; and thus the appropriate fuzzy UMS is established. Then, an adaptive robust controller with a leakage-type adaptation law is presented to tackle the uncertainty appropriately. To further improve the system performance considering the trade-off among the transient performance; the steady-state performance; and the control cost; a Stackelberg game-based dual-parameter optimization method is proposed to determine the Stackelberg strategy. It is rigorously proved in the developments here that the proposed controller design and parameters optimization method guarantees the desired deterministic system performance, i.e., uniform boundedness (UB) and uniform ultimate boundedness (UUB). Moreover, a series of numerical simulations based on a flywheel inverted pendulum (FIP) are performed, which illustrate the effectiveness and superiority of the proposed method.

Multi-objective optimization model for uncertain crop production under neutrosophic fuzzy environment: A case study

<abstract> <p>In real world uncertainty exist in almost every problem. Decision-makers are often unable to describe the situation accurately or predict the outcome of potential solutions due to uncertainty. To resolve these complicated situations, which include uncertainty, we use expert descriptive knowledge which can be expressed as fuzzy data. Pakistan, a country with a key geographic and strategic position in South Asia, relies heavily on irrigation for its economy, which involves careful consideration of the limits. A variety of factors can affect yield, including the weather and water availability. Crop productivity from reservoirs and other sources is affected by climate change. The project aims to optimize Kharif and Rabbi crop output in canal-irrigated areas. The optimization model is designed to maximize net profit and crop output during cropping seasons. Canal-connected farmed areas are variables in the crop planning model. Seasonal crop area, crop cultivated area, crop water requirement, canal capacity, reservoir evaporation, minimum and maximum storage, and overflow limits affect the two goals. The uncertainties associated with the entire production planning are incorporated by considering suitable membership functions and solved using the Multi-Objective Neutrosophic Fuzzy Linear Programming Model (MONFLP). For the validity and effectiveness of the technique, the model is tested for the wheat and rice production in Pakistan. The study puts forth the advantages of neutrosophic fuzzy algorithm which has been proposed, and the analyses derived can be stated to deal with yield uncertainty in the neutrosophic environments more effectively by considering the parameters which are prone to abrupt changes characterized by unpredictability.</p> </abstract>

Grey Earned Value Management: Theory and Applications

Project stakeholders always investigate possible approaches to monitor project progress closely and further, taking necessary actions during the whole phases of the project in order to manage delays. Earned value management (EVM) is one of the methods, which can forecast the required costs for accomplishment of the project. The data collected from projects undertaken in order to update the master schedule often suffer from a level of uncertainty. Ignoring these uncertainties may even lead to project failure. Fuzzy theory has been previously used in the EVM for taking uncertainties into account. A major disadvantage of using fuzzy approaches is the need for incorporating expert judgments to construct a suitable membership function for all activities in the project undertaken. A potential approach for overcoming this issue lies in Grey theory. In this article, the current study deals with the EVM method in grey systems paradigm. Also, the performance of the proposed method, called grey earned value management (EVM-G), is evaluated through some numerical examples and a case study. The results demonstrate that the proposed approach has a unique performance in highly uncertain environments when experts have become unavailable. Comparisons between EVM-G and the fuzzy earned value management approaches reveal the superior performance of EVM-G.

Dimensioning of the hydraulic gravity energy storage system using Fuzzy logic based simulation

For reasons of the intermittent nature of electricity produced by renewable power plants, the analysis and design of an efficient energy storage system (ESS) are becoming a point of interest. The current paper presents a simulation-based analysis to find the best dimensions of the gravity ESS. The considered system is the piston in cylinder pumped hydro-electric storage. Five design parameters are considered, namely, chamber height, piston height, piston diameter, return pipe length, and diameter. First, the system was modeled by the governing mathematical equations. The mathematical model is then simulated and validated by comparing its results with experimental benchmarking from literature with different cases. To find the best design parameters in function of the uncertain energy input, a Fuzzy logic controller is designed with three membership functions: triangular, trapezoidal, and Gaussian. The uncertainty of the energy produced by the renewable power plant is modeled by Fuzzy inputs for the energy and rate of energy change. The developed Fuzzy interface model is a Mamdani one. The ARMAX method was used to select the most suitable membership function to represent the Fuzzy simulation of the storage system. Results showed that the Gaussian membership function is the most suitable membership function to represent the Fuzzy model of the storage system. Using the Gaussian membership function and Centroid defuzzification, the best design parameters of the different size power plants are determined using MATLAB-Simulink.

Finding Suitable Membership Functions for Mining Fuzzy Association Rules in Web Data Using Learning Automata

Transactions in web data are huge amounts of data, often consisting of fuzzy and quantitative values. Mining fuzzy association rules can help discover interesting relationships between web data. The quality of these rules depends on membership functions, and thus, it is essential to find the suitable number and position of membership functions. The time spent by users on each web page, which shows their level of interest in those web pages, can be considered as a trapezoidal membership function (TMF). In this paper, the optimization problem was finding the appropriate number and position of TMFs for each web page. To solve this optimization problem, a learning automata-based algorithm was proposed to optimize the number and position of TMFs (LA-ONPTMF). Experiments conducted on two real datasets confirmed that the proposed algorithm enhances the efficiency of mining fuzzy association rules by extracting the optimized TMFs.

Artificial neural networks and adaptive neuro-fuzzy models for predicting WEDM machining responses of Nitinol alloy: comparative study

This article reports a comparative study of artificial neural network (ANN) and adaptive neuro-fuzzy inference system (ANFIS) models for better prediction of wire electro-discharge machining (WEDM) responses like material removal rate and surface roughness of a Nitinol alloy. Pulse on time (Ton), pulse off time (Toff), peak current (Ipeak) and gap voltage (V) were selected as input attributes. Experimental results were performed to verify the results from ANN and ANFIS models. ANN model, back-propagation with three different algorithms Levenberg–Marquardt (LM), Elman regression neural network and generalized regression neural network and ANFIS model, were developed using the same input variables. The most suitable algorithm and neuron number in the hidden layer were found as LM with 10 neurons for ANN models whereas the most suitable membership functions and number of membership functions are found to be gauss and two, respectively. The statistical validation measures such as root mean square error, mean square error and mean absolute percentage error are obtained through ANN and ANFIS models. The statistical values are given in the tables. As per the statistical measures perspective, the ANFIS model will have better accuracy for anticipation of WEDM attributes of a Nitinol alloy.

Fuzzy C-Means Based Clustering and Rule Formation Approach for Classification of Bearing Faults Using Discrete Wavelet Transform

The rolling bearings are considered as the heart of rotating machinery and early fault diagnosis is one of the biggest challenges during operation. Due to complicated mechanical assemblies, detection of the advancing fault and faults at the incipient stage is very difficult and tedious. This work presents a fuzzy rule based classification of bearing faults using Fuzzy C-means clustering method using vibration measurements. Experiments were conducted to collect the vibration signals of a normal bearing and bearings with faults in the inner race, outer race and ball fault. Discrete Wavelet Transform (DWT) technique is used to decompose the vibration signals into different frequency bands. In order to detect the early faults in the bearings, various statistical features were extracted from this decomposed signal of each frequency band. Based on the extracted features, Fuzzy C-means clustering method (FCM) is developed to classify the faults using suitable membership functions and fuzzy rule base is developed for each class of the bearing fault using labeled data. The experimental results show that the proposed method is able to classify the condition of the bearing using the extracted features. The proposed FCM based clustering and classification model provides easier interpretation and implementation for monitoring the condition of the rolling bearings at an early stage and it will be helpful to take the preventive action before a large-scale failure.

A MODEL FOR EVALUATING THE RISK EFFECTS ON CONSTRUCTION PROJECT ACTIVITIES

Cost overruns and time delays are considered to be very important challenges for the majority of construction projects. These challenges are typically attributed to their associated risks. Due to the risky and uncertain nature of construction projects, an increasing amount of attention is given to estimating and overcoming cost overruns and time delays. New techniques are being developed to help project managers to contractually complete projects within cost and time constraints. The objective of this study was to develop a new qualitative and quantitative risk analysis model that can be employed for construction projects. The proposed model, which is based on a fuzzy logic tool, consists of two modules for assessing risk factors that affect the main construction activities and computing the expected cost overruns and time delays that are associated with these risks. Using numerous logical rules, the model applies the probability of occurrences and impacts of the risks on the cost and time of the main activities. The Spearman and Kendall correlation coefficient tests are applied to verify and select a suitable membership function. Using four proposed membership functions, the results of these tests confirmed that the triangle membership function is suitable for the model. The model is verified by application to HVAC system activities in two actual construction projects, which serve as case studies. Two different methods are proposed and applied to quantify the cost overruns and time delays. The first method is based on determining the cost overruns and time delay values for each activity according to their weight in the system. Triple premise rules are proposed and applied in the second method, which is established to relate all activities. The results from the second method are more accurate compared with the first method based on actual data from the case study projects. In addition, the results demonstrated that the proposed model can be used to quantify the expected cost overrun and time delays in construction project activities and can be generalized and implemented in different construction activities.

Application of fuzzy logic and geometric average: A Cu sulfide deposits potential mapping case study from Kapsan Basin, DPR Korea

In this paper, two kinds of knowledge-driven methods, one using the fuzzy logic and another using geometric average, were applied to create the mineral potential maps for Cu sulfide deposits in the greenfield Kapsan Basin, DPR Korea. The ore geology studies for the study area have revealed that Cu sulfide deposits of hydrothermal genesis in Kapsan Basin are closely associated with Jurassic intrusions and faulting tectonics. Based on the conceptual model of Cu sulfide deposits and the available spatial datasets in the study area, we used five independent evidential maps for Cu sulfide deposits potential mapping. They include: (1) faults; (2) aeromagnetic anomaly; (3) Cu geochemical data; (4) Pb geochemical data; and (5) Zn geochemical data. The evidential map values were transformed into continuous values of the [0, 1] range using the non-linear fuzzy membership functions; logistic sigmoid and fuzzy Gaussian functions. Because the fuzzy logic and geometric average methods can use the same fuzzification methodology based on suitable membership functions, it is very economic and efficient to simultaneously apply two predictive models for mineral potential mapping of the study area. The preparation of these evidential layers were performed using spatial analyses supported in ArcGIS 10.4 GIS platform based on geological, geophysical and geochemical spatial datasets.The validation and comparative analysis results for the two predictive models demonstrated that most of known mineral occurrences are distributed in areas with high potential values. The target areas classified by the fuzzy logic occupy 15% of the study area and contain 78% of the total number of known mineral occurrences. Compared with the fuzzy logic, the resulting areas by the geometric average occupy 13% of the study area, but contain 93% of the total number of known mineral occurrences. Although the total number of known mineral occurrences is relatively low for the application of ROC (receiver operating characteristics) technique, the areas under the ROC curve (AUC) obtained by two predictive models were greater than 0.5, suggesting that both predictive models and their resulting potential maps are useful for evaluating the prospectivity of Cu sulfide deposits in Kapsan Basin.

Modeling neutrosophic variables based on particle swarm optimization and information theory measures for forest fires

Recently, neutrosophic systems modeling gained great attention in indeterminacy handling. Generating suitable membership, indeterminacy and non-membership functions for neutrosophic temperature variable in the forest fires data is a challenging problem. This paper presents a platform for integrating information theory measures with particle swarm optimization as a meta-heuristic algorithm to model a neutrosophic variable out of concrete data. Particle swarm optimization is an efficient population-based algorithm presented to find membership, indeterminacy and non-membership optimal partitions. The integrated hybrid model of information theory measures and particle swarm optimization is presented. The experimental comparisons are applied between the proposed technique and the Fuzzy C-Mean MATLAB tool for generating fuzzy membership functions. Extensive graphical results of the temperature variable of the forest fires data set and the conversion rates are provided to demonstrate the effectiveness of the proposed paradigm. The conversion rates of the proposed technique show that the optimal membership and non-membership functions are concluded after 10 and 17 iterations, respectively, which is a feasibly fast rate. Also, the proposed technique generated relatively similar functions’ subsets for the forest fire temperature variable, while the Fuzzy C-Mean clearly shifted the functions’ subsets during the experiments. Calculating the indeterminacy of the temperature in forest fire data will contribute to forecasting these fires accurately.

SEGMENTATION OF IMAGES OF EYE GROUND BLOOD VESSELS INVOLVING APPLICATION OF FUZZY IMAGING

Segmentation of images is an important task while processing images. Among the most widespread methods are methods based on pixel clustering, histogram methods, morphological methods, watershed segmentation, multiscale segmentation, and others. A promising trend in image processing is the use of fuzzy logic methods and the fuzzy set theory. Their application makes it possible to improve the quality of processing by providing information in a fuzzy form. The article proposes a new method for images segmentation involving boundaries detection based on the fuzzy representation of the image and fuzzy pixels. The membership functions are proposed for describing fuzzy pixels, and the requirements for their form and type are provided. The most suitable membership functions for fuzzy imaging are the s-function and the π-function. A description of a new method for boundaries detection based on the Sobel operator and the developed fuzzy type of image is described. In this case, standard calculations of the image brightness gradient are supplemented with their fuzzy versions which are then combined to obtain the final result. The experimental verification of the developed method is carried out using the example of eyeground images. In addition to the fuzzy detection of boundaries for the detection of blood vessels, the images were subjected to pre-processing (halftone imaging, mask matching, contrasting), morphological operators (thinning of boundaries, dilatation), and an algorithm for removing small details was applied. During testing, the developed algorithm showed acceptable results in terms of segmentation of blood vessels. In the future, a fuzzy image model can be extended to use fuzzy features of the second and higher types

Adaptive neuro-fuzzy approach for modeling equilibrium speed–density relationship

ABSTRACTThis paper endeavors to model the equilibrium speed–density relationship using a new fuzzy logic-based approach. To capture the randomness in traffic flow dynamics, it develops a single-input Adaptive Neuro-Fuzzy Inference System (ANFIS) trained with a hybrid algorithm. Furthermore, it proposes a new Premise-Consequent Conjugate Effect (PCCE) relationship to estimate fundamental diagram (FD) parameters from the ANFIS model. Several options (i.e. optimal split of data into training and testing data, selection of suitable membership function) offered by ANFIS are then illustrated. An experiment is performed to determine the maximum achievable goodness of fit without the occurrence of overfitting phenomenon by changing the number of clusters. The calibrated ANFIS model is compared against traditional and advanced speed–density models for five different freeway locations. Results show that the proposed model outperforms other preceding models by attaining goodness-of-fit values within a range of 0.82–0.92. Finally, the proposed PCCE relationship shows the ANFIS model's robustness in accurate estimation of FD parameters for all freeway locations.

Finding suitable membership functions for fuzzy temporal mining problems using fuzzy temporal bees method

Data are usually stored in databases with temporal components and quantitative values. In temporal databases, each data item has its own exhibition periods. In other word, early items have a longer lifespan. Item lifespan must be considered in mining process for fair and accurate results. Otherwise, extracted patterns may not exhibit the associations of items correctly enough. Some efficient algorithms have been thus proposed for extracting fuzzy association rules from quantitative temporal transactions. However, the membership functions affect rule discovery. Learning or tuning the membership functions for fuzzy temporal association rules mining is thus necessary. In this paper, we combined fuzzy temporal mining concepts into a fuzzy-evolutionary approach and designed an efficient fuzzy temporal-evolutionary mining process based on bees algorithm to find suitable membership functions for fuzzy temporal mining problems before searching for temporal frequent itemsets and fuzzy associations. Experimental results show that the proposed method exhibits good performance with respect to the effectiveness of the obtained solution.

Identification of suitable membership and kernel function for FCM based FSVM classifier model

Support vector machine (SVM) a new machine learning algorithm emerging as an effective classification technique than neural networks and other statistical classifiers. Its performance is however, observed to be affected by presence of outliers or noise in the dataset. So, to enhance the generalization capability of SVM, researchers focussed on developing fuzzy support vector machines (FSVM), which works on the concept of fuzzy membership to data points for learning the best decision hyperplane to classify data points. In such case, the choice of membership for data points is critical for good behaviour of the model. So, this paper focuses on finding right membership using optimal fuzzy index parameter (m) and demonstrating its significance in membership and classifier performance. This research work focussed on finding membership values using FCM and final membership from derived low value and high value membership function and incorporating them in FSVM model, to find the best membership value from analysis of five datasets used in the study. For linear classification of data, as kernel functions can show its influence, the study also performed analysis on selection of appropriate kernel function that can best fit the FCM based FSVM classifier model and prove enhancement of generalization performance with our research findings. High and low membership functions are compared through computational analyses and the membership that results in better generalization capability in the model is proven in the study.

On solving multi-objective linear bi-level multi-follower programming problem

Multi-objective linear bi-level multi-follower programming problem (MOBLMFPP) is a special case of two level hierarchical programming problems in which the second level hierarchy includes multiple followers. This paper presents an alternate method based on fuzzy goal programming approach for the solution of multi-objective linear bi-level multi-follower programming (MOLBMFP) problem in which there is no sharing of information among followers. In the proposed FGP model formulation, each of objective functions of each level (leader and follower's level of MOLBMFPP) as well as decision variables at each level are characterised into fuzzy goals. Suitable linear membership functions are defined for each objective function and decision variables. Then minimising the sum of the negative deviational variables of both levels, the highest membership value of each of fuzzy goals is obtained. Existence of compromise optimal solution for MOLBMFPP is established in context of proposed approach. A numerical example is illustrated in support of proposed methodology. A comparative analysis is also carried out in order to show efficiency of proposed method over earlier method for the solution of MOLBMFPP.

Takagi–Sugeno Dynamic Neuro-Fuzzy Controller of Uncertain Nonlinear Systems

The identification problem incorporated in feedback control of uncertain nonlinear systems exhibiting complex behavior has been solved in different ways. Some of these solutions have used artificial intelligence methods like fuzzy logic and neural networks. However, their individual implementation suffers from certain drawbacks, such as the black-box nature of neural network and the problem of finding suitable membership functions for fuzzy systems. These weaknesses can be avoided by implementing a hybrid structure combining these two approaches, the so-called neuro-fuzzy system. In this paper, a neuro-fuzzy system that implements differential neural networks (DNNs) as consequences of Takagi–Sugeno (T–S) fuzzy inference rules is proposed. The DNNs substitute the local linear systems that are used in the common T–S method. In this paper, DNNs are used to provide an effective instrument for dealing with the identification of the uncertain nonlinear system, while the T–S rules are used to provide the framework of previous knowledge of the system. The main idea is to carry out an online identification process of an uncertain nonlinear system with the aim to design a closed-loop trajectory tracking controller. The methodology developed in this study that supports the identification and trajectory control designs is based on the Lyapunov formalism. The DNN implementation results in a time-varying T–S system. As a consequence, the solution of two time-varying Riccati equations was used to adjust the learning laws in the DNN as well as to adjust the gains of the controller. Two results were provided to justify the existence of positive-definite solutions for the class of Riccati equations used in the learning laws of DNNs. A complete description of the learning laws used for the set of DNN identifiers is also obtained. An autonomous underwater vehicle system is used to demonstrate the performance of the controller on tracking a desired 3-D path by this combination of the DNN and the T–S system.

ANFIS CONTROLLER FOR AN ACTIVE MAGNETIC BEARINGS TO SUSPEND A FLYWHEEL ENERGY STORAGE SYSTEM

<p><span style="font-family: Times New Roman;">A flywheel energy storage system (FESS) is an effective energy-saving device. It works by accelerating a rotor flywheel disc at a very high speed and maintaining the energy in the system as rotational energy. Active magnetic bearings (AMBs) are ideally suited for use at high-speed and are so used in FESSs. This work develops a mathematical model of the electromagnet force and rotor dynamic of a flywheel. The systems for controlling the position and velocity of the flywheel are designed based on the emerging approaches of an adaptive neuro-fuzzy inference system (ANFIS). Fuzzy logic has occurred as a mathematical tool to deal with the uncertainties in human perception. It also provides a framework for applying approximate human reasoning capabilities to knowledge-based systems. Additionally, ANFIS has emerged as an intelligent controller with learning and adaptive capabilities. ANFIS is combined the fuzzy logic controller (FLC) and neural networks (NNs). In the method that is developed herein, the control model uses Takagi-Sugeno fuzzy logic, in which the back-propagation algorithm processes information from neural networks to adjust suitably the parameters of the fuzzy controller, and the output control signal tracks the input signal. This method can be applied to improve the control performance of nonlinear systems. The output signal responses transient performance of systems use an ANFIS that must be trained through a learning process to yield suitable membership functions and weightings. The results of the FESS indicated that the system responds with satisfactory control performance to reduce overshoot, a zero-error steady-state, and short rise time. The proposed controller can be feasibly applied to FESS with various external disturbances, and the effectiveness of the ANFIS with self-learning and self-improving capacities is proven.</span></p>

ANFIS CONTROLLER FOR AN ACTIVE MAGNETIC BEARINGS TO SUSPEND A FLYWHEEL ENERGY STORAGE SYSTEM

<p><span style="font-family: Times New Roman;">A flywheel energy storage system (FESS) is an effective energy-saving device. It works by accelerating a rotor flywheel disc at a very high speed and maintaining the energy in the system as rotational energy. Active magnetic bearings (AMBs) are ideally suited for use at high-speed and are so used in FESSs. This work develops a mathematical model of the electromagnet force and rotor dynamic of a flywheel. The systems for controlling the position and velocity of the flywheel are designed based on the emerging approaches of an adaptive neuro-fuzzy inference system (ANFIS). Fuzzy logic has occurred as a mathematical tool to deal with the uncertainties in human perception. It also provides a framework for applying approximate human reasoning capabilities to knowledge-based systems. Additionally, ANFIS has emerged as an intelligent controller with learning and adaptive capabilities. ANFIS is combined the fuzzy logic controller (FLC) and neural networks (NNs). In the method that is developed herein, the control model uses Takagi-Sugeno fuzzy logic, in which the back-propagation algorithm processes information from neural networks to adjust suitably the parameters of the fuzzy controller, and the output control signal tracks the input signal. This method can be applied to improve the control performance of nonlinear systems. The output signal responses transient performance of systems use an ANFIS that must be trained through a learning process to yield suitable membership functions and weightings. The results of the FESS indicated that the system responds with satisfactory control performance to reduce overshoot, a zero-error steady-state, and short rise time. The proposed controller can be feasibly applied to FESS with various external disturbances, and the effectiveness of the ANFIS with self-learning and self-improving capacities is proven.</span></p>

Synchronization and stabilization of fractional order nonlinear systems with adaptive fuzzy controller and compensation signal

In this paper, a direct adaptive fuzzy controller with compensation signal is presented to control and stabilize a class of fractional order systems with unknown nonlinearities. Based on a Lyapunov function candidate the global Mittag–Leffler stability is proved and a new fractional order adaptation law is derived. The adaptation law adjusts free parameters of the fuzzy controller and bounds them by utilizing a novel fractional order projection algorithm. Furthermore, due to the use of compensation term, the proposed approach does not demand suitable membership functions in the fuzzy system. In addition, the stability of the closed-loop system is guaranteed by utilizing a supervisory controller. Numerical simulations show the validity and effectiveness of the introduced scheme for various fractional order nonlinear models that perturbed by disturbance and uncertainty.