Fuzzy Function Research Articles

Adaptive fuzzy ABLF function fixed‐time tracking control method for nonlinear fault‐tolerant control systems with event triggering mechanism

SummaryAs the nonlinearities of many real physical controlled systems become stronger and stronger, they are difficult to be described by accurate mathematical models. For a class of nonlinear single‐input single‐output systems with all‐state constraints and actuator faults, an event‐triggered adaptive fuzzy ABLF function fixed‐time tracking control method is proposed. The asymmetric barrier Lyapunov function (ABLF) combined with backward step technique is used to ensure the boundedness of the closed‐loop system output. In order to solve the problem of limited bandwidth resources, this paper adopts the event trigger mechanism and fuzzy control technology to approximate the unknown function to ensure the convergence of the system in a fixed time. After theoretical analysis, it is proved that the tracking error of the system converges in the small neighborhood of the origin, and it still has good tracking performance when the actuator faults. Finally, by comparing the proposed method with the asymmetric barrier Lyapunov function, which verifies the effectiveness of the proposed method.

Prediction of Energy Demand of Customer Microgrid Based on Multi-Level Fuzzy Evaluation Model

In order to improve the accuracy of customer microgrid energy demand forecasting, this paper conducts an in-depth study on the prediction of customer microgrid energy demand based on a multi-level fuzzy evaluation model. Firstly, a multi-level fuzzy evaluation index system was constructed, the multi-level fuzzy evaluation indicators were clarified, and the correlation characteristics, interactive characteristics and data dimension characteristics of customer energy consumption data were analyzed. Then, according to the principle of multi-level fuzzy evaluation method, the factor set and the evaluation set are clarified, and the multi-level fuzzy relationship is induced by the multi-level fuzzy mapping, the fuzzy matrix is obtained, and the identification parameters are processed to complete the prediction of the customer's microgrid energy demand. The simulation results show that the overall prediction accuracy of the proposed model is high, the prediction time is short, and the practicability is strong.

Admissibility and event-triggered dissipative observer-based stabilization of discrete-time T-S fuzzy singular systems via fuzzy Lyapunov functions

Admissibility and event-triggered dissipative observer-based stabilization of discrete-time T-S fuzzy singular systems via fuzzy Lyapunov functions

Using an Intuitionistic Fuzzy Map to Evaluate the Effect of Risk Variables in Supply Chain Management

Using an Intuitionistic Fuzzy Map to Evaluate the Effect of Risk Variables in Supply Chain Management

Fixed-Point Theorems for Fuzzy Mappings

Since the 1970s and 1980s, significant contributions have been made by Weiss, Butnariu, Heilpern, Chitra, Subrahmanyam, and others, extending fixed-point theorems to fuzzy mappings and topological spaces. This paper provides two generalizations of two important fixed-point theorems, one provided by Butnariu and the other provided by Chitra. The first generalization ensures that, under certain conditions, an acyclic fuzzy mapping has a fixed point. The second result ensures the existence of a point in the intersection of two or more fuzzy mappings considering contractible finite dimensional ANR spaces, which is a generalization of the statement provided by Chitra.

Analyze the Stochastic Solid Fuzzy Transportation Problem with Mixed Constraints through Weibull Distribution

This paper proposed a general formulation of the stochastic solid transportation problem (SSTP) with mixed constraints such as supply, demand and conveyance capacity taken as uncertain under stochastic environment, following the Weibull distribution (WD). The aim of this study is to minimize the transportation cost includes probabilistic constraints have inequalities of stochastic solid transportation problem (SSTP). SSTP with probabilistic constraints is represented as a chance constrained programming problem. Obtain alpha cut representation from cost coefficient of the fuzzy objective function. We have developed four models for stochastic solid transportation problem. The suggested models are demonstrated by taken as numerical example. A sensitivity analysis is performed to understand parameter’s sensitivity in the proposed model. Introduction: In system of transportation, goods are moved from various sources to destinations using different vehicles and organizational systems, involving both technology and human efforts. Efficient resource allocation in transportation system is crucial for industries and imprecision from factors like fluctuating demand, unreliable supply chains and unpredictable traffic. To address these complexities, advanced mathematical models are needed to manage stochasticity, fuzziness and mixed constraints. The study explores the stochastic solid fuzzy transportation problem with mixed constraint by utilizing the Weibull distribution to model uncertainties inherent in transportation systems. This research addresses the complexity introduced by stochastic variables and fuzzy parameters, particularly in situations where demand, supply and cost of transportation are not deterministic. Objectives: The aim of this study is to minimize the cost of transportation includes probabilistic constraints have inequalities of stochastic solid transportation problem (SSTP). Methods: Obtain alpha cut representation form the cost coefficient of the fuzzy objective function and four models are developed for stochastic solid transportation problem. These models are demonstrating by using a numerical example and a sensitivity analysis is conducted to understand the sensitivity of the parameters in the propose model. Results: Obtained optimal solutions for developed four models of SFSTPMC and sensitivity analysis shows that cost of transportation and flow of unit are sensitive to change in probabilities of demand. Improve transportation system by understanding sensitivity patterns that help decision maker choose appropriate supply availability probabilities. Conclusions: This study presented an approach for solving the SFSTPMC using the Weibull distribution for probabilistic constraints and fuzzy objective functions for transportation cost. Developed and optimized four models, focusing on stochastic parameters. Sensitivity analysis demonstrated the impact of these parameters on transportation cost and unit flow. The results validate the model’s effectiveness in practical resource allocation and decision making under uncertainty.

Solving Intuitionistic Fuzzy Unconstrained Optimization Problems Using Interval Newton’s Method

This research work studies the optimization of intuitionistic fuzzy valued functions in unconstrained problems. The Interval Newton's Method using an Intuitionistic approach addresses both single and multivariable optimization problems. The study incorporates a mathematical comprehension of interval intuitionistic fuzzy valued problems, as well as real-world examples to demonstrate their effectiveness. Furthermore, MATLAB code is provided to demonstrate the implementation of the Interval Newton's Method.

Some New Estimations of Ostrowski-Type Inequalities for Harmonic Fuzzy Number Convexity via Gamma, Beta and Hypergeometric Functions

This paper demonstrates several of Ostrowski-type inequalities for fuzzy number functions and investigates their connections with other inequalities. Specifically, employing the Aumann integral and the Kulisch–Miranker order, as well as the inclusion order on the space of real and compact intervals, we establish various Ostrowski-type inequalities for fuzzy-valued mappings (F·V·Ms). Furthermore, by employing diverse orders, we establish connections with the classical versions of Ostrowski-type inequalities. Additionally, we explore new ideas and results rooted in submodular measures, accompanied by examples and applications to illustrate our findings. Moreover, by using special functions, we have provided some applications of Ostrowski-type inequalities.

Holiday Peak Load Forecasting Using Grammatical Evolution-Based Fuzzy Regression Approach

Peak load forecasting plays an important role in electric utilities. However, the daily peak load forecasting problem, especially for holidays, is fuzzy and highly nonlinear. In order to address the nonlinearity and fuzziness of the holiday load behaviors, a grammatical evolution-based fuzzy regression approach is proposed in this paper. The proposed hybrid approach is based on the theorem that fuzzy polynomial regression can model all fuzzy functions. It employs the rules of the grammatical evolution to generate fuzzy nonlinear structures in polynomial form. Then, a two-stage fuzzy regression approach is used to determine the coefficients and calculate the fitness of the fuzzy functions. An artificial bee colony algorithm is used as the evolution system to update the elements of the grammatical evolution system. The process is repeated until a fuzzy model that best fits the load data is found. After that, the developed fuzzy nonlinear model is applied to forecast holiday peak load. Considering that different holidays possess different load patterns, a separate forecaster model is built for each holiday. Test results on real load data show that an averaged absolute percent error less than 2% can be achieved, which significantly outperforms existing methods involved in the comparison.

A Recurrent Neural Network Approach for Constrained Distributed Fuzzy Convex Optimization.

This article investigates a class of constrained distributed fuzzy convex optimization problems, where the objective function is the sum of a set of local fuzzy convex objective functions, and the constraints include partial order relation and closed convex set constraints. In undirected connected node communication network, each node only knows its own objective function and constraints, and the local objective function and partial order relation functions may be nonsmooth. To solve this problem, a recurrent neural network approach based on differential inclusion framework is proposed. The network model is constructed with the help of the idea of penalty function, and the estimation of penalty parameters in advance is eliminated. Through theoretical analysis, it is proven that the state solution of the network enters the feasible region in finite time and does not escape again, and finally reaches consensus at an optimal solution of the distributed fuzzy optimization problem. Furthermore, the stability and global convergence of the network do not depend on the selection of the initial state. A numerical example and an intelligent ship output power optimization problem are given to illustrate the feasibility and effectiveness of the proposed approach.

Robust Picture Fuzzy Regression Functions Approach Based on M-Estimators for the Forecasting Problem

AbstractA picture fuzzy regression function approach is a fuzzy inference system method that uses as input the lagged variables of a time series and the positive, negative and neutral membership values obtained by picture fuzzy clustering method. In a picture fuzzy regression functions method, the parameter estimation is also obtained by ordinary least squares method. Since the picture fuzzy regression functions approach is based on the ordinary least squares method, the forecasting performance decreases when there are outliers in the time series. In this study, a picture fuzzy regression function approach that can be used even in the presence of outliers in a time series is proposed. In the proposed method, the parameter estimation for the picture fuzzy regression function approach is performed based on robust regression with Bisquare, Cauchy, Fair, Huber, Logistic, Talwar and Welsch functions. The forecasting performance of the proposed method is evaluated on the time series of the Spanish and the London stock exchange time series. The forecasting performance of these time series are evaluated separately for both the original and outlier cases. Besides, the proposed method is compared with several different fuzzy regression function approaches and a neural network method. Based on the results of the analysis, it is concluded that the proposed method outperforms the other methods even when the time series contains both original and outliers.

Characterization results of generalized differentiabilities of fuzzy functions

This paper presents new properties of the so-called generalized derivatives of fuzzy and interval-valued functions, which consist of the gH, gH⁎, and g-derivatives of these functions. Several recent properties concerning the generalized derivatives of interval-valued functions are extended to the fuzzy case, for which general characterization results are provided. Lastly, in order to illustrate the presented results, a study on the Malthusian growth and decay models considering fuzzy populations is made.

A membership function-dependent L∞ performance based output interval estimation method for discrete-time Takagi–Sugeno fuzzy systems applied to fault detection

In the design process of traditional L∞ methods, a common performance index is often considered in different fuzzy rules. This may introduce a certain conservatism when most of the whole fuzzy system operates under a certain fuzzy rule or a few fuzzy rules. Therefore, this paper focuses on robust interval estimation problems for discrete-time Takagi–Sugeno fuzzy systems, and a new membership function-dependent L∞ performance index is proposed. The proposed L∞ performance can ensure better results in practical engineering applications by using the priori knowledge that the model works most of the time on some specific fuzzy systems to design performance index. By means of fuzzy basis-dependent Lyapunov functions, a membership function-dependent L∞ performance based interval estimation strategy applied to fault detection is presented. Through simulation analysis, the L∞ index in this paper is better than the conventional one and a tighter interval estimation is obtained leading to more accurate and faster fault detection effects. Meanwhile, the simulation results of this method also reflect the membership function-dependent property well.

A Comprehensive Fuzzy Model for Understanding Neuronal Calcium Distribution in Presence of VGCC, Na+/Ca2+ Exchanger, Buffer, and ER Fluxes.

Free Calcium ions in the cytosol are essential for many physiological and physical functions. The free calcium ions are commonly regarded as a second messenger, are an essential part of brain communication. Numerous physiological activities, such as calcium buffering and calcium ion channel flow, etc. influence the cytosolic calcium concentration. In light of the above, the primary goal of this study is to develop a model of calcium distribution in neuron cells when a Voltage-Gated Calcium Channel and Sodium Calcium Exchanger are present. As we know, decreased buffer levels and increased calcium activity in the Voltage-Gated Calcium Channel and Sodium Calcium Exchanger lead to Alzheimer's disease. Due to these changes, the calcium diffusion in that location becomes disrupted and impacted by Alzheimer's disease. The model has been constructed by considering key factors like buffers and ER fluxes when Voltage-Gated Calcium Channels and Sodium Calcium Exchangers are present. Based on the physiological conditions of the parameters, appropriate boundary conditions have been constructed in the fuzzy environment. This model is considered a fuzzy boundary value problem with the source term and initial boundary conditions are modeled by triangular fuzzy functions. In this, paper we observed the approximate solution of the mathematical model which was investigated by the fuzzy undetermined coefficient method. The solution has been performed through MATLAB and numerical results have been computed using simulation. The observation made that the proper operation of the Voltage-Gated Calcium Channel and Sodium Calcium Exchanger is critical for maintaining the delicate equilibrium of calcium ions, which regulates vital cellular activities. Dysregulation of Voltage-Gated Calcium Channel and Sodium Calcium Exchanger activity has been linked to neurodegenerative illnesses like Alzheimer's disease.

Security concern and fuzzy output sliding mode load frequency control of power systems

This research investigates the concern of security in multi-area load frequency control (LFC) power systems using fuzzy logic output sliding mode control approach. A motion-trends-based deception attack model is proposed to represent malicious threats from communication networks. The motion-trends-based deception attack model enhances the destructive effect of deception attacks by monitoring changes in system state. In terms of the security concerns, a fuzzy-based output sliding mode control (OSMC) is applied to protect the power system from potential cyber-attacks. We provide sufficient conditions for the power systems stability under motion-trends-based deception attack. An initial controller gain that will bring the power system to stability is calculated by linear matrix inequalities (LMIs). The obtained controller gain provides a design method for fuzzy logic member functions. Fuzzy logic is then used to adjust the resulting controller gain to improve power system performance. The simulation results validate the usefulness of the proposed strategy, and indicate the viability of fuzzy logic in fine-tuning controllers.

Modeling brain information flow dynamics with multidimensional fuzzy inference systems

In this paper, we introduce a pioneering approach to analyzing brain information flow dynamics through the development of a novel hybrid system. The key innovation of our system lies in the integration of n-cell fuzzy functions into a multidimensional fuzzy inference system (MFIS). Fuzzy n-cell numbers, representing a significant advancement in fuzzy mathematics, are central to our methodology, showcasing their capability to handle intricate uncertainties and imprecise data within multidimensional scenarios. Our proposed system aligns seamlessly with neuroscience principles, providing a robust framework to unravel the complexities of brain information flow dynamics. We leverage Transfer Entropy and Granger Causality measures for a quantitative assessment of influence strengths, enabling a systematic exploration of temporal information flow within the brain. Furthermore, the utilization of fuzzy n-cell numbers enhances the system's adaptability to the intricacies of neural processes. The outcomes of our study are presented through a series of visualizations, offering a comprehensive demonstration of the effectiveness and adaptability of the proposed hybrid system. This article aims to contribute to the understanding of brain dynamics by presenting a versatile and complex methodology that combines fuzzy mathematics, neuroscience principles, and advanced analytical techniques.

Solution Strategy and Associated Results for Fuzzy Mellin Transformation

The advancement of the Mellin transformation is that it exhibits a scale-invariant nature, and thus, it is widely used in computer science. Fuzzy Mellin transformation can provide the purposes of such problems in uncertain phenomena. The existing studies on fuzzy Mellin transformation consider the fuzzy level cut approach, which makes the fuzzy valued functions into their crisp couplets regarding lower and uppercuts. This paper reconstructs a distinct mathematical frame for the Mellin transformation in a completely fuzzy environment in the sense of a modified Hukuhara derivative. In this paper, we mean the complete fuzzy environment to describe fuzzy phenomena where the conversion of the fuzzy function to its crisp counterpart is annulled. The proposed fuzzy Mellin transformation’s superiority over existing approaches is that it can be used directly to fuzzy valued functions without converting them to their crisp counter version. The challenges to dealing with fuzzy valued mappings and inputs without its level cut representation have been tackled in this manuscript. Furthermore, numerical examples, possible applications, and further extensions of this work are hinted at in this paper.

Interval-valued spherical fuzzy quality function deployment methodology: Metaverse collaborative system design application

The metaverse collaborative system (MCS) aims to assist users in working together, sharing resources, and participating in joint activities in the metaverse environment. As an interactive system, identifying essential design requirements (DRs) based on user requirements (URs) during the conceptual design phase of MCS is challenging as it directly affects the cost of subsequent optimization design and user experimentation, as well as user experience. Therefore, we propose a quality function deployment (QFD) method based on multi-criteria decision-making (MCDM) in an interval-valued spherical fuzzy (IVSF) environment to establish a mapping relationship between URs and DRs and identify human-centered DRs to guide the conceptual design process of MCS. Firstly, we summarized the URs for MCS design and calculated their weights using IVSF based analytical hierarchy process (IVSF-AHP). Afterward, we determined DRs based on URs and established the UR-DR relationship in the house of quality of QFD. Next, we integrate the VlseKriterijuska Optimizacija I Komoromisno Resenje (VIKOR) method in the IVSF environment into QFD to determine the priority of DRs. The results indicate that system functional design, scene layout design, compatibility design, and multi-user collaborative optimization design are important DRs for designing MCS and should be carefully considered. We validated the applicability of this method through an urban planning MCS design example and provided a conceptual design case including functional blueprints. Sensitivity analysis and comparative analysis demonstrate the flexibility and reliability of the proposed method.

On Atangana–Baleanu fractional granular calculus and its applications to fuzzy economic models in market equilibrium

Based on relative-distance-measure fuzzy interval arithmetic (RDM-FIA), this paper presents several new concepts for fuzzy functions in fractional granular calculus with Mittag-Leffler (ML) kernels, such as Atangana–Baleanu (AB) fractional granular integrals, AB fractional granular derivatives, and AB Caputo fractional granular derivatives. Then, we present formulas for the fuzzy granular Laplace transform on Caputo fractional granular derivatives, AB fractional granular derivatives, and AB Caputo fractional granular derivatives. In addition to the relationship between AB fractional granular derivatives and AB Caputo fractional granular derivatives established using the granular Laplace transform, new fractional Newton–Leibniz-type formulas for both derivatives and integrals are proved. As applications, we obtain the fundamental solutions for the fuzzy economic models in the framework of Caputo fractional granular derivatives and AB Caputo fractional granular derivatives by utilizing the fuzzy granular Laplace transform. Several graphical explanations are given to show the dynamic behavior of the fundamental solutions. Moreover, we illustrate the advantages of using RDM-FIA to study the mathematical models of uncertain dynamical systems. The results of this study are illustrated using several numerical examples.

Feature-weight and cluster-weight learning in fuzzy c-means method for semi-supervised clustering

Semi-supervised clustering aims to guide the clustering by utilizing auxiliary information about the class labels. Among the semi-supervised clustering categories, the constraint-based approach uses the available pairwise constraints in some steps of the clustering procedure, usually by adding new terms to the objective function. Considering this category, Semi-supervised FCM (SSFCM) is a semi-supervised version of the fuzzy c-means algorithm, which takes advantage of fuzzy logic and auxiliary class distribution knowledge. Despite the performance enhancement caused by incorporating this extra knowledge in the clustering process, semi-supervised fuzzy approaches still suffer from some problems. All the data attributes in the feature space are assumed to have equal importance in the cluster formation, while some features may be more informative than others. Thus the feature importance issue is not addressed in the semi-supervised category. This paper proposes a novel Semi-Supervised Fuzzy c-means approach, which is designed based on Feature-Weight, and Cluster-Weight learning, named SSFCM-FWCW. Inspired by the SSFCM, a fuzzy objective function is presented, which is composed of (1) a semi-supervised term representing the external class knowledge; (2) a feature weighting; and (3) a cluster weighting. Both feature weights and cluster weights are determined adaptively during the clustering. Considering these two techniques leads to insensitivity to the initial center selection, insensitivity to noise, and consequently helps to form an optimal clustering structure. Experimental comparisons are carried out on several benchmark datasets to evaluate the proposed approach's performance, and promising results are achieved. The Matlab implementation source code of the proposed method is publicly accessible at https://github.com/Amin-Golzari-Oskouei/SSFCM-FWCW.