General Fuzzy Systems Research Articles

Scikit-ANFIS: A Scikit-Learn Compatible Python Implementation for Adaptive Neuro-Fuzzy Inference System

The Adaptative neuro-fuzzy inference system (ANFIS) has shown great potential in processing practical data from control, prediction, and inference applications, reflecting advantages in both high performance and system interpretability as a result of the hybridization of neural networks and fuzzy systems. Matlab has been a prevalent platform that allows to utilize and deploy ANFIS conveniently. On the other hand, due to the recent popularity of machine learning and deep learning, which are predominantly Python-based, implementations of ANFIS in Python have attracted recent attention. Although there are a few Python-based ANFIS implementations, none of them are directly compatible with scikit-learn, one of the most frequently used libraries in machine learning. As such, this paper proposes Scikit-ANFIS, a novel scikit-learn compatible Python implementation for ANFIS by adopting a uniform format such as fit() and predict() functions to provide the same interface as scikit-learn. Our Scikit-ANFIS is designed in a user-friendly way to not only manually generate a general fuzzy system and train it with the ANFIS method but also to automatically create an ANFIS fuzzy system. We also provide four kinds of representative cases to show that Scikit-ANFIS represents a valuable addition to the scikit-learn compatible Python software that supports ANFIS fuzzy reasoning. Experimental results on four datasets show that our Scikit-ANFIS outperforms recent Python-based implementations while achieving parallel performance to ANFIS in Matlab, a standard implementation officially realized by Matlab, which indicates the performance advantages and application convenience of our software.

Literature Review on Development of Feature Selection and Learning Mechanism for Fuzzy Rule-Based System

Abstract: This research is being conducted to study fuzzy system with improved rule base. Rule base is an important part of any fuzzy inference system designed. Rules of a fuzzy system depend on the number of features selected. Selecting an optimized number of features is called feature selection. All features (parameters) play an important role in the input to the system, but they have a different impact on the system performance. Some features do not even have a positive impact of classifier on multiple classes. Reduced features, depending on the objective to be achieved require fewer training rules, Thereby, improving the accuracy of the system. Learning is an important mechanism to automate fuzzy systems. The overall purpose of the research is to design a general fuzzy expert system with improvements in the relationship between interpretability and accuracy by improving the feature selection and learning mechanism processes through nature-inspired techniques or innovating new methodologies for the same.

NARX Deep Convolutional Fuzzy System for Modelling Nonlinear Dynamic Processes

This paper presents a new approach for modelling nonlinear dynamic processes (NDP). It is based on a nonlinear autoregressive with exogenous (NARX) inputs model structure and a deep convolutional fuzzy system (DCFS). The DCFS is a hierarchical fuzzy structure, which can overcome the deficiency of general fuzzy systems when facing high dimensional data. For relieving the curse of dimensionality, as well as improving approximation performance of fuzzy models, we propose combining the NARX with the DCFS to provide a good approximation of the complex nonlinear dynamic behavior and a fast-training algorithm with ensured convergence. There are three NARX DCFS structures proposed, and the appropriate training algorithm is adapted. Evaluations were performed on a popular benchmark—Box and Jenkin’s gas furnace data set and the four nonlinear dynamic test systems. The experiments show that the proposed NARX DCFS method can be successfully used to identify nonlinear dynamic systems based on external dynamics structures and nonlinear static approximators.

Carbon emission sensitive deteriorating inventory model with trade credit under volumetric fuzzy system

This study explores carbon emission issues over production manufacturing system with the context of joint inventory control and sustainable trade credit financing for deteriorating items in a supplier–retailer–customer model under volumetric fuzzy system. Here, the supplier offers an upstream full trade credit to the retailer and the retailer in turn, provides a downstream partial trade credit to her/his customers. Demand rate is assumed to be functionally depends upon carbon emissions, unit selling price and the length of the credit period offered by the retailer to their customers explicitly. Taking a case study, we develop a profit maximization crisp problem first then considering the unit selling price and the carbon elasticity as fuzzy numbers we formulate an equivalent fuzzy problem. After that, the fuzzy problem has been defuzzified with the help of general fuzzy system and the new volumetric fuzzy system under two different scenarios namely with and without carbon emissions also. The aim of this study is to develop a best methodology over fuzzy system on trade credit financing problem so as to optimize the average profit function over the optimum design variables the credit period, the order quantity, the cycle time, the partial shortage time and minimum carbon emissions, respectively. Finally, numerical illustrations and its comparative study with the existing methods, sensitivity analysis and graphical illustrations are performed to justify the novelty of the proposed approach.

Solving an EOQ model under fuzzy reasoning

In general fuzzy system, the nature of a fuzzy number is usually viewed as specific and deterministic but, in fuzzy reasoning the membership function is developed in a randomized​ sense. Here, we have studied the concept of fuzzy approximate reasoning over the modelling of a cost minimization classical economic order quantity (EOQ) inventory management problem. We have developed the model where all parameters assume randomized fuzzy set by means of fuzzy approximate reasoning. First of all, considering the probability density function of the fuzzy variable, utilizing possibility measures of fuzzy sets within we have formulated the expectations of the fuzzy membership functions then we split the model into seven several sub models accordingly. To defuzzify the fuzzy functions the traditional α-cuts and its dual k-cuts have been utilized over several feasible spaces of dual spaces of fuzzy variables. Numerical illustrations under different α-dual k cuts of the objective functions are done with the help of LINGO software via solution algorithm. Moreover, a comparative study has been done with the existing general fuzzy solution. Managerial insights are also highlighted by showing the superiority of the proposed approach. Finally, sensitivity analysis and graphical illustrations are made to justify the model.

An Adaptive Fuzzy Feedforward-Feedback Control System Applied to a Saccharification Process

Abstract In industrial bioprocess control, disturbance sources typically influences process variable regulation. These disturbances may reduce a system control performance or even affect the final bioproduct quality. Therefore, feedforward control is desired because it anticipates the effects caused by these disturbances in an attempt to keep the process variable at the setpoint value. However, designing a feedforward control law requires process modeling, which can be a tough task when dealing with bioprocesses that are intrinsically nonlinear and multivariable systems. Thus, an adaptive feedforward control law or other advanced control system is needed for satisfactory disturbance rejection. For this reason, a general fuzzy feedforward control system is proposed in this paper to replace the classical feedforward control, making it easier to implement the feedforward control action by avoiding nonlinear and multivariable process modeling. The adaptive fuzzy feedforward-feedback (A4FB) system was applied to a product concentration control loop in an enzymatic reactor, to reject disturbances caused by variations in the substrate and enzymatic solutions feed concentration. The results showed that the A4FB controller rejected much more disturbance effects than classical feedforward control law, demonstrating its advantage, supported by not only its simple implementation, but also its improved disturbance rejection.

On numerical solution of general fuzzy type-1 and type-2 arbitrary order dynamical systems

On numerical solution of general fuzzy type-1 and type-2 arbitrary order dynamical systems

A discussion on “solving fuzzy complex system of linear equations”

Recently, Behera and Chakraverty [Solving fuzzy complex system of linear equations, Information Sciences 277 (2014) 154–162] proposed a new method for solving general fuzzy complex linear systems. However, in this note, we show that there are some problems in their method. Also, we present a modified version of their method which can avoid these problems for solving a fuzzy complex linear system.

Global exponential stability of general A-monotone implicit fuzzy proximal dynamical systems in Banach spaces

The purpose of this paper is to introduce the notion of general A-monotone operators in Banach spaces. Under some suitable conditions and using generalized proximal mapping technique, neural network technique and Gronwall's inequality, a new class of general A-monotone implicit fuzzy proximal dynamical systems in Banach spaces are also proposed and analyzed through a recurrent neural network with a one-layer structure, and the existence of the solutions of the proximal dynamical systems is shown. Further, the global exponential stability of these dynamical systems is proved. The results presented in this paper improve and generalize the corresponding results in the literature.

Postmodern Fuzzy System Theory: A Deconstruction Approach Based on Kabbalah

Modern general system theory proposed a holistic integrative approach based on input-state-output dynamics as opposed to the traditional reductionist detail based approach. Information complexity and uncertainty required a fuzzy system theory, based on fuzzy sets and fuzzy logic. While successful in dealing with analysis, synthesis and control of technical engineering systems, general system theory and fuzzy system theory could not fully deal with humanistic and human-like intelligent systems which combine technical engineering components with human or human-like components characterized by their cognitive, emotional/motivational and behavioral/action levels of operation. Such humanistic systems are essential in artificial intelligence, cognitive and behavioral science applications, organization management and social systems, man-machine systems or human factor systems, behavioral knowledge based economics and finance applications. We are introducing here a “postmodern fuzzy system theory” for controlled state dynamics and output fuzzy systems and fuzzy rule based systems using our earlier postmodern fuzzy set theory and a Kabbalah possible worlds model of modal logic and semantics type. In order to create a postmodern fuzzy system theory, we “deconstruct” a fuzzy system in order to incorporate in it the cognitive, emotional and behavioral actions and expressions levels characteristic for humanistic systems. Kabbalah offers a structural, fractal and hierarchic model for integrating cognition, emotions and behavior. We obtain a canonic deconstruction for a fuzzy system into its cognitive, emotional and behavioral fuzzy subsystems.

Solving fuzzy complex system of linear equations

This paper proposes a new and simple method for solving general fuzzy complex system of linear equations. In the original system, the elements of unknown variable vector and right hand side vector are considered as complex fuzzy number. Initially the general system is solved by adding and subtracting the left and right bounds of the fuzzy complex unknown and right hand side fuzzy complex vector respectively. Subsequently above obtained solutions are used to get the final solution of the general fuzzy complex system of linear equations. Two theorems are stated and proved related to the proposed method. A mathematical example and an application problem based on electrical circuit are solved using the present method. Results obtained are compared with the known results and are found to be in good agreement.

Fuzzy system of linear equations with crisp coefficients

This paper proposed a new and simple solution approach to solve a general fuzzy system of linear equations. In the original system, the coefficient matrix is considered as real crisp whereas the unknown variable vector and right hand side vector are considered as fuzzy. Here, three different types of fuzzy numbers viz. triangular, trapezoidal and Gaussian fuzzy numbers are taken into consideration. First the system is solved in term of fuzzy centre then this solution is used with width to get the final solution of the original system. Solution theorems are stated and proved related to the proposed method. The method is also used to solve some known example problems found in the references and the results are compared to show the efficacy of the proposed method.

Linear System Of Equations With Trapezoidal Fuzzy Numbers

A general fuzzy linear system of equations is investigated using embedding approach. In the literature of fuzzy systems two types of linear systems are more important: 1) Fuzzy Linear Systems, 2) Fully Fuzzy Linear Systems. In both class of these systems usually the authors considered triangular type of fuzzy numbers. In this paper, we introduce a linear system of equations with trapezoidal fuzzy numbers as an extension of the fuzzy linear systems which was first introduced by Friedman et al. [6]. Conditions for the existence of a trapezoidal fuzzy solution to n×n linear system are derived and a numerical procedure for calculating the solution is illustrated with some examples.

Bounded and symmetric solutions of fully fuzzy linear systems in dual form

Abstract Linear systems have important applications in many branches of science and engineering in many applications, at least some of the parameters of the system are represented by fuzzy rather than crisp numbers. So, it is immensely important to develop a numerical procedure that would appropriately treat general fuzzy linear systems and solve them. In this paper, we propose bounded and symmetric solutions of fully fuzzy linear systems in the dual form (DFFLS) based on a 1-cut expansion. To this end, we solve the 1-cut of a DFFLS (we assumed that the 1-cut of a DFFLS is a crisp linear system), then some unknown symmetric spreads are allocated to each row of a 1-cut of a DFFLS. So, after some manipulations, the original DFFLS is transformed to solving 2∗ n linear equations to find the symmetric spreads. However, our method always give us a fuzzy vector solution. Moreover, we show that the bounded and symmetric solution of the DFFLS will be placed in the tolerable solution set (TSS) and in the controllable solution set (CSS), respectively. Also, an economic example is solved to illustrate the ability of proposed method and a new pattern is suggested for comparing the obtained solutions.

H.INF. Output Feedback Control for A Class of General Fuzzy Systems

This paper is concerned with H∞ output feedback control design for a general class of fuzzy systems. When the premise variable is the measurement output, the control design is easy but a fuzzy system representation is limited. On the other hand, when it is the state variable, a fuzzy system describes a general class of nonlinear systems but the control design becomes difficult. This is because a conventional parallel distributed compensator (PDC) is not feasible any more. In this paper, we introduce an output feedback controller whose premise variable is independent of the premise variable of an original fuzzy system. Then, we formulate an H∞ control problem for a general class of fuzzy system where the premise variable is not always available. Our control design method is based on a set of strict LMI conditions. No scaling parameter is necessary a priori to solve LMI conditions for designing an H∞ controller. Our method includes matrices that can tune control gain matrices in an H∞ controller and hence it can reflect the control performance of the resulting closed-loop system. A numerical example is given to illustrate our H∞ output feedback control design.

Bisimulations for Fuzzy-Transition Systems

There has been a long history of using fuzzy-language equivalence to compare the behavior of fuzzy systems; however, the comparison at this level is too coarse. Recently, a finer behavioral measure, i.e., bisimulation, has been introduced to fuzzy-finite automata. However, the results obtained are applicable only to finite-state systems. In this paper, we consider bisimulation for general fuzzy systems, which may be infinite state or infinite event, by modeling them as fuzzy-transition systems (FTSs). To help understand and check bisimulation, we characterize it in three ways by enumerating whole transitions, comparing individual transitions, and using a monotonic function. In addition, we address composition operations, subsystems, quotients, and homomorphisms of FTSs and discuss their properties connected with bisimulation. The results presented here are useful to compare the behavior of general fuzzy systems. In particular, this makes it possible to relate an infinite fuzzy system to a finite one, which is easier to analyze, with the same behavior.

Solving fuzzy linear systems

In this paper, the main aim is to develop a method for solving an arbitrary general fuzzy linear system by using the embedding approach. Considering the existing and uniqueness of fuzzy solution to n × n linear fuzzy system is done. Numerical examples are presented to illustrate the proposed model.

Generalized Receding Horizon Control of Fuzzy Systems Based on Numerical Optimization Algorithm

The optimal control of fuzzy systems with constraints is still an open problem. Our focus concerns the optimal control problem of fuzzy systems derived from receding horizon control (RHC) schemes. We consider methods to numerically compute the value function for general fuzzy systems. The numerical method that is developed using the finite difference with sigmoidal transformation is a stable and convergent algorithm for the Hamilton-Jacobi-Bellman (HJB) equation. An optimization procedure is developed to increase the calculation accuracy with less computation time. A parallel-processing method is employed in the optimization procedure. The optimization results are applied to the controller design of general fuzzy dynamic systems. Employing the principle of conventional RHC schemes, RHC-form controllers are designed for some classes of fuzzy dynamic systems. The basic ideas are as follows. First, the value function is calculated by numerical methods. Then, the value function is used as controller-design parameters to redesign RHC controllers for fuzzy systems, which is motivated by the inverse Lyapunov function design method. It is proven that the closed-loop system is asymptotically stable. An engineering implementation of the controller redesign scheme is discussed. Meanwhile, the parallel-processing framework that can improve the closed-loop performance is also introduced.

Linear Formed General Fuzzy Linear Systems

Abstract According to the structured element method, the article investigates a method of solving the general fuzzy linear systems and dual general fuzzy linear systems, which are linear formed by fuzzy structured element, and gives the necessary and sufficient condition for the existence of a fuzzy solution. At the same time, some examples in engineering and economics are designed.

Solution to General Fuzzy Linear System and Its Necessary and Sufficient Condition

This paper investigates general fuzzy linear systems of the form Ax = y and general dual fuzzy linear systems of the form Ax + y = Bx + z with A, B matrices of crisp coefficients and x, y fuzzy number vectors. The aim of this paper is twofold. First, by the unique least Euclidean norm solution we solve the systems with non-full rank matrices A, B. Second, we give the new necessary and sufficient condition for a strong fuzzy solution existence. Moreover, some numerical examples are designed.