Antecedent Membership Functions Research Articles

A composite framework coupling FCRM, LSSVM and improved hybrid IHHOMFO optimization for Takagi–Sugeno fuzzy model identification

This paper proposes a novel Takagi–Sugeno fuzzy model identification method by combining fuzzy c-regression model clustering (FCRM), least squares support vector machine (LSSVM) and intelligent optimization algorithm. Firstly, in order to improve the performance of FCRM for the complex nonlinear dataset, in this paper the method of FCRM based on LSSVM (FCRM-LSSVM) is proposed to discover the data structure and obtain the antecedent parameters. And then, a newly developed intelligent optimization algorithm by hybridizing Harris hawks optimization and moth-flame optimization algorithm (IHHOMFO) is proposed to further optimize the antecedent membership function parameters obtained by the FCRM-LSSVM. Finally, the proposed novel T-S fuzzy model identification combines FCRM, LSSVM and IHHOMFO for solving actual model identification problems. Experiments on five different datasets demonstrate that the proposed method is more efficient than conventional methods, such as T-S model identification based on fuzzy c-means (FCM), FCRM and FCRM-LSSVM, in standard measurement indexes. This study thus demonstrates that the proposed method is a credible and competitive fuzzy model identification method. The novel method contributes not only to the theoretical aspects of fuzzy model, but is also widely applicable in data mining, image recognition and prediction problems.

Open Circuit Fault Diagnosis in Five-Level Cascaded H-Bridge Inverter

The development of power electronic converter, especially multilevel converter, is remarkable for several decades. The complex switching and increased power of semiconductor devices are prime reasons for faults in multilevel inverters and have raised question about reliability. To improve the reliability, a cost-effective solution in terms of fault diagnosis is essential. In this context, this study proposed an open circuit fault (OCF) diagnosis technique for a switching device in a five-level cascaded H-bridge multilevel inverter using fuzzy logic control. The OCF features like output voltage total harmonic distortion (THD) and normalized average output voltage are fuzzed as input variables of the fuzzy logic controller. These input variables are divided into various triangular antecedent membership function (MF). The output produced by the fuzzy controller as consequent MFs is divided into different levels to identify the faulty switch. In order to make a complete fault-tolerant structure, a reduced modulation index-based postfault control is suggested to get a balanced output voltage. The MATLAB/Simulink results and prototype results are the evidence to support the proposed fault diagnosis technique.

A new imputation-based incomplete data-driven fuzzy modeling for accuracy improvement in ubiquitous computing applications

Purpose In real-world decision-making, high accuracy data analysis is essential in a ubiquitous environment. However, we encounter missing data while collecting user-related data information because of various privacy concerns on account of a user. This paper aims to deal with incomplete data for fuzzy model identification, a new method of parameter estimation of a Takagi–Sugeno model in the presence of missing features. Design/methodology/approach In this work, authors proposed a three-fold approach for fuzzy model identification in which imputation-based linear interpolation technique is used to estimate missing features of the data, and then fuzzy c-means clustering is used for determining optimal number of rules and for the determination of parameters of membership functions of the fuzzy model. Finally, the optimization of the all antecedent and consequent parameters along with the width of the antecedent (Gaussian) membership function is done by gradient descent algorithm based on the minimization of root mean square error. Findings The proposed method is tested on two well-known simulation examples as well as on a real data set, and the performance is compared with some traditional methods. The result analysis and statistical analysis show that the proposed model has achieved a considerable improvement in accuracy in the presence of varying degree of data incompleteness. Originality/value The proposed method works well for fuzzy model identification method, a new method of parameter estimation of a Takagi–Sugeno model in the presence of missing features with varying degree of missing data as compared to some well-known methods.

Critical Thinking About Explainable AI (XAI) for Rule-Based Fuzzy Systems

This article is about explainable artificial intelligence (XAI) for rule-based fuzzy systems [that can be expressed generically, as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$y({{\bf x}}) = f({{\bf x}})$</tex-math></inline-formula> ]. It explains why it is <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">not valid</i> to explain the output of Mamdani or Takagi–Sugeno–Kang rule-based fuzzy systems using IF-THEN rules, and why it <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">is valid</i> to explain the output of such rule-based fuzzy systems as an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">association</i> of the compound antecedents of a small subset of the original larger set of rules, using a phrase such as “these linguistic antecedents are <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">symptomatic</i> of this output.” Importantly, it provides a novel multi-step approach to obtain such a small subset of rules for three kinds of fuzzy systems, and illustrates it by means of a very comprehensive example. It also explains why the choice for antecedent membership function shapes may be more critical for XAI than before XAI, why linguistic approximation and similarity are essential for XAI, and, it provides a way to estimate the quality of the explanations.

Identification of Circulating Fluidized Bed Boiler Bed Temperature Based on Hyper-Plane-Shaped Fuzzy C-Regression Model

Bed temperature in dense-phase zone is the key parameter of circulating fluidized bed (CFB) boiler for stable combustion and economic operation. It is difficult to establish an accurate bed temperature model as the complexity of circulating fluidized bed combustion system. T-S fuzzy model was widely applied in the system identification for it can approximate complex nonlinear system with high accuracy. Fuzzy c-regression model (FCRM) clustering based on hyper-plane-shaped distance has the advantages in describing T-S fuzzy model, and Gaussian function was adapted in antecedent membership function of T-S fuzzy model. However, Gaussian fuzzy membership function was more suitable for clustering algorithm using point to point distance, such as fuzzy c-means (FCM). In this paper, a hyper-plane-shaped FCRM clustering algorithm for T-S fuzzy model identification algorithm is proposed. The antecedent membership function of proposed identification algorithm is defined by a hyper-plane-shaped membership function and an improved fuzzy partition method is applied. To illustrate the efficiency of the proposed identification algorithm, the algorithm is applied in four nonlinear systems which shows higher identification accuracy and simplified identification process. At last, the algorithm is used in a circulating fluidized bed boiler bed temperature identification process, and gets better identification result.

An evolving T–S fuzzy model identification approach based on a special membership function and its application on pump-turbine governing system

Hyper-plane-shaped clustering (HPSC) has been proved to be more effective in Takagi–Sugeno (T–S) fuzzy model identification compared with hyper-sphere-shaped clustering (HSSC). However, there is no special membership function matching HPSC in fuzzy modeling, and the commonly used bell-shaped Gaussian function is more suitable for HSSC. In this paper, a novel T–S fuzzy model identification method is adopted, in which a new fuzzy membership function designed for HSPC is designed. In this approach, a fuzzy c-regression model based clustering method is used to partition the fuzzy space firstly; and then a new HPSC fuzzy membership function is designed to identify the antecedent membership function (MF) parameters; finally the gravitational search algorithm is applied to optimize the MF parameters further. Experimental results on several benchmark problems show that modeling accuracies have been promoted significantly. The proposed approach has been applied in fuzzy modeling of pump-turbine governing system (PTGS). The comparative experimental results reveal that the proposed approach could achieve high accuracy and would be an effective modeling tool for complicated nonlinear system in engineering applications.

Quaternion neuro-fuzzy learning algorithm for generation of fuzzy rules

The conventional neuro-fuzzy learning algorithm with Gaussian membership functions based on the gradient descent method can generate or tune fuzzy rules. However, increasing the number of inputs greatly increases the number of parameters. Thus, representing fuzzy rule tables is difficult, learning time increases, and learning accuracy decreases. To overcome these problems, we have developed a complex-valued neuro-fuzzy learning algorithm that extends the neuro-fuzzy learning algorithm to complex numbers. In the method, the inputs, antecedent membership functions, and consequent singletons are complex numbers, but the outputs are real. For converting complex to real numbers, we proposed two types of activation function. The complex-valued method reduced the parameter numbers and showed better learning accuracy. In this paper, we extend the method to the quaternion domain. In the quaternion neuro-fuzzy learning algorithm, the inputs, antecedent membership functions, and consequent singletons are quaternion, and the outputs are real. For parameter tuning, we derived the quaternion back propagation of quaternion neural networks that outputs real values for quaternion-valued inputs. The quaternion back propagation shows better learning convergence and accuracy than the conventional back propagation, and the tuning process is more complex, although it benefits from the quaternion back propagation. The method assigns a four-dimensional real number to one real and three imaginary parts of a quaternion number, which is used as a single quaternion input. This process greatly reduces the number of tuned parameters, leading to better learning than the conventional method. We compare the proposed and conventional methods using several function identification problems, and show that the proposed method outperforms its counterpart, making it a useful tool for learning in a fuzzy system model. In the best cases, the number of epochs was reduced to one-fortieth and the error to one-thirtieth of those in the conventional method.

A systematic design of interval type-2 fuzzy logic system using extreme learning machine for electricity load demand forecasting

This paper presents a novel design of interval type-2 fuzzy logic systems (IT2FLS) by utilizing the theory of extreme learning machine (ELM) for electricity load demand forecasting. ELM has become a popular learning algorithm for single hidden layer feed-forward neural networks (SLFN). From the functional equivalence between the SLFN and fuzzy inference system, a hybrid of fuzzy-ELM has gained attention of the researchers. This paper extends the concept of fuzzy-ELM to an IT2FLS based on ELM (IT2FELM). In the proposed design the antecedent membership function parameters of the IT2FLS are generated randomly, whereas the consequent part parameters are determined analytically by the Moore–Penrose pseudo inverse. The ELM strategy ensures fast learning of the IT2FLS as well as optimality of the parameters. Effectiveness of the proposed design of IT2FLS is demonstrated with the application of forecasting nonlinear and chaotic data sets. Nonlinear data of electricity load from the Australian National Electricity Market for the Victoria region and from the Ontario Electricity Market are considered here. The proposed model is also applied to forecast Mackey-glass chaotic time series data. Comparative analysis of the proposed model is conducted with some traditional models such as neural networks (NN) and adaptive neuro fuzzy inference system (ANFIS). In order to verify the structure of the proposed design of IT2FLS an alternate design of IT2FLS based on Kalman filter (KF) is also utilized for the comparison purposes.

Observer-based adaptive interval type-2 fuzzy control of uncertain MIMO nonlinear systems with unknown asymmetric saturation actuators

An observer-based controller is proposed for a class of uncertain MIMO nonlinear systems preceded by actuators with unknown asymmetric saturations. The gains of actuators are unknown nonlinear functions that are generally functions of states. The unknown nonlinearities are approximated by interval type-2 fuzzy systems where the consequent parameters are tuned adaptively. Meanwhile, the interval type-2 fuzzy systems due to their structure can cope with the uncertainties in antecedent membership functions. The states of the system do not need to be known. To tackle the unavailability of states an observer based on strictly positive real (SPR) theory is designed to estimate the states. To cope with the fuzzy approximation error and external disturbances an adaptive continuous robust structure is proposed. The continuous robust structure due to adaptiveness does not need the bound of uncertainties and due to continuity does not have the chattering problem. All the adaptive laws are derived via Lyapunov synthesis approach. Meanwhile, all the signals in the closed-loop system are guaranteed to be semi-globally uniformly ultimately bounded (SGUUB) and the mean-square tracking error can be made arbitrarily small. Simulation results reveal the effectiveness of the proposed method in terms of coping well with the uncertainties and saturation actuators while maintaining satisfactory performance.

Hydraulic turbine governing system identification using T–S fuzzy model optimized by chaotic gravitational search algorithm

Hydraulic turbine governing system (HTGS) is a complicated nonlinear system that controls the frequency and power output of hydroelectric generating unit (HGU). The modeling of HTGS is an important and difficult task, because some components, like hydraulic turbine and governor actuator, are with strong nonlinearity. In this paper, a novel Takagi–Sugeno (T–S) fuzzy model identification method based on chaotic gravitational search algorithm (CGSA) is proposed and applied in the modeling of HTGS. In the proposed method, fuzzy c-regression model clustering algorithm is used to partition the input space and identify the coarse antecedent membership function (MF) parameters at first. And then, a novel CGSA is proposed to search better MF parameters around the coarse results, in which chaotic search has been embedded in the iteration of basic GSA to search and replace the current best solution of GSA. The performance of the proposed fuzzy model identification method is validated by benchmark problems, and the results show that the accuracies of identified models have been improved significantly compared with the other existing models. Finally, the proposed approach has been applied to approximate the dynamic behaviors of HTGS of a HGU in a hydropower station of Jiangxi Province of China. The experimental results show that our approach can identify the HTGS satisfactorily with acceptable accuracy.

Estimation of deformation modulus of rock masses by using fuzzy clustering-based modeling

The paper describes a method which incorporates Takagi–Sugeno (TS) fuzzy modeling with two data clustering approaches including fuzzy- c-means (FCM) clustering and subtractive clustering to estimate the rock mass modulus of deformation. For this aim, a database including 120 cases collected from several galleries of dam sites locations was established. The information returned by fuzzy clustering was initially used to define the number of rules and antecedent membership functions and afterwards linear least squares estimation implemented to obtain fuzzy consequent parameters. An adaptive neuro-fuzzy inference system (ANFIS) was applied to modify the pre-determined TS clustering-based model structures to improve the generalization performance of those. For evaluation of the performance, root mean square error ( RMSE) and variance account for ( VAF) values have been utilized as performance criteria. It can be said, that ANFIS approach enhances the performances of fuzzy clustering-based models in predicting modulus of deformation of rock masses successfully.

Identification of a research nuclear reactor using computational intelligence techniques

In the field of nuclear engineering, the development of methods to identify the non-linear dynamic behaviour of nuclear reactors is an important area of research. In this paper, the design and performance of a multi-input multi-output Takagi?Sugeno (TS) neuro-fuzzy network are presented. The aim of this network is to identify the temporal behaviour of the power, the fuel temperature and the core reactivity of a TRIGA Mark III type nuclear research reactor. The tuning of the parameters corresponding to the antecedent membership functions is carried out by means of descent gradient algorithms with stable training, whereas the consequent parameters are identified using a Kalman estimator. Genetic algorithms are used to define the best input selection in the model. The results of the simulations show that the identification system converges rapidly and with high accuracy to both the training data and the test data. Thus, in the absence of real reactor data, the identified system can be used for tuning purposes of reactor power control schemes.

Design of Hybrid Genetic- SVD ANFIS Networks for Fatigue Life Modelling of GRP Composites

Genetic algorithm (GA) and singular value decomposition (SVD) are deployed for the optimal design of both Gaussian membership functions of antecedents and the vector of linear coefficients of consequents, respectively, of adaptive neuro-fuzzy inference systems (ANFIS) networks that are used for fatigue life modelling and prediction of unidirectional GRP Composites. The aim of such modelling is to show how the fatigue life varies with the variation of important parameters namely, maximum stress, stress ratio, fiber angle. It is demonstrated that SVD can be effectively used to optimally find the vector of linear coefficients of conclusion parts in ANFIS models and their Gaussian membership functions in premise parts are determined by GA.

Design of ANFIS networks using hybrid genetic and SVD methods for modeling and prediction of rubber engine mount stiffness

Genetic Algorithm (GA) and Singular Value Decomposition (SVD) are deployed for optimal design of both the Gaussian membership functions of antecedents and the vector of linear coefficients of consequents, respectively, of ANFIS networks. These networks are used for stiffness modelling and prediction of rubber engine mounts. The aim of such modelling is to show how the stiffness of an engine mount changes with variations in geometric parameters. It is demonstrated that SVD can be optimally used to find the vector of linear coefficients of conclusion parts using ANFIS (Adaptive Neuro-Fuzzy Inference Systems) models. In addition, the Gaussian membership functions in premise parts can be determined using a GA. In this study, the stiffness training data of 36 different bush type engine mounts were obtained using the finite element analysis (FEA).

C-Means 클러스터링 기반의 Type-2 퍼지 논리 시스템을 이용한 비선형 모델 설계

This paper deal with uncertainty problem by using Type-2 fuzzy logic set for nonlinear system modeling. We design Type-2 fuzzy logic system in which the antecedent and the consequent part of rules are given as Type-2 fuzzy set and also analyze the performance of the ensuing nonlinear model with uncertainty. Here, the apexes of the antecedent membership functions of rules are decided by C-means clustering algorithm and the apexes of the consequent membership functions of rules are learned by using back-propagation based on gradient decent method. Also, the parameters related to the fuzzy model are optimized by means of particle swarm optimization. The proposed model is demonstrated with the aid of two representative numerical examples, such as mathematical synthetic data set and Mackey-Glass time series data set and also we discuss the approximation as well as generalization abilities for the model.

Fuzzy model identification based on cluster estimation for reservoir inflow forecasting

AbstractFuzzy theory appears to be extremely effective at handling dynamic, non‐linear and noisy data, especially when the underlying physical relationships are not fully understood. Since hydrologists are still uncertain about many of the aspects of the physical processes in the watershed, fuzzy theory has proved to be a very attractive tool enabling them to investigate such problems. The effectiveness of the fuzzy model lies in the identification of the antecedent membership function (MF), which is generally addressed through a fuzzy clustering approach. Most of the applications of fuzzy computing in hydrology seem to have selected the clustering algorithm quite arbitrarily. However, it is apparent that, as the antecedent parameters are based solely on the identified clusters, the method used for clustering should certainly have an impact on the overall performance of the model. This paper presents the results of a study conducted to investigate the impact of choice of clustering algorithm on the overall performance of a fuzzy‐based hydrologic model. The research is illustrated through a case study of developing a Takagi–Sugeno fuzzy model for reservoir inflow forecasting in the Narmada basin, India. The model was developed using two popular clustering techniques, namely Gustafson–Kessel (GK) and subtractive clustering (SC), and was extensively evaluated for performance based on various statistical indices. The results show that the model performance is comparable at a 1 h lead forecast. However, it is observed that the GK approach results in a better performance than the SC approach in computing forecasts at higher lead times. The analysis suggest that the GK method clusters the input space based on the actual pattern, since it uses a membership‐grade weighted‐distance measure as the measure of closeness, whereas the SC method classifies the input space more logically according to the magnitude of flow available in the data set. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Application of evolving Takagi–Sugeno fuzzy model to nonlinear system identification

In this paper, a new encoding scheme is presented for learning the Takagi–Sugeno (T–S) fuzzy model from data by genetic algorithms (GAs). In the proposed encoding scheme, the rule structure (selection of rules and number of rules), the input structure (selection of inputs and number of inputs), and the antecedent membership function (MF) parameters of the T–S fuzzy model are all represented in one chromosome and evolved together such that the optimisation of rule structure, input structure, and MF parameters can be achieved simultaneously. The performance of the developed evolving T–S fuzzy model is first validated by studying the benchmark Box–Jenkins nonlinear system identification problem and nonlinear plant modelling problem, and comparing the obtained results with other existing results. Then, it is applied to approximate the forward and inverse dynamic behaviours of a magneto-rheological (MR) damper of which identification problem is significantly difficult due to its inherently hysteretic and highly nonlinear dynamics. It is shown by the validation applications that the developed evolving T–S fuzzy model can identify the nonlinear system satisfactorily with acceptable number of rules and appropriate inputs.

Evolving Compact and Interpretable Takagi–Sugeno Fuzzy Models With a New Encoding Scheme

Developing Takagi-Sugeno fuzzy models by evolutionary algorithms mainly requires three factors: an encoding scheme, an evaluation method, and appropriate evolutionary operations. At the same time, these three factors should be designed so that they can consider three important aspects of fuzzy modeling: modeling accuracy, compactness, and interpretability. This paper proposes a new evolutionary algorithm that fulfills such requirements and solves fuzzy modeling problems. Two major ideas proposed in this paper lie in a new encoding scheme and a new fitness function, respectively. The proposed encoding scheme consists of three chromosomes, one of which uses unique chained possibilistic representation of rule structure. The proposed encoding scheme can achieve simultaneous optimization of parameters of antecedent membership functions and rule structures with the new fitness function developed in this paper. The proposed fitness function consists of five functions that consider three evaluation criteria in fuzzy modeling problems. The proposed fitness function guides evolutionary search direction so that the proposed algorithm can find more accurate compact fuzzy models with interpretable antecedent membership functions. Several evolutionary operators that are appropriate for the proposed encoding scheme are carefully designed. Simulation results on three modeling problems show that the proposed encoding scheme and the proposed fitness functions are effective in finding accurate, compact, and interpretable Takagi-Sugeno fuzzy models. From the simulation results, it is shown that the proposed algorithm can successfully find fuzzy models that approximate the given unknown function accurately with a compact number of fuzzy rules and membership functions. At the same time, the fuzzy models use interpretable antecedent membership functions, which are helpful in understanding the underlying behavior of the obtained fuzzy models.

Constraint-Based Fuzzy Models for an Environment with Heterogeneous Information-Granules

A novel framework for fuzzy modeling and model-based control design is described. Based on the theory of fuzzy constraint processing, the fuzzy model can be viewed as a generalized Takagi-Sugeno (TS) fuzzy model with fuzzy functional consequences. It uses multivariate antecedent membership functions obtained by granular-prototype fuzzy clustering methods and consequent fuzzy equations obtained by fuzzy regression techniques. Constrained optimization is used to estimate the consequent parameters, where the constraints are based on control-relevant a priori knowledge about the modeled process. The fuzzy-constraint-based approach provides the following features. 1) The knowledge base of a constraint-based fuzzy model can incorporate information with various types of fuzzy predicates. Consequently, it is easy to provide a fusion of different types of knowledge. The knowledge can be from data-driven approaches and/or from control-relevant physical models. 2) A corresponding inference mechanism for the proposed model can deal with heterogeneous information granules. 3) Both numerical and linguistic inputs can be accepted for predicting new outputs.The proposed techniques are demonstrated by means of two examples: a nonlinear function-fitting problem and the well-known Box-Jenkins gas furnace process. The first example shows that the proposed model uses fewer fuzzy predicates achieving similar results with the traditional rule-based approach, while the second shows the performance can be significantly improved when the control-relevant constraints are considered.

A SATURATION BASED INTERPOLATION METHOD FOR FUZZY SYSTEMS

This paper presents an alternative inference-defuzzification algorithm for Takagi-Sugeno fuzzy systems that preserves local-model interpretation and convexity properties. The linear model in the rule consequent is saturated outside the core set of the antecedent membership functions. This allows the interpretation of the consequents of fuzzy rules as a local linearization of the model restricted to the subset where it is valid. The setting has readability advantages over Takagi-Sugeno frameworks, and it is simpler that other interpolation proposals. Some examples illustrate the approach.