Takagi-Sugeno-Kang Fuzzy Model Research Articles

A novel multi-task TSK fuzzy system modeling method based on multi-task fuzzy clustering

The traditional multi-task Takagi-Sugeno-Kang (TSK) fuzzy system modeling methods pay more attention to utilizing the inter-task correlation to learn the consequent parameters but ignore the importance of the antecedent parameters of the model. To this end, we propose a novel multi-task TSK fuzzy system modeling method based on multi-task fuzzy clustering. This method first proposes a novel multi-task fuzzy c-means clustering method that learns multiple specific clustering centers for each task and some common clustering centers for all tasks. Secondly, for the consequent parameters of the fuzzy system, the novel low-rank and row-sparse constraints are proposed to better implement multi-task learning. The experimental results demonstrate that the proposed model shows better performance compared with other existing methods.

Multi-View Fuzzy Classification With Subspace Clustering and Information Granules

Multi-view learning becomes increasingly attractive and promising because multimodal or multi-view data are commonly encountered in real-world applications. In this study, we develop a novel multi-view Takagi-Sugeno-Kang (TSK) fuzzy system framework to handle classification problems for such data. We propose an anchor and graph subspace clustering strategy to discover and represent the actual latent data distribution for each view separately. In this way, the discriminate anchors (landmarks) are learned to capture the main structure of the multi-view data. This strategy also provides a computationally efficient clustering algorithm with respect to the number of instances. These resulting anchors are formed as the prototypes of information granules (IGs) for fuzzy modeling. Then we construct an information-granule-based multi-view TSK fuzzy classification model inherited from the natural interpretability of fuzzy rule-based systems. Concretely, the relationship between the multi-view input and label output spaces is depicted by IGs-oriented fuzzy rules. The experimental studies involve various commonly used benchmark datasets, which indicate that our proposed method achieves comparable or better performance compared to the state-of-the-art algorithms.

Fuzzy Model Parameter and Structure Optimization Using Analytic, Numerical and Heuristic Approaches

Fuzzy systems are widely used in most fields of science and engineering, mainly because the models they produce are robust, accurate, easy to evaluate and capture real-world uncertainty better than do the classical alternatives. We propose a new methodology for structure and parameter tuning of Takagi–Sugeno–Kang fuzzy models using several optimization techniques. The output parameters are determined analytically, by finding the minimum of the root-mean-square error (RMSE) for a properly defined error function. The membership functions are simplified by considering symmetry and equispacing, to reduce the optimization problem of finding their parameters, and allow it to be carried out using the numerical method of gradient descent. Both algorithms are fast enough to finally implement a strategy based on the hill climbing approach to finding the optimal structure (number and type of membership functions) of the fuzzy system. The effectiveness of the proposed strategy is shown by comparing its performance, using four case studies found in current relevant works, to the popular adaptive network-based fuzzy inference system (ANFIS), and to other recently published strategies based on evolutionary fuzzy models. In terms of the RMSE, performance was at least 28% better in all case studies.

Ranking of Z-Numbers Based on the Developed Golden Rule Representative Value

Real-world decision-making is based on human cognitive information, which is characterized by fuzziness and partial reliability. In order to better describe such information, Zadeh proposed the concept of Z-number. Ranking the Z-number is an indispensable step in solving the decision-making problem under the Z-number-based information. Golden rule representative value is a new concept introduced by Yager to rank interval values. This article expands it and proposes a new golden rule representative value for fuzzy numbers, and then, apply it to the ranking of the Z-number. Some new rules involving the centroid and fuzziness of fuzzy numbers are constructed to capture the preference of decision-makers. The Takagi–Sugeno–Kang fuzzy model is used to implement these rules. The obtained Rep function is used to construct a new golden rule representative value fuzzy subset of the Z-number and associate this new fuzzy subset with a scalar value. This fuzzy subset not only implies the fuzzy aspect of the Z-number but also contains the information in the hidden probability distribution. The scalar value is regarded as the golden rule representative value of the Z-number to participate in the ranking. The proposed method greatly retains the original information of the Z-number and can overcome the shortcomings of the existing methods. Some numerical examples are used to describe the specific process of the proposed method. The comparative analysis and discussion with existing methods clarify the advantages of the proposed method.

IRI: An intelligent resistivity inversion framework based on fuzzy wavelet neural network

To acquire high-quality resistivity inversion results, an intelligent resistivity inversion (IRI) framework based on a fuzzy wavelet neural network (FWNN) is proposed in this paper. In the IRI framework, first, the FWNN is applied to build an interpretable inversion model for analyzing the apparent resistivity data. Takagi–Sugeno–Kang (TSK) fuzzy model is introduced to FWNN to explain the rules of the inversion results, and the wavelet neural network (WNN) is applied to construct the consequent part for each fuzzy rule and enhance the high-order nonlinear fitting ability of TSK fuzzy model. Then, to enhance the generalization and robustness, an elastic gradient descent (EGD) method is designed, which is used to update the linear parameters of the FWNN. Next, a differential and adaptive whale optimization algorithm (DAWOA) is introduced to train the nonlinear parameters of the FWNN for avoiding local optimum. Moreover, in the proposed DAWOA, a differential foraging strategy and an adaptive predation strategy are introduced to improve global exploration and local exploitation. All these measures can improve the inversion accuracy and accelerate the training process of FWNN for ERI inversion. Several simulation results demonstrate the feasibility and applicability of the IRI framework.

Structure Identification of Recursive TSK Particle Filtering via Type-2 Intuitionistic Fuzzy Decision

To reduce the dimension and realize the adaptive adjustment of the fuzzy rules in the Takagi–Sugeno–Kang (TSK) model, a recursive TSK particle filtering algorithm based on type-2 intuitionistic fuzzy decision is proposed. In the proposed algorithm, we mainly emphasize on the structure identification of the TSK fuzzy model. Firstly, the similarity of features is calculated by the ridge reduction distribution function. Secondly, the type-2 fuzzy membership degree is applied to redefined a new intuitionistic fuzzy set, which is used to calculate the decision score (type-2 intuitionistic fuzzy membership degree) of each attribute feature. And we select the appropriate features (premise variables) by the threshold decision principle. Thirdly, the importance density function of particle filtering is constructed with the estimation results of the proposed TSK fuzzy model, which can effectively reduce the particle degradation phenomenon. Finally, the results of experiments on datasets show that the proposed algorithm is efficient and is excellent at handling the nonlinear non-Gaussian problem in the maneuvering target tracking system.

Assessment of a Takagi-Sugeno-Kang fuzzy model assembly for examination of polyphasic loglinear allometry.

BackgroundThe traditional allometric analysis relies on log- transformation to contemplate linear regression in geometrical space then retransforming to get Huxley’s model of simple allometry. Views assert this induces bias endorsing multi-parameter complex allometry forms and nonlinear regression in arithmetical scales. Defenders of traditional approach deem it necessary since generally organismal growth is essentially multiplicative. Then keeping allometry as originally envisioned by Huxley requires a paradigm of polyphasic loglinear allometry. A Takagi-Sugeno-Kang fuzzy model assembles a mixture of weighted sub models. This allows direct identification of break points for transition between phases. Then, this paradigm is seamlessly appropriate for efficient allometric examination of polyphasic loglinear allometry patterns. Here, we explore its suitability.MethodsPresent fuzzy model embraces firing strength weights from Gaussian membership functions and linear consequents. Weights are identified by subtractive clustering and consequents through recursive least squares or maximum likelihood. Intersection of firing strength factors set criterion to estimate breakpoints. A multi-parameter complex allometry model follows by adapting firing strengths by composite membership functions and linear consequents in arithmetical space.ResultsTakagi-Sugeno-Kang surrogates adapted complexity depending on analyzed data set. Retransformation results conveyed reproducibility strength of similar proxies identified in arithmetical space. Breakpoints were straightforwardly identified. Retransformed form implies complex allometry as a generalization of Huxley’s power model involving covariate depending parameters. Huxley reported a breakpoint in the log–log plot of chela mass vs. body mass of fiddler crabs (Uca pugnax), attributed to a sudden change in relative growth of the chela approximately when crabs reach sexual maturity. G.C. Packard implied this breakpoint as putative. However, according to present fuzzy methods existence of a break point in Huxley’s data could be validated.ConclusionsOffered scheme bears reliable analysis of zero intercept allometries based on geometrical space protocols. Endorsed affine structure accommodates either polyphasic or simple allometry if whatever turns required. Interpretation of break points characterizing heterogeneity is intuitive. Analysis can be achieved in an interactive way. This could not have been obtained by relying on customary approaches. Besides, identification of break points in arithmetical scale is straightforward. Present Takagi-Sugeno-Kang arrangement offers a way to overcome the controversy between a school considering a log-transformation necessary and their critics claiming that consistent results can be only obtained through complex allometry models fitted by direct nonlinear regression in the original scales.

Multitask TSK Fuzzy System Modeling by Jointly Reducing Rules and Consequent Parameters

Existing multitask Takagi-Sugeno-Kang (TSK) fuzzy modeling methods always produce high complex fuzzy models with numerous redundant rules and consequent parameters. To this end, we propose a novel multitask TSK fuzzy modeling method called mtSparseTSK, which learns a compact set of fuzzy rules and shared consequent parameters across tasks in a unified procedure. Specifically, we consider the fuzzy rule reduction and consequent parameter selection across tasks by devising novel group sparsity regularizations in the learning criterion of the model. We also integrate the intertask relations in the proposed TSK model for multitask learning. We fully utilize the block structure in the TSK fuzzy models in formulating a joint block sparse optimization problem and develop a procedure for alternating direction method of multipliers (ADMMs) to find the optimal solution of the problem. Experiments on the synthetic and real-world datasets demonstrate the distinctive performance of the proposed methods over the existing ones on multitask fuzzy system modeling.

Indirect adaptive fuzzy control of nonlinear descriptor systems

This paper focuses on indirect adaptive fuzzy control of nonlinear descriptor systems described by both uncertain algebraic and differential equations aiming to guarantee asymptotic tracking of a regular and impulse-free descriptor reference model. The proposed controller exploits the universal approximation capability of Takagi–Sugeno–Kang (TSK) fuzzy models for the identification of the unknown system dynamics. More specifically, it is assumed that only the system order is known while all the dynamical equations of the system are completely unknown. In the proposed method, the asymptotic tracking of the reference model is guaranteed by suitable adaptation laws for the parameters of the TSK fuzzy model. Simulation results are presented to demonstrate the effectiveness of the proposed method.

Aggregation framework for TSK fuzzy and association rules: interpretability improvement on a traffic accidents case

The number and diversity of machine learning applications causes an increasing need for understanding computational models and used data. This paper deals with a framework design of easily interpretable rules of the Takagi-Sugeno-Kang (TSK) fuzzy model. The proposed framework aggregates TSK fuzzy rules and association rules by calculating overlapping value intervals of variables appearing in both antecedent and consequent parts of fuzzy and association rules. Besides a simple insight into rule interconnections of the rule-based models, the framework provides an assessment of fuzzy rule importance, and in accordance with other rules and the complete TSK fuzzy model. The proposed framework is developed and illustrated by analysing traffic accidents with pedestrian involvement. It provides a deeper understanding of the built rule-based model, as well as more readable identification of significant accident causes. The framework can be used in many domains of analysis modelling and decision making processes where computational model understanding is crucial.

Wavelet-TSK-Type Fuzzy Cerebellar Model Neural Network for Uncertain Nonlinear Systems

In this paper, a novel fuzzy neural network structure for uncertain nonlinear systems is proposed. This network is called wavelet Takagi–Sugeno–Kang (TSK) fuzzy cerebellar model neural network, which includes the framework of a cerebellar model neural network (CMNN) and the wavelet-function-based TSK fuzzy inference model. In order to effectively solve the uncertainty problem of nonlinear systems, a new structure is proposed where the wavelet function is used in the consequent parts of TSK-type fuzzy CMNN instead of the linear combination of the input variables in the traditional TSK fuzzy systems. This structure combines the advantages of the wavelet function, the CMNN and the TSK fuzzy inference system; thus, it is a more effective model for the uncertain nonlinear systems. In order to provide fast training, parameter update laws of the proposed model are derived based on the gradient descent method in which the learning-rates are online adapted. Furthermore the Lyapunov function is used to analyze the convergence of the considered systems. Finally, four different types of applications are applied to demonstrate the effectiveness of the proposed model. The simulation comparisons with other neural network models have verified the effectiveness of the new model.

Bayesian Takagi–Sugeno–Kang Fuzzy Model and Its Joint Learning of Structure Identification and Parameter Estimation

In this paper, a novel B ayesian T akagi– S ugeno– K ang (BTSK) fuzzy model and its j oint l earning method BTSK-JL of structure identification and parameter estimation are proposed for regression tasks from the perspective of Bayesian inference framework with a prior assumption about the number of fuzzy rules. Unlike most of existing TSK fuzzy systems where both their structure identification and parameter estimation of each fuzzy rule involved in them are learnt in a separate manner, BTSK-JL can determine the number of fuzzy rules and antecedent/consequent parameters of rules simultaneously in the proposed model. In order to guarantee their optimal solutions, BTSK-JL adopts a particle filter method to find the maximum-a-posterior value of the parameters. Due to taking into account the subtle interaction between input and output spaces, BTSK-JL can obtain good predictive performance and a set of compact fuzzy rules. Fuzziness and probability can work complementarily rather than competitively for such a TSK fuzzy system modeling. Experimental results on four time-series datasets and a glutamic acid fermentation process dataset have shown the validity and effectiveness of the proposed model.

Boundedness and convergence of split complex gradient descent algorithm with momentum and regularizer for TSK fuzzy models

Abstract This paper investigates the split complex gradient descent based neuro-fuzzy algorithm with self-adaptive momentum and L2 regularizer for training TSK (Takagi–Sugeno–Kang) fuzzy inference models. The major threat for disposing complex data with fuzzy system is contradiction of boundedness and analyticity in the complex domain, as expressed by Liouville’s theorem. The proposed algorithm operates a couple of real-valued functions and splits the complex variables into real part and imaginary part. Dynamical momentum is included in the learning mechanism to promote learning speed. L2 regularizer is also added to control the magnitude of the weight parameters. Furthermore, a detailed convergence analysis of the proposed algorithm is fully studied. The monotonic decreasing property of the error function and convergence of the weight sequence are guaranteed. Plus a mild condition, strong convergence of the weight sequence is deduced. Finally, the simulation results are also demonstrated to verify the theoretical analysis results.

Sugeno-Fuzzy Expert System Modeling for Quality Prediction of Non-Contact Machining Process

Modeling can be categorised into four main domains: prediction, optimisation, estimation and calibration. In this paper, the Takagi-Sugeno-Kang (TSK) fuzzy logic method is examined as a prediction modelling method to investigate the taper quality of laser lathing, which seeks to replace traditional lathe machines with 3D laser lathing in order to achieve the desired cylindrical shape of stock materials. Three design parameters were selected: feed rate, cutting speed and depth of cut. A total of twenty-four experiments were conducted with eight sequential runs and replicated three times. The results were found to be 99% of accuracy rate of the TSK fuzzy predictive model, which suggests that the model is a suitable and practical method for non-linear laser lathing process.

Robust Q-parametrisation control for nonlinear magnetic bearing systems with imbalance based on TSK fuzzy model

This paper presents a methodology for designing a robust gain scheduled Takagi-Sugeno-Kang (TSK) fuzzy Q-parametrisation controller for nonlinear magnetic bearing systems subjected to imbalance sinusoidal disturbance. First, the mathematical model of nonlinear magnetic bearing is presented. Second, a set of Q-parametrisation observer based stabilising controllers is obtained based on linearisation of the nonlinear system at different operating points. Third, the structure that combines the Q-parametrisation observer based controller (OBC) with TSK fuzzy modelling to overcome the model nonlinearity and expand the operating envelopes is explained. Fourth, the proposed controller is applied to a nonlinear magnetic bearing system. Finally, the simulation results are presented. The results clearly show that the proposed controller is able to merge the intelligence of fuzzy systems with robustness of Q-parametrisation control to extend operating range up to more than 80% of gap length and reject imbalance sinusoidal disturbances at different operating speeds.

New Formulation for Representing Higher Order TSK Fuzzy Systems

This paper introduces a recursive relation between an existing Takagi–Sugeno–Kang (TSK) fuzzy system of order $n$ and systems of order $n-1$ and $0$ . The presented method helps to realize higher order TSK fuzzy systems using first- and zero-order TSK fuzzy system models, which are available in simulation software (e.g., MATLAB and its toolboxes). This method has two major benefits: 1) easier and faster simulation of higher order TSK systems using available software; and 2) possibility of tuning high-order TSK systems by methods like ANFIS, which can only tune first- and zero-order TSK systems. The final section of this paper includes a comparison of simulation speed if the proposed method is used or not and nonlinear function estimation by applying ANFIS on high-order TSK system using the proposed method. Simulation results prove the superiority and usefulness of the proposed method.

Stability Analysis of Electricity Markets Using TSK Fuzzy Modeling

This paper discusses the stability properties of electricity markets and their implications for the deregulated market environment. We formulate a Takagi–Sugeno–Kang (TSK) fuzzy model for electricity markets to account for their uncertainty and nonlinearity and obtain realistic stability results. The paper provides sufficient conditions for the stability and instability of electricity markets. All the proposed stability and instability conditions can be tested using linear matrix inequalities. Simulation results demonstrate the application of the stability and instability tests to electric power market under different market conditions. The tests also demonstrate the importance of the including nonlinearities and uncertainty in the electricity market when investigating its stability.

Data-based multiple-model prediction of the production rate for hematite ore beneficiation process

The prediction of the production rate of the hematite ore beneficiation process is important to plant-wide optimization. This paper presents a data-based multi-model approach to predict the production rate with multiple operating modes. The inputs of the predictive model are the performance indices of each unit process, and the output is the global production index (the production rate) of the hematite ore beneficiation process. The multiple models are developed by integrating the fuzzy clustering algorithm and machine learning algorithm. A global model, Takagi–Sugeno–Kang fuzzy model, and multiple neural network model were compared using the data obtained from a practical industrial process, and the effectiveness of the proposed algorithm was proven.

Experimental forward and inverse modelling of magnetorheological dampers using an optimal Takagi–Sugeno–Kang fuzzy scheme

An evolving encoding scheme is presented in this article for a fuzzy-based nonlinear system identification scheme, using the subtractive fuzzy C-mean clustering and a modified version of non-dominated sorting genetic algorithm. This method is able to automatically select the best inputs as well as the structure of the fuzzy model such as rules and membership functions. Moreover, three objective functions are considered to satisfy both accuracy and compactness of the model. The developed method is then employed to identify both forward and inverse models of a highly nonlinear structural control device, that is, magnetorheological damper. Experimental results showed that the proposed evolving Takagi–Sugeno–Kang fuzzy model can identify and grasp the nonlinear behaviour of magnetorheological damper very well with minimal number of inputs and fuzzy rules.

Nature-inspired optimal tuning of input membership functions of Takagi-Sugeno-Kang fuzzy models for Anti-lock Braking Systems

This paper suggests a synergy of fuzzy logic and nature-inspired optimization in terms of the nature-inspired optimal tuning of the input membership functions of a class of Takagi-Sugeno-Kang (TSK) fuzzy models dedicated to Anti-lock Braking Systems (ABSs). A set of TSK fuzzy models is proposed by a novel fuzzy modeling approach for ABSs. The fuzzy modeling approach starts with the derivation of a set of local state-space models of the nonlinear ABS process by the linearization of the first-principle process model at ten operating points. The TSK fuzzy model structure and the initial TSK fuzzy models are obtained by the modal equivalence principle in terms of placing the local state-space models in the rule consequents of the TSK fuzzy models. An operating point selection algorithm to guide modeling is proposed, formulated on the basis of ranking the operating points according to their importance factors, and inserted in the third step of the fuzzy modeling approach. The optimization problems are defined such that to minimize the objective functions expressed as the average of squared modeling errors over the time horizon, and the variables of these functions are a part of the parameters of the input membership functions. Two representative nature-inspired algorithms, namely a Simulated Annealing (SA) algorithm and a Particle Swarm Optimization (PSO) algorithm, are implemented to solve the optimization problems and to obtain optimal TSK fuzzy models. The validation and the comparison of SA and PSO and of the new TSK fuzzy models are carried out for an ABS laboratory equipment. The real-time experimental results highlight that the optimized TSK fuzzy models are simple and consistent with both training data and validation data and that these models outperform the initial TSK fuzzy models.