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.     

Performance Enhancement for T–S Fuzzy Control Using Neural Networks

A new control scheme is proposed to improve the system performance for Takagi–Sugeno (T–S) fuzzy system using control grade functions tuned by neural networks. First, systematic modeling method is introduced to construct the exact T–S fuzzy model for a nonlinear control system. For the T–S fuzzy model, the system uncertainty affects only the membership functions. To cope with this problem, the grade functions, resulting from the membership functions of the control rules, are tuned by a back-propagation network. On the other hand, the feedback gains of the control rules are determined by solving a set of linear matrix inequalities (LMIs) which satisfy sufficient conditions of the closed-loop stability. As a result, both stability guarantee and better performance are concluded. The scheme is applied to a ball-and-beam system example verified by numerical simulations.     

基于VISIT算法设计模糊分类器的进化方法

介绍了在没有数据分布先验知识的情况下，用进化方法直接从训练数据中建立紧致模糊分类系统的方法。使用VISIT算法获取每个个体模糊系统，再用遗传算法从中搜索最优的模糊系统。规则和隶属函数是在进化过程中自动建立和优化的。为了同时有效地评价系统的精度和紧致性，用一个模糊专家系统作适应度函数。在2个基准分类问题上的实验结果表明了新方法的有效性。     

Structured Learning and Decomposition of Fuzzy Models for Robotic Control Applications

In this paper a specially designed structured-optimization procedure is used for learning the parameters of the Takagi–Sugeno (TS) type fuzzy models. It is well-known that the number of learning parameters increases exponentially with the number of model inputs. Therefore an appropriate learning scheme with preliminary structuring of the learning parameters into two groups: antecedent parameters and consequent parameters can be helpful for speeding-up the learning process. Two different optimization algorithms for tuning the antecedent and consequent parameters respectively are used in a sequence of repetitive loops (epochs). The stop criterion is defined as a number of repetitions of the loops or as a desired minimal error. Random walk algorithm with variable step size is used in this paper for tuning the antecedent parameters of the membership functions. For tuning the consequent parameters of the singletons, a specially proposed local learning algorithm is used. The problem of dimensionality reduction in fuzzy modeling is also considered in the paper from another viewpoint, namely as a hierarchical fuzzy model structure. It is accomplished by a decomposition of the complete fuzzy model into a feedforward hierarchical structure of sub-models called partial fuzzy models each one with two inputs and one output. Then the local models are learned separately in a preliminary specified and repetitive order. Such decomposition scheme has a potential for a significant reduction of the number of model parameters to be tuned thus reducing the total learning time. It has been experimentally shown that both concepts for dimensionality reduction in learning fuzzy models have benefits in learning speed and accuracy. A comparison with simultaneous optimization of all parameters of a single fuzzy model is also given. It shows that the proposed structured learning as well as the decomposition of the fuzzy model into a hierarchical fuzzy model structure lead to reducing the learning time and creating more accurate fuzzy models. Finally an application for learning a fuzzy controller of a two-link robot motion is shown and analysed.     

A Proposal for the Genetic Lateral Tuning of Linguistic Fuzzy Systems and Its Interaction With Rule Selection

Linguistic fuzzy modeling allows us to deal with the modeling of systems by building a linguistic model which is clearly interpretable by human beings. However, since the accuracy and the interpretability of the obtained model are contradictory properties, the necessity of improving the accuracy of the linguistic model arises when complex systems are modeled. To solve this problem, one of the research lines in recent years has led to the objective of giving more accuracy to linguistic fuzzy modeling without losing the interpretability to a high level. In this paper, a new postprocessing approach is proposed to perform an evolutionary lateral tuning of membership functions, with the main aim of obtaining linguistic models with higher levels of accuracy while maintaining good interpretability. To do so, we consider a new rule representation scheme base on the linguistic 2-tuples representation model which allows the lateral variation of the involved labels. Furthermore, the cooperation of the lateral tuning together with fuzzy rule reduction mechanisms is studied in this paper, presenting results on different real applications. The obtained results show the good performance of the proposed approach in high-dimensional problems and its ability to cooperate with methods to remove unnecessary rules.     

GSETSK: a generic self-evolving TSK fuzzy neural network with a novel Hebbian-based rule reduction approach

Takagi–Sugeno–Kang (TSK) fuzzy systems have been widely applied for solving function approximation and regression-centric problems. Existing dynamic TSK models proposed in the literature can be broadly classified into two classes. Class I TSK models are essentially fuzzy systems that are limited to time-invariant environments. Class II TSK models are generally evolving systems that can learn in time-variant environments. This paper attempts to address the issues of achieving compact, up-to-date fuzzy rule bases and interpretable knowledge bases in TSK models. It proposes a novel rule pruning method which is simple, computationally efficient and biologically plausible. This rule pruning algorithm applies a gradual forgetting approach and adopts the Hebbian learning mechanism behind the long-term potentiation phenomenon in the brain. It also proposes a merging approach which is used to improve the interpretability of the knowledge bases. This approach can prevent derived fuzzy sets from expanding too many times to protect their semantic meanings. These two approaches are incorporated into a generic self-evolving Takagi–Sugeno–Kang fuzzy framework (GSETSK) which adopts an online data-driven incremental-learning-based approach.Extensive experiments were conducted to evaluate the performance of the proposed GSETSK against other established evolving TSK systems. GSETSK has also been tested on real world dataset using the high-way traffic flow density and Dow Jones index time series. The results are encouraging. GSETSK demonstrates its fast learning ability in time-variant environments. In addition, GSETSK derives an up-to-date and better interpretable fuzzy rule base while maintaining a high level of modeling accuracy at the same time.     

Enhanced Fuzzy System Models With Improved Fuzzy Clustering Algorithm

Although traditional fuzzy models have proven to have high capacity of approximating the real-world systems, they have some challenges, such as computational complexity, optimization problems, subjectivity, etc. In order to solve some of these problems, this paper proposes a new fuzzy system modeling approach based on improved fuzzy functions to model systems with continuous output variable. The new modeling approach introduces three features: i) an improved fuzzy clustering (IFC) algorithm, ii) a new structure identification algorithm, and iii) a nonparametric inference engine. The IFC algorithm yields simultaneous estimates of parameters of c-regression models, together with fuzzy c-partitioning of the data, to calculate improved membership values with a new membership function. The structure identification of the new approach utilizes IFC, instead of standard fuzzy c-means clustering algorithm, to fuzzy partition the data, and it uses improved membership values as additional input variables along with the original scalar input variables for two different choices of regression methods: least squares estimation or support vector regression, to determine ldquofuzzy functionsrdquo for each cluster. With novel IFC, one could learn the system behavior more accurately compared to other FSM models. The nonparametric inference engine is a new approach, which uses the alike -nearest neighbor method for reasoning. Empirical comparisons indicate that the proposed approach yields comparable or better accuracy than fuzzy or neuro-fuzzy models based on fuzzy rules bases, as well as other soft computing methods.     

Evolutionary design of a fuzzy classifier from data

Genetic algorithms show powerful capabilities for automatically designing fuzzy systems from data, but many proposed methods must be subjected to some minimal structure assumptions, such as rule base size. In this paper, we also address the design of fuzzy systems from data. A new evolutionary approach is proposed for deriving a compact fuzzy classification system directly from data without any a priori knowledge or assumptions on the distribution of the data. At the beginning of the algorithm, the fuzzy classifier is empty with no rules in the rule base and no membership functions assigned to fuzzy variables. Then, rules and membership functions are automatically created and optimized in an evolutionary process. To accomplish this, parameters of the variable input spread inference training (VISIT) algorithm are used to code fuzzy systems on the training data set. Therefore, we can derive each individual fuzzy system via the VISIT algorithm, and then search the best one via genetic operations. To evaluate the fuzzy classifier, a fuzzy expert system acts as the fitness function. This fuzzy expert system can effectively evaluate the accuracy and compactness at the same time. In the application section, we consider four benchmark classification problems: the iris data, wine data, Wisconsin breast cancer data, and Pima Indian diabetes data. Comparisons of our method with others in the literature show the effectiveness of the proposed method.     

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.     

Fuzzy rule-base driven orthogonal approximation

In this study, orthogonal approximation concept is applied to fuzzy systems. We propose a new useful model adapted from the well-known Sugeno type fuzzy system. The proposed fuzzy model is a generalization of the zero-order Sugeno fuzzy system model. Instead of linear functions in standard Sugeno model, we use nonlinear functions in the consequent part. The nonlinear functions are selected from a trigonometric orthogonal basis. Orthogonal function parameters are trained along with the Sugeno fuzzy system. The proposed model is demonstrated using three simulations—a nonlinear piecewise-continuous scalar function modeling and filtering, nonlinear dynamic system identification, and time series prediction. Finally some performance comparisons are carried out.     

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.     

Modified Gath-Geva fuzzy clustering for identification of Takagi-Sugeno fuzzy models

The construction of interpretable Takagi-Sugeno (TS) fuzzy models by means of clustering is addressed. First, it is shown how the antecedent fuzzy sets and the corresponding consequent parameters of the TS model can be derived from clusters obtained by the Gath-Geva (GG) algorithm. To preserve the partitioning of the antecedent space, linearly transformed input variables can be used in the model. This may, however, complicate the interpretation of the rules. To form an easily interpretable model that does not use the transformed input variables, a new clustering algorithm is proposed, based on the expectation-maximization (EM) identification of Gaussian mixture models. This new technique is applied to two well-known benchmark problems: the MPG (miles per gallon) prediction and a simulated second-order nonlinear process. The obtained results are compared with results from the literature.     

A novel data reduction method for Takagi–Sugeno fuzzy system design based on statistical design of experiment

This paper introduces a simple, systematic and effective method for designing Takagi–Sugeno (T–S) fuzzy systems utilizing a significantly smaller training data set versus existing methods. Creating proper training data is usually not an easy task and requires spending considerable time and resources. The proposed method first uses a three-level factorial design to partition the output space. Next the least square technique is used to estimate each of the partitioned output spaces. The membership functions are introduced with only three variables (min, max and number of membership functions). Fuzzy rules are generated with respect to the partitioned output surfaces and the membership functions. The proposed method is applied to two benchmark problems, controlling an inverted pendulum as well as modeling a nonlinear function. In the case of the inverted pendulum simulation results demonstrate significant improvement. In the case of nonlinear function modeling we demonstrated sufficient accuracy with only 9 training data, which represents 98% reduction in the number of training data compare to other method. Additionally, the proposed method offers extremely low computation time allowing it to be used with adaptive type systems.     

A new evolving compact optimised Takagi–Sugeno fuzzy model and its application to nonlinear system identification

A new encoding scheme is presented for a fuzzy-based nonlinear system identification methodology, using the subtractive clustering and non-dominated sorting genetic algorithm. The proposed method consists of two parts. The first part is related to the selection of most relevant or influencing inputs to the system and the second one is related to the tuning of fuzzy rules and parameters of the membership functions. The main purpose of the proposed scheme is to reduce the complexity and increase the accuracy of the model. In particular, three objectives are considered in the process of optimisation, namely, the number of inputs, number of rules and the root mean square of the modelling error. The performance of the developed method is validated by identifying the Box–Jenkins nonlinear benchmark system, and to the modelling of the forward and inverse dynamic behaviours of a magneto-rheological (MR) damper. The latter is also a challenging problem due to the inherent hysteretic and highly nonlinear dynamics of the MR damper. It is shown that the developed evolving Takagi–Sugeno (T–S) fuzzy model can identify and grasp the nonlinear dynamics of both systems very well, while a small number of inputs and fuzzy rules are required for this purpose.     

A systematic approach to a self-generating fuzzy rule-table forfunction approximation

In this paper, a systematic design is proposed to determine fuzzy system structure and learning its parameters, from a set of given training examples. In particular, two fundamental problems concerning fuzzy system modeling are addressed: 1) fuzzy rule parameter optimization and 2) the identification of system structure (i.e., the number of membership functions and fuzzy rules). A four-step approach to build a fuzzy system automatically is presented: Step 1 directly obtains the optimum fuzzy rules for a given membership function configuration. Step 2 optimizes the allocation of the membership functions and the conclusion of the rules, in order to achieve a better approximation. Step 3 determines a new and more suitable topology with the information derived from the approximation error distribution; it decides which variables should increase the number of membership functions. Finally, Step 4 determines which structure should be selected to approximate the function, from the possible configurations provided by the algorithm in the three previous steps. The results of applying this method to the problem of function approximation are presented and then compared with other methodologies proposed in the bibliography.     

基于Sugeno型神经模糊系统的交通流状态预测算法

从交通流状态的模糊特性出发,设计基于Sugeno型神经模糊系统的交通流状态预测算法.选择交通流状态的影响指标作为模糊推理系统的输入、交通流状态作为输出;据经验对输入、输出划分模糊子集,给出相应的隶属度函数并制定模糊规则;建立具有5层结构的神经模糊推理系统,利用神经网络优化调整模糊推理系统的隶属度函数和模糊规则.仿真实验表明,神经网络可直接优化模糊推理系统的隶属度函数,通过对连接权值的训练间接优化模糊规则,故Sugeno型神经模糊系统相比常规模糊系统具有更好的交通流状态预测性能.     

Genetic fuzzy system for data-driven soft sensors design

This paper proposes a new method for soft sensors (SS) design for industrial applications based on a Takagi–Sugeno (T–S) fuzzy model. The learning of the T–S model is performed from input/output data to approximate unknown nonlinear processes by a coevolationary genetic algorithm (GA). The proposed method is an automatic tool for SS design since it does not require any prior knowledge concerning the structure (e.g. the number of rules) and the database (e.g. antecedent fuzzy sets) of the T–S fuzzy model, and concerning the selection of the adequate input variables and their respective time delays for the prediction setting. The GA approach is composed by five hierarchical levels and has the global goal of maximizing the prediction accuracy. The first level consists in the selection of the set of input variables and respective delays for the T–S fuzzy model. The second level considers the encoding of the membership functions. The individual rules are defined at the third level, the population of the set of rules is treated in fourth level, and a population of fuzzy systems is handled at the fifth level. To validate and demonstrate the performance and effectiveness of the proposed algorithm, it is applied on two prediction problems. The first is the Box–Jenkins benchmark problem, and the second is the estimation of the flour concentration in the effluent of a real-world wastewater treatment system. Simulation results are presented showing that the developed evolving T–S fuzzy model can identify the nonlinear systems satisfactorily with appropriate input variables and delay selection and a reasonable number of rules. The proposed methodology is able to design all the parts of the T–S fuzzy prediction model. Moreover, presented comparison results indicate that the proposed method outperforms other previously proposed methods for the design of prediction models, including methods previously proposed for the design of T–S models.     

An improved fuzzy neural network based on T–S model

An improved fuzzy neural network based on Takagi–Sugeno (T–S) model is proposed in this paper. According to characteristics of samples spatial distribution the number of linguistic values of every input and the means and deviations of corresponding membership functions are determined. So the reasonable fuzzy space partition is got. Further a subtractive clustering algorithm is used to derive cluster centers from samples. With the parameters of linguistic values the cluster centers are fuzzified to get a more concise rule set with importance for every rule. Thus redundant rules in the fuzzy space are deleted. Then antecedent parts of all rules determine how a fuzzification layer and an inference layer connect. Next, weights of the defuzzification layer are initialized by a least square algorithm. After the network is built, a hybrid method combining a gradient descent algorithm and a least square algorithm is applied to tune the parameters in it. Simultaneous, an adaptive learning rate which is identified from input-state stability theory is adopted to insure stability of the network. The improved T–S fuzzy neural network (ITSFNN) has a compact structure, high training speed, good simulation precision, and generalization ability. To evaluate the performance of the ITSFNN, we experiment with two nonlinear examples. A comparative analysis reveals the proposed T–S fuzzy neural network exhibits a higher accuracy and better generalization ability than ordinary T–S fuzzy neural network. Finally, it is applied to predict markup percent of the construction bidding system and has a better prediction capability in comparison to some previous models.     

A highly interpretable form of Sugeno inference systems

We present a form of fuzzy inference systems (FISs) that is highly interpretable and easy to manipulate. The form is based on a judicious choice of membership functions that have strong locality and differentiability properties and on a modification of the Sugeno and generalized Sugeno forms of the consequent polynomials so as to make them rule centered. Under these conditions, the coefficients in the consequent polynomials can be exactly interpreted as Taylor series coefficients. Besides the intuitive interpretation thus bestowed on the coefficients, we show that the new form allows easy design, manipulation, testing, training, and combination of the resulting fuzzy inference systems. The rudiments of a calculus of fuzzy inference systems are then introduced     

云环境下基于模糊隶属度的虚拟机放置算法

虚拟机放置问题是云数据中心资源调度的核心问题之一,它对数据中心的性能、资源利用率和能耗有着重要的影响。针对此问题,以降低数据中心能耗、改善资源利用率和保证服务质量(QoS)为优化目标,借助模糊聚类的思想提出了一种基于模糊隶属度的虚拟机放置算法。首先,结合物理主机过载概率和虚拟机与物理主机之间的相适性放置关系,提出了新的距离度量方法;然后,根据模糊隶属度函数计算得出虚拟机与物理主机之间的相适性模糊隶属度矩阵;最后,借助能耗感知机制,在模糊隶属度矩阵中进行局部搜索从而获得迁移虚拟机的最优放置方案。仿真实验结果表明,提出的算法在降低云数据中心能耗、改善资源利用率和保证QoS方面表现比较优异。