Robust TSK fuzzy modeling for function approximation with outliers

The Takagi-Sugeno-Kang (TSK) type of fuzzy models has attracted a great attention of the fuzzy modeling community due to their good performance in various applications. Most approaches for modeling TSK fuzzy rules define their fuzzy subspaces based on the idea of training data being close enough instead of having similar functions. Besides, training data sets algorithms often contain outliers, which seriously affect least-square error minimization clustering and learning algorithms. A robust TSK fuzzy modeling approach is presented. In the approach, a clustering algorithm termed as robust fuzzy regression agglomeration (RFRA) is proposed to define fuzzy subspaces in a fuzzy regression manner with robust capability against outliers. To obtain a more precision model, a robust fine-tuning algorithm is then employed. Various examples are used to verify the effectiveness of the proposed approach. From the simulation results, the proposed robust TSK fuzzy modeling indeed showed superior performance over other approaches     

TSK fuzzy systems types II and III stability analysis: continuous case

We propose a new approach for the stability analysis of continuous Sugeno Types II and III dynamic fuzzy systems. We introduce the concept of fuzzy positive definite and fuzzy negative definite systems and use them in arguments similar to those of traditional Lyapunov stability theory to derive new conditions for stability and asymptotic stability for continuous Type II/III dynamic fuzzy systems. To demonstrate the new approach, we apply it to numerical examples.     

SISO Mamdani模糊系统作为函数逼近器的必要条件

模糊系统已被证明是通用逼近器,但实现高精度通常需要大量规则．模糊系统满足给定精度的必要条件能指导最优系统的构造,如输入输出模糊集、模糊规则的选取．研究了单输入单输出（SISO）Mamdani模糊系统在给定逼近精度下作为函数逼近器的必要条件．由于通用型SISO Mamdani模糊系统在划分子区间单调,所以模糊系统的最优配置是输入域的划分数至少为系统输出的单调性变化次数．当模糊系统满足给定逼近精度时,通过分析目标函数的局部特性,基于目标函数的极点,建立了SISO Mamdani模糊系统的必要条件．更重要的是证明了现有的必要条件仅仅是该文结论的一种特例．最后,使用数值实例来验证该文的结论,分析模糊系统作为函数逼近器的优劣．     

Multi-objective optimization of TSK fuzzy models

In this paper we propose a hybrid algorithm to optimize the structure of TSK type fuzzy model using backpropagation (BP) learning algorithm and non-dominated sorting genetic algorithm (NSGA-II). In a first step, BP algorithm is used to optimize the parameters of the model (parameters of membership functions and fuzzy rules). NSGA-II is used in a second phase, to optimize the number of fuzzy rules and to fine tune the parameters. A well known benchmark is used to evaluate performances of the proposed modelling approach, and compare it with other modelling approaches.     

TSK fuzzy model with minimal parameters

In this paper, an efficient genetic algorithm (GA) to generate a simple and well defined TSK model is proposed. The approach is derived from the use of the improved Strength Pareto Evolutionary Algorithm (SPEA-2), where the genes of the chromosome are arranged into control genes and parameter genes. These genes are in a hierarchical form so that the control genes can manipulate the parameter genes in a more effective manner. In our approach, we first apply the back-propagation algorithm to optimize the parameters of the model (parameters of membership functions and fuzzy rules), then we apply the SPEA-2 to optimize the number of fuzzy rules, the number of parameters and to fine tune these parameters.Two well-known dynamic benchmarks are used to evaluate the performance of the proposed algorithm. Simulation results show that our modeling approach outperforms some methods proposed in previous work.     

Bayesian zero-order TSK fuzzy system modeling

Different from the existing TSK fuzzy system modeling methods, a novel zero-order TSK fuzzy modeling method called Bayesian zero-order TSK fuzzy system (B-ZTSK-FS) is proposed from the perspective of Bayesian inference in this paper. The proposed method B-ZTSK-FS constructs zero-order TSK fuzzy system by using the maximum a posteriori (MAP) framework to maximize the corresponding posteriori probability. First, a joint likelihood model about zero-order TSK fuzzy system is defined to derive a new objective function which can assure that both antecedents and consequents of fuzzy rules rather than only their antecedents of the most existing TSK fuzzy systems become interpretable. The defined likelihood model is composed of three aspects: clustering on the training set for antecedents of fuzzy rules, the least squares (LS) error for consequent parameters of fuzzy rules, and a Dirichlet prior distribution for fuzzy cluster memberships which is considered to not only automatically match the “sum-to-one” constraints on fuzzy cluster memberships, but also make the proposed method B-ZTSK-FS scalable for large-scale datasets by appropriately setting the Dirichlet index. This likelihood model indeed indicates that antecedent and consequent parameters of fuzzy rules can be linguistically interpreted and simultaneously optimized by the proposed method B-ZTSK-FS which is based on the MAP framework with the iterative sampling algorithm, which in fact implies that fuzziness and probability can co-jointly work for TSK fuzzy system modeling in a collaborative rather than repulsive way. Finally, experimental results on 28 synthetic and real-world datasets are reported to demonstrate the effectiveness of the proposed method B-ZTSK-FS in the sense of approximation accuracy, interpretability and scalability.     

The minimum worst case error of fuzzy approximators

The approximation capability of fuzzy systems is an important topic of research when the systems are regarded as input-output maps. By using the notion of information-based complexity (IBC), we derive the minimum worst case error of a fuzzy approximator, which is independent of the detailed construction of the fuzzy rule bases     

模糊系统作为通用逼近器的10 年历程

总结了模糊系统作为通用逼近器在存在性、充分性和必要性3个方面所作过的主要理论研究，并分析了这些理论成果在工程上的若干应用，最后就该理论的未来发展方向作了大胆而理性的分析和预测。     

Fuzzy function approximators with ellipsoidal regions

This paper discusses two types of fuzzy function approximators that dynamically generate fuzzy rules with ellipsoidal regions: a function approximator based on Takagi-Sugeno type model with the center-of-gravity defuzzification and a function approximator based on a radial basis function network. Hereafter the former is called FACG and the latter is called FALC. In FACG, for each training datum the number of the training data that are within the specified distance is calculated and the training datum which has the maximum number of the training data is selected as the center of a fuzzy rule and the covariance matrix is calculated using the training data around the center. Then the parameters of the linear equation that defines the output value of the fuzzy rule are determined by the least-squares method using the training data around the center. In FALC, the training datum with the maximum approximation error is selected as the center of a fuzzy rule. Then using the training data around the center, the covariance matrix is calculated, and the parameters of a linear equation that determines the output value are calculated by the least-squares method. Performance of FACG and FALC is compared with that of multilayered neural networks and other fuzzy function approximators for the data generated by the Mackey-Glass differential equation and the data from a water purification plant.     

Knowledge-based parameter identification of TSK fuzzy models

Linear/1st order Takagi–Sugeno–Kang (TSK) fuzzy models are widely used to identify static nonlinear systems from a set of input–output pairs. The synergetic integration of TSK fuzzy models with artificial neural networks (ANN) has led to the emergence of hybrid neuro-fuzzy models that can have excellent adaptability and interpretability at the same time. One drawback of these hybrid models is that they tend to have more black-box characteristics of ANN than the transparency of fuzzy systems. If the quality of training data is questionable then it may lead to a fuzzy model with poor interpretability. In an attempt to remediate this problem, we propose a parameter identification technique for TSK models that relies on a-priori available qualitative domain knowledge. The technique is devised for rule-centered TSK models in which the consequent polynomial can be interpreted as the 1st order Taylor series approximation of the underlying nonlinear function that is being modeled. The resulting neuro-fuzzy model is named as a-priori knowledge-based fuzzy model (APKFM). We have shown that besides being reasonably accurate, APKFM has excellent interpretability and extrapolation capability. The effectiveness of APKFM is shown using two examples of static systems. In the first example, a toy nonlinear function is chosen for approximation by an APKFM. In the second example, a real world problem pertaining to the maintenance cost estimation of electricity distribution networks is addressed.     

Stability analysis of recurrent type-2 TSK fuzzy systems with nonlinear consequent part

Neural Computing and Applications - A necessary condition for stability of a class of recurrent type-2 TSK fuzzy systems is presented. In this system, the antecedent part is indeed represented by...     

布尔型模糊系统逼近的充分条件

为了保证布尔模糊系统逼近定义在紧集上任意实值连续函数的逼近精度．给出一个估计布尔模糊系统的输入变量与输出变量各自需要构造的模糊集个数的公式,讨论如何设计布尔模糊系统．以便实现逼近任给的实值连续函数到需要的逼近精度．最后通过一个例子展示了如何设计布尔模糊系统来逼近所给的连续函数的具体方法．     

Sufficient and necessary conditions for Boolean fuzzy systems as universal approximators

A formula is first presented to compute the lower upper bounds on the number of fuzzy sets to achieve pre-specified approximation accuracy for an arbitrary multivariate continuous function. The necessary condition for Boolean fuzzy systems as universal approximators with minimal system configurations is then discussed. Two examples are provided to demonstrate how to design a Boolean fuzzy system in order to approximate a given continuous function with a required approximation accuracy.     

并行集成具有高可解释的TSK模糊分类器

针对分层Takagi-Sugeno-Kang (TSK)模糊分类器可解释性差,以及当增加或删除一个TSK模糊子分类器时Boosting模糊分类器需要重新训练所有TSK模糊子分类器等问题,提出一种并行集成具有高可解释的TSK模糊分类器EP-Q-TSK.该集成模糊分类器每个TSK模糊子分类器可以使用最小学习机(LLM)被并行地快速构建.作为一种新的集成学习方式,该分类器利用每个TSK模糊子分类器的增量输出来扩展原始验证数据空间,然后采用经典的模糊聚类算法FCM获取一系列代表性中心点,最后利用KNN对测试数据进行分类.在标准UCI数据集上,分别从分类性能和可解释性两方面验证了EP-Q-TSK的有效性.     

Hybrid compensation control for affine TSK fuzzy control systems

The paper proposes a way of designing state feedback controllers for affine Takagi-Sugeno-Kang (TSK) fuzzy models. In the approach, by combining two different control design methodologies, the proposed controller is designed to compensate all rules so that the desired control performance can appear in the overall system. Our approach treats all fuzzy rules as variations of a nominal rule and such variations are individually dealt with in a Lyapunov sense. Previous approaches have proposed a similar idea but the variations are dealt with as a whole in a robust control sense. As a consequence, when fuzzy rules are distributed in a wide range, the stability conditions may not be satisfied. In addition, the control performance of the closed-loop system cannot be anticipated in those approaches. Various examples were conducted in our study to demonstrate the effectiveness of the proposed control design approach. All results illustrate good control performances as desired.     

Hybrid robust approach for TSK fuzzy modeling with outliers

This study proposes a hybrid robust approach for constructing Takagi–Sugeno–Kang (TSK) fuzzy models with outliers. The approach consists of a robust fuzzy C-regression model (RFCRM) clustering algorithm in the coarse-tuning phase and an annealing robust back-propagation (ARBP) learning algorithm in the fine-tuning phase. The RFCRM clustering algorithm is modified from the fuzzy C-regression models (FCRM) clustering algorithm by incorporating a robust mechanism and considering input data distribution and robust similarity measure into the FCRM clustering algorithm. Due to the use of robust mechanisms and the consideration of input data distribution, the fuzzy subspaces and the parameters of functions in the consequent parts are simultaneously identified by the proposed RFCRM clustering algorithm and the obtained model will not be significantly affected by outliers. Furthermore, the robust similarity measure is used in the clustering process to reduce the redundant clusters. Consequently, the RFCRM clustering algorithm can generate a better initialization for the TSK fuzzy models in the coarse-tuning phase. Then, an ARBP algorithm is employed to obtain a more precise model in the fine-tuning phase. From our simulation results, it is clearly evident that the proposed robust TSK fuzzy model approach is superior to existing approaches in learning speed and in approximation accuracy.     

TSK模糊模型的协同进化学习方法

针对TSK模糊模型的学习是多约束和多目标优化问题,提出TSK模糊模型分解为两类不同的种群,协作共同进化的模型学习方法．论述了所涉及的相关问题,包括各种群的编码及其不同的进化计算,各种群个体的合作及其适应值评估策略,模型的后件参数估计方法．该方法要求先验知识少,收敛速度快,能形成简洁的模糊模型,最后以函数近似为例说明了该方法的有效性．     

Adaptive nonlinear control of pH neutralization processes using fuzzy approximators

In this paper, an adaptive control scheme, based on fuzzy logic systems, for pH control is addressed. For implementation of the proposed scheme no composition measurement is required. Stability of the closed-loop system is established and it is shown that the solution of the closed-loop system is uniformly ultimately bounded and under a certain condition, asymptotical stability is achieved. Effectiveness of the proposed controller is tested through simulation and experimental studies. Results indicate that the proposed controller has good performances in set-point tracking and load rejection and much better than that of a tuned PI controller.     

基于TSK模糊系统的生化变量预估模型

提出了一种利用TSK(Takagi-Sugeno-Kang)模糊系统建立生化变量预估模型的方法,用于生化过程的工艺参数预测。利用TSK模糊系统的非线性逼近能力,以谷氨酸发酵过程为研究对象,建立了基于TSK模糊逻辑系统的生化变量预估器。以工厂现场采集的发酵过程参数大量数据为样本,训练TSK模糊系统生化变量预估器,并对模型的预估精度进行检验。仿真结果显示了预估模型的有效性,可以有效地预测谷氨酸发酵过程中生化变量的估计值。TSK模糊系统生化变量预估模型能预估生化过程中工业发酵罐的放罐时间,预估模型的状态预报对正常罐批具有足够高的预报精度,预报误差如果偏大亦可作为异常罐批的早期警示信息。有鉴于此,TSK模糊逻辑系统可望开辟生化过程参数预估的新途径。     

A two-stage evolutionary process for designing TSK fuzzy rule-basedsystems

Nowadays, fuzzy rule-based systems are successfully applied to many different real-world problems. Unfortunately, relatively few well-structured methodologies exist for designing and, in many cases, human experts are not able to express the knowledge needed to solve the problem in the form of fuzzy rules. Takagi-Sugeno-Kang (TSK) fuzzy rule-based systems were enunciated in order to solve this design problem because they are usually identified using numerical data. In this paper we present a two-stage evolutionary process for designing TSK fuzzy rule-based systems from examples combining a generation stage based on a (mu, lambda)-evolution strategy, in which the fuzzy rules with different consequents compete among themselves to form part of a preliminary knowledge base, and a refinement stage in which both the antecedent and consequent parts of the fuzzy rules in this previous knowledge base are adapted by a hybrid evolutionary process composed of a genetic algorithm and an evolution strategy to obtain the final Knowledge base whose rules cooperate in the best possible way. Some aspects make this process different from others proposed until now: the design problem is addressed in two different stages, the use of an angular coding of the consequent parameters that allows us to search across the whole space of possible solutions, and the use of the available knowledge about the system under identification to generate the initial populations of the Evolutionary Algorithms that causes the search process to obtain good solutions more quickly. The performance of the method proposed is shown by solving two different problems: the fuzzy modeling of some three-dimensional surfaces and the computing of the maintenance costs of electrical medium line in Spanish towns. Results obtained are compared with other kind of techniques, evolutionary learning processes to design TSK and Mamdani-type fuzzy rule-based systems in the first case, and classical regression and neural modeling in the second.