Analysis and design of hierarchical fuzzy systems

In this letter, the hierarchical fuzzy systems are analyzed and designed. In the analysis part, we prove that the hierarchical fuzzy systems are universal approximators and analyze the sensitivity of the fuzzy system output with respect to small perturbations in its inputs. In the design part, we derive a gradient descent algorithm for tuning the parameters of the hierarchical fuzzy system to match the input-output pairs. The algorithm is simulated for two examples and the results show that the algorithm is effective and the hierarchical structure gives good approximation accuracy     

On the applicability of the minimal configured hierarchical fuzzy control and its relevance to function approximation

This paper presents an approach to minimal hierarchical fuzzy control systems design. It maintains the idea of distributing behaviors inside a fuzzy structure, while the minimal hierarchical fuzzy control is an effective procedure for dealing with systems complexity and their control. The MCHFLS, minimal configured hierarchical fuzzy logic systems, is minimizing the fuzzy inferences parameters by adjusting essentially the variables associations and levels structures. This approach performs function approximation under conditions and it was tested on textile mill control.     

Approximation Capabilities of Hierarchical Fuzzy Systems

Derived from practical application in location analysis and pricing, and based on the approach of hierarchical structure analysis of continuous functions, this paper investigates the approximation capabilities of hierarchical fuzzy systems. By first introducing the concept of the natural hierarchical structure, it is proved that continuous functions with natural hierarchical structure can be naturally and effectively approximated by hierarchical fuzzy systems to overcome the curse of dimensionality in both the number of rules and parameters. Then, based on Kolmogorov's theorem, it is shown that any continuous function can be represented as a superposition of functions with the natural hierarchical structure and can then be approximated by hierarchical fuzzy systems to achieve the universal approximation property. Further, the conditions under which the hierarchical fuzzy approximation is superior to the standard fuzzy approximation in overcoming the curse of dimensionality are analyzed     

Hierarchical Fuzzy Systems for Function Approximation on Discrete Input Spaces With Application

This paper investigates the capabilities of hierarchical fuzzy systems to approximate functions on discrete input spaces. First, it is shown that any function on a discrete space has an arbitrary separable hierarchical structure and can be naturally approximated by hierarchical fuzzy systems. As a by-product of this result, a discrete version of Kolmogorov's theorem is obtained; second, it is proven that any function on a discrete space can be approximated to any degree of accuracy by hierarchical fuzzy systems with any desired separable hierarchical structure. That is, functions on discrete spaces can be approximated more simply and flexibly than those on continuous spaces; third, a hierarchical fuzzy system identification method is proposed in which human knowledge and numerical data are combined for system construction and identification. Finally, the proposed method is applied to the market condition performance modeling problem in site selection decision support and shows the better performance in both accuracy and interpretability than the regression and neural network approaches. In additions, the reason and mechanism why hierarchical fuzzy systems outperform regression and neural networks in this type of application are analyzed.      

单路口交通实时模糊控制的一种方法

根据城市交通系统的实际状况，依据单路口４个方向车流的来车信息，提出了一种单路口交通模糊控制方法－多级模糊控制方法，设计了多级模糊控制器，并对该控制器进行了仿真研究，结果令人满意。     

核动力装置蒸汽发生器水位的分层模糊自适应控制

针对压水堆核动力装置蒸汽发生器的水位控制提出一种分层模糊自适应控制方案，该方案中2个模糊控制器分层连接，每个模糊控制均采用典型模糊控制单元，使得模糊规则个数和可调参数个数大大减少，便于在线学习和实时控制，给出了分层模糊控制器的解析表达式及可调参数的在线学习方法，在快速加负荷和突然甩负荷的仿真实验中，该方案的控制效果明显优于已有的变参数PID控制，验证了该方案的有效性。     

Selection and impact of different topologies in multi-layered hierarchical fuzzy systems

An evolutionary algorithm based approach for selection of topologies in hierarchical fuzzy systems (HFS) is presented. Coupling fuzzy system with evolutionary algorithm provides a solution to the automated acquisition of the fuzzy rule base. It is difficult to study the problem of hierarchical decomposition for a large class of fuzzy systems but it is possible to analyse such architectures on the example of a particular fuzzy system, such as inverted pendulum. Topology of the HFS must be selected according to the physical properties of the dynamical system under consideration. Different HFS topologies for an inverted pendulum system are investigated and analysed to address the problem of how input configuration in multi-layered structure affects the controller performance. The experiments are conducted to test controller performance for different topologies of the hierarchical fuzzy system. The impact of different topologies on control process is discussed. The results from the case study of inverted pendulum can be extended to other dynamical systems.     

广义递阶Mamdani模糊系统及其泛逼近性

从解决模糊系统的“规则爆炸”问题出发,本文首先给出广义递阶M amdan i模糊系统的定义,然后证明其与具有中间变量的广义M amdan i模糊系统等价,并借助方形分片线性函数构造性的证明了在最大模和积分模意义下该系统是泛逼近器.最后仿真实例证实了该系统的有效性.     

Hierarchical unsupervised fuzzy clustering

A recursive algorithm for hierarchical fuzzy partitioning is presented. The algorithm has the advantages of hierarchical clustering, while maintaining fuzzy clustering rules. Each pattern can have a nonzero membership in more than one subset of the data in the hierarchy. Optimal feature extraction and reduction is optionally reapplied for each subset. Combining hierarchical and fuzzy concepts is suggested as a natural feasible solution to the cluster validity problem of real data. The convergence and membership conservation of the algorithm are proven. The algorithm is shown to be effective for a variety of data sets with a wide dynamic range of both covariance matrices and number of members in each class     

Integrating fuzzy DEMATEL and fuzzy hierarchical TOPSIS methods for truck selection

In the present paper, a novel fuzzy Multiple Attribute Decision Making (MADM) model is proposed for modeling and solving truck selection problem of a land transportation company. As truck selection is a very crucial problem for the land transportation companies, a systematic and scientific approach is necessary for its solution. A systematic methodology is proposed in the present study by integrating “fuzzy DEMATEL” and “fuzzy hierarchical TOPSIS”. The proposed method makes use of fuzzy DEMATEL method for evaluating the weights of the criteria and hierarchical fuzzy TOPSIS method for assessing the alternatives according to criteria. The steps of the method are described first then a case study is presented in the paper.     

Application of Fuzzy Logic for the Solution of Inverse Kinematics and Hierarchical Controls of Robotic Manipulators

In this paper, hierarchical control techniques is used for controlling a robotic manipulator. The proposed method is based on the establishment of a non-linear mapping between Cartesian and joint coordinates using fuzzy logic in order to direct each individual joint. The hierarchical control will be implemented with fuzzy logic to improve the robustness and reduce the run time computational requirements. Hierarchical control consists of solving the inverse kinematic equations using fuzzy logic to direct each individual joint. A commercial Microbot with three degrees of freedom is utilized to evaluate this methodology. A decentralized fuzzy controller is used for each joint, with a Fuzzy Associative Memories (FAM) performing the inverse kinematic mapping in a supervisory mode. The FAM determines the inverse kinematic mapping which maps the desired Cartesian coordinates to the individual joint angles. The individual fuzzy controller for each joint generates the required control signal to a DC motor to move the associated link to the new position. The proposed hierarchical fuzzy controller is compared to a conventional controller. The simulation experiments indeed demonstate the effectiveness of the proposed method.     

复杂系统的递阶模糊辨识

针对Takagi_Sugeno模糊模型 (T_S模型 )严重的维数灾问题, 借鉴GMDH算法, 提出了一种新的复杂系统递阶模糊辨识方法. 本文首先详细描述了由两输入变量的特殊T_S模型所组成的递阶模糊模型 ;然后提出了具体的辨识该递阶模糊模型的方法. 该方法的特点是 :a)在结构辨识阶段, 用FCM模糊聚类方法评价系统中每个输入变量的重要性, 以便构造合理的递阶模糊模型 ;b)预先合理地确定了所要辨识的参数的初始值, 用扩展卡尔曼滤波方法可很快地得到这些参数. 最后, 给出的仿真实例说明了本文辨识方法的有     

遗传算法在电液伺服系统分层模糊控制中的应用

针对模糊控制中规则数和可调参数多的问题,对遗传算法在分层模糊控制系统中的应用进行了研究,考虑遗传算法具有强大的全局搜索能力,提出了基于遗传算法的分层模糊控制的设计方案。该方案采用2个模糊控制器分层连接,即第一层采用基于遗传算法优化的模糊控制器,第二层采用典型的模糊控制器,从而大大减少了模糊规则和可调参数的个数,便于实时控制。电液伺服系统的实际仿真结果验证了该方案的有效性。     

应用进化策略的城域交叉路口分级模糊控制

提出一种面向城域单交叉路口的自适应分级模糊控制系统，采用进化策略对分级模糊控制器的模糊隶属度函数进行离线优化调整．该控制系统不仅具有分级模糊控制的优点，同时能使模糊隶属度函数根据不同的交通情况自适应地变化，从而改善控制效果．对一个具有直行和左转车流运动的四向交叉路口进行的仿真表明了该方法能比定时控制和隶属度函数固定的分级模糊控制取得更好的控制效果．     

基于二叉树型分层的广义混合模糊系统推理规则数的缩减

为避免广义混合模糊系统因输入变量个数的增加而引起规则爆炸现象,应用二叉树型分层方法给出混合推理规则,进而对广义混合模糊系统的输入实施二叉树型分层,从理论上获得了该系统分层后的输入输出表达式和推理规则总数的计算公式.此外,通过实例对该系统分层和不分层的规则总数进行了比较和分析,结果表明分层后广义混合模糊系统可大幅度缩减推理规则总数,并可有效地避免规则爆炸.     

A new method for fuzzy information retrieval based on fuzzy hierarchical clustering and fuzzy inference techniques

In this paper, we extend the work of Kraft et al. to present a new method for fuzzy information retrieval based on fuzzy hierarchical clustering and fuzzy inference techniques. First, we present a fuzzy agglomerative hierarchical clustering algorithm for clustering documents and to get the document cluster centers of document clusters. Then, we present a method to construct fuzzy logic rules based on the document clusters and their document cluster centers. Finally, we apply the constructed fuzzy logic rules to modify the user's query for query expansion and to guide the information retrieval system to retrieve documents relevant to the user's request. The fuzzy logic rules can represent three kinds of fuzzy relationships (i.e., fuzzy positive association relationship, fuzzy specialization relationship and fuzzy generalization relationship) between index terms. The proposed fuzzy information retrieval method is more flexible and more intelligent than the existing methods due to the fact that it can expand users' queries for fuzzy information retrieval in a more effective manner.     

基于减法聚类和自适应神经模糊推理系统的递阶模糊系统的设计

提出了一种设计递阶模糊系统的简易而有效的方法.在得到一个单级模糊系统的基础上,用灵敏度分析法对每一个输入变量的重要性进行排序,从而确定每一级子系统的输入变量.利用减法聚类和自适应神经 模糊推理系统逐级对子系统进行训练.所得到的递阶模糊系统可进一步得到简化.仿真实例证实了设计方法的有效性.     

Intermediate Variable Normalization for Gradient Descent Learning for Hierarchical Fuzzy System

When applying gradient descent learning methods to hierarchical fuzzy systems, there is great difficulty in handling the intermediate variables introduced by the hierarchical structures, as the intermediate variables may go outside their definition domain that makes gradient descent learning invalid. To overcome this difficulty, this paper proposes a learning scheme that integrates a normalization process for intermediate variables into gradient descent learning. This ensures that gradient descent methods are applicable to, and correctly used for, learning general hierarchical fuzzy systems. Benchmark datasets are used to demonstrate the validity and advantages of the proposed learning scheme over other existing methods in terms of better accuracy, better transparency, and fewer fuzzy rules and parameters.     

基于分层模糊系统的直接自适应控制

为解决模糊控制器中规则数目随系统变量呈指数增长的问题，利用分层模糊系统设计了一类非线性系统的直接自适应模糊控制器，并证明了所提出的设计方法不但能保证闭环系统的一致有界性。而且可使跟踪误差收敛到原点的小领域内，通过对倒立摆控制的仿真研究验证了该方法的有效性。     

A fuzzy hierarchical operator in the grey wolf optimizer algorithm

The main goal of this paper is to study the performance of the Grey Wolf Optimizer (GWO) algorithm when a new hierarchical operator is introduced in the algorithm. This new operator is basically a hierarchical transformation that is inspired in the hierarchical social pyramid of the grey wolf. This proposed operator is applied to the simulation of the hunting process in the algorithm and has 5 variants that are explained in more detail in this paper (centroid, weighted, based on the fitness and two variants using fuzzy logic). Notably the variants having the greatest impact in the GWO performance are based on the use of fuzzy logic. We also present the motivation and results of experiments, as well as the benchmark functions that were used for the tests that are presented. In addition we are presenting a comparison among all methods for 30, 64 and 128 dimensions and we conclude that the performance of the Hierarchical GWO algorithm is better when using a fuzzy variant of the hierarchical operator.