Fuzzy model reference control with adaptation mechanism

An improved approach to adaptation in fuzzy model reference learning control (FMRLC) will be introduced in this paper. The main idea of the presented method consists in including into adaptation process the input membership functions in the fuzzy controller. In comparison with original FMRLC algorithm the proposed method can be started with smaller number of input membership functions and reduces amount of penalization after few steps that results in convergent rule base and better and more reliable behavior of the closed loop that is shown on an simulation example of control of a nonlinear time-varying system.     

一种复杂模糊系统生成方法

生成模糊系统传统方法的工作量往往随输入变量数的增长而爆炸性也增加，用于抽取模糊规则的神经网络的规模迅速地增加且能量的极小值点也迅速地增多。针对这一问题，本文发展了一种新的模糊系统生成方法，将复杂系统的模糊输入，输出关系分解成简单的模糊输入，输出关系叠加，采用了一种新的网络优化的方法－基于浮点编码的遗传算法来生成该系统。     

Genetic tuning of fuzzy inference within fuzzy classifier systems

Abstract: In generating a suitable fuzzy classifier system, significant effort is often placed on the determination and the fine tuning of the fuzzy sets. However, in such systems little thought is given to the way in which membership functions are combined within the fuzzy rules. Often traditional fuzzy inference strategies are used which consequently provide no control over how strongly or weakly the inference is applied within these rules. Furthermore such strategies will allow no interaction between grades of membership. A number of theoretical fuzzy inference operators have been proposed for both regression and classification problems but they have not been investigated in the context of real-world applications. In this paper we propose a novel genetic algorithm framework for optimizing the strength of fuzzy inference operators concurrently with the tuning of membership functions for a given fuzzy classifier system. Each fuzzy system is generated using two well-established decision tree algorithms: C4.5 and CHAID. This will enable both classification and regression problems to be addressed within the framework. Each solution generated by the genetic algorithm will produce a set of fuzzy membership functions and also determine how strongly the inference will be applied within each fuzzy rule. We investigate several theoretical proven fuzzy inference techniques (T-norms) in the context of both classification and regression problems. The methodology proposed is applied to a number of real-world data sets in order to determine the effects of the simultaneous tuning of membership functions and inference parameters on the accuracy and robustness of fuzzy classifiers.     

Avoiding exponential parameter growth in fuzzy systems

For standard fuzzy systems where the input membership functions are defined on a grid on the input space, and all possible combinations of rules are used, there is an exponential growth in the number of parameters of the fuzzy system as the number of input dimensions increases. This “curse of dimensionality” effect leads to problems with design of fuzzy controllers (e.g., how to tune all these parameters), training of fuzzy estimators (e.g., complexity of a gradient algorithm for training, and problems with “over parameterization” that lead to poor convergence properties), and with computational complexity in the implementation for practical problems. We introduce a fuzzy system whose number of parameters grows linearly depending upon the number of inputs, even though it is constructed by using all possible combinations of the membership functions in defining the rules. We prove that this fuzzy system is equivalent to the standard fuzzy system as long as its parameters are specified in a certain way. Then, we show that it still holds the universal approximator property by using the Stone-Welerstrass theorem. Finally, we illustrate the performance of the fuzzy system via an application     

Orthogonal Fuzzy Rule-Based Systems: Selection of Optimum Rules

In this paper, the concept of orthogonal fuzzy rule-based systems is introduced. Orthogonal rules are an extension to the definition of orthogonal vectors when the vectors are vectors of membership functions in the antecedent part of rules. The number and combination of rules in a fuzzy rule-based system will be optimised by applying orthogonal rules. The number of rules, and subsequently the complexity of the fuzzy rule-based systems, are directly associated with the number of input variables and distinguishable membership functions for each individual input variable. A subset of rules can be used if it is known which subset provides closer behaviour to the case when all rules are used. Orthogonal fuzzy rule-based systems are proposed as a judgment as to whether the optimal rules are selected. The application of orthogonal fuzzy rules becomes essential when fuzzy rule-based systems containing many inputs are used. An illustrative example is presented to create a model for the solder paste printing stage of surface mount tech-nology.     

Optimization of multiple input–output fuzzy membership functions using clonal selection algorithm

A clonal selection algorithm (CLONALG) inspires from clonal selection principle used to explain the basic features of an adaptive immune response to an antigenic stimulus. It takes place in various scientific applications and it can be also used to determine the membership functions in a fuzzy system. The aim of the study is to adjust the shape of membership functions and a novice aspect of the study is to determine the membership functions. Proposed method has been implemented using a developed CLONALG program for a multiple input–output (MI–O) fuzzy system. In this study, GA and binary particle swarm optimization (BPSO) are used for implementing the proposed method as well and they are compared. It has been shown that using clonal selection algorithm is advantageous for finding optimum values of fuzzy membership functions     

A systematic method for design of multivariable fuzzy logic controlsystems

This paper proposes a systematic method to design a multivariable fuzzy logic controller for large-scale nonlinear systems. In designing a fuzzy logic controller, the major task is to determine fuzzy rule bases, membership functions of input/output variables, and input/output scaling factors. In this work, the fuzzy rule base is generated by a rule-generated function, which is based on the negative gradient of a system performance index; the membership functions of isosceles triangle of input/output variables are fixed in the same cardinality and only the input/output scaling factors are generated from a genetic algorithm based on a fitness function. As a result, the searching space of parameters is narrowed down to a small space, the multivariable fuzzy logic controller can quickly constructed, and the fuzzy rules and the scaling factors can easily be determined. The performance of the proposed method is examined by computer simulations on a Puma 560 system and a two-inverted pendulum system     

An efficient quantum neuro-fuzzy classifier based on fuzzy entropy and compensatory operation

In this paper, a quantum neuro-fuzzy classifier (QNFC) for classification applications is proposed. The proposed QNFC model is a five-layer structure, which combines the compensatory-based fuzzy reasoning method with the traditional Takagi–Sugeno–Kang (TSK) fuzzy model. The compensatory-based fuzzy reasoning method uses adaptive fuzzy operations of neuro-fuzzy systems that can make the fuzzy logic system more adaptive and effective. Layer 2 of the QNFC model contains quantum membership functions, which are multilevel activation functions. Each quantum membership function is composed of the sum of sigmoid functions shifted by quantum intervals. A self-constructing learning algorithm, which consists of the self-clustering algorithm (SCA), quantum fuzzy entropy and the backpropagation algorithm, is also proposed. The proposed SCA method is a fast, one-pass algorithm that dynamically estimates the number of clusters in an input data space. Quantum fuzzy entropy is employed to evaluate the information on pattern distribution in the pattern space. With this information, we can determine the number of quantum levels. The backpropagation algorithm is used to tune the adjustable parameters. The simulation results have shown that (1) the QNFC model converges quickly; (2) the QNFC model has a higher correct classification rate than other models.     

Adaptive-tree-structure-based fuzzy inference system

A new fuzzy inference system named adaptive-tree-structure-based fuzzy inference system (ATSFIS) is proposed, which is abbreviated as fuzzy tree (FT). The fuzzy partition of input data set and the membership function of every subset are obtained by means of the fuzzy binary tree structure based algorithm. Two structures of FT, FT-I, and FT-II, are presented. The characteristics of FT are: 1) The parameters of antecedent and consequent for a Takagi-Sugeno fuzzy model are learned simultaneously; and 2) The fuzzy partition of input data set is adaptive to the pattern of data distribution to optimize the number of the subsets automatically. The main advantage of FT is more suitable to solve the problems, for which the number of input dimension is large, since by using the fuzzy binary tree, every farther set will be partitioned into only two subsets no matter how large the input dimension is. Therefore, in some sense the "rule explosion" will be avoided possibly. In comparison with some existing fuzzy inference systems, it is shown that the FT is also of less computation and high accuracy. The advantages of FT are illustrated by simulation results.     

A Genetic Approach for Simultaneous Design of Membership Functions and Fuzzy Control Rules

Based on the genetic algorithm (GA), an approach is proposed for simultaneous design of membership functions and fuzzy control rules since these two components are interdependent in designing a fuzzy logic controller (FLC). With triangular membership functions, the left and right widths of these functions, the locations of their peaks, and the fuzzy control rules corresponding to every possible combination of input linguistic variables are chosen as parameters to be optimized. By using a proportional scaling method, these parameters are then transformed into real-coded chromosomes, over which the offspring are generated by rank-based reproduction, convex crossover, and nonuniform mutation. Meanwhile, the concept of enlarged sampling space is used to expedite the convergence of the evolutionary process. To show the feasibility and validity of the proposed method, a cart-centering example will be given. The simulation results will show that the designed FLC can drive the cart system from any given initial state to the desired final state even when the cart mass varies within a wide range.     

模糊神经网络变幅液压系统故障诊断

针对变幅液压系统复杂性、不确定性、模糊性的特点,提出基于故障树的模糊神经网络作为变幅液压系统故障诊断的方法。该方法利用故障树知识提取变幅液压系统故障诊断的输入变量和输出变量,引入模糊逻辑的概念,采用模糊隶属函数来描述这些故障的程度,利用Levenberg-Marquardt优化算法对神经网络进行训练,系统推理速度快,容错能力强,并通过实例分析验证了变幅液压系统模糊神经网络故障诊断的有效性。     

Development of a systematic methodology of fuzzy logic modeling

This paper proposes a systematic methodology of fuzzy logic modeling for complex system modeling. It has a unified parameterized reasoning formulation, an improved fuzzy clustering algorithm, and an efficient strategy of selecting significant system inputs and their membership functions. The reasoning mechanism introduces 4 parameters whose variation provides a continuous range of inference operation. As a result, we are no longer restricted to standard extremes in any step of reasoning. The fuzzy model itself can then adjust the reasoning process by optimizing the inference parameters based on input-output data. The fuzzy rules are generated through fuzzy c-means (FCM) clustering. Major bottlenecks are addressed and analytical solutions are suggested. We also address the classification process to extend the derived fuzzy partition to the entire output space. In order to select suitable input variables among a finite number of candidates (unlike traditional approaches) we suggest a new strategy through which dominant input parameters are assigned in one step and no iteration process is required. Furthermore, a clustering technique called fuzzy fine clustering is introduced to assign the input membership functions. In order to evaluate the proposed methodology, two examples-a nonlinear function and a gas furnace dynamic procedure-are investigated in detail. The significant improvement of the model is concluded compared to other fuzzy modeling approaches     

A NEURO‐FUZZY SYSTEM DESIGN METHODOLOGY FOR VIBRATION CONTROL

Fuzzy system has been known to provide a framework for handling uncertainties and imprecision by taking linguistic information from human experts. However, difficulties arise in determining effectively the fuzzy system configuration, i.e., the number of rules, input and output membership functions. A neuro‐fuzzy system design methodology by combining neural network and fuzzy logic is developed in this paper to adaptively adjust the fuzzy membership functions and dynamically optimize the linguistic‐fuzzy rules. The structure of a five‐layer feedforward network is shown to determine systematically the correct fuzzy logic rules, tune optimally (in the sense of local region) the parameters of the membership functions, and perform accurately the fuzzy inference. It is shown both numerically and experimentally that engineering applications of the neuro‐fuzzy system to vibration control have been very successful.     

模糊聚类辨识算法

采用模糊输入聚类算法来辨识系统的模型,通过两个模糊聚类准则函数求得聚类中心和模糊规则数,然后求出各个子窨支态模型,再用隶属函数光滑地把他们连接成一个全局动态模糊模型,这种模型可以转化成状态空间模型,从而进行控制器的设计和稳定性分析。     

模糊逻辑系统的快速构造与优化

提出一种直接利用均匀分布于待逼近系统输入空间的Ｉ／Ｏ数据,快速构造满足一定精度要求的模糊逻辑系统方法,并从理论上证明了该方法的可行笥。在此基础上采用一种新型的ＧＡ＋ＢＰ混合自治对模糊逻辑系统进行优化,以求用最少的规则数实现满意的精度。数字仿真结果表明这种快速构造和优化方法是可行和高效的。     

一种基于聚类的超闭球模糊神经网络

针对一类不确定非线性多输入多输出复杂系统,根据系统的输入输出数据对,提出一种基于聚类的超闭球模糊神经网络系统．该系统通过改进的模糊聚类方法（FCM）确定模糊规则数,采用高维隶属度函数取代常规的单维隶属度函数,并对隶属度函数中心值和隶属度函数参数采用一步通过算法,所提方法可降低系统的模糊规则数,简化网络计算．此外,当系统的输入输出发生变化时,可实现模糊规则库的在线修改．仿真实例验证了所提方法的有效性．     

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.     

An ART-based fuzzy adaptive learning control network

This paper addresses the structure and an associated online learning algorithm of a feedforward multilayer neural net for realizing the basic elements and functions of a fuzzy controller. The proposed fuzzy adaptive learning control network (FALCON) can be contrasted with traditional fuzzy control systems in network structure and learning ability. An online structure/parameter learning algorithm, FALCON-ART, is proposed for constructing FALCON dynamically. It combines backpropagation for parameter learning and fuzzy ART for structure learning. FALCON-ART partitions the input state space and output control space using irregular fuzzy hyperboxes according to the data distribution. In many existing fuzzy or neural fuzzy control systems, the input and output spaces are always partitioned into “grids”. As the number of variables increases, the number of partitioned grids grows combinatorially. To avoid this problem in some complex systems, FALCON-ART partitions the I/O spaces flexibly based on data distribution. It can create and train FALCON in a highly autonomous way. In its initial form, there is no membership function, fuzzy partition, and fuzzy logic rule. They are created and begin to grow as the first training pattern arrives. Thus, the users need not give it any a priori knowledge or initial information. FALCON-ART can online partition the I/O spaces, tune membership functions, find proper fuzzy logic rules, and annihilate redundant rules dynamically upon receiving online data     

Fuzzy local linearization and local basis function expansion innonlinear system modeling

Fuzzy local linearization is compared with local basis function expansion for modeling unknown nonlinear processes. First-order Takagi-Sugeno fuzzy model and the analysis of variance (ANOVA) decomposition are combined for the fuzzy local linearization of nonlinear systems, in which B-splines are used as membership functions of the fuzzy sets for input space partition. A modified algorithm for adaptive spline modeling of observation data (MASMOD) is developed for determining the number of necessary B-splines and their knot positions to achieve parsimonious models. This paper illustrates that fuzzy local linearization models have several advantages over local basis function expansion based models in nonlinear system modeling.     

自适应神经网络模糊推理系统最优参数的研究

模糊规则的提取和隶属度函数的学习是模糊系统设计中重要而困难的问题。自适应神经网络模糊推理系统（ANFIS）能基于数据建模,无须专家经验,自动产生模糊规则和调整隶属度函数。在建立一个初始系统进行训练时,其隶属度函数的类型、隶属度函数的数日以及训练次数都是待定的,这三个参数的选择直接影响系统训练后的效果,它们的确定方法有待研究。该文应用自适应神经网络模糊推理系统的方法对一个典型系统进行建模仿真,并阐述这三个参数的寻优方法。