Genetic reinforcement learning through symbiotic evolution forfuzzy controller design

An efficient genetic reinforcement learning algorithm for designing fuzzy controllers is proposed in this paper. The genetic algorithm (GA) adopted in this paper is based upon symbiotic evolution which, when applied to fuzzy controller design, complements the local mapping property of a fuzzy rule. Using this Symbiotic-Evolution-based Fuzzy Controller (SEFC) design method, the number of control trials, as well as consumed CPU time, are considerably reduced when compared to traditional GA-based fuzzy controller design methods and other types of genetic reinforcement learning schemes. Moreover, unlike traditional fuzzy controllers, which partition the input space into a grid, SEFC partitions the input space in a flexible way, thus creating fewer fuzzy rules. In SEFC, different types of fuzzy rules whose consequent parts are singletons, fuzzy sets, or linear equations (TSK-type fuzzy rules) are allowed. Further, the free parameters (e.g., centers and widths of membership functions) and fuzzy rules are all tuned automatically. For the TSK-type fuzzy rule especially, which put the proposed learning algorithm in use, only the significant input variables are selected to participate in the consequent of a rule. The proposed SEFC design method has been applied to different simulated control problems, including the cart-pole balancing system, a magnetic levitation system, and a water bath temperature control system. The proposed SEFC has been verified to be efficient and superior from these control problems, and from comparisons with some traditional GA-based fuzzy systems.     

T–S fuzzy model identification based on a novel fuzzy c-regression model clustering algorithm

This paper proposes a novel approach for identification of Takagi–Sugeno (T–S) fuzzy model, which is based on a new fuzzy c-regression model (FCRM) clustering algorithm. The clustering prototype in fuzzy space partition is hyper-plane, so FCRM clustering technique is more suitable to be applied in premise parameters identification of T–S fuzzy model. A new FCRM clustering algorithm (NFCRMA) is presented, which is deduced from the fuzzy clustering objective function of FCRM with Lagrange multiplier rule, possessing integrative and concise structure. The proposed approach consists mainly of two steps: premise parameter identification and consequent parameter identification. The NFCRMA is utilized to partition the input–output data and identify the premise parameters, which can discover the real structure of the training data; on the other hand, orthogonal least square is exploited to identify the consequent parameters. Finally, some examples are given to verify the validity of the proposed modeling approach, and the results show the new approach is very efficient and of high accuracy.     

典型T-S模糊系统是通用逼近器

研究的多输入单输出T－S模糊系统采用输入变量的线性函数作为规则后件,称为典型T－S模糊系统,在每个输入变量的模糊子集满足一致性以及隶属函数连续且分段可微的条件下,证明了典型T－S模糊系统是通和逼近器,以此为基础,提出当典型T－S模糊系统采用BP算法进行在线学习时,可以仅调节规则后件的参数,而同时仍然能够确保通用逼所性。     

A comparative study on similarity-based fuzzy reasoning methods

If the given fact for an antecedent in a fuzzy production rule (FPR) does not match exactly with the antecedent of the rule, the consequent can still be drawn by technique such as fuzzy reasoning. Many existing fuzzy reasoning methods are based on Zadeh's Compositional Rule of Inference (CRI) which requires setting up a fuzzy relation between the antecedent and the consequent part. There are some other fuzzy reasoning methods which do not use Zadeh's CRI. Among them, the similarity-based fuzzy reasoning methods, which make use of the degree of similarity between a given fact and the antecedent of the rule to draw the conclusion, are well known. In this paper, six similarity-based fuzzy reasoning methods are compared and analyzed. Two of them are newly proposed by the authors. The comparisons are two-fold. One is to compare the six reasoning methods in drawing appropriate conclusions for a given set of FPRs. The other is to compare them based on five issues: 1) types of FPR handled by these methods; 2) the complexity of the methods; 3) the accuracy of the conclusion drawn; 4) the accuracy of the similarity measure; and 5) the multi-level reasoning capability. The results have shed some lights on how to select an appropriate fuzzy reasoning method under different environments.     

A reduction approach for fuzzy rule bases of fuzzy controllers

In this paper, a new approach to reducing the number of rules in a given fuzzy rule base of a fuzzy controller is presented. The fuzzy mechanism of the fuzzy controller under consideration consists of the product-sum inference, singleton output consequents and centroid defuzzification. The output consequents in the cells of the rule table are collected and represented as an output consequent matrix. The feature of the output consequent matrix is extracted by the singular values of the matrix. The output consequent matrix is reasonably approximated with a dominant consequent matrix. Also, the elements of the dominant consequent matrix is determined to minimize the approximation error function. Then the size of the dominant consequent matrix (the size of the fuzzy rule base) is reduced through the rule combination approach. The scaling factors for the fuzzy controller with the reduced rule table are adjusted to have the control system satisfy the performance indices. The effectiveness of the proposed approach is shown using simulation and experimental results.     

Type-2 fuzzy neural networks with fuzzy clustering and differential evolution optimization

In many real-world problems involving pattern recognition, system identification and modeling, control, decision making, and forecasting of time-series, available data are quite often of uncertain nature. An interesting alternative is to employ type-2 fuzzy sets, which augment fuzzy models with expressive power to develop models, which efficiently capture the factor of uncertainty. The three-dimensional membership functions of type-2 fuzzy sets offer additional degrees of freedom that make it possible to directly and more effectively account for model’s uncertainties. Type-2 fuzzy logic systems developed with the aid of evolutionary optimization forms a useful modeling tool subsequently resulting in a collection of efficient “If-Then” rules.The type-2 fuzzy neural networks take advantage of capabilities of fuzzy clustering by generating type-2 fuzzy rule base, resulting in a small number of rules and then optimizing membership functions of type-2 fuzzy sets present in the antecedent and consequent parts of the rules. The clustering itself is realized with the aid of differential evolution.Several examples, including a benchmark problem of identification of nonlinear system, are considered. The reported comparative analysis of experimental results is used to quantify the performance of the developed networks.     

A Self-Evolving Interval Type-2 Fuzzy Neural Network With Online Structure and Parameter Learning

This paper proposes a self-evolving interval type-2 fuzzy neural network (SEIT2FNN) with online structure and parameter learning. The antecedent parts in each fuzzy rule of the SEIT2FNN are interval type-2 fuzzy sets and the fuzzy rules are of the Takagi–Sugeno–Kang (TSK) type. The initial rule base in the SEIT2FNN is empty, and the online clustering method is proposed to generate fuzzy rules that flexibly partition the input space. To avoid generating highly overlapping fuzzy sets in each input variable, an efficient fuzzy set reduction method is also proposed. This method independently determines whether a corresponding fuzzy set should be generated in each input variable when a new fuzzy rule is generated. For parameter learning, the consequent part parameters are tuned by the rule-ordered Kalman filter algorithm for high-accuracy learning performance. Detailed learning equations on applying the rule-ordered Kalman filter algorithm to the SEIT2FNN consequent part learning, with rules being generated online, are derived. The antecedent part parameters are learned by gradient descent algorithms. The SEIT2FNN is applied to simulations on nonlinear plant modeling, adaptive noise cancellation, and chaotic signal prediction. Comparisons with other type-1 and type-2 fuzzy systems in these examples verify the performance of the SEIT2FNN.      

A hardware digital fuzzy inference engine using standard integrated circuits

The paper describes a general-purpose board-level fuzzy inference engine intended primarily for experimental and educational applications. The components are all standard TTL integrated circuits (7400 series) and CMOS RAMs (CY7C series). The engine processes 16 rules in parallel with two antecedents and one consequent per rule. The design may easily be scaled to accommodate more or fewer rules. Static RAMs are used to store membership functions of both antecedent and consequent variables. “Min-max” composition is used for inferencing, and for defuzzification, the mean of maxima strategy is used. Simulation on VALID CAE software predicts that the engine is capable of performing up to 1.56 million fuzzy logic inferences per second.     

Rule chaining in fuzzy expert systems

A fuzzy expert system must do rule chaining differently than a nonfuzzy expert system. In particular, any rule that can fire with a particular linguistic variable in its consequent must fire before any rule whose antecedent conditions depend upon the resultant fuzzy set value of the consequent linguistic variable is allowed to fire. The dependent rules would be considered in a chain with the fuzzy rules which generate or assert the needed fuzzy linguistic variable. A recent paper by J. Pan et al. (1998) points out that a version of the FuzzyCLIPS expert system shell does not operate with chained fuzzy rules as one would expect. They introduce FuzzyShell which is described as the only known shell to have the expected fuzzy rule chaining performance. We show several approaches to obtaining the desired behavior in FuzzyCLIPS. Further, a potential pitfall with the FuzzyShell approach to dealing with chaining is pointed out     

Knowledge Representation Using High-Level Fuzzy Petri Nets

This correspondence presents a high-level fuzzy Petri net (HLFPN) model to represent the fuzzy production rules of a knowledge-based system, where a fuzzy production rule is the one that describes the fuzzy relation between the antecedent and the consequent. The HLFPN can be used to model fuzzy IF-THEN rules and IF-THEN-ELSE rules, where the fuzzy truth values of the propositions are restricted to [0, 1]. Based on the HLFPN model, an efficient algorithm is proposed to automatically reason about imprecise and fuzzy information. In this correspondence, a novel model to represent fuzzy knowledge is developed. When compared with other related models, the HLFPN model preserves several significant advantages. Finally, main results are presented in the form of eight properties and are supported by a comparison with other existing algorithms     

基于模糊子空间聚类的0 阶岭回归TSK 模糊系统

经典数据驱动型TSK模糊系统在利用高维数据训练模型时,由于规则前件采用的特征过多,导致规则的解释性和简洁性下降.对此,根据模糊子空间聚类算法的子空间特性,为TSK模型添加特征抽取机制,并进一步利用岭回归实现后件的学习,提出一种基于模糊子空间聚类的0阶岭回归TSK模型构建方法.该方法不仅能为规则抽取出重要子空间特征,而且可为不同规则抽取不同的特征.在模拟和真实数据集上的实验结果验证了所提出方法的优势.     

Adaptive Fuzzy Neural Networks as identifiers of discrete-time nonlinear dynamic systems

An adaptive supervised learning scheme is proposed in this paper for training Fuzzy Neural Networks (FNN) to identify discrete-time nonlinear dynamical systems. The FNN constructs are neural-network-based connectionist models consisting of several layers that are used to implement the functions of a fuzzy logic system. The fuzzy rule base considered here consists of Takagi-Sugeno IF-THEN rules, where the rule outputs are realized as linear polynomials of the input components. The FNN connectionist model is functionally partitioned into three separate parts, namely, the premise part, which provides the truth values of the rule preconditional statements, the consequent part providing the rule outputs, and the defuzzification part computing the final output of the FNN construct. The proposed learning scheme is a two-stage training algorithm that performs both structure and parameter learning, simultaneously. First, the structure learning task determines the proper fuzzy input partitions and the respective precondition matching, and is carried out by means of the rule base adaptation mechanism. The rule base adaptation mechanism is a self-organizing procedure which progressively generates the proper fuzzy rule base, during training, according to the operating conditions. Having completed the structure learning stage, the parameter learning is applied using the back-propagation algorithm, with the objective to adjust the premise/consequent parameters of the FNN so that the desired input/output representation is captured to an acceptable degree of accuracy. The structure/parameter training algorithm exhibits good learning and generalization capabilities as demonstrated via a series of simulation studies. Comparisons with conventional multilayer neural networks indicate the effectiveness of the proposed scheme.     

General SISO Takagi-Sugeno fuzzy systems with linear ruleconsequent are universal approximators

Takagi-Sugeno (TS) fuzzy systems have been employed as fuzzy controllers and fuzzy models in successfully solving difficult control and modeling problems in practice. Virtually all the TS fuzzy systems use linear rule consequent. At present, there exist no results (qualitative or quantitative) to answer the fundamentally important question that is especially critical to TS fuzzy systems as fuzzy controllers and models, “Are TS fuzzy systems with linear rule consequent universal approximators?” If the answer is yes, then how can they be constructed to achieve prespecified approximation accuracy and what are the sufficient renditions on systems configuration? In this paper, we provide answers to these questions for a general class of single-input single-output (SISO) fuzzy systems that use any type of continuous input fuzzy sets, TS fuzzy rules with linear consequent and a generalized defuzzifier containing the widely used centroid defuzzifier as a special case. We first constructively prove that this general class of SISO TS fuzzy systems can uniformly approximate any polynomial arbitrarily well and then prove, by utilizing the Weierstrass approximation theorem, that the general TS fuzzy systems can uniformly approximate any continuous function with arbitrarily high precision. Furthermore, we have derived a formula as part of sufficient conditions for the fuzzy approximation that can compute the minimal upper bound on the number of input fuzzy sets and rules needed for any given continuous function and prespecified approximation error bound, An illustrative numerical example is provided     

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.     

FINDING SIMPLIFIED FUZZY IF-THEN RULES FOR FUNCTION APPROXIMATION PROBLEMS USING A FUZZY DATA MINING APPROACH

ABSTRACT

A fuzzy if-then rule whose consequent part is a real number is referred to as a simplified fuzzy rule. Since no defuzzification is required for this rule type, it has been widely used in function approximation problems. Furthermore, data mining can be used to discover useful information by exploring and analyzing data. Therefore, this paper proposes a fuzzy data mining approach to discover simplified fuzzy if-then rules from numerical data in order to approximate an unknown mapping from input to output. Since several pre-specified parameters for deriving fuzzy rules are not easily specified, they are automatically determined by the genetic algorithm with binary chromosomes. To evaluate performance of the proposed method, computer simulations are performed on various numerical data sets, showing that the fitting ability and the generalization ability of the proposed method are comparable to the known fuzzy rule-based methods.     

A comparative study on sufficient conditions for Takagi-Sugenofuzzy systems as universal approximators

Universal approximation is the basis of theoretical research and practical applications of fuzzy systems. Studies on the universal approximation capability of fuzzy systems have achieved great progress in recent years. In this paper, linear Takagi-Sugeno (TS) fuzzy systems that use linear functions of input variables as rule consequent and their special case, named simplified fuzzy systems that use fuzzy singletons as rule consequent, are investigated. On condition that overlapped fuzzy sets are employed, new sufficient conditions for simplified fuzzy systems and linear TS fuzzy systems as universal approximators are given, respectively. Then, a comparative study on existing sufficient conditions is carried out with numeric examples     

基于核映射的高阶Takagi-Sugeno模糊模型

本文研究规则后件为非线性函数的高阶Takagi-Sugeno(TS)模糊系统.为求解规则后件的函数表达式,首先通过一个核映射将原输入空间映射到高维特征空间,使原空间的非线性子模型转化为高维特征空间的线性子模型,获得了规则后件的非线性函数的计算公式.然后,给出了用核模糊聚类和最小二乘支持向量机设计模糊系统的一种新算法.最后通过4个公开数据集上的仿真实验验证了所提算法的逼近能力、推广能力和鲁棒性能.     

Simplification of a fuzzy mechanism with rule combination

 In this paper, a systematic approach to reduce the complexity of a fuzzy controller with the rule combination of a fuzzy rule base is presented. The complexity of a fuzzy controller is defined to be the computation load in this work. The proposed rule combination approach can be applied to the fuzzy mechanisms with product–sum and min–max inferences. With the input membership functions indexed in sequence for each input variable, the n-dimensional fuzzy rule table is represented as vectors so that the combination of the fuzzy rule base is realizable. Then the adjacent fuzzy rules with the same output consequent are combined to have smaller size of fuzzy rule base. The fuzzy mechanism with the combined rule table is shown to have the same output with the original fuzzy mechanism (without rule combination). Thus, in many applications, the rule combination approach presented in this paper can be used to reduce the complexity of the fuzzy mechanism without degrading the performances. Moreover, the Don't Care fuzzy rules are defined and it is indicated that the number of the necessary fuzzy rules might be decreased when the Don't Care fuzzy rules are taken into consideration. Further, the properties of the simplification approach for the fuzzy rule base of the fuzzy mechanism are discussed.     

SIGNIFICANT FUZZY RULES EXTRACTION BY AN SVD–QR-BASED APPROACH

ABSTRACT

In this article, an SVD–QR-based approach is proposed to extract the important fuzzy rules from a rule base with several fuzzy rule tables to design an appropriate fuzzy system directly from some input-output data of the identified system. A fuzzy system with fuzzy rule tables is defined to approach the input-output pairs of an identified system. In the rule base of the defined fuzzy system, each fuzzy rule table corresponds to a partition of an input space. In order to extract the important fuzzy rules from the rule base of the defined fuzzy system, a firing strength matrix determined by the membership functions of the premise fuzzy sets is constructed. According to the firing strength matrix, the number of important fuzzy rules is determined by the Singular Value Decomposition SVD, and the important fuzzy rules are selected by the SVD–QR-based method. Consequently, a reconstructed fuzzy rule base composed of significant fuzzy rules is determined by the firing strength matrix. Furthermore, the recursive least-squares method is applied to determine the consequent part of the reconstructed fuzzy system according to the gathered input-output data so that a fine fuzzy system is determined by the proposed method. Finally, three nonlinear systems illustrate the efficiency of the proposed method.     

Application of fuzzy Naive Bayes and a real-valued genetic algorithm in identification of fuzzy model

We present a method to identify a fuzzy model from data by using the fuzzy Naive Bayes and a real-valued genetic algorithm. The identification of a fuzzy model is comprised of the extraction of “if–then” rules that is followed by the estimation of their parameters. The involved parameters include those which determine the membership function of fuzzy sets and the certainty factors of fuzzy if–then rules. In our method, as long as the fuzzy partition in the input–output space is given, the certainty factor of each rule is computed with the fuzzy conditional probability of the consequent conditioned on the antecedent by using the fuzzy Naive Bayes, which is a generalization of Naive Bayes. The fuzzy model involves the rules characterized by the highest values of certainty factors. The certainty factor of each rule is the fuzzy conditional probability, and it reflects the inner relationship between the antecedent and the consequent. In order to improve the accuracy of the fuzzy model, the real-valued genetic algorithm is incorporated into our identification process. This process concerns the optimization of the membership functions occurring in the rules. We just involve the parameters of membership function of the fuzzy sets into the real-valued genetic algorithm, since the certainty factor of each rule can be computed automatically. The performance of the model is shown for the backing-truck problem and the prediction of Mackey–Glass time series.