A generalized concept for fuzzy rule interpolation

The concept of fuzzy rule interpolation in sparse rule bases was introduced in 1993. It has become a widely researched topic in recent years because of its unique merits in the topic of fuzzy rule base complexity reduction. The first implemented technique of fuzzy rule interpolation was termed as /spl alpha/-cut distance based fuzzy rule base interpolation. Despite its advantageous properties in various approximation aspects and in complexity reduction, it was shown that it has some essential deficiencies, for instance, it does not always result in immediately interpretable fuzzy membership functions. This fact inspired researchers to develop various kinds of fuzzy rule interpolation techniques in order to alleviate these deficiencies. This paper is an attempt into this direction. It proposes an interpolation methodology, whose key idea is based on the interpolation of relations instead of interpolating /spl alpha/-cut distances, and which offers a way to derive a family of interpolation methods capable of eliminating some typical deficiencies of fuzzy rule interpolation techniques. The proposed concept of interpolating relations is elaborated here using fuzzy- and semantic-relations. This paper presents numerical examples, in comparison with former approaches, to show the effectiveness of the proposed interpolation methodology.     

Additive Fuzzy Systems: From Generalized Mixtures to Rule Continua

A generalized probability mixture density governs an additive fuzzy system. The fuzzy system's if‐then rules correspond to the mixed probability densities. An additive fuzzy system computes an output by adding its fired rules and then averaging the result. The mixture's convex structure yields Bayes theorems that give the probability of which rules fired or which combined fuzzy systems fired for a given input and output. The convex structure also results in new moment theorems and learning laws and new ways to both approximate functions and exactly represent them. The additive fuzzy system itself is just the first conditional moment of the generalized mixture density. The output is a convex combination of the centroids of the fired then‐part sets. The mixture's second moment defines the fuzzy system's conditional variance. It describes the inherent uncertainty in the fuzzy system's output due to rule interpolation. The mixture structure gives a natural way to combine fuzzy systems because mixing mixtures yields a new mixture. A separation theorem shows how fuzzy approximators combine with exact Watkins‐based two‐rule function representations in a higher‐level convex sum of the combined systems. Two mixed Gaussian densities with appropriate Watkins coefficients define a generalized mixture density such that the fuzzy system's output equals any given real‐valued function if the function is bounded and not constant. Statistical hill‐climbing algorithms can learn the generalized mixture from sample data. The mixture structure also extends finite rule bases to continuum‐many rules. Finite fuzzy systems suffer from exponential rule explosion because each input fires all their graph‐cover rules. The continuum system fires only a special random sample of rules based on Monte Carlo sampling from the system's mixture. Users can program the system by changing its wave‐like meta‐rules based on the location and shape of the mixed densities in the mixture. Such meta‐rules can help mitigate rule explosion. The meta‐rules grow only linearly with the number of mixed densities even though the underlying fuzzy if‐then rules can have high‐dimensional if‐part and then‐part fuzzy sets.     

Fuzzy if... then rule models and their transformation into oneanother

The two most essential fuzzy rule-based models used in the literature and in the industrial applications are briefly described. The way of reasoning in these models is shown. Interpolative reasoning for the case of sparse rule bases is also discussed. Rule base compression by eliminating redundant rules whose information can be reconstructed within a set accuracy interval from the remaining rules by using the previous interpolation method is shown. A general fuzzy model is discussed that contains the previous fuzzy models (as well as the nonfuzzy one) as special cases. Ways of transforming the different approximative models into each other via the general model and interpolation are presented     

Robust neurofuzzy rule base knowledge extraction and estimation using subspace decomposition combined with regularization and D-optimality.

A new robust neurofuzzy model construction algorithm has been introduced for the modeling of a priori unknown dynamical systems from observed finite data sets in the form of a set of fuzzy rules. Based on a Takagi-Sugeno (T-S) inference mechanism a one to one mapping between a fuzzy rule base and a model matrix feature subspace is established. This link enables rule based knowledge to be extracted from matrix subspace to enhance model transparency. In order to achieve maximized model robustness and sparsity, a new robust extended Gram-Schmidt (G-S) method has been introduced via two effective and complementary approaches of regularization and D-optimality experimental design. Model rule bases are decomposed into orthogonal subspaces, so as to enhance model transparency with the capability of interpreting the derived rule base energy level. A locally regularized orthogonal least squares algorithm, combined with a D-optimality used for subspace based rule selection, has been extended for fuzzy rule regularization and subspace based information extraction. By using a weighting for the D-optimality cost function, the entire model construction procedure becomes automatic. Numerical examples are included to demonstrate the effectiveness of the proposed new algorithm.     

Fuzzy rule interpolation for multidimensional input spaces with applications: a case study

Fuzzy rule based systems have been very popular in many engineering applications. However, when generating fuzzy rules from the available information, this may result in a sparse fuzzy rule base. Fuzzy rule interpolation techniques have been established to solve the problems encountered in processing sparse fuzzy rule bases. In most engineering applications, the use of more than one input variable is common, however, the majority of the fuzzy rule interpolation techniques only present detailed analysis to one input variable case. This paper investigates characteristics of two selected fuzzy rule interpolation techniques for multidimensional input spaces and proposes an improved fuzzy rule interpolation technique to handle multidimensional input spaces. The three methods are compared by means of application examples in the field of petroleum engineering and mineral processing. The results show that the proposed fuzzy rule interpolation technique for multidimensional input spaces can be used in engineering applications.     

Interpolation and extrapolation of fuzzy quantities revisited – an axiomatic approach

 This paper deals with the problem of rule interpolation and rule extrapolation for fuzzy and possibilistic systems. Such systems are used for representing and processing vague linguistic If-Then-rules, and they have been increasingly applied in the field of control engineering, pattern recognition and expert systems. The methodology of rule interpolation is required for deducing plausible conclusions from sparse (incomplete) rule bases. For this purpose the well-known fuzzy inference mechanisms have to be extended or replaced by more general ones. The methods proposed so far in the literature for rule interpolation are mainly conceived for the application to fuzzy control and miss certain logical characteristics of an inference. First, a set of axioms is proposed in this paper. With this, a definition is given for the notion of interpolation, extrapolation, linear interpolation and linear extrapolation of fuzzy rules. The axioms include all the conditions that have been of interest in the previous attempts and others which either have logical characteristics or try to capture the linearity of the interpolation. A new method for linear interpolation and extrapolation of compact fuzzy quantities of the real line is suggested and analyzed in the spirit of the given definition. The method is extended to non-linear interpolation and extrapolation as well.     

两类模糊系统具有插值性的充要条件

当模糊系统具有插值性时,它必具有泛逼近性.因此,由插值性可以分析模糊系统的逼近能力.本文讨论了由“交”和“并”的方式聚合推理规则所生成的两类模糊系统的插值性问题.首先,通过分析由“单点”模糊化方法、CRI(com positional ru le of inference)算法以及“重心法”构造的模糊系统,指出模糊系统是否具有插值性关键取决于模糊蕴含算子的第二个变量为0和1时的表达式或取值.在此基础上,得到两类模糊系统具有插值性的充要条件.最后给出了满足这两个充要条件的一些常用的蕴涵算子.     

Fuzzy Interpolation and Extrapolation: A Practical Approach

Fuzzy interpolation does not only help to reduce the complexity of fuzzy models, but also makes inference in sparse rule-based systems possible. It has been successfully applied to systems control, but limited work exists for its applications to tasks like prediction and classification. Almost all fuzzy interpolation techniques in the literature make strong assumptions that there are two closest adjacent rules available to the observation, and that such rules must flank the observation for each attribute. Also, some interpolation approaches cannot handle fuzzy sets whose membership functions involve vertical slopes. To avoid such limitations and develop a more practical approach, this paper extends the work of Huang and Shen. The result enables both interpolation and extrapolation which involve multiple fuzzy rules, with each rule consisting of multiple antecedents. Two realistic applications, namely truck backer-upper control and computer activity prediction, are provided in this paper to demonstrate the utility of the extended approach. Experiment-based comparisons to the most commonly used Mamdani fuzzy reasoning mechanism, and to other existing fuzzy interpolation techniques are given to show the significance and potential of this research.     

Decomposition of a fuzzy controller based on the inference break-up method

A concept called the decomposition of multivariable control rules is presented. Fuzzy control is the application of the compositional rule of inference and it is shown how the inference of the rule base with complex rules can be reduced to the inference of a number of rule bases with simple rules. A fuzzy logic based controller is applied to a simple magnetic suspension system. The controller has proportional, integral and derivative separate parts which are tuned independently. This means that all parts have their own rule bases. By testing it was found that the fuzzy PID controller gives better performance over a typical operational range then a traditional linear PID controller. The magnetic suspension system and the contact-less optical position measurement system have been developed and applied for the comparative analysis of the real-time conventional PID control and the fuzzy control.     

可理解模糊分类系统的分层演化学习

针对模糊系统的可理解性要求,结合微粒群算法和遗传算法各自的演化特点,采用两阶段学习策略,对模糊分类系统进行分层演化。首先利用微粒群算法优化各输入变量的语言值数目及对应的模糊集参数,形成候选规则集,再应用遗传算法选择规则,得到可理解的和精确的模糊分类系统。该方法几乎无需先验知识,可直接从实值数据获取模糊分类系统,应用典型分类问题为例说明其有效性。     

Model construction, rule reduction, and robust compensation forgeneralized form of Takagi-Sugeno fuzzy systems

This paper presents a systematic procedure of fuzzy control system design that consists of fuzzy model construction, rule reduction, and robust compensation for nonlinear systems. The model construction part replaces the nonlinear dynamics of a system with a generalized form of Takagi-Sugeno fuzzy systems, which is newly developed by us. The generalized form has a decomposed structure for each element of A_i and B_i matrices in consequent parts. The key feature of this structure is that it is suitable for constructing IF-THEN rules and reducing the number of IF-THEN rules. The rule reduction part provides a successive procedure to reduce the number of IF-THEN rules. Furthermore, we convert the reduction error between reduced fuzzy models and a system to model uncertainties of reduced fuzzy models. The robust compensation part achieves the decay rate controller design guaranteeing robust stability for the model uncertainties. Finally, two examples demonstrate the utility of the systematic procedure developed     

一种快速模糊推理系统

提出一种新的模糊推理系统，其模糊知识库具有紧致模糊规则库，即规则集为仅存储规则后的完全规则集，推理过程中可以根据当前输入信号值直接寻址被激励的模糊规则，从而只是有选择地执行被激励的规则，其优点是可以提高模糊推理速度，减少规则库存储容量，针对模糊芯片的VLSI实现，提出了可以根据输入信号值直接寻址被激励规则的电路。     

MULTITUNING FUZZY CONTROL SYSTEMS DESIGN

A multituning fuzzy control system structure that involves two simple, but effective tuning mechanisms, is proposed: one is called fuzzy control rule tuning mechanism (FCRTM); the other is called dynamic scalar tuning mechanism (DSTM). In FCRTM, it is used to generate the necessary control rules with a center extension method. In DSTM, it contains three fuzzy IF-THEN rules for determining the appropriate scaling factors for the fuzzy control system. In this paper, a method based on the genetic algorithm (GA) is proposed to simultaneously choose the appropriate parameters in FCRTM and DSTM. That is, the proposed GA-based method can automatically generate the required rule base of fuzzy controller and efficiently determine the appropriate map for building the dynamic scalars of fuzzy controller. A multiobjective fitness function is proposed to determine an appropriate parameter set such that not only the selected fuzzy control structure has fewer fuzzy rules, but also the controlled system has a good control performance. Finally, an inverted pendulum control problem is given to illustrate the effectiveness of the proposed control scheme.     

Online identification of evolving Takagi–Sugeno–Kang fuzzy models for crane systems

This paper suggests new evolving Takagi–Sugeno–Kang (TSK) fuzzy models dedicated to crane systems. A set of evolving TSK fuzzy models with different numbers of inputs are derived by the novel relatively simple and transparent implementation of an online identification algorithm. An input selection algorithm to guide modeling is proposed on the basis of ranking the inputs according to their important factors after the first step of the online identification algorithm. The online identification algorithm offers rule bases and parameters which continuously evolve by adding new rules with more summarization power and by modifying existing rules and parameters. The potentials of new data points are used with this regard. The algorithm is applied in the framework of the pendulum–crane system laboratory equipment. The evolving TSK fuzzy models are tested against the experimental data and a comparison with other TSK fuzzy models and modeling approaches is carried out. The comparison points out that the proposed evolving TSK fuzzy models are simple and consistent with both training data and testing data and that these models outperform other TSK fuzzy models.     

On rule self-generating for fuzzy control

It has been proved that fuzzy control is a powerful tool to control a complicated system. But, sometimes it has still suffered from collecting fuzzy control rules which is its critical part. In this article, inspired by the control strategy of the conventional PID control, we propose a rule self-generating method for fuzzy control. With the help of the proposed self-generating algorithm, we can obtain the fuzzy rules for a fuzzy controller easily. the numerical results confirm the effectivity of the designed algorithm compared with PID controller and a common Fuzzy Controller whose rules are derived from the experts' experience and knowledge by use of a practical temperature process. © 1994 John Wiley & Sons, Inc.     

Hierarchical fuzzy control

In a conventional rule based fuzzy control system, the rules are of the following form: if (condition) then (action), and all rules are essentially in a random order. The number of rules increases exponentially as the number of the system variables, on which the fuzzy rules are based, is increased. In this paper, the rules are structured in a hierarchical way so that the total number of rules will be a linear function of the system variables. The hierarchical fuzzy control algorithm developed in this paper is applied to control the feedwater flow to a steam generator of a power plant. The simulation results show that the hierarchical fuzzy controller yields superior performance over the conventional PID controller.     

Fuzzy classifier system architectures for mobile robotics: An experimental comparison

We present an experimental comparison between two approaches to optimization of the rules for a fuzzy controller. More specifically, the problem is autonomous acquisition of an “investigative” obstacle avoidance competency for a mobile robot. We report on results from investigating two alternative approaches to the use of a Learning Classifier System (LCS) to optimize the fuzzy rule base. One approach operates at the level of whole rule bases, the “Pittsburgh” LCS. The other approach operates at the level of individual rules, the “Michigan” LCS. In this work, both of these Fuzzy Classifier Systems were designed to operate only on the rules of fuzzy controllers, with predefined fuzzy membership functions. There are two main results from this work. First, both approaches were capable of producing fuzzy controllers with subtle interactions between rules leading to competencies exceeding that of the hand‐coded fuzzy controller presented in this article. Second, the Michigan approach suffered more seriously than the Pittsburgh approach from the well‐known LCS “cooperation/competition” problem, which is accentuated here by the structural combination of Evolutionary Computation and a fuzzy system. This problem was alleviated a little by the combination of a clustered subpopulation niche system and a fitness‐sharing scheme applied to the Michigan approach, but still remains. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 993–1019, 2007.     

Rule base reduction for knowledge-based fuzzy controllers with application to a vacuum cleaner

A rule base reduction and tuning algorithm is proposed as a design tool for the knowledge-based fuzzy control of a vacuum cleaner. Given a set of expert-based control rules in a fuzzy rule base structure, proposed algorithm computes the inconsistencies and redundancies in the overall rule set based on a newly proposed measure of equality of the individual fuzzy sets. An inconsistency and redundancy measure is proposed and computed for each rule in the rule base. Then the rules with high inconsistency and redundancy levels are removed from the fuzzy rule base without affecting the overall performance of the controller. The algorithm is successfully tested experimentally for the control of a commercial household vacuum cleaner. Experimental results demonstrate the effective use of the proposed algorithm.     

Multi-variable fuzzy forecasting based on fuzzy clustering and fuzzy rule interpolation techniques

In this paper, we present a new method for multi-variable fuzzy forecasting based on fuzzy clustering and fuzzy rule interpolation techniques. First, the proposed method constructs training samples based on the variation rates of the training data set and then uses the training samples to construct fuzzy rules by making use of the fuzzy C-means clustering algorithm, where each fuzzy rule corresponds to a given cluster. Then, we determine the weight of each fuzzy rule with respect to the input observations and use such weights to determine the predicted output, based on the multiple fuzzy rules interpolation scheme. We apply the proposed method to the temperature prediction problem and the Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) data. The experimental results show that the proposed method produces better forecasting results than several existing methods.     

FUZZY SLIDING MODE CONTROLLER DESIGN: INDIRECT ADAPTIVE APPROACH

In this paper, using the concept of sliding mode control SMC, a fuzzy sliding mode controller FSMC, which is synthesized by linguistic control rules, is proposed. Two sets of fuzzy rule bases are utilized to represent the controlled system. The membership functions of the THEN-part, which is used to construct a suitable equivalent control of SMC, are changed according to adaptive law. In particular, only one adaptive factor is characterized to adapt the membership functions instead of several ones in conventional adaptive approaches. Under this design scheme, we not only maintain the distribution of membership functions over state space but also reduce considerably computing time. The proposed indirect adaptive FSMC is synthesized through the following stages. First, we construct the fuzzy rule bases according to the common sense of SMC to describe the model of the controlled system, and define the fuzzy sets whose membership functions are equally distributed in state space. Then, the derived adaptive law is used to adjust the membership functions of the THEN-part to approximate an equivalent control without knowing the mathematical model of the controlled system. Third, a hitting control is developed to guarantee the stability of the control system. Finally, we smooth the hitting control via proposed heuristic control rules. We apply this FSMC to controlling a nonlinear inverted pendulum system to confirm the validity of the proposed approach.