Fundamental Study Of Probabilistic Fuzzy Logic System With Application In Complex Modeling

There always exist various uncertainties in real-world applications, which are usually come from the coarse measurement or imprecise perception of the process due to the harsh industrial environment, missing dynamics (dimensions/factors) and insufficient sampling data, etc. Though these uncertainties come from various sources, they share some common features, either in stochastic nature or non-stochastic nature, which should be properly captured for process modeling, decision making or control. Unfortunately, these two kinds of uncertainties are essentially different. The stochastic uncertainty can be captured by stochastic modeling methods. Mature statistics theories make it feasible for the stochastic modeling method to tackle stochastic information well. However, this kind of method may not be able to process the incomplete information and cognitive vagueness. On the other hand, since the fuzzy logic system is not inherently designed for the stochastic variation, it could hardly work under complex stochastic environment. It would be valuable if the statistical analysis method and the fuzzy system can be integrated into a unified platform to process uncertainties that contain both non-stochastic and stochastic features.Recently, probabilistic fuzzy logic system (PFLS) was developed to handle complex process with uncertainties from different physical spaces. It uses a three-dimensional probabilistic fuzzy set which introduces the probability dimension into the traditional fuzzy domain. Consequently, the probabilistic fuzzy logic system has the ability to deal with non-stochastic uncertainty and stochastic uncertainty simultaneously. However, research about probabilistic fuzzy logic system still remains at the beginning phase. There are still many challenges for PFLS development. For examples, the original PFLS has to be decomposed into a large number of ordinary fuzzy logic systems for reasoning, which will lead to a huge computational burden and information loss. Moreover, the probabilistic information in the fuzzy domain is hard to extract and the previous PFLS can not be applied to the decision making and pattern recognition field directly. Thus, based on the problems that exist in the previous PFLS, this paper aims to perform a fundamental study of the probabilistic fuzzy logic system and develop a practical and systematic design framework of the PFLS for the complex process modeling applications. In detail, these works consider the following problems:how to make the previous PFLS have faster computation and have a capacity to produce fuzzy output in more complete probabilistic distribution; how to use PFLS in the decision making (pattern recognition) field; how to avoid complex derivation of the probability density function under Bayesian probability framework and deal with the incomplete probabilistic information; how to develope a systematic design framework for the complex process modeling.The contents of the fundamental research in this paper mainly include the following four parts:1) A unified PFLS is built.A novel unified PFLS is introduced based on a continuous (secondary) probability density function (PDF). Using a continuous PDF, the fuzzy grades in the PFLS can be taken as stochastic variables that follow the distributions defined by PDFs. The unified inference and defuzzification operations based on the stochastic fuzzy variables can be easily carried out using the probability theory. This effort results in a faster computation and more complete probabilistic distribution in the fuzzy domain. This complete information can provide bound information of the approximation error.2) The probabilistic fuzzy decision making system is built.Because the decision making problems (classification, pattern recognition) in the real-world application always involve uncertainties in both stochastic and fuzzy nature, this paper proposes the novel probabilistic fuzzy decision making frameworks based on probability vote and probabilistic unnormalized output. They can generate the probability output information which is not achieved in other fuzzy rule based decision systems.3) A Dempster-Shafer based PFLS is built.The Dempster-Shafer theory (D-S) is introduced in the probabilistic fuzzy logic system in a very unique way. Through use of three-sigma rule popularly used in quality control, a fuzzy grade based belief structure (information granule) will be constructed to replace the secondary probability density functions in the probabilistic fuzzy set. And then a novel fuzzy set is proposed based on probabilistic information granule of fuzzy grade. Further, a novel D-S based inference framework is introduced based on the extensional Dempser's rule of combination for modeling. The obvious advantage of this novel approach is that it avoids the complicated derivation of secondary probability density function so that it can make a quick and effective inference for any kind of primary membership functions.4) A systematic design framework of PFLS is built.The fundamental philosophy is discussed in the paper for designing three major components of PFLS, based on the concepts of decomposition principle-Taylor expansion. Design the rules to accommodate the dominant dynamics, design primary fuzzy sets to handle the deterministic uncertainties, and tune the probabistic distribution in the fuzzy domain to approximate the stochastic variations. And then a systematic design framework is provided for complex process modeling. The effectiveness of this intelligent modeling framework is demonstrated by the wind speed prediction and the modeling of an industrial curing process.