An extended Takagi–Sugeno–Kang inference system (TSK+) with fuzzy interpolation and its rule base generation

A rule base covering the entire input domain is required for the conventional Mamdani inference and Takagi-Sugeno-Kang (TSK) inference. Fuzzy interpolation enhances conventional fuzzy rule inference systems by allowing the use of sparse rule bases by which certain inputs are not covered. Given that almost all of the existing fuzzy interpolation approaches were developed to support the Mamdani inference, this paper presents a novel fuzzy interpolation approach that extends the TSK inference. This paper also proposes a data-driven rule base generation method to support the extended TSK inference system. The proposed system enhances the conventional TSK inference in two ways: 1) workable with incomplete or unevenly distributed data sets or incomplete expert knowledge that entails only a sparse rule base, and 2) simplifying complex fuzzy inference systems by using more compact rule bases for complex systems without the sacrificing of system performance. The experimentation shows that the proposed system overall outperforms the existing approaches with the utilisation of smaller rule bases

TSK Inference with Sparse Rule Bases

The Mamdani and TSK fuzzy models are fuzzy inference engines which have been most widely applied in real-world problems. Compared to the Mamdani approach, the TSK approach is more convenient when the crisp outputs are required. Common to both approaches, when a given observation does not overlap with any rule antecedent in the rule base (which usually termed as a sparse rule base), no rule can be fired, and thus no result can be generated. Fuzzy rule interpolation was proposed to address such issue. Although a number of important fuzzy rule interpolation approaches have been proposed in the literature, all of them were developed for Mamdani inference approach, which leads to the fuzzy outputs. This paper extends the traditional TSK fuzzy inference approach to allow inferences on sparse TSK fuzzy rule bases with crisp outputs directly generated. This extension firstly calculates the similarity degrees between a given observation and every individual rule in the rule base, such that the similarity degrees between the observation and all rule antecedents are greater than 0 even when they do not overlap. Then the TSK fuzzy model is extended using the generated matching degrees to derive crisp inference results. The experimentation shows the promising of the approach in enhancing the TSK inference engine when the knowledge represented in the rule base is not complete

Interval Type-2 TSK+ Fuzzy Inference System

Type-2 fuzzy sets and systems can better handle uncertainties compared to its type-1 counterpart, and the widely applied Mamdani and TSK fuzzy inference approaches have been both extended to support interval type-2 fuzzy sets. Fuzzy interpolation enhances the conventional Mamdani and TKS fuzzy inference systems, which not only enables inferences when inputs are not covered by an incomplete or sparse rule base but also helps in system simplification for very complex problems. This paper extends the recently proposed fuzzy interpolation approach TSK+ to allow the utilization of interval type-2 TSK fuzzy rule bases. One illustrative case based on an example problem from the literature demonstrates the working of the proposed system, and the application on the cart centering problem reveals the power of the proposed system. The experimental investigation confirmed that the proposed approach is able to perform fuzzy inferences using either dense or sparse interval type-2 TSK rule bases with promising results generated

Fuzzy Interpolation Systems and Applications

Fuzzy inference systems provide a simple yet effective solution to complex non-linear problems, which have been applied to numerous real-world applications with great success. However, conventional fuzzy inference systems may suffer from either too sparse, too complex or imbalanced rule bases, given that the data may be unevenly distributed in the problem space regardless of its volume. Fuzzy interpolation addresses this. It enables fuzzy inferences with sparse rule bases when the sparse rule base does not cover a given input, and it simplifies very dense rule bases by approximating certain rules with their neighbouring ones. This chapter systematically reviews different types of fuzzy interpolation approaches and their variations, in terms of both the interpolation mechanism (inference engine) and sparse rule base generation. Representative applications of fuzzy interpolation in the field of control are also revisited in this chapter, which not only validate fuzzy interpolation approaches but also demonstrate its efficacy and potential for wider applications