FITSK: online local learning with generic fuzzy input Takagi-Sugeno-Kang fuzzy framework for nonlinear system estimation

Existing Takagi-Sugeno-Kang (TSK) fuzzy models proposed in the literature attempt to optimize the global learning accuracy as well as to maintain the interpretability of the local models. Most of the proposed methods suffer from the use of offline learning algorithms to globally optimize this multi-criteria problem. Despite the ability to reach an optimal solution in terms of accuracy and interpretability, these offline methods are not suitably applicable to learning in adaptive or incremental systems. Furthermore, most of the learning methods in TSK-model are susceptible to the limitation of the curse-of-dimensionality. This paper attempts to study the criteria in the design of TSK-models. They are: 1) the interpretability of the local model; 2) the global accuracy; and 3) the system dimensionality issues. A generic framework is proposed to handle the different scenarios in this design problem. The framework is termed the generic fuzzy input Takagi-Sugeno-Kang fuzzy framework (FITSK). The FITSK framework is extensible to both the zero-order and the first-order FITSK models. A zero-order FITSK model is suitable for the learning of adaptive system, and the bias-variance of the system can be easily controlled through the degree of localization. On the other hand, a first-order FITSK model is able to achieve higher learning accuracy for nonlinear system estimation. A localized version of recursive least-squares algorithm is proposed for the parameter tuning of the first-order FITSK model. The local recursive least-squares is able to achieve a balance between interpretability and learning accuracy of a system, and possesses greater immunity to the curse-of-dimensionality. The learning algorithms for the FITSK models are online, and are readily applicable to adaptive system with fast convergence speed. Finally, a proposed guideline is discussed to handle the model selection of different FITSK models to tackle the multi-criteria design problem of applying the TSK-model. Extensive simulations were conducted using the proposed FITSK models and their learning algorithms; their performances are encouraging when benchmarked against other popular fuzzy systems.