A Nested Tensor Product Model Transformation

The tensor product model transformation (TPMT) is an emerging numerical framework of the Takagi-Sugeno (T-S) fuzzy (or polytopic) system modeling for a linear matrix inequality based system control design. A nested TPMT (NTPMT) is proposed in this paper, which merges the dimensions of the tensors and performs the TPMT iteratively. The resultant fuzzy model is in a multilevel nested tensor product (TP) structure. The vertex tensor obtained by NTPMT has fewer dimension results than the original TPMT so the number of vertices or fuzzy rules, which have been the main bottleneck for further application of the TPMT in higher dimensional systems, is expected to decrease significantly. It is also proven that the NTPMT contains the hierarchical fuzzy logic, meaning that the NTPMT is capable of conducting hierarchical fuzzy modeling and reduction. Furthermore, because the inclusion of multiple TPMTs is prone to augment the conservativeness of the resultant fuzzy model, a suboptimal convex hull rectification algorithm for the TPMT is developed based on a newly defined tightness measure, and then extended to render the NTPMT as less conservative as possible. Finally, numerical simulations on two real physical systems (two- and four-parameter dimension) are verified to demonstrate the performance of the methods.