The Generalized TP Model Transformation for T–S Fuzzy Model Manipulation and Generalized Stability Verification

This paper integrates various ideas about the tensor product (TP) model transformation into one conceptual framework and formulates it in terms of the Takagi–Sugeno (T–S) fuzzy model manipulation and control design framework. Several new extensions of the TP model transformation are proposed, such as the quasi and “full,” compact and rank-reduced higher order singular-value-decomposition-based canonical form of T–S fuzzy models, and the bilinear-, multi, pseudo-, convex-, partial TP model transformations. All of these extensions together form the generalized TP model transformation, which provides an effective tool to freely and readily manipulate the antecedent sets and rules of T–S fuzzy models and also provides main fuzzy rule component analysis, as well as a means for complexity and accuracy tradeoffs. It is demonstrated in this paper that the proposed manipulation forms a new, effective, and necessary optimization step of T–S fuzzy or polytopic models and linear-matrix-inequality-based control design, and can also decrease conservativeness. Identification techniques are typically constructed according to the available data and measurement set, as well as the type of system to be identified. As a result, they may not always provide good representations for control design frameworks. This paper demonstrates that the proposed TP model transformation is unique in that it bridges between various soft-computing-based identification techniques and T–S fuzzy model-based approaches. Finally, this paper proposes the multi-TP model transformation, which is a tractable and nonheuristic framework to verify the stability of the result of fuzzy or various soft-computing-based control designs. The multi-TP model transformation could provide an answer to the frequently emerging criticisms regarding the lack of mathematical stability verification techniques in the soft-computing-based control design. Control examples are provided in this paper.