Relaxed LMI conditions for closed-loop fuzzy systems with tensor-product structure

Current fuzzy control research tries to obtain the less conservative conditions to prove stability and performance of fuzzy control systems. In many fuzzy models, membership functions with multiple arguments are defined as the product of simpler ones, where all possible combinations of such products conform a fuzzy partition. In particular, such situation arises with widely used fuzzy modelling techniques for nonlinear systems. These type of fuzzy models will be denoted as tensor-product (TP) fuzzy systems, because its expressions can be understood as operations on multi-dimensional arrays. This paper discusses the generalisation to TP fuzzy systems of the results in Kim and Lee [2000. New approaches to relaxed quadratic stability condition of fuzzy control systems. IEEE Transactions on Fuzzy Systems 2, 1571–1582] and Xiaodong and Qinling [2003. New approaches to H∞H∞ controller designs based on fuzzy observers for T–S fuzzy systems via LMI. Automatica 39, 1571–1582]. The procedures here will allow to set up linear matrix inequality conditions which are less conservative than the cited ones, by exploiting the TP structure of the membership functions. A numerical example illustrates the achieved improvement.