Takagi-Sugeno models in a tensor product approach: exploiting the representation - Víctor Costa da Silva Campos (2015)
Takagi-Sugeno(TS) fuzzy analysis and control techniques extend several results from linear robust control theory to non linear systems. However, in many cases,the conditions used do not explore the membership functions information aside from the fact that they belong to the standard unit simplex. In addition, since we are dealing with nonlinear systems, the use of quadratic Lyapunov functions carries a certain conservatism.In this work, we propose different ways to exploit the membership functions information in parameter dependent Linear Matrix Inequalities (LMIs) as well as new classes of candidate Lyapunov functions that cover other existing classes in the literature.Asymptotically necessary conditions for parameter dependent LMIs are proposed based on partitioning the membership functions image space. Such partitions are also used to propose a switched control law and we present ways to modify the conditions of this controller such that a continuous controller can be recovered. An alternative way to reduce the number of rules of a given Takagi-Sugeno model, based on the Higher Order Singular Value Decomposition (HOSVD), is presented as well as how to model the uncertainty introduced by this rule reduction scheme. We propose the use of local transformations of the membership functions in conjunction with piece wise fuzzy Lyapunov functions so as to nd more relaxed stability conditions than others available in the literature. A novel way to deal with the time derivative of the membership functions, in cases where they are functions of the states, is proposed. This new proposition avoids the use of direct bounds over the membership functions time-derivative, and new stability and stabilization conditions are presented. Finally, we propose new LMI synthesis conditions based on piecewise-like Lyapunov functions. This class of Lyapunov functions behaves similarly to piecewise functions (in the sense that we can understand that a different function is valid for each region) while using a fuzzy formulation. By adapting recent conditions to deal with bounds on the Lyapunov functions membership functions time derivative, new synthesis conditions are presented to guarantee local stabilization