Linear Matrix Inequalities (LMI) optimization problems became the tool of choice for fuzzy 
control in the 1990's. Many nonlinear systems can be modelled as fuzzy systems, so fuzzy 
control may be considered as a nonlinear control technique. Useful results have been 
obtained using LMIs, however some sources of conservativeness remain when compared to 
other nonlinear approaches. This thesis deals with such issues of conservativeness and 
discusses some ideas on overcoming them.

The main advantage of Linear Matrix Inequalities formulations is that they can ensure 
stability and performance of a nonlinear system modelled by a Takagi-Sugeno fuzzy system. 
The system is described by fuzzy IF-THEN rules which present ``local'' linear systems of the 
nonlinear plant. These rules are numerically represented by a set of membership functions. 
As a drawback, current linear matrix inequalities methodologies do not include the shape of 
the membership functions. Therefore the stability is proved for any set of rules with any 
membership function that can be described by these linear models. This is a source of 
conservatism that can be reduced.

In this thesis some methodologies are presented which include the membership function 
information into the linear matrix inequalities stability and performance problem. We 
propose two main contributions in this area. The first method introduces a set of relaxation 
matrices that incorporates the information on the membership functions. The other uses the 
description of a wide class of Takagi-Sugeno fuzzy systems, labelled as Tensor-Product 
Takagi-Sugeno fuzzy systems. In these systems, each membership function is the product of 
several membership functions.

On the other hand, the problem of stability and performance for Takagi-Sugeno fuzzy 
systems is based on Lyapunov stability conditions which are not equivalent to the linear 
matrix inequalities optimization problem, the second one is conservative compared to the 
first. That is why one of the main contributions of the thesis is a set of progressively less 
conservative sufficient conditions in order to hold stability or performance conditions for 
Takagi-Sugeno fuzzy systems. These conditions are asymptotically equivalent. But the 
problem complexity increases as the conditions are closer.