This paper provides convex conditions with certificates of convergence for the design of state feedback controllers that quadratically stabilize and also ensure optimal H2 and H∞ performances under quadratic stability for Takagi-Sugeno continuous-time fuzzy systems. The proposed conditions are formulated as parameter-dependent linear matrix inequalities (LMIs) that have extra variables from Finsler's lemma and parameters belonging to the unit simplex. The fuzzy control law is written as a state feedback gain that is a homogeneous polynomial of degree g, encompassing the parallel distributed compensator as a special case when g = 1. Algebraic properties of the system parameters and recent results of positive polynomials are used to construct LMI relaxations which, differently from most relaxations in the literature, asymptotically converge to a solution whenever such solution exists. Due to the degree of freedom obtained with the extra variables, the conditions presented in the paper are an improvement over earlier results based on Pólya's theorem and can be viewed as an alternative to the use of techniques based on the relaxation of quadratic forms. The numerical efficiency in terms of precision and computational effort is demonstrated by means of comparisons with other methods from the literature.
Takagi-Sugeno fuzzy systems; Linear matrix inequalities; Convergent relaxations; H2 control; H∞ control
