Necessary and sufficient LMI conditions to compute quadratically stabilizing state feedback controllers for Takagi-Sugeno systems

This paper provides necessary and sufficient finite dimensional linear matrix inequality conditions to compute linearly parameter-dependent state feedback controllers ensuring quadratic stability for Takagi-Sugeno fuzzy systems. The proposed conditions are stated as progressively less conservative sets of linear matrix inequalities based on an extension of Polya's theorem, allowing to obtain a solution for the quadratic stabilizability problem whenever a solution exists. An additional design condition is also given, relying on the use of slack matrix variables to recover the control gains. Problems of decentralized control and control with a prescribed decay rate are also addressed, being the results illustrated by means of numerical examples.