On the definition of non quadratic Lyapunov function for continuous Takagi Sugeno fuzzy models through their discretized forms

Since a few years, LMIs conditions associated to the control of continuous Takagi Sugeno (TS) fuzzy models have used non quadratic Lyapunov functions. Indeed they are much more general than classical quadratic functions. However, there are requirements about the derivative of the membership functions appearing in the LMIs. Whereas, this problem doesn´t exist with discrete time models. This study tries to put a bridge between the continuous and discrete cases for the class of continuous Takagi Sugeno fuzzy models which can be exactly discretized. Indeed, for this particular class, ones the stability of the discrete model is ensured, the same control law applied to the continuous model will ensure the stability too. The interest of such an approach is that complex control laws can be applied with no hypothesis on the membership functions. Simulation examples illustrate the effectiveness of the proposed approach.