Observer for Lipschitz nonlinear systems: Mean Value Theorem and sector nonlinearity transformation

The problem of observer design for nonlinear Lipschitz systems is dealt with in this work. An emphasis is put on the maximization of the admissible Lipschitz constant for which the observer design is possible. This problem is tackled using a Takagi-Sugeno modeling approach. The idea is to rewrite the state estimation error dynamics as an autonomous Takagi-Sugeno system, using the Mean Value Theorem and the sector nonlinearity transformation. State estimation error dynamics stability is studied with the Lyapunov theory by choosing a non-quadratic Lyapunov function and by computing its variation between m consecutive samples. The interest of these manipulations is to obtain LMI conditions admitting solutions for large values of the Lipschitz constant. Finally, illustrative examples are provided in order to highlight the performances of the proposed approach.