# LYAPUNOV FUNCTIONS FOR TIME-VARYING SYSTEMS SATISFYING GENERALIZED CONDITIONS OF MATROSOV THEOREM

Topic: Stabilization | All

5 décembre 2005, Salle R05, au Centre Automatique et Systèmes, Fontainebleau

14h : Frédéric MAZENC, INRA.

The classical Matrosov theorem concludes uniform stability of time varying systems via a weak Lyapunov function (positive definite, descrescent, with negative semi-definite derivative along the solutions) and another auxiliary function with derivative that is strictly non-zero where the derivative of the Lyapunov function is zero.

Recently, several generalizations of the classical Matrosov theorem that use a finite number of Lyapunov-like functions have been reported. None of these results provide with a construction of a strong Lyapunov function (positive definite, descrescent, with negative definite derivative along the solutions) that is a very useful analysis and controller design tool for nonlinear systems. We provide a construction of a strong Lyapunov function via an appropriate weak Lyapunov function and a set of Lyapunov-like functions whose derivatives along solutions of the system satisfy inequalities that have a particular triangular structure. Our results will be very useful in a range of situations where strong Lyapunov functions are needed, such as robustness analysis and Lyapunov function based controller redesign. We illustrate our results by constructing a strong Lyapunov function for a class of Euler-Lagrange systems (e.g. robotic manipulators) controlled by the well known Li-Slotine model reference adaptive controller.

14h : Frédéric MAZENC, INRA.

The classical Matrosov theorem concludes uniform stability of time varying systems via a weak Lyapunov function (positive definite, descrescent, with negative semi-definite derivative along the solutions) and another auxiliary function with derivative that is strictly non-zero where the derivative of the Lyapunov function is zero.

Recently, several generalizations of the classical Matrosov theorem that use a finite number of Lyapunov-like functions have been reported. None of these results provide with a construction of a strong Lyapunov function (positive definite, descrescent, with negative definite derivative along the solutions) that is a very useful analysis and controller design tool for nonlinear systems. We provide a construction of a strong Lyapunov function via an appropriate weak Lyapunov function and a set of Lyapunov-like functions whose derivatives along solutions of the system satisfy inequalities that have a particular triangular structure. Our results will be very useful in a range of situations where strong Lyapunov functions are needed, such as robustness analysis and Lyapunov function based controller redesign. We illustrate our results by constructing a strong Lyapunov function for a class of Euler-Lagrange systems (e.g. robotic manipulators) controlled by the well known Li-Slotine model reference adaptive controller.