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Lyapunov functions for time-varying systems satisfying generalized conditions of Matrosov theorem

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Abstract

The classical Matrosov theorem concludes uniform asymptotic stability of time-varying systems via a weak Lyapunov function (positive definite, decrescent, with negative semi-definite derivative along solutions) and another auxiliary function with derivative that is strictly nonzero where the derivative of the Lyapunov function is zero (Mastrosov in J Appl Math Mech 26:1337–1353, 1962). Recently, several generalizations of the classical Matrosov theorem have been reported in Loria et al. (IEEE Trans Autom Control 50:183–198, 2005). None of these results provides a construction of a strong Lyapunov function (positive definite, decrescent, with negative definite derivative along solutions) which is a very useful analysis and controller design tool for nonlinear systems. Inspired by generalized Matrosov conditions in Loria et al. (IEEE Trans Autom Control 50:183–198, 2005), 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 simple Euler-Lagrange system controlled by an adaptive controller and use this result to determine an ISS controller.

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Authors and Affiliations

  1. Projet MERE INRIA-INRA, UMR Analyse des Systèmes, et Biométrie, INRA 2, pl. Viala, 34060, Montpellier, France

    Frédéric Mazenc

  2. Department of Electrical and Electronic Engineering, The University of Melbourne, Melbourne, VIC, 3052, Australia

    Dragan Nesic

Authors
  1. Frédéric Mazenc
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  2. Dragan Nesic
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Correspondence to Frédéric Mazenc.

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Mazenc, F., Nesic, D. Lyapunov functions for time-varying systems satisfying generalized conditions of Matrosov theorem. Math. Control Signals Syst. 19, 151–182 (2007). https://doi.org/10.1007/s00498-007-0015-7

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  • DOI: https://doi.org/10.1007/s00498-007-0015-7

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