4.7 Article

Cone-Copositive Lyapunov Functions for Complementarity Systems: Converse Result and Polynomial Approximation

Journal

IEEE TRANSACTIONS ON AUTOMATIC CONTROL
Volume 67, Issue 3, Pages 1253-1268

Publisher

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TAC.2021.3061557

Keywords

Lyapunov methods; Numerical stability; Trajectory; Dynamical systems; Optimization; Tools; Stability criteria; Constrained systems; converse Lyapunov theorem; hybrid systems; sums-of-squares optimization

Funding

  1. ANR project CONVAN [ANR-17-CE40-0019-01]
  2. Agence Nationale de la Recherche (ANR) [ANR-17-CE40-0019] Funding Source: Agence Nationale de la Recherche (ANR)

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This article establishes the existence of Lyapunov functions and describes algorithms for their numerical computation. By constructing cone-copositive Lyapunov functions, the stability analysis of state-constrained systems is addressed. The article proves that exponentially stable complementarity systems always admit a continuously differentiable cone-copositive Lyapunov function, and further investigates the approximation methods for this function.
This article establishes the existence of Lyapunov functions for analyzing the stability of a class of state-constrained systems, and it describes algorithms for their numerical computation. The system model consists of a differential equation coupled with a set-valued relation that introduces discontinuities in the vector field at the boundaries of the constraint set. In particular, the set-valued relation is described by the subdifferential of the indicator function of a closed convex cone, which results in a cone-complementarity system. The question of analyzing the stability of such systems is addressed by constructing cone-copositive Lyapunov functions. As a first analytical result, we show that exponentially stable complementarity systems always admit a continuously differentiable cone-copositive Lyapunov function. Putting some more structure on the system vector field, such as homogeneity, we can show that the aforementioned functions can be approximated by a rational function of cone-copositive homogeneous polynomials. This latter class of functions is seen to be particularly amenable for numerical computation as we provide two types of algorithms for precisely that purpose. These algorithms consist of a hierarchy of either linear or semidefinite optimization problems for computing the desired cone-copositive Lyapunov function. Some examples are given to illustrate our approach.

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