Chebyshev approximation and higher order derivatives of Lyapunov functions for estimating the domain of attraction.

Abstract

Estimating the Domain of Attraction (DA) of non-polynomial systems is a challenging problem. Taylor expansion is widely adopted for transforming a nonlinear analytic function into a polynomial function, but the performance of Taylor expansion is not always satisfactory. This paper provides solvable ways for estimating the DA via Chebyshev approximation. Firstly, for Chebyshev approximation without the remainder, higher order derivatives of Lyapunov functions are used for estimating the DA, and the largest estimate is obtained by solving a generalized eigenvalue problem. Moreover, for Chebyshev approximation with the remainder, an uncertain polynomial system is reformulated, and a condition is proposed for ensuring the convergence to the largest estimate with a selected Lyapunov function. Numerical examples demonstrate that both accuracy and efficiency are improved compared to Taylor approximation.

Bib

@inproceedings{DBLP:conf/cdc/HanP17a,
  author       = {Dongkun Han and
                  Dimitra Panagou},
  title        = {Chebyshev approximation and higher order derivatives of Lyapunov functions
                  for estimating the domain of attraction},
  booktitle    = {56th {IEEE} Annual Conference on Decision and Control, {CDC} 2017,
                  Melbourne, Australia, December 12-15, 2017},
  pages        = {1181--1186},
  publisher    = {{IEEE}},
  year         = {2017},
  url          = {https://doi.org/10.1109/CDC.2017.8263816},
  doi          = {10.1109/CDC.2017.8263816},
  timestamp    = {Fri, 04 Mar 2022 13:29:55 +0100},
  biburl       = {https://dblp.org/rec/conf/cdc/HanP17a.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}