Data driven verification of positive invariant sets for discrete, nonlinear systems

Amy K. Strong, Leila J. Bridgeman
Proceedings of the 6th Annual Learning for Dynamics & Control Conference, PMLR 242:1477-1488, 2024.

Abstract

Invariant sets are essential for understanding the stability and safety of nonlinear systems. However, certifying the existence of a positive invariant set for a nonlinear model is difficult and often requires knowledge of the system’s dynamic model. This paper presents a data driven method to certify a positive invariant set for an unknown, discrete, nonlinear system. A triangulation of a subset of the state space is used to query data points. Then, linear programming is used to create a continuous piecewise affine function that fulfills the criteria of the Extended Invariant Set Principle by leveraging an inequality error bound that uses the Lipschitz constant of the unknown system. Numerical results demonstrate the program’s ability to certify positive invariant sets from sampled data.

Cite this Paper

BibTeX
@InProceedings{pmlr-v242-strong24a, title = {Data driven verification of positive invariant sets for discrete, nonlinear systems}, author = {Strong, Amy K. and Bridgeman, Leila J.}, booktitle = {Proceedings of the 6th Annual Learning for Dynamics & Control Conference}, pages = {1477--1488}, year = {2024}, editor = {Abate, Alessandro and Cannon, Mark and Margellos, Kostas and Papachristodoulou, Antonis}, volume = {242}, series = {Proceedings of Machine Learning Research}, month = {15--17 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v242/strong24a/strong24a.pdf}, url = {https://proceedings.mlr.press/v242/strong24a.html}, abstract = {Invariant sets are essential for understanding the stability and safety of nonlinear systems. However, certifying the existence of a positive invariant set for a nonlinear model is difficult and often requires knowledge of the system’s dynamic model. This paper presents a data driven method to certify a positive invariant set for an unknown, discrete, nonlinear system. A triangulation of a subset of the state space is used to query data points. Then, linear programming is used to create a continuous piecewise affine function that fulfills the criteria of the Extended Invariant Set Principle by leveraging an inequality error bound that uses the Lipschitz constant of the unknown system. Numerical results demonstrate the program’s ability to certify positive invariant sets from sampled data.} }
Endnote
%0 Conference Paper %T Data driven verification of positive invariant sets for discrete, nonlinear systems %A Amy K. Strong %A Leila J. Bridgeman %B Proceedings of the 6th Annual Learning for Dynamics & Control Conference %C Proceedings of Machine Learning Research %D 2024 %E Alessandro Abate %E Mark Cannon %E Kostas Margellos %E Antonis Papachristodoulou %F pmlr-v242-strong24a %I PMLR %P 1477--1488 %U https://proceedings.mlr.press/v242/strong24a.html %V 242 %X Invariant sets are essential for understanding the stability and safety of nonlinear systems. However, certifying the existence of a positive invariant set for a nonlinear model is difficult and often requires knowledge of the system’s dynamic model. This paper presents a data driven method to certify a positive invariant set for an unknown, discrete, nonlinear system. A triangulation of a subset of the state space is used to query data points. Then, linear programming is used to create a continuous piecewise affine function that fulfills the criteria of the Extended Invariant Set Principle by leveraging an inequality error bound that uses the Lipschitz constant of the unknown system. Numerical results demonstrate the program’s ability to certify positive invariant sets from sampled data.
APA
Strong, A.K. & Bridgeman, L.J.. (2024). Data driven verification of positive invariant sets for discrete, nonlinear systems. Proceedings of the 6th Annual Learning for Dynamics & Control Conference, in Proceedings of Machine Learning Research 242:1477-1488 Available from https://proceedings.mlr.press/v242/strong24a.html.

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