Mesh-Based Affine Abstraction of Nonlinear Systems with Tighter Bounds

Kanishka Raj Singh, Qiang Shen, Sze Yong

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

In this paper, we consider the problem of piecewise affine abstraction of nonlinear systems, i.e., the over-approximation of its nonlinear dynamics by a pair of piecewise affine functions that 'includes' the dynamical characteristics of the original system. As such, guarantees for controllers or estimators based on the affine abstraction also apply to the original nonlinear system. Our approach consists of solving a linear programming (LP) problem that over-approximates the nonlinear function at only the grid points of a mesh with a given resolution and then accounting for the entire domain via an appropriate correction term. To achieve a desired approximation accuracy, we also iteratively subdivide the domain into subregions. Our method applies to nonlinear functions with different degrees of smoothness, including Lipschitz continuous functions, and improves on existing approaches by enabling the use of tighter bounds. Finally, we compare the effectiveness of our approach with existing optimization-based methods in simulation and illustrate its applicability for estimator design.

Original languageEnglish (US)
Title of host publication2018 IEEE Conference on Decision and Control, CDC 2018
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages3056-3061
Number of pages6
ISBN (Electronic)9781538613955
DOIs
StatePublished - Jan 18 2019
Event57th IEEE Conference on Decision and Control, CDC 2018 - Miami, United States
Duration: Dec 17 2018Dec 19 2018

Publication series

NameProceedings of the IEEE Conference on Decision and Control
Volume2018-December
ISSN (Print)0743-1546

Conference

Conference57th IEEE Conference on Decision and Control, CDC 2018
CountryUnited States
CityMiami
Period12/17/1812/19/18

Fingerprint

Nonlinear Function
Nonlinear systems
Nonlinear Systems
Mesh
Estimator
Subdivide
Affine Function
Lipschitz Function
Approximation
Nonlinear Dynamics
Linear programming
Smoothness
Continuous Function
Entire
Grid
Controller
Optimization
Term
Controllers
Simulation

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Modeling and Simulation
  • Control and Optimization

Cite this

Singh, K. R., Shen, Q., & Yong, S. (2019). Mesh-Based Affine Abstraction of Nonlinear Systems with Tighter Bounds. In 2018 IEEE Conference on Decision and Control, CDC 2018 (pp. 3056-3061). [8618714] (Proceedings of the IEEE Conference on Decision and Control; Vol. 2018-December). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/CDC.2018.8618714

Mesh-Based Affine Abstraction of Nonlinear Systems with Tighter Bounds. / Singh, Kanishka Raj; Shen, Qiang; Yong, Sze.

2018 IEEE Conference on Decision and Control, CDC 2018. Institute of Electrical and Electronics Engineers Inc., 2019. p. 3056-3061 8618714 (Proceedings of the IEEE Conference on Decision and Control; Vol. 2018-December).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Singh, KR, Shen, Q & Yong, S 2019, Mesh-Based Affine Abstraction of Nonlinear Systems with Tighter Bounds. in 2018 IEEE Conference on Decision and Control, CDC 2018., 8618714, Proceedings of the IEEE Conference on Decision and Control, vol. 2018-December, Institute of Electrical and Electronics Engineers Inc., pp. 3056-3061, 57th IEEE Conference on Decision and Control, CDC 2018, Miami, United States, 12/17/18. https://doi.org/10.1109/CDC.2018.8618714
Singh KR, Shen Q, Yong S. Mesh-Based Affine Abstraction of Nonlinear Systems with Tighter Bounds. In 2018 IEEE Conference on Decision and Control, CDC 2018. Institute of Electrical and Electronics Engineers Inc. 2019. p. 3056-3061. 8618714. (Proceedings of the IEEE Conference on Decision and Control). https://doi.org/10.1109/CDC.2018.8618714
Singh, Kanishka Raj ; Shen, Qiang ; Yong, Sze. / Mesh-Based Affine Abstraction of Nonlinear Systems with Tighter Bounds. 2018 IEEE Conference on Decision and Control, CDC 2018. Institute of Electrical and Electronics Engineers Inc., 2019. pp. 3056-3061 (Proceedings of the IEEE Conference on Decision and Control).
@inproceedings{e4707712f1d34afc9b457f73f7bbc7ed,
title = "Mesh-Based Affine Abstraction of Nonlinear Systems with Tighter Bounds",
abstract = "In this paper, we consider the problem of piecewise affine abstraction of nonlinear systems, i.e., the over-approximation of its nonlinear dynamics by a pair of piecewise affine functions that 'includes' the dynamical characteristics of the original system. As such, guarantees for controllers or estimators based on the affine abstraction also apply to the original nonlinear system. Our approach consists of solving a linear programming (LP) problem that over-approximates the nonlinear function at only the grid points of a mesh with a given resolution and then accounting for the entire domain via an appropriate correction term. To achieve a desired approximation accuracy, we also iteratively subdivide the domain into subregions. Our method applies to nonlinear functions with different degrees of smoothness, including Lipschitz continuous functions, and improves on existing approaches by enabling the use of tighter bounds. Finally, we compare the effectiveness of our approach with existing optimization-based methods in simulation and illustrate its applicability for estimator design.",
author = "Singh, {Kanishka Raj} and Qiang Shen and Sze Yong",
year = "2019",
month = "1",
day = "18",
doi = "10.1109/CDC.2018.8618714",
language = "English (US)",
series = "Proceedings of the IEEE Conference on Decision and Control",
publisher = "Institute of Electrical and Electronics Engineers Inc.",
pages = "3056--3061",
booktitle = "2018 IEEE Conference on Decision and Control, CDC 2018",

}

TY - GEN

T1 - Mesh-Based Affine Abstraction of Nonlinear Systems with Tighter Bounds

AU - Singh, Kanishka Raj

AU - Shen, Qiang

AU - Yong, Sze

PY - 2019/1/18

Y1 - 2019/1/18

N2 - In this paper, we consider the problem of piecewise affine abstraction of nonlinear systems, i.e., the over-approximation of its nonlinear dynamics by a pair of piecewise affine functions that 'includes' the dynamical characteristics of the original system. As such, guarantees for controllers or estimators based on the affine abstraction also apply to the original nonlinear system. Our approach consists of solving a linear programming (LP) problem that over-approximates the nonlinear function at only the grid points of a mesh with a given resolution and then accounting for the entire domain via an appropriate correction term. To achieve a desired approximation accuracy, we also iteratively subdivide the domain into subregions. Our method applies to nonlinear functions with different degrees of smoothness, including Lipschitz continuous functions, and improves on existing approaches by enabling the use of tighter bounds. Finally, we compare the effectiveness of our approach with existing optimization-based methods in simulation and illustrate its applicability for estimator design.

AB - In this paper, we consider the problem of piecewise affine abstraction of nonlinear systems, i.e., the over-approximation of its nonlinear dynamics by a pair of piecewise affine functions that 'includes' the dynamical characteristics of the original system. As such, guarantees for controllers or estimators based on the affine abstraction also apply to the original nonlinear system. Our approach consists of solving a linear programming (LP) problem that over-approximates the nonlinear function at only the grid points of a mesh with a given resolution and then accounting for the entire domain via an appropriate correction term. To achieve a desired approximation accuracy, we also iteratively subdivide the domain into subregions. Our method applies to nonlinear functions with different degrees of smoothness, including Lipschitz continuous functions, and improves on existing approaches by enabling the use of tighter bounds. Finally, we compare the effectiveness of our approach with existing optimization-based methods in simulation and illustrate its applicability for estimator design.

UR - http://www.scopus.com/inward/record.url?scp=85062183896&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85062183896&partnerID=8YFLogxK

U2 - 10.1109/CDC.2018.8618714

DO - 10.1109/CDC.2018.8618714

M3 - Conference contribution

T3 - Proceedings of the IEEE Conference on Decision and Control

SP - 3056

EP - 3061

BT - 2018 IEEE Conference on Decision and Control, CDC 2018

PB - Institute of Electrical and Electronics Engineers Inc.

ER -