A Chebyshev interval method for nonlinear dynamic systems under uncertainty

Wu, J; Zhang, Y; Chen, L; Luo, Z

A Chebyshev interval method for nonlinear dynamic systems under uncertainty

Wu, J

Zhang, Y

Chen, L Luo, Z

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Publication Type:: Journal Article
Citation:: Applied Mathematical Modelling, 2013, 37 (6), pp. 4578 - 4591
Issue Date:: 2013-03-15

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Field	Value	Language
dc.contributor.author	Wu, J https://orcid.org/0000-0002-4925-4242	en_US
dc.contributor.author	Zhang, Y https://orcid.org/0000-0003-0059-5369	en_US
dc.contributor.author	Chen, L	en_US
dc.contributor.author	Luo, Z https://orcid.org/0000-0002-6953-3986	en_US
dc.date.issued	2013-03-15	en_US
dc.identifier.citation	Applied Mathematical Modelling, 2013, 37 (6), pp. 4578 - 4591	en_US
dc.identifier.issn	0307-904X	en_US
dc.identifier.uri	http://hdl.handle.net/10453/27422
dc.description.abstract	This paper proposes a new interval analysis method for the dynamic response of nonlinear systems with uncertain-but-bounded parameters using Chebyshev polynomial series. Interval model can be used to describe nonlinear dynamic systems under uncertainty with low-order Taylor series expansions. However, the Taylor series-based interval method can only suit problems with small uncertain levels. To account for larger uncertain levels, this study introduces Chebyshev series expansions into interval model to develop a new uncertain method for dynamic nonlinear systems. In contrast to the Taylor series, the Chebyshev series can offer a higher numerical accuracy in the approximation of solutions. The Chebyshev inclusion function is developed to control the overestimation in interval computations, based on the truncated Chevbyshev series expansion. The Mehler integral is used to calculate the coefficients of Chebyshev polynomials. With the proposed Chebyshev approximation, the set of ordinary differential equations (ODEs) with interval parameters can be transformed to a new set of ODEs with deterministic parameters, to which many numerical solvers for ODEs can be directly applied. Two numerical examples are applied to demonstrate the effectiveness of the proposed method, in particular its ability to effectively control the overestimation as a non-intrusive method. © 2012 Elsevier Inc.	en_US
dc.relation.ispartof	Applied Mathematical Modelling	en_US
dc.relation.isbasedon	10.1016/j.apm.2012.09.073	en_US
dc.subject.classification	Mechanical Engineering & Transports	en_US
dc.title	A Chebyshev interval method for nonlinear dynamic systems under uncertainty	en_US
dc.type	Journal Article
utslib.citation.volume	6	en_US
utslib.citation.volume	37	en_US
utslib.for	0902 Automotive Engineering	en_US
utslib.for	0102 Applied Mathematics	en_US
utslib.for	0103 Numerical and Computational Mathematics	en_US
utslib.for	0801 Artificial Intelligence and Image Processing	en_US
pubs.embargo.period	Not known	en_US
pubs.organisational-group	/University of Technology Sydney
pubs.organisational-group	/University of Technology Sydney/Faculty of Engineering and Information Technology
pubs.organisational-group	/University of Technology Sydney/Faculty of Engineering and Information Technology/School of Mechanical and Mechatronic Engineering
utslib.copyright.status	open_access
pubs.issue	6	en_US
pubs.publication-status	Published	en_US
pubs.volume	37	en_US

Abstract:

This paper proposes a new interval analysis method for the dynamic response of nonlinear systems with uncertain-but-bounded parameters using Chebyshev polynomial series. Interval model can be used to describe nonlinear dynamic systems under uncertainty with low-order Taylor series expansions. However, the Taylor series-based interval method can only suit problems with small uncertain levels. To account for larger uncertain levels, this study introduces Chebyshev series expansions into interval model to develop a new uncertain method for dynamic nonlinear systems. In contrast to the Taylor series, the Chebyshev series can offer a higher numerical accuracy in the approximation of solutions. The Chebyshev inclusion function is developed to control the overestimation in interval computations, based on the truncated Chevbyshev series expansion. The Mehler integral is used to calculate the coefficients of Chebyshev polynomials. With the proposed Chebyshev approximation, the set of ordinary differential equations (ODEs) with interval parameters can be transformed to a new set of ODEs with deterministic parameters, to which many numerical solvers for ODEs can be directly applied. Two numerical examples are applied to demonstrate the effectiveness of the proposed method, in particular its ability to effectively control the overestimation as a non-intrusive method. © 2012 Elsevier Inc.

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http://hdl.handle.net/10453/27422