A Chebyshev interval method for nonlinear dynamic systems under uncertainty |
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Authors: | Jinglai Wu Yunqing Zhang Liping Chen Zhen Luo |
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Institution: | 1. National Engineering Research Center for CAD, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China;2. School of Electrical, Mechanical and Mechatronic Systems, The University of Technology, Sydney, Ultimo, NSW 2007, Australia |
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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. |
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Keywords: | Interval model Chebyshev polynomial series Dynamic response of nonlinear systems Ordinary differential equations (ODEs) |
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