Sparse regression Chebyshev polynomial interval method for nonlinear dynamic systems under uncertainty

Abstract

• A non-intrusive method for nonlinear systems with interval uncertainties is proposed. • The sparse regression is employed in the construction of interval inclusion function. • The lower-degree Chebyshev polynomial base functions are used to increase efficiency. • The problem is transformed into a set of deterministic ones with interval analysis. • The proposed method needs fewer number of original nonlinear systems evaluations. This paper proposes a new higher-efficiency interval method for the response bound estimation of nonlinear dynamic systems, whose uncertain parameters are bounded. This proposed method uses sparse regression and Chebyshev polynomials to help the interval analysis applied on the estimation. It is also a non-intrusive method which needs much fewer evaluations of original nonlinear dynamic systems than the other Chebyshev polynomials based interval methods. By using the proposed method, the response bound estimation of nonlinear dynamic systems can be performed more easily, even if the numerical simulation in nonlinear dynamic systems is costly or the number of uncertain parameters is higher than usual. In our approach, the sparse regression method “elastic net” is adopted to improve the sampling efficiency, but with sufficient accuracy. It alleviates the sample size required in coefficient calculation of the Chebyshev inclusion function in the sampling based methods. Moreover, some mature technologies are adopted to further reduce the sample size and to guarantee the accuracy of the estimation. So that the number of sampling, which solves the certain ordinary differential equations (ODEs), can be reduced significantly in the Chebyshev interval method. Three numerical examples are presented to illustrate the efficiency of proposed interval method. In particular, the last two examples are high dimension uncertain problems, which can further exhibit the ability to reduce the computational cost.