A random-bound Chebyshev method for uncertainty propagation of nonlinear dynamics under imprecise probabilities

Abstract

Abstract The distribution-free P-box is an effective quantification model for uncertainties with only imprecise probabilistic information. However, its application to nonlinear dynamical systems is limited due to a lack of efficient uncertainty propagation (UP) methods. To end this, this work develops a random-bound Chebyshev (RBC) UP method based on the framework of the interval Monte Carlo (IMC) method. First, the Chebyshev method is applied to solve the interval analysis in the IMC simulations. Here, the bounds of intervals can be regarded as random variables whose cumulative density functions (CDFs) are the CDF bounds of the P-box. Since the CDF bounds of distribution-free P-boxes are always arbitrary and non-parameterized, the data-driven polynomial chaos expansion (DD-PCE), which only requires the information of statistical moments, is introduced to solve the problem of random bounds. Then a sparse-regression strategy is utilized to deal with the ‘curse of dimensionality’ of the DD-PCE for high-dimensional problems. As a result, the RBC method efficiently achieves a non-intrusive UP of nonlinear dynamics with distribution-free P-boxes. The method is also effective for hybrid UP problems with random, interval, and P-box variables. Then the RBC method is validated based on test cases, including a duffing oscillator, a vehicle ride, and an engineering application of launch-vehicle trajectory. The results verify the ability of the method to deal with complex black-box problems. In comparison with the reference solutions based on the IMC simulations, with relative errors of less than 1%, the proposed method requires less than 0.0004% sample size and 0.015% calculation time.