An efficient uncertainty propagation method for nonlinear dynamics with distribution-free P-box processes

The distribution-free P-box process serves as an effective quantification model for time-varying uncertainties in dynamical systems when only imprecise probabilistic information is available. However, its application to nonlinear systems remains limited due to excessive computation. This work develops an efficient method for propagating distribution-free P-box processes in nonlinear dynamics. First, using the Covariance Analysis Describing Equation Technique (CADET), the dynamic problems with P-box processes are transformed into interval Ordinary Differential Equations (ODEs). These equations provide the Mean-and-Covariance (MAC) bounds of the system responses in relation to the MAC bounds of P-box-process excitations. They also separate the previously coupled P-box analysis and nonlinear-dynamic simulations into two sequential steps, including the MAC bound analysis of excitations and the MAC bounds calculation of responses by solving the interval ODEs. Afterward, a Gaussian assumption of the CADET is extended to the P-box form, i.e., the responses are approximate parametric Gaussian P-box processes. As a result, the probability bounds of the responses are approximated by using the solutions of the interval ODEs. Moreover, the Chebyshev method is introduced and modified to efficiently solve the interval ODEs. The proposed method is validated based on test cases, including a duffing oscillator, a vehicle ride, and an engineering black-box problem of launch vehicle trajectory. Compared to the reference solutions based on the Monte Carlo method, with relative errors of less than 3%, the proposed method requires less than 0.2% calculation time. The proposed method also possesses the ability to handle complex black-box problems.