A polynomial chaos expansion approach for nonlinear dynamic systems with interval uncertainty

This paper proposes a non-intrusive interval uncertainty analysis method for estimation of the dynamic response bounds of nonlinear systems with uncertain-but-bounded parameters using polynomial chaos expansion. The conventional interval arithmetic and Taylor series methods usually lead to large overestimation because of the intrinsic wrapping effect, especially for the multidimensional and non-monotonic problems. To overcome this drawback, a novel polynomial chaos inclusion function, based on the truncated polynomial chaos expansion, is proposed in the present work to evaluate interval functions. In this method, the Legendre polynomial in interval space is employed as the trial basis to expand the interval processes, and the polynomial coefficients are calculated through the collocation method. Two examples show that the polynomial chaos inclusion function is capable of determining tighter enclosures of the true solutions and effectively dealing with the wrapping effect. The response of nonlinear systems with respect to interval variables is approximated by the polynomial chaos inclusion function, through which the supremum and infimum of the dynamic responses over all time iteration steps can be easily estimated by an appropriate numerical solver. Four dynamics examples described by ordinary differential equations demonstrate the effectiveness, feasibility, and efficiency of the proposed interval uncertainty analysis method compared with other methods.