Quantum mechanics

Function (biology)

Control (management)

CHAOS (operating system)

Biology

Statistics

Applied mathematics

Computer security

Computer science

Mathematical optimization

Taylor series

Bounded function

Polynomial

Evolutionary biology

Physics

Nonlinear system

Polynomial chaos

Mathematics

Mathematical analysis

Artificial intelligence

Interval (graph theory)

Infimum and supremum

Combinatorics

Monte Carlo method

Control theory (sociology)

Interval arithmetic

Nonlinear Dynamics

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.

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