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.

本文提出了一种非侵入式区间不确定性分析方法，用于利用多项式混沌展开估计参数不确定但有界的非线性系统的动态响应范围。传统的区间算术和泰勒级数方法通常会由于固有的包裹效应而导致较大的高估，尤其是对于多维和非单调问题。为了克服这个缺点，在本工作中提出了一种基于截断多项式混沌展开的新颖的多项式混沌包含函数来评估区间函数。该方法以区间空间中的勒让德多项式为试验依据，扩展了区间过程，并通过搭配法计算出多项式系数。两个例子表明，多项式混沌包含函数能够确定更紧密的真实解的包围，并有效地处理包裹效应。非线性系统对区间变量的响应可以通过多项式混沌包含函数来近似，通过该函数，可以通过适当的数值求解器轻松估算所有时间迭代步骤中动态响应的最大值和最小值。由常微分方程描述的四个动力学示例证明了与其他方法相比，所提出的区间不确定性分析方法的有效性，可行性和效率。非线性系统对区间变量的响应可以通过多项式混沌包含函数来近似，通过该函数，可以通过适当的数值求解器轻松估算所有时间迭代步骤中动态响应的最大值和最小值。由常微分方程描述的四个动力学示例证明了与其他方法相比，所提出的区间不确定性分析方法的有效性，可行性和效率。非线性系统对区间变量的响应可以通过多项式混沌包含函数来近似，通过该函数，可以通过适当的数值求解器轻松估算所有时间迭代步骤中动态响应的最大值和最小值。由常微分方程描述的四个动力学示例证明了与其他方法相比，所提出的区间不确定性分析方法的有效性，可行性和效率。