A validated integration algorithm for nonlinear ODEs using Taylor models and ellipsoidal calculus
A validated integration algorithm for nonlinear ODEs using Taylor models and ellipsoidal calculus
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DOI：
10.1109/cdc.2013.6759928
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发表时间：
2013-12
期刊：
影响因子：
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通讯作者：
B. Houska;M. E. Villanueva;B. Chachuat
中科院分区：
文献类型：
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作者：
B. Houska;M. E. Villanueva;B. Chachuat
提出了一种新的含参非线性微分方程可达集的定界算法。该算法基于一种先离散后定界的方法，通过传播具有椭球余项的泰勒模型来封闭可达集，并且它考虑了离散化所固有的截断误差。与现有的算法进行两个阶段的先验外壳阶段，紧接着的阶段，所提出的算法首先预测一个连续时间的外壳，然后寻求一个最大的步长，预测的外壳的有效性可以建立。结果表明，这种反向的方法导致一个自然的步长控制机制，它不再依赖于先验外壳的可用性。文中还介绍了该算法在ACADO Toolkit中的开源实现。一个简单的数值算例研究，以说明该算法的性能和稳定性。
This paper presents a novel algorithm for bounding the reachable set of parametric nonlinear differential equations. This algorithm is based on a first-discretize-then-bound approach to enclose the reachable set via propagation of a Taylor model with ellipsoidal remainder, and it accounts for truncation errors that are inherent to the discretization. In contrast to existing algorithms that proceed in two phases-an a priori enclosure phase, followed by a tightening phase-the proposed algorithm first predicts a continuous-time enclosure and then seeks a maximal step-size for which validity of the predicted enclosure can be established. It is shown that this reversed approach leads to a natural step-size control mechanism, which no longer relies on the availability of an a priori enclosure. Also described in the paper is an open-source implementation of the algorithm in ACADO Toolkit. A simple numerical case study is presented to illustrate the performance and stability of the algorithm.