Calculation of positive invariant sets for nonlinear systems using efficient novel eigenvalue bounds.

使用有效的新特征值界计算非线性系统的正不变集。

基本信息

批准号：
123554934
负责人：
Professor Dr.-Ing. Martin Mönnigmann
金额：
--
依托单位：
Lehrstuhl für Regelungstechnik und Systemtheorie
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2009
资助国家：
德国
起止时间：
2008-12-31 至 2014-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/123554934?language=en
关键词：
Calculation positive invariant sets nonlinear

项目摘要

We developed a new method that calculates Hessian matrix eigenvalue bounds. Because the method deliberately avoids the calculation of Hessian matrices, it requires a surprisingly low computational effort. The new approach, which we refer to as eigenvalue arithmetic, was subsequently applied in the numerical computation of positive invariant sets of nonlinear dynamical systems. Our results show that the eigenvalue arithmetic can compete with existing methods as far as the tightness of the eigenvalue bounds is concerned. Its computational effort, however, is lower than that of existing approaches. As a result, the eigenvalue arithmetic is an attractive alternative for algorithms that spend a considerable fraction of their computational time on the calculation of Hessian eigenvalue bounds, such as certain variants of global optimization. The subsequent project aims at exploiting sparsity in the eigenvalue arithmetic. While sparsity is usually used to reduce the number of operations in numerical algorithms, it can here be used in a more fundamental way to tighten the eigenvalue bounds. Sparse intermediate matrices naturally appear in the eigenvalue arithmetic even if the function of interest results in a dense Hessian. Consequently, the tightening effect is expected to play a role for a large class of functions, among them those which do not have sparse Hessian matrices.The improved eigenvalue bounds are anticipated to result in enlarged positive invariant domains. As a final test, these domains will be used in stability criteria for model predictive control for a number of benchmark and technical nonlinear dynamical systems.

我们开发了一种计算 Hessian 矩阵特征值界限的新方法。由于该方法刻意避免了 Hessian 矩阵的计算，因此所需的计算量出奇地低。这种新方法，我们称之为特征值算术，随后被应用于非线性动力系统的正不变集的数值计算。我们的结果表明，就特征值界限的紧密性而言，特征值算法可以与现有方法竞争。然而，它的计算量低于现有方法。因此，对于将大量计算时间用于计算 Hessian 特征值界限的算法（例如全局优化的某些变体）来说，特征值算术是一种有吸引力的替代方案。随后的项目旨在利用特征值算术中的稀疏性。虽然稀疏性通常用于减少数值算法中的运算数量，但这里可以以更基本的方式使用稀疏性来收紧特征值界限。即使感兴趣的函数产生稠密的 Hessian 矩阵，稀疏中间矩阵也会自然地出现在特征值算术中。因此，紧缩效应预计将对一大类函数发挥作用，其中包括那些没有稀疏 Hessian 矩阵的函数。改进的特征值界限预计会导致扩大的正不变域。作为最终测试，这些域将用于许多基准和技术非线性动力系统的模型预测控制的稳定性标准。