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矩阵特征值边界的新方法。由于该方法故意避免了Hessian矩阵的计算，因此它需要令人惊讶的计算工作。我们称为特征值算术的新方法随后应用于非线性动力学系统的正不变集集合中。我们的结果表明，就特征值界限的紧密度而言，特征值算术可以与现有方法竞争。但是，它的计算工作低于现有方法。结果，特征值算术是将其计算时间大部分时间用于计算Hessian特征值界限的算法的有吸引力的替代方法，例如某些全局优化的变体。随后的项目旨在利用特征值算术中的稀疏性。尽管稀疏性通常用于减少数值算法中的操作数量，但在这里可以使用更基本的方式来拧紧特征值边界。稀疏的中间矩阵自然出现在特征值算术中，即使感兴趣的功能导致了密集的黑森。因此，预计收紧效果将对大量功能发挥作用，其中包括没有稀疏的黑森矩阵的功能。预计改进的特征值界限会导致阳性不变域的扩大。作为最终测试，这些域将用于许多基准和技术非线性动力学系统的模型预测控制稳定性标准。