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

我们开发了一种计算Hessian矩阵特征值界的新方法。由于该方法有意避免了Hessian矩阵的计算，因此它需要的计算工作量非常低。新的方法，我们称之为特征值算法，随后被应用于数值计算的非线性动力系统的正不变集。我们的研究结果表明，特征值算法可以竞争与现有的方法，就本征值界的紧密性而言。然而，它的计算工作量低于现有的方法。因此，特征值算法是一个有吸引力的替代算法，花了相当大的一部分计算时间的海森特征值界的计算，如某些变种的全局优化。随后的项目旨在利用稀疏的特征值算法。虽然稀疏性通常用于减少数值算法中的运算次数，但它可以以更基本的方式用于收紧特征值边界。稀疏中间矩阵自然出现在特征值算法中，即使感兴趣的函数导致稠密Hessian。因此，紧缩效应预计将发挥作用的一大类功能，其中包括那些没有稀疏Hessian矩阵。改进的特征值界预计会导致扩大的正不变域。作为最后的测试，这些域将用于一些基准和技术非线性动态系统的模型预测控制的稳定性准则。

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.