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

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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.