Large-Scale and Global Maximization of the Distance to Instability
This work provides the first globally convergent algorithms for maximizing the distance to instability, which is crucial for ensuring robust stability in dynamical systems with uncertainties.
The paper tackles the maximization of the distance to instability for parameter-dependent matrices, a nonconvex and nonsmooth optimization problem. It proposes a globally convergent algorithm for small-scale problems and a subspace framework for large-scale problems, achieving superlinear convergence with respect to subspace dimension.
The larger the distance to instability from a matrix is, the more robustly stable the associated autonomous dynamical system is in the presence of uncertainties and typically the less severe transient behavior its solution exhibits. Motivated by these issues, we consider the maximization of the distance to instability of a matrix dependent on several parameters, a nonconvex optimization problem that is likely to be nonsmooth. In the first part we propose a globally convergent algorithm when the matrix is of small size and depends on a few parameters. In the second part we deal with the problems involving large matrices. We tailor a subspace framework that reduces the size of the matrix drastically. The strength of the tailored subspace framework is proven with a global convergence result as the subspaces grow and a superlinear rate-of-convergence result with respect to the subspace dimension.