Nearest $$\varOmega $$-stable matrix via Riemannian optimization

Abstract

We study the problem of finding the nearest $$\varOmega $$ Ω -stable matrix to a certain matrix A, i.e., the nearest matrix with all its eigenvalues in a prescribed closed set $$\varOmega $$ Ω . Distances are measured in the Frobenius norm. An important special case is finding the nearest Hurwitz or Schur stable matrix, which has applications in systems theory. We describe a reformulation of the task as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices. The problem can then be solved using standard methods from the theory of Riemannian optimization. The resulting algorithm is remarkably fast on small-scale and medium-scale matrices, and returns directly a Schur factorization of the minimizer, sidestepping the numerical difficulties associated with eigenvalues with high multiplicity.

Solving matrix nearness problems via Hamiltonian systems, matrix factorization, and optimization

“…However, in the unstructured case, the two approaches perform similarly, as reported in [53]. [96] recently proposed a highly efficient approach to compute the nearest Ω-stable matrix. They parametrize X with its complex Schur factorization/decomposition, X = U T U * where U is unitary (that is, U U * = I) and T is upper triangular.…”

“…But from the system theoretical point of view it is interesting to find a value or a bound for the smallest perturbation (r P (Σ)) that makes the system achieves property P. In general, determining the minimal perturbation for system properties, such as stability or passivity, is very challenging. Instead, one uses (non-convex) optimization methods to estimate r P (Σ), see, e.g., [98,44,53,96] for distance to stability (when P = stability) and [36,50,107,1,45] for distance to passivity (when P = passivity). Another closely related problem is that of finding the closest stable polynomial to a given unstable one [91].…”