Distance to the Nearest Stable Metzler Matrix
For control theorists and engineers, this work provides a computational method to quantify the distance to stability for Metzler matrices, which are important for scalable linear systems.
This paper addresses the non-convex problem of finding the nearest stable Metzler matrix to a given unstable matrix, using dissipative Hamiltonian theory to develop a block coordinate descent algorithm. The algorithm combines a quadratic program and a semidefinite program, leveraging diagonal dominance for improved tractability.
This paper considers the non-convex problem of finding the nearest Metzler matrix to a given possibly unstable matrix. Linear systems whose state vector evolves according to a Metzler matrix have many desirable properties in analysis and control with regard to scalability. This motivates the question, how close (in the Frobenius norm of coefficients) to the nearest Metzler matrix are we? Dropping the Metzler constraint, this problem has recently been studied using the theory of dissipative Hamiltonian (DH) systems, which provide a helpful characterization of the feasible set of stable matrices. This work uses the DH theory to provide a block coordinate descent algorithm consisting of a quadratic program with favourable structural properties and a semidefinite program for which recent diagonal dominance results can be used to improve tractability.