Distance to the nearest stable Metzler matrix

Abstract

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.

Additional Information

© 2017 IEEE. This work was funded by NSF CNS award 1545096, ECCS award 1619352, CCF award 1637598, and ARPA-E award GRID DATA. I would like to thank Riley Murray at Caltech for many helpful discussions regarding the dual problem. The outcome of which is currently being written up.

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Submitted - 1709.02461

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