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We propose and study an algorithm for computing a nearest passive system to agiven non-passive linear time-invariant system (with much freedom in the choiceof the metric defining `nearest', which may be restricted to structuredperturbations), and also a closely related algorithm for computing thestructured distance of a given passive system to non-passivity. Both problemsare addressed by solving eigenvalue optimization problems for Hamiltonianmatrices that are constructed from perturbed system matrices. The proposedalgorithms are two-level methods that optimize the Hamiltonian eigenvalue ofsmallest positive real part over perturbations of a fixed size in the inneriteration, using a constrained gradient flow. They optimize over theperturbation size in the outer iteration, which is shown to convergequadratically in the typical case of a defective coalescence of simpleeigenvalues approaching the imaginary axis. For large systems, we propose avariant of the algorithm that takes advantage of the inherent low-rankstructure of the problem. Numerical experiments illustrate the behavior of theproposed algorithms.

Finding the nearest passive or non-passive system via Hamiltonian eigenvalue optimization