Finding the Nearest Passive or Nonpassive System via Hamiltonian Eigenvalue Optimization

We propose and study an algorithm for computing a nearest passive system to a given nonpassive linear time-invariant system (with much freedom in the choice of the metric defining “nearest,” which may be restricted to structured perturbations), and also a closely related algorithm for computing the structured distance of a given passive system to nonpassivity. Both problems are addressed by solving eigenvalue optimization problems for Hamiltonian matrices that are constructed from perturbed system matrices. The proposed algorithms are two-level methods that optimize the Hamiltonian eigenvalue of the smallest positive real part over perturbations of a fixed size in the inner iteration, using a constrained gradient flow. They optimize over the perturbation size in the outer iteration, which is shown to converge quadratically in the typical case of a defective coalescence of simple eigenvalues approaching the imaginary axis. For large systems, we propose a variant of the algorithm that takes advantage of the inherent low-rank structure of the problem. Numerical experiments illustrate the behavior of the proposed algorithms.