On computing the complex passivity radius

We characterize the complex passivity radius of a rational transfer matrix G(s):=C(sI
n
−A)
⁻¹
B+D and propose an approach to compute it. The method depends on computing the smallest structured indefinite perturbation to a Hermitian matrix that makes it singular. We consider both additive and multiplicative perturbations, giving details for the additive case. In both cases, the smallest indefinite perturbation can be efficiently computed by solving a unimodal optimization problem in a real parameter. The passivity radius can be computed by minimizing the smallest singularity-inducing multiplicative indefinite perturbation of a frequency-dependent matrix over the imaginary axis.