Efficient Localization Methods for Passivity Enforcement of Linear Dynamical Models

This paper describes a novel approach for passivity enforcement of compact dynamical models of electrical interconnects. The proposed approach is based on a parameterization of general state-space scattering models with fixed poles. We formulate the passivity constraints as a unity boundedness condition on the H∞ norm of the system transfer function. When this condition is not verified, we use it as an explicit constraint within an iterative perturbation loop of the system state-space matrices. Since the resulting optimization framework is convex but non-smooth, we solve it via localization based algorithms, such as the ellipsoid and the cutting plane methods. Such methods are guaranteed to converge to the global optimum, namely the passive system that is closest to the original system in a suitable system norm. We provide a systematic way of initializing the localization methods by defining an initial feasible set that is guaranteed to contain the optimal solution. Also, we provide a lower bound for the cost function, which becomes tighter at each iteration, allowing to bracket the global optimum within a prescribed accuracy threshold. The proposed technique solves two critical bottleneck issues of the existing approaches for passivity enforcement of linear macromodels. Compared to quasi-optimal schemes based on singular value or Hamiltonian eigenvalue perturbation, we are able to guarantee convergence to the optimal solution. Compared to convex formulations based on direct Bounded Real Lemma constraints, we are able to reduce both the memory and time requirements by orders of magnitude. We demonstrate the effectiveness of our approach on a number of cases for which existing algorithms either fail or exhibit very slow convergence.