# A Perturbation Scheme for Passivity Verification and Enforcement of Parameterized Macromodels

## TL;DR

This paper introduces an algorithm that verifies and enforces passivity in parameterized macromodels of LTI systems, ensuring models are stable and reliable across varying parameters, which is crucial for design and optimization tasks.

## Contribution

It presents a novel perturbation scheme combined with eigenvalue analysis and adaptive sampling to correct passivity violations in parameter-dependent models.

## Key findings

- Effective identification of passivity violations across parameter space
- Successful correction of models to ensure passivity
- Numerical examples demonstrate robustness and accuracy

## Abstract

This paper presents an algorithm for checking and enforcing passivity of behavioral reduced-order macromodels of LTI systems, whose frequency-domain (scattering) responses depend on external parameters. Such models, which are typically extracted from sampled input-output responses obtained from numerical solution of first-principle physical models, usually expressed as Partial Differential Equations, prove extremely useful in design flows, since they allow optimization, what-if or sensitivity analyses, and design centering. Starting from an implicit parameterization of both poles and residues of the model, as resulting from well-known model identification schemes based on the Generalized Sanathanan-Koerner iteration, we construct a parameter-dependent Skew-Hamiltonian/Hamiltonian matrix pencil. The iterative extraction of purely imaginary eigenvalues ot fhe pencil, combined with an adaptive sampling scheme in the parameter space, is able to identify all regions in the frequency-parameter plane where local passivity violations occur. Then, a singular value perturbation scheme is setup to iteratively correct the model coefficients, until all local passivity violations are eliminated. The final result is a corrected model, which is uniformly passive throughout the parameter range. Several numerical examples denomstrate the effectiveness of the proposed approach.