Passivity enforcement is an important issue for macro-modeling for passive systems from measured or simulated data. Existing convex programming based methods are too expensive and thus are ruled out for realistic application. Other methods based on iteratively fixing the passivity through perturbing the eigenvalues of the Hamiltonian matrix either suffer from convergence issue or lack optimality which will sometimes lead to unacceptable error. In this paper we propose a novel framework for macro-modeling. In addition to the traditional two-stage (fixing plus enforcement) schemes, we propose a post-enforcement optimization, which takes a passive, while potentially not-so-accurate model, as the starting point, and performs local search to find the local optimum with passivity constraint or build-in passivity guarantee. A simple yet stable passive modeling generator is proposed to produce the starting model for optimization. Two algorithms are proposed for performing constrained and unconstrained optimizations. Experiments show that the accuracy of passivity-fixed model can be significantly improved with the proposed methods.   The main problems of such algorithms are 1) The perturbation process does not guarantee convergence. Each step of perturbation, though will move some imaginary eigenvalues away from the axis, will potentially move other eigenvalues to the imaginary axis and thus introduce new passivity violations. 2) Even if the enforcement procedure stops with a passive model, the error could be large. This is because each step of the enforcement is aiming at minimizing the perturbation instead of minimizing the model error. After a few steps of iterations, the model error will generally be accumulated, and eventually it could be quite large.   In this paper we propose to perform an additional refinement step to improve the accuracy of a passive model while still guarantee the passivity. The input to the proposed algorithms is a passive, while not-so-accurate starting model. This is different from existing passivity enforcement schemes, in which the input is an accurate, but not passive system. The starting passive model can be provided with any of the existing enforcement techniques, or with some simpler (and more stable) approaches that will be introduced later.   The rest of this paper is organized as follows. In Section 2 we introduce the mathematical background of the problem, and review exiting methods for passivity constrained macro-modeling. Section 3 presents the framework for the proposed three-stage passive macro-modeling. Section 4 and 5 presents two schemes for postenforcement optimization. Results and conclusions will be given in Sections 6 and 7.
['Zuochang Ye', 'Yang Li', 'Mingzhi Gao', 'Zhiping Yu']
A novel framework for passive macro-modeling