Guaranteed passive parameterized macromodeling by using Sylvester state-space realizations

A novel state-space realization for parameterized macromodeling is proposed in this paper. A judicious choice of the state-space realization is required in order to account for the assumed smoothness of the state-space matrices with respect to the design parameters. This technique is used in combination with suitable interpolation schemes to interpolate a set of state-space matrices, and hence the poles and residues indirectly, in order to build accurate parameterized macromodels. The key points of the novel state-space realizations are the choice of a proper pivot matrix and a well-conditioned solution of a Sylvester equation. Stability and passivity are guaranteed by construction over the design space of interest. Pertinent numerical examples validate the proposed Sylvester realization for parameterized macromodeling

Passive parametric macromodeling by using Sylvester state-space realizations

A judicious choice of the state-space realization is required in order to account for the assumed smoothness of the state-space matrices with respect to the design parameters. The direct parameterization of poles and residues may be not appropriate, due to their possible non-smooth behavior with respect to design parameters. This is avoided in the proposed technique, by converting the pole-residue description to a Sylvester description which is computed for each root macromodel. This technique is used in combination with suitable parameterizing schemes for interpolating a set of state-space matrices, and hence the poles and residues indirectly, in order to build accurate parametric macromodels. The key features of the present approach are first the choice of a proper pivot matrix and second, finding a well-conditioned solution of a Sylvester equation. Stability and passivity are guaranteed by construction over the design space of interest. Pertinent numerical examples validate the proposed Sylvester technique for parametric macromodeling

A Perturbation Scheme for Passivity Verification and Enforcement of Parameterized Macromodels

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.Comment: Submitted to the IEEE Transactions on Components, Packaging and Manufacturing Technology on 13-Apr-201

A Novel Framework for Parametric Loewner Matrix Interpolation

The generation of black-box macromodels of passive components at the chip, package, and board levels has become an important step of the electronic design automation (EDA) workflow. The vector fitting (VF) scheme is a very popular method for the extraction of such macromodels, and several multivariate extensions are now available for embedding external parameters in the model structure, thus enabling model-based variability analysis and design optimization. The Loewner matrix interpolation framework was recently suggested as an effective and promising alternative macromodeling approach to VF. In this article, we propose a parametric version of Loewner interpolation, which embeds orthogonal polynomials as an integral part of the parameterization framework. This approach is shown to be efficient and accurate and presents various advantages with respect to competing multivariate rational interpolation methods. These advantages include better control of model smoothness in the parameter space and a particularly efficient implementation of the singular value decomposition, which is the core of the model extraction scheme. These advantages are confirmed through several examples relevant for signal and power integrity applications

Bivariate macromodeling with guaranteed uniform stability and passivity

This paper extends the well-established macromod- eling flows based on rational fitting and passivity enforcement to the bivariate case, where the model response depends on frequency and on some additional design parameter. We propose a black-box model identification algorithm that is able to guarantee uniform stability and passivity throughout the parameter range. The resulting models, which can be cast as parameterized SPICE subnetworks, may be used to construct parameterized component libraries for design optimization, what-if analyses and fast parametric sweeps in frequency or time domain

Machine learning for the performance assessment of high-speed links

This paper investigates the application of support vector machine to the modeling of high-speed interconnects with largely varying and/or highly uncertain design parameters. The proposed method relies on a robust and well-established mathematical framework, yielding accurate surrogates of complex dynamical systems. An identification procedure based on the observation of a small set of system responses allows generating compact parametric relations, which can be used for design optimization and/or stochastic analysis. The feasibility and strength of the method are demonstrated based on a benchmark function and on the statistical assessment of a realistic printed circuit board interconnect, highlighting the main features and benefits of this technique over state-of-the-art solutions. Emphasis is given to the effects of the initial sample size and of input noise on the model estimation

A Compressed Multivariate Macromodeling Framework for Fast Transient Verification of System-Level Power Delivery Networks

This paper discusses a reduced-order modeling and simulation approach for fast transient power integrity verifi- cation at full system level. The reference structure is a com- plete power distribution network (PDN) from platform voltage regulator module (VRM) to multiple cores, including board, package, decoupling capacitors, and per-core fully integrated voltage regulators (FIVR). All blocks are characterized and known through high-fidelity models derived from first-principle solvers (full-wave electromagnetic and circuit-level extractions). The complexity of such detailed characterization grows very large and becomes intractable, especially for power integrity verification of massive multicore platforms subjected to real workload scenarios. We approach this problem by exploiting a multi-stage macromodeling and compression process, leading to a compact representation of the system dynamics in terms of a linearized state-space structure with multiple feedback loops from the FIVR controllers. The PDN macromodel is obtained through a data-driven approach starting from reference small- signal frequency responses, obtaining a sparse and structured representation specifically designed to match the behavior of the reference system. The resulting compact model is then solved in time-domain very efficiently. Results on mobile and enterprise server benchmarks demonstrate a speedup in runtime up to 50× with respect to HSPICE, with negligible loss of accuracy

On stabilization of parameterized macromodeling

We propose an algorithm for the identification of guaranteed stable parameterized macromodels from sampled frequency responses. The proposed scheme is based on the standard Sanathanan-Koerner iteration in its parameterized form, which is regularized by adding a set of inequality constraints for enforcing the positiveness of the model denominator at suitable discrete points. We show that an ad hoc aggregation of such constraints is able to stabilize the iterative scheme by significantly improving its convergence properties, while guaranteeing uniformly stable model poles as the parameter(s) change within their design range