Parameterized macromodeling of passive and active dynamical systems

L'abstract è presente nell'allegato / the abstract is in the attachmen

Fast LDO Simulations via Parameter-Varying Linearized Macromodels

An approach for generating time-varying linearized macromodels of analog circuit blocks is presented. These models can be used to perform fast small-signal analyses characterized by nonstationary operating conditions, thanks to their certified stability. We validate the proposed approach by performing post-layout simulations of a Low DropOut (LDO) voltage regulator, in view of power integrity assessment applications

Addressing Load Sensitivity of Rational Macromodels

Behavioral models are effective tools used to relieve the computational burden of large-scale system-level simulations. In electrical and electronic applications, the Vector Fitting (VF) iteration often represents the algorithm of choice for generating low order equivalent circuits for complex multiport components in a data-driven setting. Although accurate and reliable in general, macromodels generated via VF are inherently represented in terms of a rational approximation of one specific input-output transfer function of the structure under modeling, e.g., its scattering matrix. However, accuracy in the scattering representation does not necessarily imply a good accuracy when solving the macromodel in a system-level setting, under different termination conditions. In fact, the sensitivity of the macromodel with respect to its loading conditions may be large and needs to be addressed and controlled. In this work, we present a modified VF scheme that overcomes this issue, by introducing in the rational approximation algorithm the requirement that the macromodel remains accurate when interconnected with a known class of admissible networks. The proposed formulation is based on an augmentation of the cost function minimized at each VF iteration; further, it does not require additional expensive data gathering steps when compared to standard approaches. The effectiveness of the scheme is tested over a set of relevant examples, in particular for Power Integrity applications

A Vector Fitting Approach for the Automated Estimation of Lumped Boundary Conditions of 1D Circulation Models

Purpose: The choice of appropriate boundary conditions is a crucial step in the development of cardiovascular models for blood flow simulations. The three-element Windkessel model is usually employed as a lumped boundary condition, providing a reduced order representation of the peripheral circulation. However, the systematic estimation of the Windkessel parameters remains an open problem. Moreover, the Windkessel model is not always adequate to model blood flow dynamics, which often require more elaborate boundary conditions. In this study, we propose a method for the estimation of the parameters of high order boundary conditions, including the Windkessel model, from pressure and flow rate waveforms at the truncation point. Moreover, we investigate the effect of adopting higher order boundary conditions, corresponding to equivalent circuits with more than one storage element, on the accuracy of the model. Method: The proposed technique is based on Time-Domain Vector Fitting, a modeling algorithm that, given samples of the input and output of a system, such as pressure and flow waveforms, can derive a differential equation approximating their relation. Results: The capabilities of the proposed method are tested on a 1D circulation model consisting of the 55 largest human systemic arteries, to demonstrate its accuracy and its usefulness to estimate boundary conditions with order higher than the traditional Windkessel models. The proposed method is compared to other common estimation techniques, and its robustness in parameter estimation is verified in presence of noisy data and of physiological changes of aortic flow rate induced by mental stress. Conclusion: Results suggest that the proposed method is able to accurately estimate boundary conditions of arbitrary order. Higher order boundary conditions can improve the accuracy of cardiovascular simulations, and Time-Domain Vector Fitting can automatically estimate them

Enforcing passivity of parameterized LTI macromodels via Hamiltonian-driven multivariate adaptive sampling

We present an algorithm for passivity verification and enforcement of multivariate macromodels whose state-space matrices depend in closed form on a set of external or design parameters. Uniform passivity throughout the parameter space is a fundamental requirement of parameterized macromodels of physically passive structures, that must be guaranteed during model generation. Otherwise, numerical instabilities may occur, due to the ability of non-passive models to generate energy. In this work, we propose the first available algorithm that, starting from a generic parameter-depedent state-space model, identifies the regions in the frequency-parameter space where the model behaves locally as a non-passive system. The approach we pursue is based on an adaptive sampling scheme in the parameter space, which iteratively constructs and perturbs the eigenvalue spectrum of suitable Skew-Hamiltonian/Hamiltonian (SHH) pencils, with the objective of identifying the regions where some of these eigenvalues become purely imaginary, thus pinpointing local passivity violations. The proposed scheme is able to detect all relevant violations. An outer iterative perturbation method is then applied to the model coefficients in order to remove such violations and achieve uniform passivity. Although a formal proof of global convergence is not available, the effectiveness of the proposed implementation of the passivity verification and enforcement schemes is demonstrated on several examples

Uniformly Stable Parameterized Macromodeling through Positive Definite Basis Funtions

Reduced-order models are widely used to reduce the computational cost required by the numerical assessment of electrical performance during the design cycle of electronic circuits and systems. Although standard macromodeling algorithms can be considered to be well consolidated, the generation of macromodels that embed in a closed form some dependence on the design variables still presents considerable margins for improvement. One of these aspects is enforcement of uniform stability throughout the parameter space of interest. This paper proposes a novel parameterized macromodeling strategy, which enforces by construction that all macromodel poles are stable for any combination of possibly several independent design variables. The key enabling factor is adoption of positive definite multivariate basis functions for the representation of model variations induced by the parameters. This representation leads to robust model generation from tabulated frequency responses, at a computational cost that is dramatically reduced with respect to competing approaches. This result arises from a number of algebraic constraints for stability enforcement that depends on the model complexity (number of basis functions) and not on the model behavior as a function of the parameters. As a byproduct, the proposed strategy lends itself to much improved scaling with the dimension of parameter space, allowing to circumvent the curse of dimensionality that may occur when the number of independent parameters grows beyond few units. To this end, we exploit representations based on positive definite radial basis functions. The benefits of the proposed approach are demonstrated through an extensive experimental campaign applied to both passive and active devices and components, comparing the performance of different model parameterizations in terms of accuracy, time requirements and model compactness

Multivariate macromodeling with stability and passivity constraints

We present a general framework for the construction of guaranteed stable and passive multivariate macromodels from sampled frequency responses. The obtained macromodels embed in closed form the dependence on external parameters, through a data-driven approximation of input data samples based on orthogonal polynomial bases. The key novel contribution of this work is an extension to the multivariate and possibly high-dimensional case of Hamiltonian-based passivity check and enforcement algorithms, which can be applied to enforce both uniform stability and uniform passivity of the models. The modeling flow is demonstrated on a representative interconnect example

An Adaptive Sampling Process for Automated Multivariate Macromodeling Based on Hamiltonian-Based Passivity Metrics

This paper introduces a fully automated greedy algorithm for the construction of parameterized behavioral models of electromagnetic structures, targeting at the same time uniform model stability and passivity. The proposed algorithm is able to determine a small set of parameter configurations for which an external solver provides on the fly the sampled scattering parameters of the structure over a predetermined frequency band. These samples are subjected to a multivariate rational/polynomial fitting process, which iteratively leads to a parameterized descriptor realization of the model. The main novel contribution in this work is the adoption of a model-based approach for the adaptive augmentation of an initially small set of frequency responses, each corresponding to a randomly-selected parameter configuration. In particular, the locations of the in-band passivity violations of intermediate macromodels constructed at each iteration are used as a proxy for the model-data error in those regions where input data are not available. This physics-based consistency check, which is enabled by recent developments in multivariate passivity characterization based on Skew-Hamiltonian-Hamiltonian (SHH) spectra, is combined with standard space exploration metrics to obtain a small-size and automatically-determined distribution of points in the parameter space, leading to the construction of an accurate macromodel with a very limited number of external field solver runs. The embedded passivity check and enforcement process guarantees that either the final model is passive throughout the parameter space, or the residual violations, if present, are negligible for practical purposes. Several examples validate the proposed approach for up to three concurrent parameters

Handling Initial Conditions in Vector Fitting for Real Time Modeling of Power System Dynamics

This paper develops a predictive modeling algorithm, denoted as Real-Time Vector Fitting (RTVF), which is capable of approximating the real-time linearized dynamics of multi-input multi-output (MIMO) dynamical systems via rational transfer function matrices. Based on a generalization of the well-known Time-Domain Vector Fitting (TDVF) algorithm, RTVF is suitable for online modeling of dynamical systems which experience both initial-state decay contributions in the measured output signals and concurrently active input signals. These adaptations were specifically contrived to meet the needs currently present in the electrical power systems community, where real-time modeling of low frequency power system dynamics is becoming an increasingly coveted tool by power system operators. After introducing and validating the RTVF scheme on synthetic test cases, this paper presents a series of numerical tests on high-order closed-loop generator systems in the IEEE 39-bus test system