Pade-Type Model Reduction of Second-Order and Higher-Order Linear Dynamical Systems

A standard approach to reduced-order modeling of higher-order linear dynamical systems is to rewrite the system as an equivalent first-order system and then employ Krylov-subspace techniques for reduced-order modeling of first-order systems. While this approach results in reduced-order models that are characterized as Pade-type or even true Pade approximants of the system's transfer function, in general, these models do not preserve the form of the original higher-order system. In this paper, we present a new approach to reduced-order modeling of higher-order systems based on projections onto suitably partitioned Krylov basis matrices that are obtained by applying Krylov-subspace techniques to an equivalent first-order system. We show that the resulting reduced-order models preserve the form of the original higher-order system. While the resulting reduced-order models are no longer optimal in the Pade sense, we show that they still satisfy a Pade-type approximation property. We also introduce the notion of Hermitian higher-order linear dynamical systems, and we establish an enhanced Pade-type approximation property in the Hermitian case

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

On the Generation of Large Passive Macromodels for Complex Interconnect Structures

This paper addresses some issues related to the passivity of interconnect macromodels computed from measured or simulated port responses. The generation of such macromodels is usually performed via suitable least squares fitting algorithms. When the number of ports and the dynamic order of the macromodel is large, the inclusion of passivity constraints in the fitting process is cumbersome and results in excessive computational and storage requirements. Therefore, we consider in this work a post-processing approach for passivity enforcement, aimed at the detection and compensation of passivity violations without compromising the model accuracy. Two complementary issues are addressed. First, we consider the enforcement of asymptotic passivity at high frequencies based on the perturbation of the direct coupling term in the transfer matrix. We show how potential problems may arise when off-band poles are present in the model. Second, the enforcement of uniform passivity throughout the entire frequency axis is performed via an iterative perturbation scheme on the purely imaginary eigenvalues of associated Hamiltonian matrices. A special formulation of this spectral perturbation using possibly large but sparse matrices allows the passivity compensation to be performed at a cost which scales only linearly with the order of the system. This formulation involves a restarted Arnoldi iteration combined with a complex frequency hopping algorithm for the selective computation of the imaginary eigenvalues to be perturbed. Some examples of interconnect models are used to illustrate the performance of the proposed technique

Block oriented model order reduction of interconnected systems

Unintended and parasitic coupling effects are becoming more relevant in currently designed, small-scale/highfrequency RFICs. Electromagnetic (EM) based procedures must be used to generate accurate models for proper verification of system behaviour. But these EM methodologies may take advantage of structural sub-system organization as well as information inherent to the IC physical layout, to improve their efficiency. Model order reduction techniques, required for fast and accurate evaluation and simulation of such models, must address and may benefit from the provided hierarchical information. System-based interconnection techniques can handle some of these situations, but suffer from some drawbacks when applied to complete EM models. We will present an alternative methodology, based on similar principles, that overcomes the limitations of such approaches. The procedure, based on structure-preserving model order reduction techniques, is proved to be a generalization of the interconnected system based framework. Further improvements that allow a trade off between global error and block size, and thus allow a better control on the reduction, will be also presented

Bordered Block-Diagonal Preserved Model-Order Reduction for RLC Circuits

This thesis details the research of the bordered block-diagonal preserved model-order reduction (BVOR) method and implementation of the corresponding tool designed for facilitating the simulation of industrial, very large sized linear circuits or linear sub-circuits of a nonlinear circuit. The BVOR tool is able to extract the linear RLC parts of the circuit from any given typical SPICE netlist and perform reduction using an appropriate algorithm for optimum efficiency. The implemented algorithms in this tool are bordered block-diagonal matrix solver and bordered block-diagonal matrix based block Arnoldi method

Model order reduction of time-delay systems using a laguerre expansion technique

The demands for miniature sized circuits with higher operating speeds have increased the complexity of the circuit, while at high frequencies it is known that effects such as crosstalk, attenuation and delay can have adverse effects on signal integrity. To capture these high speed effects a very large number of system equations is normally required and hence model order reduction techniques are required to make the simulation of the circuits computationally feasible. This paper proposes a higher order Krylov subspace algorithm for model order reduction of time-delay systems based on a Laguerre expansion technique. The proposed technique consists of three sections i.e., first the delays are approximated using the recursive relation of Laguerre polynomials, then in the second part, the reduced order is estimated for the time-delay system using a delay truncation in the Laguerre domain and in the third part, a higher order Krylov technique using Laguerre expansion is computed for obtaining the reduced order time-delay system. The proposed technique is validated by means of real world numerical examples

Krylov subspaces associated with higher-order linear dynamical systems

A standard approach to model reduction of large-scale higher-order linear dynamical systems is to rewrite the system as an equivalent first-order system and then employ Krylov-subspace techniques for model reduction of first-order systems. This paper presents some results about the structure of the block-Krylov subspaces induced by the matrices of such equivalent first-order formulations of higher-order systems. Two general classes of matrices, which exhibit the key structures of the matrices of first-order formulations of higher-order systems, are introduced. It is proved that for both classes, the block-Krylov subspaces induced by the matrices in these classes can be viewed as multiple copies of certain subspaces of the state space of the original higher-order system