An all-purpose dispersive multiconductor interconnect modelcompatible with PRIMA

This paper describes a new, transmission line-based discrete model for interconnects with per-unit-length, frequency-dependent resistance and inductance. The proposed model is developed in such a manner that the resulting discrete form of Telegrapher's equations constitutes a passive system, and is compatible with passive, reduced-order macromodeling algorithms. The validity of the proposed model is demonstrated through numerical examples     

Hankel-type model reduction for linear repetitive processes: differential and discrete cases

This paper investigates a Hankel-type model reduction problem for linear repetitive processes. Both differential and discrete cases are considered. For a given stable along the pass process, our attention is focused on the construction of a reduced-order stable along the pass process, which guarantees the corresponding error process to have a specified Hankel-type error performance. The Hankel-type performances are first established for differential and discrete linear repetitive processes, respectively, and the corresponding model reduction problems are solved by using the projection approach. Since these obtained conditions are not expressed in linear matrix inequality (LMI) form, the cone complementary linearization (CCL) method is exploited to cast them into sequential minimization problems subject to LMI constraints, which can be solved efficiently. Three numerical examples are provided to demonstrate the proposed theory. This work was partially supported by RGC HKU 7028/04P.     

Rules for robust generation of accurate reduced-order models forhigh-speed coupled interconnections

Recently, robust algorithms have been established fur passive order reduction of electrical models of complex interconnection networks. However, very little is known about the way the order of the reduced model should be chosen to ensure accuracy in subsequent transient simulation studies, in this paper, a rule is derived for the selection of the order of the reduced model for interconnections modeled as transmission lines. It is shown that pulse rise time, interconnection length, and physical properties impact the order of the reduced model. The proposed rule is validated through numerical studies involving both analytic and numerical results from the frequency- and time-domain response of multiconductor transmission line circuits     

Circuit-Based Analysis of Electromagnetic Field Coupling With Nonuniform Transmission Lines

This paper presents a new algorithm for simulating electromagnetic (EM) field coupling with nonuniform multiconductor transmission lines in a circuit simulation environment. The proposed algorithm is based on the concept of passive model-order reduction, whereby an algorithmically developed passive reduced-order model, coupled with a set of equivalent sources representing the incident filed, are shown to accurately capture the behavior of the transmission line under EM excitation. The reduced-order model is developed independently from the particular shape of the incident field pulse, in the sense that, in constructing the model, one does not need prior knowledge about the waveform of the incident pulse of the EM field. In addition, it is also shown that the model developed can be used to simulate the transmission line in the absence of the EM field. The derived equivalent sources, representing the field coupling, are given directly in the time domain, thereby making simulation under nonlinear circuit terminations an easy task. Although the proposed work is aimed mainly at simulating nonuniform transmission lines, it can be applied to uniform lines as a special case. The proposed algorithm has been validated numerically with several examples.     

Fast Simulation of steady-state temperature distributions in electronic components using multidimensional model reduction

In this paper, a model reduction technique is applied to the thermal modeling of electronic components and devices with complex geometries. The reduced-order model is capable of predicting a complete detailed three-dimensional temperature distribution in the original model. The small size and the simplicity of the reduced model allows for the very quick simulation of the device under a wide range of input parameters, such as different boundary conditions and power distributions. Use of the reduced-order model in a thermal design cycle can have a significant effect on both prediction accuracy and simulation efficiency. In the paper, the usefulness of this technique is demonstrated through examples from different electronic devices and packages. Accuracy of the reduced-order model is validated by comparison with the solution to a detailed numerical model.     

Generalized $${\varvec{H}_{\infty }}$$ model reduction for stable two-dimensional discrete systems

For model reduction, the approximation performance sometimes needs to be enhanced over a specific frequency range. Motivated by this fact, the paper investigates generalized \(H_{\infty }\) model reduction for stable two-dimensional (2-D) discrete systems represented by the Roesser model and the Fornasini–Machesini local state-space model, respectively. The generalized \(H_{\infty }\) norm of 2-D systems is introduced to evaluate the approximation error over a specific finite frequency (FF) domain. In light of the 2-D generalized Kalman–Yakubovich–Popov lemmas, sufficient conditions in terms of linear matrix inequalities are derived for the existence of a stable reduced-order model satisfying a specified generalized \(H_{\infty }\) level. Several examples are provided to illustrate the effectiveness and advantages of the proposed method. Compared with most of the existing 2-D model reduction results, the proposed method has the following merits: (1) The generalized \(H_\infty \) model reduction problems are considered for both important types of 2-D models, and no structural assumption is made for the plant model, so that our method has a broader applicable scope. (2) The reduced-order model is guaranteed to be stable, and an upper bound on the generalized \(H_{\infty }\) error is provided. Moreover, no frequency weighting function is needed. (3) The proposed method is applicable for 2-D model reduction with multiple FF specifications.     

线性逆问题求解的多尺度降阶模型

基于相对误差协方差矩阵信息,该文给出线性逆问题求解的多尺度降阶模型,把高阶模型的求解问题转化为一个近似的低阶模型再进行求解．利用该降阶模型可以得到与完全模型相当的估计效果,同时又能大大降低逆算法的计算量,从而有效地解决逆问题求解中计算复杂度过高的难题,增强逆问题求解算法的可实施性,采用降阶模型进行求解还可以增加那些提供显著信息的点的估计精度。     

Efficient model reduction of linear periodically time-varying systems via compressed transient system function

This paper presents a new algorithm to obtain reduced-order model of linear periodically time-varying (LPTV) systems. The proposed algorithm introduces the approach of projection-based reduction algorithms, which have been used for reduction of linear time-invariant (LTI) systems, to the domain of LPTV systems. The key idea in the proposed approach is the utilization of integrated congruence transform to project the original LPTV system matrices onto the Hilbert subspace spanned by the time-dependent derivatives (or moments) of the transfer function. We prove that the transfer function of the resulting reduced-order model has the same derivatives as that of the original system in the Laplace-domain. The new approach presents a computationally efficient method to generate the orthogonal transformation operator (used in the integrated congruence transform) through expanding the time-varying transfer function in the right-half plane of the Laplace-domain. This enables using numerical time-domain integration for a very short transient period to generate the orthogonal transformation operator.     

Time-Delay Dependent H∞ Model Reduction for?Uncertain Stochastic Systems: Continuous-Time Case

The problem of robust H _∞ model reduction for uncertain stochastic systems with time delay is investigated in this paper. The attention of this paper is focused on the construction of a reduced-order model for a given stable system. Several sufficient conditions are obtained for ensuring the existence of the reduced-order model by means of linear matrix inequalities and a coupling non-convex rank constraint condition. Under the sufficient conditions, the desired reduced-order model can be constructed, and the error system between the original model and the reduced-order model is asymptotically stable and has a prescribed H _∞ performance. Based on the proposed method, the construction approaches of reduced-order models with special structures, such as the zeroth-order model, the delay-free model, and the no-parameter-uncertainties model, are also developed. Finally, the effectiveness of the proposed model reduction method is illustrated by a numerical simulation example.     

H/sub /spl infin// mode reduction for two-dimensional discrete state-delayed systems

The problem of H_infin model reduction for two-dimensional (2-D) discrete systems with delay in state is considered. The mathematical model of 2-D systems is established on the basis of the well-known Fornasini-Marchesini local state-space. First, conditions are established to guarantee the asymptotic stability and a prescribed noise attenuation level in the H_infin sense for the underlying system. For a given stable system, attention is focused on the construction of a reduced-order model, which approximates the original system well in an H_infin norm sense. Sufficient conditions are proposed for the existence of admissible reduced-order solutions. Since these obtained conditions are not expressed as strict linear matrix inequalities (LMIs), the cone complementary linearisation method is exploited to cast them into sequential minimisation problems subject to LMI constraints, which can be readily solved using standard numerical software. These obtained results are further extended to more general cases whose system states contain multiple delays. Two numerical examples are provided to demonstrate the effectiveness of the proposed techniques     

A new finite element model for reduced order electromagneticmodeling

This paper introduces a new formulation suitable for direct model order reduction of finite element approximations of electromagnetic systems using Krylov subspace methods. The proposed formulation utilizes a finite element model of Maxwell's curl equations to generate a state-space representation of the electromagnetic system most suitable for the implementation of model order reduction techniques based on Krylov subspaces. It is shown that, with a proper selection of the finite element interpolation functions for the fields, the proposed formulation is equivalent to the commonly used approximation of the vector wave equation with tangentially continuous vector finite elements. This equivalence is exploited to improve the computational efficiency of the model order reduction process     

Simulation and sensitivity analysis of nonuniform transmission lines using integrated congruence transform

This paper presents a new algorithm for computing sensitivity information of nonuniform multiconductor transmission lines with respect to arbitrary physical parameters. The proposed algorithm provides sensitivity information through a reduced-order system that has a simple representation in the time-domain. This feature enables computing sensitivity in the presence of nonlinear terminations. The proposed algorithm is based on the concept of passive model-order reduction using integrated congruence transform. It also addresses the problem of sensitivity analysis for nonuniform multiconductor transmission lines without having to resort to any discretization techniques     

Reduced-order modeling for hyperthermia control

This paper analyzes the feasibility of using reduced-order modeling techniques in the design of multiple-input, multiple-output (MIMO) hyperthermia temperature controllers. State space thermal models are created based upon a finite difference expansion of the bioheat transfer equation model of a scanned focused ultrasound system (SFUS). These thermal state space models are reduced using the balanced realization technique, and an order reduction criterion is tabulated. Results show that a drastic reduction in model dimension can be achieved using the balanced realization. The reduced-order model is then used to design a reduced-order optimal servomechanism controller for a two-scan input, two thermocouple output tissue model. In addition, a full-order optimal servomechanism controller is designed for comparison and validation purposes. These two controllers are applied to a variety of perturbed tissue thermal models to test the robust nature of the reduced-order controller. A comparison of the two controllers validates the use of open-loop balanced reduced-order models in the design of MIMO hyperthermia controllers.     

Least-squares model order reduction enhancements

Two enhancements to the least-squares (LS) discrete-time model order reduction (MOR) method are presented: scaling and frequency response matching. Scaling generally improves the low-frequency fit between the reduced-order model (ROM) and the original model. For exact gains at specific frequencies, optional frequency response constraints can easily be added to the LS MOR method. An example is presented that illustrates these enhancements. The example model is reduced with the Hankel norm, weighted impulse response gramian, and LS MOR methods. Plots of error versus frequency are given for each of the three MOR methods     

An efficient terminal and model order reduction algorithm

The paper proposes an efficient terminal and model order reduction method for compact modeling of interconnect circuits with many terminals. The new method is inspired by the recently proposed terminal reduction method, SVDMOR [P. Feldmann, F. Liu, Sparse and efficient reduced order modeling of linear subcircuits with large number of terminals, in: Proceedings of the International Conference on Computer Aided Design (ICCAD), 2004, pp. 88-92]. But different from SVDMOR, the new method considers higher order moment information for terminal responses during the terminal reduction and separately applies singular value decomposition (SVD) on both input and output terminals for low-rank approximations. This is in contrast to the SVDMOR method where input and output terminal responses are approximated by SVD at the same time, which can lead to large errors when the numbers of inputs and outputs are quite different. We analyze the passivity requirements for SVD-based terminal and model order reduction and show that the combined passive terminal and MOR using SVD method will not lead an effective terminal reduction in general. Our experimental results show that the proposed ESVDMOR method outperforms the SVDMOR method in terms of accuracy for the same reduced model sizes when the numbers of input and output terminals are quite different.     

Broadband Rational Macromodeling Based on the Adaptive Frequency Sampling Algorithm and the Partial Element Equivalent Circuit Method

The increasing operating frequencies in modern designs call for broadband macromodeling techniques. The problem of computing high-accuracy simulation models for high-speed interconnects is of great importance in the modeling arena. Nowadays, many full-wave numerical techniques are available that provide high accuracy, often at a significant cost in terms of memory storage and computing time. Furthermore, designers are usually only interested in a few electrical quantities such as port voltages and currents. So, model order reduction techniques are commonly used to achieve accurate results in a reasonable time. This paper presents a new technique, based on the partial element equivalent circuit method, which allows to generate reduced-order models by adaptively selecting the complexity (order) of the macromodel and suitable frequency samples. Thus, the proposed algorithm allows to limit the computing time while preserving the accuracy. Validation examples are given.     

Guaranteed passive direct lumped-element modeling of transmission lines

A novel approach to the reduced-order modeling of the frequency-dependent skin-effect (R-L) parameters of a transmission-line interconnect is proposed. A lumped-element circuit model for a short length of the lines is obtained directly from the finite-element method matrix equations. Only the most "important" eigenvalues and eigenvectors of the finite-element matrix are used, based on the contribution they make to the line parameters. The resulting reduced-order equivalent circuit is always passive and stable without the need for special modifications. The model can thus be immediately applied to problems where the transmission line is connected to nonlinear circuits. The model should find practical use in the design of flexible interconnect circuits as well as in other areas of high-speed digital electronics.     

Authors' Reply [to "Corrections and Comments to 'Model Reduction of Discrete Linear Systems Via Frequency-Domain Balanced Structure'"]

For original paper, see N. L. Prajapathy et al., ibid, vol. 54, no. 3, pp. 682-683, (2007). Reply to the comments on model reduction technique for discrete linear time invariant systems are presented. The proposed technique is based on a conceptual view point of the controllability and observability Grammians balancing of a system in an arbitrary frequency range. It can be considered as the generalization of the Moore's (1981) balance structure approach in a specific frequency range of operation. Two modified Lyapunov equations are derived for the proposed frequency domain balancing. The transfer function of the sixth-order Cheby-shev type 1 filter is considered. The Nyquist plots for the original filter as well as the reduced-order filters based on the Moore's balanced technique and the proposed method.     

<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Model Reduction of 2-D Singular Roesser Models

This paper discusses the problem of H_∞ model reduction for linear discrete time 2-D singular Roesser models (2-D SRM). A condition for bounded realness is established for 2-D SRM in terms of linear matrix inequalities (LMIs). Based on this, a sufficient condition for the solvability of the H_∞ model reduction problem is obtained via a group of LMIs and a set of coupling non-convex rank constraints. An explicit parameterization of the desired reduced-order models is presented. Particularly, a simple LMI condition without rank constraints is proposed for the zeroth-order H_∞ approximation problem. Finally, a numerical example is given to illustrate the applicability of the proposed approach.     

An aggregating based model order reduction method for power grids

Simulation of power girds has become increasingly computationally expensive. In this paper, we propose a Model Order Reduction (MOR) method for power grids by extending the existing Aggregating based MOR (AMOR) method. In the proposed method, besides resistors and capacitors, current sources are also aggregated to improve MOR efficiency. Moreover, pre-partition and parallelization techniques are employed to decrease reduction time. Numerical results demonstrate that compared to original circuits, the scale of power grids is greatly reduced without much loss of accuracy. The reduced-order models are especially useful in the multiple simulations of different working modes or different environment corners.