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Showing papers on "Model order reduction published in 2008"


Journal ArticleDOI
TL;DR: In this article, a model order reduction method is developed and applied to 1D diffusion systems with negative real eigenvalues, and residues with similar eigen values are grouped together to reduce the model order.
Abstract: A model order reduction method is developed and applied to 1D diffusion systems with negative real eigenvalues. Spatially distributed residues are found either analytically (from a transcendental transfer function) or numerically (from a finite element or finite difference state space model), and residues with similar eigenvalues are grouped together to reduce the model order. Two examples are presented from a model of a lithium ion electrochemical cell. Reduced order grouped models are compared to full order models and models of the same order in which optimal eigenvalues and residues are found numerically. The grouped models give near-optimal performance with roughly 1/20 the computation time of the full order models and require 1000―5000 times less CPU time for numerical identification compared to the optimization procedure.

119 citations


Book ChapterDOI
01 Jan 2008
TL;DR: It is argued that much more complex problems can be addressed by making use of current computing technology and advanced algorithms, but that there is a need for model order reduction in order to cope with even morecomplex problems.
Abstract: In this first section we present a high level discussion on computational science, and the need for compact models of phenomena observed in nature and industry. We argue that much more complex problems can be addressed by making use of current computing technology and advanced algorithms, but that there is a need for model order reduction in order to cope with even more complex problems. We also go into somewhat more detail about the question as to what model order reduction is.

89 citations


Journal ArticleDOI
TL;DR: An improved frequency domain interval Gramian-based model reduction scheme for discrete time systems is presented and not only yields stable reduced-order models but also has easily computable frequency response error bounds.
Abstract: An improved frequency domain interval Gramian-based model reduction scheme for discrete time systems is presented. It is first shown that two of the main results presented in the model reduction method of are incorrect. Improved methods which overcomes these shortcomings are then presented. Improved methods not only yields stable reduced-order models but also have easily computable frequency response error bounds. The method is further extended to 2-D separable denominator system approximation. The simulation results show the effectiveness of the proposed scheme.

66 citations


Journal ArticleDOI
TL;DR: PIMTAP model yields the same form of the original state equations and preserves the passivity of parameterized R LC networks like the well-known method passive reduced-order interconnect macromodeling algorithm for nonparameterized RLC networks.
Abstract: This paper presents a multiparameter moment-matching-based model order reduction technique for parameterized interconnect networks via a novel two-directional Arnoldi process (TAP). It is referred to as a Parameterized Interconnect Macromodeling via a TAP (PIMTAP) algorithm. PIMTAP inherits the advantages of previous multiparameter moment-matching algorithms and avoids their shortfalls. It is numerically stable and adaptive. PIMTAP model yields the same form of the original state equations and preserves the passivity of parameterized RLC networks like the well-known method passive reduced-order interconnect macromodeling algorithm for nonparameterized RLC networks.

59 citations


Journal ArticleDOI
TL;DR: An optimization based model order reduction (MOR) framework is proposed that involves setting up a quasi-convex program that explicitly minimizes a relaxation of the optimal H/sub /spl infin// norm MOR problem.
Abstract: In this paper, an optimization-based model order reduction (MOR) framework is proposed. The method involves setting up a quasi-convex program that solves a relaxation of the optimal Hinfin norm MOR problem. The method can generate guaranteed stable and passive reduced models and is very flexible in imposing additional constraints such as exact matching of specific frequency response samples. The proposed optimization-based approach is also extended to solve the parameterized model-reduction problem (PMOR). The proposed method is compared to existing moment matching and optimization-based MOR methods in several examples. PMOR models for large RF inductors over substrate and power-distribution grid are also constructed.

54 citations


Proceedings ArticleDOI
10 Nov 2008
TL;DR: A model order reduction approach, named maniMOR, which extends the linear projection framework to a general nonlinear projection framework and constructs a nonlinear manifold which captures important system responses and defines the corresponding non linear projection operator.
Abstract: Previous model order reduction methods fit into the framework of identifying the low-order linear subspace and using the linear projection to project the full state space into the low-order subspace. Despite its simplicity, the macromodel might automatically include redundancies. In this paper, we present a model order reduction approach, named maniMOR, which extends the linear projection framework to a general nonlinear projection framework. The two key ideas of maniMOR are (1) it explicitly separates the construction of the low-order subspace and projection operation; (2) it constructs a nonlinear manifold which captures important system responses and defines the corresponding nonlinear projection operator. The low-order manifold subspace in maniMOR is identified by stitching together the low-order linear subspaces around a set of sample points on the manifold. After the manifold is determined, it is embedded into a global nonlinear coordinate system. The projection function is defined in a piece-wise linear manner, and the model evaluation is conducted directly in the manifold subspace using cheap matrix-vector product computations. As a result, a compact model is generated by pre- computing all the functions and Jacobians and storing them in a look-up table. We apply maniMOR on two analog circuits and a bio-chemical system to validate its correctness. Extensive comparisons with the results of the full model and other macromodels are provided. Experimental results show that maniMOR manages to obtain a huge reduction - e.g., from 52 to 5 for the I/O buffer circuit and from 304 to 30 for yeast pheromone pathway system. This is less than half of the size of the TPWL model with the same accuracy. With great promise to capture important system responses, maniMOR presents a novel and powerful paradigm for nonlinear model reduction, and casts inspirations for further researches.

53 citations


Journal ArticleDOI
TL;DR: 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, and allows to limit the computing time while preserving the accuracy.
Abstract: 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.

51 citations


Journal ArticleDOI
TL;DR: A new approach based on generating low-dimensional simulation models that approximate the original spatially discretized models of electromagnetic field and their variations under conditions of component movement and material nonlinearity is presented.
Abstract: We present a new approach for generating fast simulation models of electromagnetic (EM) devices that contain moving components and magnetic materials with nonlinear properties. Our approach is based on generating low-dimensional simulation models that approximate the original spatially discretized models of electromagnetic field and their variations under conditions of component movement and material nonlinearity. The movement of the modeled device components is simulated by coupling the reduced-order EM field models weakly to the mechanical equations. We have successfully used our approach to generate a fast simulation model of a simple electromagnetic device with a moving component and nonlinear material properties.

45 citations


Book ChapterDOI
01 Jan 2008
TL;DR: In this paper, the authors give an overview of model order reduction techniques for coupled systems and discuss model reduction of linear time-invariant control systems that are coupled through input-output relations.
Abstract: In this paper we give an overview of model order reduction techniques for coupled systems. We consider linear time-invariant control systems that are coupled through input-output relations and discuss model reduction of such systems using moment matching approximation and balanced truncation. Structure-preserving approaches to model order reduction of coupled systems are also presented. Numerical examples are given.

45 citations


Journal ArticleDOI
TL;DR: The tree-cotree splitting technique is applied to edge elements to account for the decoupling between the electric and magnetic fields at low frequencies, which is the main reason for the low-frequency problem.
Abstract: In this study the tree-cotree splitting technique is applied for improving the finite-element matrix conditioning for the analysis of high-speed circuits. A well-known issue is that at low frequencies a full-wave solver yields less accurate solutions and may even breakdown due to ill-conditioned system matrices. To enhance the capability and reliability of the conventional finite element method in broadband full-wave analyses, we apply the tree-cotree splitting to edge elements to account for the decoupling between the electric and magnetic fields at low frequencies, which is the main reason for the low-frequency problem. The algorithm for finding a minimum spanning tree when there exist wave ports, lumped ports, or for a PEC-free structure are described. Besides, a model order reduction method, called the solution space projection, is applied for a fast broadband analysis. We further propose an expansion to available solution bases for a better approximation to low-frequency fields, so that a simulation can be extended to extremely low frequencies. The application is focused on the simulation of high-speed circuits, of which both low and high frequency characteristics are of equal importance. © 2008 Wiley Periodicals, Inc. Microwave Opt Technol Lett 50: 1476–1481, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.23403

44 citations


Journal ArticleDOI
TL;DR: The present article gives a detailed description of the fully coupled tube scanner dynamics over a wide frequency range modeled by finite element (FE) analysis using the commercially available software package ANSYS.
Abstract: Piezoelectric tube scanners are widely used in scanning probe microscopes to position the sample or the probe. Fast and accurate scanning requires the suppression of dominant low-frequency resonances as well as the compensation of dynamics-coupling effects. The present article gives a detailed description of the fully coupled tube scanner dynamics over a wide frequency range modeled by finite element (FE) analysis using the commercially available software package ANSYS. The effect of a sample mass attached to the top of the tube is investigated by considering its added mass and local stiffening. A model order reduction scheme is applied to obtain a low order model that describes the lateral and vertical deflections as well as the voltage induced on quadrant electrodes. Comparison to experimental data demonstrates a good agreement for both the full FE model and reduced order model.

Proceedings ArticleDOI
16 May 2008
TL;DR: In this article, the authors proposed a methodology to obtain unterminated frequency response functions for dc-dc converters in the presence of source and load coupled FRF measurements, and presented a model order reduction technique to enable the simulation of dc distributed power systems with a large number of converters.
Abstract: Black-box terminal characterization models are constructed from "un-terminated" frequency response functions (FRF) measured at the converter terminals without requiring explicit knowledge of any of the converter parameters; however to some extent, these measurements are always coupled with the source and load dynamics which reduces the fidelity of the final models obtained. This paper analyzes this problem and proposes a methodology to obtain un-terminated FRFs for dc-dc converters in the presence of source and load coupled FRF measurements. Furthermore, it presents a model order reduction technique to enable the simulation of dc distributed power systems with a large number of converters, applied to the calculated un-terminated FRFs that constitute the black-box models in question. Experimental results are presented to verify the theoretical analysis and the high accuracy obtained with the black-box models built.

Journal ArticleDOI
TL;DR: Second-order Gramian approximation (SOGA) version of SBPOR is proposed, to mitigate high computational cost of solving Lyapunov equation and demonstrate that SBPOR and SOGA are globally more accurate than the Krylov subspace based approaches.
Abstract: In this paper, we propose a novel model-order reduction (MOR) approach, second-order balanced truncation (BT) for passive-order reduction (SBPOR), which is the first second-order BT method proposed for passive reduction of RLCK circuits. By exploiting the special structure information in the circuit formulation, second-order Gramians are defined based on a symmetric first-order realization in descriptor from. As a result, SBPOR can perform the traditional balancing with passivity-preserving congruency transformation at the cost of solving one generalized Lyapunov equation. Owing to the second-order formulation, SBPOR also preserves the structure information inherent to RLCK circuits. We further propose, second-order Gramian approximation (SOGA) version of SBPOR , to mitigate high computational cost of solving Lyapunov equation. Experimental results demonstrate that SBPOR and SOGA are globally more accurate than the Krylov subspace based approaches.

Proceedings ArticleDOI
17 Mar 2008
TL;DR: This work utilizes Kalman filter (KF) for temperature estimation and for elimination of sensing inaccuracies as well and typically reduces the standard deviation and maximum value of temperature estimation errors by about an order of magnitude.
Abstract: In this work we present a method for accurate estimation of temperature at various locations on a chip considering the inaccuracies in thermal sensor readings due to limitations mainly due to thermal sensor placement and sensor noise. This technique enables accurate estimation of temperature at different locations on the chip with only a limited number of sensors in an efficient way. We utilize Kalman filter (KF) for temperature estimation and for elimination of sensing inaccuracies as well. The computational complexity is reduced by using steady state Kalman filter during normal operation of the chip and reducing the order of the thermal model by a projection based model order reduction method. Our experimental results show that this technique typically reduces the standard deviation and maximum value of temperature estimation errors by about an order of magnitude.

Book ChapterDOI
01 Jan 2008
TL;DR: The problem of structure-preserving model order reduction of general RCL circuits is described, and two state-of-the-art algorithms, PRIMA and SPRIM, are discussed for the solution of this problem.
Abstract: In recent years, order-reduction techniques based on Krylov subspaces have become the methods of choice for generating macromodels of large multi-port RCL circuits. Despite the success of these techniques and the extensive research efforts in this area, for general RCL circuits, the existing Krylov subspace-based reduction algorithms do not fully preserve all essential structures of the given large RCL circuit. In this paper, we describe the problem of structure-preserving model order reduction of general RCL circuits, and we discuss two state-of-the-art algorithms, PRIMA and SPRIM, for the solution of this problem. Numerical results are reported that illustrate the higher accuracy of SPRIM vs. PRIMA. We also mention some open problems.

Journal ArticleDOI
TL;DR: This algorithm requires recursive computations with respect to the order of the Taylor series in which the authors need to solve linear equations with unknown parameters in each step to achieve nonlinear balanced realization and model reduction.

Journal ArticleDOI
TL;DR: Theoretical foundations of global Krylov subspace methods for model order reductions for multiple-inputs multiple-outputs (MIMO) systems are presented and two algorithms, including the adaptive-order rational global Arnoldi (AORGA) algorithm and theadapt-order global Lanczos (AOGL) algorithm, are developed in detail.

Journal ArticleDOI
TL;DR: The contribution will give an overview of the symbolic tool Analog Insydes algorithms for extraction of dominant behavior of linear systems, e.g. formulas for poles and zeros as well as algorithms for generating behavioral models from nonlinear DAEs.

Proceedings ArticleDOI
10 Nov 2008
TL;DR: An algorithm is presented that combines the advantages of widely-used approaches such as PRIMA and TICER but avoids many of the drawbacks of both and is capable of high-order rational approximation, exploits network sparsity, and preserves passivity.
Abstract: This paper is concerned with model order reduction of large scale dynamic systems that have sparse matrix representations, particularly systems with large numbers of input/output ldquoports.rdquo We present an algorithm that combines the advantages of widely-used approaches such as PRIMA and TICER but avoids many of the drawbacks of both. The resulting algorithm is capable of high-order rational approximation, exploits network sparsity, preserves passivity, can be extended to general non-symmetric systems, and can be applied to networks with hundreds or thousands of ports. We develop a common mathematical framework that can encompass all three algorithms, show mathematical relations between them, and point out certain special cases where they are equivalent. We show examples from analysis of industrial on-chip RC/RLC networks that demonstrate performance advantages of more than three orders of magnitude.

Journal ArticleDOI
TL;DR: This paper presents a numerically stable method for the model order reduction of finite element approximations to passive microwave structures parameterized by polynomials in several variables using Krylov subspaces and extends the works of Gunupudi etal.
Abstract: In this paper we present a numerically stable method for the model order reduction of finite element (FE) approximations to passive microwave structures parameterized by polynomials in several variables. The proposed method is a projection-based approach using Krylov subspaces and extends the works of Gunupudi etal. (P. Gunupudi, R. Khazaka and M. Nakhla, Analysis of transmission line circuits using multidimensional model reduction techniques, IEEE Trans. Adv. Packaging 25 (2002), pp. 174–180) and Slone etal. (R.D. Slone, R. Lee and J.-F. Lee, Broadband model order reduction of polynomial matrix equations using single-point well-conditioned asymptotic waveform evaluation: derivations and theory, Int. J. Numer. Meth. Eng. 58 (2003), pp. 2325–2342). First, we present the multivariate Krylov space of higher order associated with a parameter-dependent right-hand-side vector and derive a general recursion for generating its basis. Next, we propose an advanced algorithm to compute such basis in a numerically st...

Journal ArticleDOI
Bjorn Gustavsen1, J. Nordstrom
TL;DR: This paper solves the problem of pole identification for the propagation function of the universal line model by basing the pole identification on trace fitting rather than mode fitting, followed by time-delay refinement and model-order reduction.
Abstract: The universal line model (ULM) is a frequency dependent transmission-line model based on the method of characteristics in the phase domain. Although the ULM is known to produce highly accurate models for both overhead lines and underground cables, situations have been encountered where the pole identification for the propagation function fails. In this paper, we overcome the problem by basing the pole identification on trace fitting rather than mode fitting. This is achieved by introducing delayed basis functions in the vector fitting algorithm, followed by time-delay refinement and model-order reduction. In situations where the modes can be fitted without difficulty, the existing approach using modes obtained by a frequency-dependent transformation matrix remains the most accurate.

DOI
01 Jan 2008
TL;DR: The objective of this PhD research is to increase the performance of Pstar, the in-house analog circuit simulator at Philips and now of NXP Semiconductors, while properties like accuracy and robustness are maintained, in particular the convergence and stability properties of newly developed multirate time-integration algorithms is studied.
Abstract: Circuit simulation is an essential step within circuit design. Because of the increasing complexity of the Integrated Circuits, electronic companies need fast and accurate simulation software and there is a constant request at the companies to further improve the simulation software. Development of new, more advanced, transient simulation algorithms is an attractive way to increase the performance of this software. Mathematics is the basis to analyze the convergence properties. The objective of this PhD research is to increase the performance of Pstar, the in-house analog circuit simulator at Philips and now of NXP Semiconductors, while properties like accuracy and robustness are maintained. In particular the convergence and stability properties of newly developed multirate time-integration algorithms is studied. Usually circuit models are large systems of differential-algebraic equations that are derived from Kirchhoff’s conservation laws for currents and voltages and the constitutive relations for the electronic components. For a transient analysis one traditionally uses implicit time-integration schemes, like Backward Difference Formulae (BDF). All these schemes discretise the time on one time-grid. In contrast multirate algorithms use more than one time-grid and compute the slowly time-varying state elements only at coarsely distributed time-points, while the fastly time-varying state elements are computed at finer distributed timepoints. This makes a multirate algorithm potentially much faster for circuits with large low-frequency parts. There are many types of multirate timeintegration methods that may differ in the order of the slow and fast integration and the treatment of the interface variables. We used a direct extension of the BDF scheme combined with Lagrange interpolation of the same order. The standard theory for multistep methods does not hold anymore for multirate algorithms. Therefore we look at properties like stability and convergence in more detail. It turns out that the method is stable if the partitioned subsystems are individually stable and if the coupling is sufficiently weak. The discretisation error for a multirate method also contains an interpolation error due to the slow unknowns at the interface. This error component is not needed for ordinary multistep methods. It is possible to control this error by independent control of the coarse and fine macro and micro time-steps, respectively. The interpolation error and the coarse discretisation error is controlled by the macro stepsize, while the micro stepsize controls the fine discretisation errors for the fast state part. For multirate it is necessary to partitioning the system into a slow and a fast part. Therefore a part of the research is spent to the development and analysis of automatic partitioning algorithms. The underlying problem is a discrete optimisation problem, that can be handled by greedy-like methods. It is also possible to change the partitioning dynamically during the simulation, which is useful for moving active parts. All algorithms are implemented in Matlab; they work satisfactorily when tested for a variety of circuit models. Furthermore a multirate implementation including error control and dynamical partitioning is created in the circuit simulator Pstar itself. Besides multirate time-integration also model order reduction is studied, which transforms the large data models into smaller and simpler models, that still give the proper accuracy, but that are much cheaper to solve. Because IC models are nonlinear, nonlinear reduction techniques are considered in particular, like POD. In particular we focused on the problem to reduce the evaluation costs of these reduced models. A proper use of multirate and model order reduction is able to speed up transient simulation in general and is significantly faster (more than an order) for redundant circuit models, while the accuracy and robustness are maintained. Redundancy occurs if the state elements have many correlations, or if the sampled state signal has correlations in time.

Journal Article
TL;DR: An embeddable Input-Output structure Preserving Order Reduction (IOPOR) technique is proposed to further preserve the structures of input and output incidence matrices and inline diagonalization and regularization techniques are specifically proposed to enhance the robustness of inductance synthesis.
Abstract: This paper aims to explore RLC equivalent circuit synthesis method for reduced-order models of interconnect circuits obtained by Krylov subspace based model order reduction (MOR) methods. To guarantee pure RLC equivalent circuits can be synthesized, both the structures of input and output incidence matrices and the block structure of the circuit matrices should be preserved in the reduced-order models. Block structure preserving MOR methods have been well established. In this paper, we propose an embeddable Input-Output structure Preserving Order Reduction (IOPOR) technique to further preserve the structures of input and output incidence matrices. By combining block structure preserving MOR methods and IOPOR technique, we develop an RLC equivalent circuit synthesis method RLCSYN (RLC SYNthesis). Inline diagonalization and regularization techniques are specifically proposed to enhance the robustness of inductance synthesis. The pure RLC model, high modeling accuracy, passivity guaranteed property and SPICE simulation robustness make RLCSYN more applicable in interconnect analysis, either for digital IC design or mixed signal IC simulation. AMS subject classifications: 94C05, 93A15, 68U07, 68U20, 41A21

Journal Article
TL;DR: In this article, a newly developed identification algorithm, called moments method, is introduced and applied to the parameter identification of a dc motor, and the simulation and experimental results are presented and compared.
Abstract: The identification process consists of estimating the unknown parameters of system dynamics. Consequently, determination of the assumed system structure is of great importance in the process of system identification. Time moments have been introduced in automatic control because of the analogy between the impulse response of a linear system and a probability function. This basic idea has generated applications in identification, model order reduction and controller design. In this paper, a newly developed identification algorithm, called moments method, is introduced and applied to the parameter identification of a dc motor. The simulation and experimental results are presented and compared.

Journal ArticleDOI
TL;DR: A novel computational intelligence technique to generate concise neural network models for distributed dynamic systems based on artificial neural network architectures that incorporate linear and nonlinear principal component analysis, combined with generalized dimensional analysis.
Abstract: Purpose – This paper seeks to present a novel computational intelligence technique to generate concise neural network models for distributed dynamic systems.Design/methodology/approach – The approach used in this paper is based on artificial neural network architectures that incorporate linear and nonlinear principal component analysis, combined with generalized dimensional analysis.Findings – Neural network principal component analysis coupled with generalized dimensional analysis reduces input variable space by about 90 percent in the modeling of oil reservoirs. Once trained, the computation time is negligible and orders of magnitude faster than any traditional discretisation schemes such as fine‐mesh finite difference.Practical implications – Finding the minimum number of input independent variables needed to characterize a system helps in extracting general rules about its behavior, and allows for quick setting of design guidelines, and particularly when evaluating changes in the physical properties o...

01 Jan 2008
TL;DR: This paper proposes a novel model reduction technique which can efficiently perform model reduction for linear systems with a large number of inputs and outputs.
Abstract: At present, almost all model order reduction methods assume single-input single-output (SISO) systems or systems with a small number of inputs and outputs. Few methods can deal with systems with a large number of inputs and outputs. Multi-input multi-output (MIMO) systems appear for example in modeling of integrated circuits. The number of inputs and outputs sometimes is very large, even close to the number of state variables. In this paper we propose a novel model reduction technique which can efficiently perform model reduction for linear systems with a large number of inputs and outputs. Motivated by the superposition property of linear systems, model order reduction is performed separately with respect to each column of the input matrix. Then, the output response of the original multi-input system is approximated by the summation of the output responses of the reduced-order single-input systems corresponding to each column of the input matrix. The proposed method applies to both multi-input single-output systems

Journal ArticleDOI
TL;DR: The proposed modeling methodology utilizes the multiple power and ground planes used in such structures as natural physical boundaries for their decomposition into a set of subdomains, each one of which is meshed and discretized separately from the rest.
Abstract: The complicated geometry of high-density interconnect structures in multilayer, planar substrates is one of the major hurdles in the application of finite element methods for their multigigahertz electromagnetic analysis. In addition to compounding the complexity in the generation of the finite element grid, the multilayer nature of the structures and their multiscale attributes result in finite element systems of very large dimension which, more often than not, are not well conditioned. This paper presents a domain decomposition methodology for overcoming these hurdles. More specifically, the proposed methodology utilizes the multiple power and ground planes used in such structures as natural physical boundaries for their decomposition into a set of subdomains, each one of which is meshed and discretized separately from the rest. The electromagnetic interaction between the domains is effected through the enforcement of tangential field continuity conditions at the voids and via holes present at the power and ground planes. In particular, a Krylov subspace model order reduction approach is used to facilitate the broadband solution of the multilayer interconnect structure. The proposed modeling methodology is demonstrated through its application to the electromagnetic analysis of several multilayer interconnect structures.

16 Dec 2008
TL;DR: The goal of this document is to provide an overview of available methods together with a classification of nonlinear problems that in principle could be handled by these methods.
Abstract: In this document we review the status of existing techniques for non-linear model order reduction by investigating how well these techniques perform for typical industrial needs. In particular the TPWL-method (Trajectory Piecewise Linear-method) and the POD-approach (Proper Orthogonal Decomposion) is taken under consideration. We address several questions that are (closely) related to both the theory and application of nonlinear model order reduction techniques. The goal of this document is to provide an overview of available methods together with a classification of nonlinear problems that in principle could be handled by these methods.

01 Jan 2008
TL;DR: In this paper, the authors give an overview of the recent work on balanced realization and the related model order reduction method based on nonlinear singular value analysis for nonlinear operators, and show how the balanced truncation method works for real-world systems.
Abstract: For linear control systems minimal realization theory and the related model reduction methods play a crucial role in understanding and handling the system. These methods are well established and have proved to be very successful. In particular the method called balanced truncation gives a good reduced order model with respect to the input-output behavior. This method relies on the relation with the system Hankel operator, which plays a central role in minimal realization theory. Specifically, the Hankel operator supplies a set of similarity invariants, the so called Hankel singular values, which can be used to quantify the importance of each state in the corresponding input-output system. The Hankel operator can also be factored into a composition of observability and controllability operators, from which Gramian matrices can be defined and the notion of balanced realization follows. This linear theory is rather complete and the relations between and interpretations in the state-space and input-output settings are fully understood. This paper gives an overview of the series of research on balanced realization and the related model order reduction method based on nonlinear singular value analysis. Section 2 explains the taken point of view on singular value analysis for nonlinear operators. Section 3 briefly reviews the linear balancing method and balanced truncation in order to show the way of thinking for the nonlinear case. Section 4 treats the state-space balancing method. Then, in Section 5 we continue with balanced realizations based on the singular value analysis of the nonlinear Hankel operator. Furthermore, in Section 6 balanced truncation based on the method of Section 5 is presented. Finally, in Section 7 a numerical simulation illustrates how the proposed model order reduction method works for real-world systems.

Patent
08 Feb 2008
TL;DR: In this article, a preliminary model associated with a system to be controlled and constructing a weighted model using one or more weighting factors is presented, where the final model has a stability margin that is greater than an uncertainty associated with the preliminary model.
Abstract: One method includes obtaining a preliminary model associated with a system to be controlled and constructing a weighted model using one or more weighting factors. The method also includes identifying a final model of the system using the preliminary and weighted models, where the final model has a stability margin that is greater than an uncertainty associated with the final model. The method further includes controlling the system using a controller designed based on the final model. Another method includes identifying a first model associated with a system to be controlled, performing model order reduction to identify a second model, and controlling the system using a controller designed based on the second model. Performing the model order reduction includes reducing a weighted coprime factor model uncertainty between the first and second models.