Showing papers on "Model order reduction published in 2010"

Parametric Model Order Reduction by Matrix Interpolation

Abstract: In this paper, a new framework for model order reduction of LTI parametric systems is introduced. After generating and reducing several local original models in the parameter space, a parametric reduced-order model is calculated by interpolating the system matrices of the local reduced models. The main task is to find compatible system representations with optimal interpolation properties. Two approaches for this purpose are presented together with several numerical simulations.

A Real-Time Thermal Model of a Permanent-Magnet Synchronous Motor

Abstract: This paper presents a real-time thermal model with calculated parameters based on the geometry of the different components of a permanent-magnet synchronous motor. The model in state-space format has been discretized and a model-order reduction has been applied to minimize the complexity. The model has been implemented in a DSP and predicts the temperature of the different parts of the motor accurately in all operating conditions, i.e., steady-state, transient, and stall torque. The results have been compared with real measurements using temperature transducers showing very good performance of the proposed thermal model.

Balanced Realization and Model Order Reduction for Nonlinear Systems Based on Singular Value Analysis

Abstract: This paper discusses balanced realization and model order reduction for both continuous-time and discrete-time general nonlinear systems based on singular value analysis of the corresponding Hankel operators. Singular value analysis clarifies the gain structure of a given nonlinear operator. Here it is proved that singular value analysis of smooth Hankel operators defined on Hilbert spaces can be characterized by simple equations in terms of their states. A balanced realization and model order reduction procedure is derived based on it, and several important properties such as stability, balanced form, Hankel norm, controllability, and observability of the original system are preserved. The work improves the earlier results of [K. Fujimoto and J. M. A. Scherpen, IEEE Trans. Automat. Control, 50 (2005), pp. 2-18] and then continues with new balancing and model reduction results.

Decentralized Control and Filtering in Interconnected Dynamical Systems

Abstract: Based on the many approaches available for dealing with large-scale systems (LSS), Decentralized Control and Filtering in Interconnected Dynamical Systems supplies a rigorous framework for studying the analysis, stability, and control problems of LSS. Providing an overall assessment of LSS theories, it addresses model order reduction, parametric un

Model order reduction for hyperelastic materials

Abstract: In this paper, we develop a novel algorithm for the dimensional reduction of the models of hyperelastic solids undergoing large strains. Unlike standard proper orthogonal decomposition methods, the proposed algorithm minimizes the use of the Newton algorithms in the search of non-linear equilibrium paths of elastic bodies.The proposed technique is based upon two main ingredients. On one side, the use of classic proper orthogonal decomposition techniques, that extract the most valuable information from pre-computed, complete models. This information is used to build global shape functions in a Ritz-like framework.On the other hand, to reduce the use of Newton procedures, an asymptotic expansion is made for some variables of interest. This expansion shows the interesting feature of possessing one unique tangent operator for all the terms of the expansion, thus minimizing the updating of the tangent stiffness matrix of the problem.The paper is completed with some numerical examples in order to show the performance of the technique in the framework of hyperelastic (Kirchhoff-Saint Venant and neo-Hookean) solids.

MMSE-Based Filtering in Presence of Non-Gaussian System and Measurement Noise

Abstract: The problem of sequential Bayesian estimation in linear non-Gaussian problems is addressed. In the Gaussian sum filter (GSF), the non-Gaussian system noise, the measurement noise, and the posterior state densities are modeled by the Gaussian mixture model (GMM). The GSF is optimal under the minimum-mean-square error (MMSE) criterion, however it is impractical due to the exponential model order growth of the system probability density function (pdf). The proposed recursive estimator, named the Gaussian mixture Kalman filter (GMKF), combines the GSF and the model order reduction procedure. The posterior state density at each iteration is approximated by a lower order density. This model order reduction procedure minimizes the estimated Kullback-Leibler divergence (KLD) of the reduced order density from the original density at each step. The estimation performance of the proposed GMKF is compared with the interactive multiple modeling (IMM), particle filter (PF), Gaussian sum PF (GSPF), and the GSF with mixture reduction (MR) method via simulations. It is shown in several examples that the proposed GMKF outperforms the other tested algorithms in terms of estimation accuracy. The superior estimation performance of the GMKF is obtained at the expense of its computational complexity, which is higher than the IMM and the MR algorithms.

Non-linear reduced order models for steady aerodynamics

Abstract: A reduced order modelling approach for predicting steady aerodynamic flows and loads data based on Computational Fluid Dynamics (CFD) and global Proper Orthogonal Decomposition (POD), that is, POD for multiple different variables of interest simultaneously, is presented. A suitable data transformation for obtaining problemadapted global basis modes is introduced. Model order reduction is achieved by parameter space sampling, reduced solution space representation via global POD and restriction of a CFD flow solver to the reduced POD subspace. Solving the governing equations of fluid dynamics is replaced by solving a non-linear least-squares optimization problem. Methods for obtaining feasible starting solutions for the optimization procedure are discussed. The method is demonstrated by computing reduced-order solutions to the compressible Euler equations for the NACA 0012 airfoil based on two different snapshot sets; one in the subsonic and one in the transonic flow regime, where shocks occur. Results are compared with those obtained by POD-based interpolation using Kriging and the Thin Plate Spline method (TPS).

Time-Varying Eigensystem Realization Algorithm

Abstract: An identification algorithm called the time-varying eigensystem realization algorithm is proposed to realize discrete-time-varying plant models from input and output experimental data. It is shown that this singular value decomposition based method is a generalization of the eigensystem realization algorithm developed to realize time invariant models from pulse response sequences. Using the results from discrete-time identification theory, the generalized Markov parameter and the generalized Hankel matrix sequences are computed via a least squares problem associated with the input-output map. The computational procedure presented in the paper outlines a methodology to extract a state space model from the generalized Hankel matrix sequence in different time-varying coordinate systems. The concept of free response experiments is suggested to identify the subspace of the unforced system response. For the special case of systems with fixed state space dimension, the free response subspace is used to construct a uniform coordinate system for the realized models at different time steps. Numerical simulation results on general systems discuss the details and effectiveness of the algorithms.

The Lanczos Method for Parameterized Symmetric Linear Systems with Multiple Right-Hand Sides

Abstract: The solution of linear systems with a parameter is an important problem in engineering applications, including structural dynamics, acoustics, and electronic circuit simulations, and in related model order reduction methods such as Pade via Lanczos. In this paper, we present a Lanczos-based method for solving parameterized symmetric linear systems with multiple right-hand sides. We show that for this class of applications, a simple deflation method can be used.

An evolutionary computation based approach for reduced order modelling of linear systems

Abstract: A new model order reduction algorithm taking the advantages of reciprocal transformation and principal pseudo break frequency estimation is presented The denominator polynomial is constructed using the approximate dominant poles obtained Ultimately the denominator polynomial formation is based on simple calculations involving high order system characteristic polynomial Numerator polynomial is then determined using a recently proposed evolutionary computation algorithm-Big Bang Big Crunch algorithm The method is simple and yields stable reduced order models Difficulty may arise in finding complex poles in the reduced order model However a modification in the algorithm by introducing search method to find the imaginary parts of such poles helps in overcoming this

An Efficient Projector-Based Passivity Test for Descriptor Systems

Abstract: An efficient passivity test based on canonical projector techniques is proposed for descriptor systems (DSs) widely encountered in circuit and system modeling. The test features a natural flow that first evaluates the index of a DS, followed by possible decoupling into its proper and improper subsystems. Explicit state-space formulations for respective subsystems are derived to facilitate further processing such as model order reduction and/or passivity enforcement. Efficient projector construction and a fast generalized Hamiltonian test for the proper-part passivity are also elaborated. Numerical examples then confirm the superiority of the proposed method over existing passivity tests for DSs based on linear matrix inequalities or skew-Hamiltonian/Hamiltonian matrix pencils.

Guaranteed Passive Parameterized Model Order Reduction of the Partial Element Equivalent Circuit (PEEC) Method

Abstract: The decrease of IC feature size and the increase of operating frequencies require 3-D electromagnetic methods, such as the partial element equivalent circuit (PEEC) method, for the analysis and design of high-speed circuits. Very large systems of equations are often produced by 3-D electromagnetic methods. During the circuit synthesis of large-scale digital or analog applications, it is important to predict the response of the system under study as a function of design parameters, such as geometrical and substrate features, in addition to frequency (or time). Parameterized model order reduction (PMOR) methods become necessary to reduce large systems of equations with respect to frequency and other design parameters. We propose an innovative PMOR technique applicable to PEEC analysis, which combines traditional passivity-preserving model order reduction methods and positive interpolation schemes. It is able to provide parametric reduced-order models, stable, and passive by construction over a user-defined range of design parameter values. Numerical examples validate the proposed approach.

Variation-aware interconnect extraction using statistical moment preserving model order reduction

Abstract: In this paper we present a stochastic model order reduction technique for interconnect extraction in the presence of process variabilities, i.e. variation-aware extraction. It is becoming increasingly evident that sampling based methods for variation-aware extraction are more efficient than more computationally complex techniques such as stochastic Galerkin method or the Neumann expansion. However, one of the remaining computational challenges of sampling based methods is how to simultaneously and efficiently solve the large number of linear systems corresponding to each different sample point. In this paper, we present a stochastic model reduction technique that exploits the similarity among the different solves to reduce the computational complexity of subsequent solves. We first suggest how to build a projection matrix such that the statistical moments and/or the coefficients of the projection of the stochastic vector on some orthogonal polynomials are preserved. We further introduce a proximity measure, which we use to determine apriori if a given system needs to be solved, or if it is instead properly represented using the currently available basis. Finally, in order to reduce the time required for the system assembly, we use the multivariate Hermite expansion to represent the system matrix. We verify our method by solving a variety of variation-aware capacitance extraction problems ranging from on-chip capacitance extraction in the presence of width and thickness variations, to off-chip capacitance extraction in the presence of surface roughness. We further solve very large scale problems that cannot be handled by any other state of the art technique.

Balanced Truncation Model Order Reduction for LTI Systems with many Inputs or Outputs

Abstract: We discuss balanced truncation (BT) based meth- ods for model order reduction (MOR) of linear time invariant (LTI) systems with many input or many output terminals. Applying BT methods makes it necessary to balance the system, which is equivalent to finding the controllability and observability Gramian of the system in a special diagonal form. The Cholesky factors of these Gramians are efficiently computable as solutions of dual Lyapunov equations for systems with only few inputs and outputs. After a brief introduction and a short recollection of basic knowledge of BT, we show a method to get the Gramians' factors also for systems with many inputs and outputs with the help of the Gauss-Kronrod quadrature formula. We show some numerical results using this quadrature rule and explain how to get the BT reduced order model out of these results. I. INTRODUCTION

Parameterized Model Order Reduction of Electromagnetic Systems Using Multiorder Arnoldi

Abstract: This paper presents an efficient algorithm to create parametric reduced order models of distributed electromagnetic systems that have arbitrary functions of frequency (due to material properties, boundary conditions, delay elements) and design parameters. The proposed method is based on a multiorder Arnoldi algorithm used to implicitly calculate the moments with respect to frequency and design parameters, as well as the cross-moments. This procedure generates parametric reduced order models that are valid over the desired parameter range without the need to redo the reduction when design parameters are changed. Numerical examples are provided to illustrate the validity of the proposed algorithm.

Efficient Algorithms for Distributed Control: A Structured Matrix Approach

Abstract: Distributed systems are all around us, and they are fascinating, and have an enormous potential to improve our lives, if their complexity can be properly harnessed. All scientists and engineers are aware of the great potential of this subject, since we witness fantastic distributed control systems every day, in the flocks of birds in the sky and fish in the sea. However, the collective behavior of millions of dynamically-coupled heterogeneous subsystems is hard to analyze and control for computational reasons. Our approach to the problems of analysis and control of distributed systems is to exploit the matrix structure of array-interconnected systems in fast iterative algorithms. Since these algorithms preserve the original matrix structure of the system, the resulting centrally optimal controller realizations have the same structure, which can conveniently be ‘redistributed’ into a set of subcontrollers linked in the same interconnection topology as the original system. For P interconnected subsystems, traditional analysis and control synthesis methods are O(P 3 ) computational complexity, but for N heterogeneous subsystems on a line, the above method is only O(N ) complexity. If the system is homogeneous with only heterogeneous boundary conditions, the complexity can be reduced to O(1), independent of the size of the homogeneous section. These results also extend to multiple spatial dimensions: for heterogeneous or homogeneous subsystems on an N ×M 2-D cartesian grid, the complexity is reduced to O(M N ) or O(1) complexity respectively, as compared to O(M 3 N 3 ) complexity of traditional techniques, an impressive improvement for very large systems N, M > 1000. Furthermore, due to the special form of the structured matrix arithmetic, the computations can actually be performed in a distributed way, on a distributed processor and memory system, with only linear complexity communication and memory requirements. Using these efficient structured techniques, one can perform stability and H2 and H? analysis to an arbitrary degree of accuracy, and sub-optimally upper-bound the structured singular value for robustness analysis. For synthesis, controllers with H2 and H? performance arbitrarily close to optimal are possible, and D-K iterations can be performed for robust design. Structure preserving model order reduction, and even system identification are also possible. It is also possible to apply this approach to analysis and synthesis of controllers for linear parameter varying(LPV) systems, and of repetitive controllers for trials with many time steps, T ,in only O(T ) complexity which would otherwise be O(T 3 ).

An XML representation of DAE systems obtained from continuous-time Modelica models

Abstract: This contribution outlines an XML format for representation of differential-algebraic equations (DAE) models obtained from continuous time Modelica models and possibly also from other equation-based modeling languages. The purpose is to offer a standardized model exchange format which is based on the DAE formalism and which is neutral with respect to model usage. Many usages of models go beyond what can be obtained from an execution interface offering evaluation of the model equations for simulation purposes. Several such usages arise in the area of control engineering, where dynamic optimization, Linear Fractional Transformations (LFTs), derivation of robotic controllers, model order reduction, and real time code generation are some examples. The choice of XML is motivated by its de facto standard status and the availability of free and efficient tools. Also, the XSLT language enables a convenient specification of the transformation of the XML model representation into other formats.

PEDS: passivity enforcement for descriptor systems via Hamiltonian-symplectic matrix pencil perturbation

Abstract: Passivity is a crucial property of macromodels to guarantee stable global (interconnected) simulation. However, weakly nonpassive models may be generated for passive circuits and systems in various contexts, such as data fitting, model order reduction (MOR) and electromagnetic (EM) macromodeling. Therefore, a post-processing passivity enforcement algorithm is desired. Most existing algorithms are designed to handle pole-residue models. The few algorithms for state space models only handle regular systems (RSs) with a nonsingular D+DT term. To the authors' best knowledge, no algorithm has been proposed to enforce passivity for more general descriptor systems (DSs) and state space models with singular D+DT terms. In this paper, a new post-processing passivity enforcement algorithm based on perturbation of Hamiltonian-symplectic matrix pencil, PEDS, is proposed. PEDS, for the first time, can enforce passivity for DSs. It can also handle all kinds of state space models (both RSs and DSs) with singular D+DT terms. Moreover, a criterion to control the error of perturbation is devised, with which the optimal passive models with the best accuracy can be obtained. Numerical examples then verify that PEDS is efficient, robust and relatively cheap for passivity enforcement of DSs with mild passivity violations.

Gramian based approach to model order-reduction for discrete-time switched linear systems

Abstract: This paper describes the problem of model orderreduction for a class of hybrid discrete-time switched linear systems composed of linear discrete-time invariant subsystems with a switching rule. The paper investigates a novel method to model reduction. The approach presents the reachability and observability Gramians of the switched systems, which allows a balanced truncation model reduction procedure. The error limit depends on the truncated singular values and balanced truncation procedure helps to keep the biggest singular values of the system. A numerical example shows the effectiveness of the proposed approach.

A Model Order Reduction Method for the Temperature Estimation in a Cylindrical Li-Ion Battery Cell

Abstract: The thermal characterization of Li-ion batteries for EVs, HEVs and PHEVs is a topic of great relevance, especially for the evaluation of the battery pack state of health (SoH) during vehicle operations and for battery life estimation. This work proposes a reduced-order model that estimates the thermal dynamics of a cylindrical Li-ion battery cell, with respect to time-varying current demands. Unlike most “black-box” dynamic models, based on system identification techniques, the proposed approach relies on the definition of a boundary-value problem for heat conduction, in the form of a linear partial differential equation. The problem is then converted into a low-order linear model by applying model-order reduction method in the frequency domain. The resulting model predicts the temperature dynamics at the center and at the external surface in relation with the rate of heat generation and the coolant temperature. In this paper, the model is applied to estimate the internal temperature of a cylindrical cell during a discharging transient. The model uses electrical data acquired from experimental tests and is validated by comparison with experimental data and 3D FEM thermal simulation.Copyright © 2010 by ASME

Model order reduction of finite element models: improved component mode synthesis

Abstract: The finite element (FE) approach constitutes an essential methodology when modelling the elastic properties of structures in various research disciplines such as structural mechanics, engine dynamics and so on. Because of increased accuracy requirements, the FE method results in discretized models, which are described by higher order ordinary differential equations, or, in FE terms, by a large number of degrees of freedom (DoF). In this regard, the application of an additional methodology, referred to as the model order reduction (MOR) or DoF condensation, is rather compulsory. Herein, a reduced dimension set of ordinary differential equations is generated, i.e. the initially large number of DoF is condensed, while aiming to keep the dynamics of the original model as intact as possible. In the commercially available FE software tools, the static and the component mode syntheses (CMS) are the only available integrated condensation methods. The latter represents the state of the art generating well-correlat...

A model order reduction method for efficient band structure calculations of photonic crystals

Abstract: The goal of this paper is to compute k-β diagrams of photonic crystals, accurately and fast. For this purpose, we propose a multipoint model-order reduction scheme for the modal analysis of periodic structures, based on the finite element method. Our numerical example demonstrates that the new approach is significantly faster than conventional finite-element solutions, while error levels are very similar. The proposed method allows for an adaptive choice of the expansion points for the reduced model.

Residual Based Sampling in POD Model Order Reduction of Drift-Diffusion Equations in Parametrized Electrical Networks

Abstract: We consider integrated circuits with semiconductors modeled by modified nodal analysis and drift-diffusion equations. The drift-diffusion equations are discretized in space using mixed finite element method. This discretization yields a high dimensional differential-algebraic equation. We show how proper orthogonal decomposition (POD) can be used to reduce the dimension of the model. We compare reduced and fine models and give numerical results for a basic network with one diode. Furthermore we discuss an adaptive approach to construct POD models which are valid over certain parameter ranges. Finally, numerical investigations for the reduction of a 4-diode rectifier network are presented, which clearly indicate that POD model reduction delivers surrogate models for the diodes involved, which depend on the position of the semiconductor in the network.

A sparse grid based collocation method for model order reduction of finite element approximations of passive electromagnetic devices under uncertainty

Abstract: A methodology is proposed for the model order reduction of finite element approximations of passive electromagnetic devices under random input conditions. In this approach, the reduced order system matrices are represented in terms of their convergent orthogonal polynomial expansions of input random variables. The coefficients of these polynomials, which are matrices, are obtained by repeated, deterministic model order reduction of finite element models generated for specific values of the input random variables. These values are chosen efficiently in a multi-dimensional grid using a Smolyak algorithm. The stochastic reduced order model is represented in the form of an augmented system which can be used for generating the desired statistics of the specific system response. The proposed method provides for significant improvement in computational efficiency over standard Monte Carlo.