Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for wh...

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A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

In this paper, we present two novel algorithms to realize a finite dimensional, linear time-invariant state-space model from input-output data. The algorithms have a number of common features. They are classified as one of the subspace model identification schemes, in that a major part of the identification problem consists of calculating specially structured subspaces of spaces defined by the input-output data. This structure is then exploited in the calculation of a realization. Another common feature is their algorithmic organization: an RQ factorization followed by a singular value decomposition and the solution of an overdetermined set (or sets) of equations. The schemes assume that the underlying system has an output-error structure and that a measurable input sequence is available. The latter characteristic indicates that both schemes are versions of the MIMO Output-Error State Space model identification (MOESP) approach. The first algorithm is denoted in particular as the (elementary MOESP scheme)...

Subspace model identification-Part 1. The output-error state-space model identification class of algorithms

Given the square matrices $A, B, D, E$ and the matrix $C$ of conforming dimensions, we consider the linear matrix equation $A{\mathbf X} E+D{\mathbf X} B = C$ in the unknown matrix ${\mathbf X}$. Our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems, which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing.

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Computational Methods for Linear Matrix Equations

We discuss the relation of a certain type of generalized Lyapunov equations to Gramians of stochastic and bilinear systems together with the corresponding energy functionals. While Gramians and energy functionals of stochastic linear systems show a strong correspondence to the analogous objects for deterministic linear systems, the relation of Gramians and energy functionals for bilinear systems is less obvious. We discuss results from the literature for the latter problem and provide new characterizations of input and output energies of bilinear systems in terms of algebraic Gramians satisfying generalized Lyapunov equations. In any of the considered cases, the definition of algebraic Gramians allows us to compute balancing transformations and implies model reduction methods analogous to balanced truncation for linear deterministic systems. We illustrate the performance of these model reduction methods by showing numerical experiments for different bilinear systems.

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Lyapunov Equations, Energy Functionals, and Model Order Reduction of Bilinear and Stochastic Systems

http://www.dct.tue.nl/New/Wouw/JSV2013_Besselink.pdf

A comparison of model reduction techniques from structural dynamics, numerical mathematics and systems and control

Basic Concepts.- to Model Order Reduction.- Linear Systems, Eigenvalues, and Projection.- Theory.- Structure-Preserving Model Order Reduction of RCL Circuit Equations.- A Unified Krylov Projection Framework for Structure-Preserving Model Reduction.- Model Reduction via Proper Orthogonal Decomposition.- PMTBR: A Family of Approximate Principal-components-like Reduction Algorithms.- A Survey on Model Reduction of Coupled Systems.- Space Mapping and Defect Correction.- Modal Approximation and Computation of Dominant Poles.- Some Preconditioning Techniques for Saddle Point Problems.- Time Variant Balancing and Nonlinear Balanced Realizations.- Singular Value Analysis and Balanced Realizations for Nonlinear Systems.- Research Aspects and Applications.- Matrix Functions.- Model Reduction of Interconnected Systems.- Quadratic Inverse Eigenvalue Problem and Its Applications to Model Updating - An Overview.- Data-Driven Model Order Reduction Using Orthonormal Vector Fitting.- Model-Order Reduction of High-Speed Interconnects Using Integrated Congruence Transform.- Model Order Reduction for MEMS: Methodology and Computational Environment for Electro-Thermal Models.- Model Order Reduction of Large RC Circuits.- Reduced Order Models of On-Chip Passive Components and Interconnects, Workbench and Test Structures.

Model Order Reduction: Theory, Research Aspects and Applications

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.

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Singular Value Analysis and Balanced Realizations for Nonlinear Systems

This technicalnote proposesa methodfor loworder H-infinity synthesis where the constraint on the order of the controller is formulated as a rational equation. The resulting nonconvex optimization problem is thensolvedby applyinga quasi-Newtonprimal-dualinterior pointmethod. The proposed method is evaluated together with a well-known method from the literature. The results indicate that the proposed method has com- parable performance and speed.

http://ieeexplore.ieee.org/iel5/9/5784391/05719547.pdf

Wilhelmus H. A. Schilders

Papers

Model Order Reduction: Theory, Research Aspects and Applications

Singular Value Analysis and Balanced Realizations for Nonlinear Systems

A Quasi-Newton Interior Point Method for Low Order H-Infinity Controller Synthesis