Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms

Laub, Alan J.; Heath, Michael T.; Paige, Christopher C.; Ward, Robert C.

doi:10.1109/tac.1987.1104549

Cited by 492 publications

(258 citation statements)

References 22 publications

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“…For this one-dimensional problem, standard algorithms for computing balanced truncation are computationally tractable. Therefore, we are able to compare the models resulting from exact balanced truncation (which for the 1-D case can be computed in MATLAB using standard algorithms [19]) to BPOD models obtained from three-dimensional simulations of the real part of the full field, Re {q(x, y, z, t)} at a particular wavenumber pair (α, β) on a large grid, similar to the comparison made by Rowley [30] for a streamwise-constant perturbation.…”

Section: Results: Single-wavenumber Perturbationsmentioning

confidence: 99%

Modeling of transitional channel flow using balanced proper orthogonal decomposition

2008

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We study reduced-order models of three-dimensional perturbations in linearized channel flow using balanced proper orthogonal decomposition (BPOD). The models are obtained from three-dimensional simulations in physical space as opposed to the traditional single-wavenumber approach, and are therefore better able to capture the effects of localized disturbances or localized actuators. In order to assess the performance of the models, we consider the impulse response and frequency response, and variation of the Reynolds number as a model parameter. We show that the BPOD procedure yields models that capture the transient growth well at a low order, whereas standard POD does not capture the growth unless a considerably larger number of modes is included, and even then can be inaccurate. In the case of a localized actuator, we show that POD modes which are not energetically significant can be very important for capturing the energy growth. In addition, a comparison of the subspaces resulting from the two methods suggests that the use of a non-orthogonal projection with adjoint modes is most likely the main reason for the superior performance of BPOD. We also demonstrate that for single-wavenumber perturbations, low-order BPOD models reproduce the dominant eigenvalues of the full system better than POD models of the same order.These features indicate that the simple, yet accurate BPOD models are a good candidate for developing model-based controllers for channel flow.

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Section: Results: Single-wavenumber Perturbationsmentioning

confidence: 99%

Modeling of transitional channel flow using balanced proper orthogonal decomposition

2008

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“…As A is assumed to be stable and thus P and Q are positive semi-definite, there exist Cholesky factorizations P = S T S and Q = R T R. In the so-called squareroot balanced truncation (SRBT) algorithms [24,25] these are used to define the truncation matrices…”

Section: Large Scale Sparse Standard State Space Systemsmentioning

confidence: 99%

Efficient balancing-based MOR for large-scale second-order systems

Benner

Saak

2011

Mathematical and Computer Modelling of Dynamical Systems

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Large-scale structure dynamics models arise in all areas where vibrational analysis is performed, ranging from control of machine tools to microsystems simulation. These models result from a finite element analysis being applied to their mechanical structure. Therefore they are in general sparse, but very large, since many details have to be resolved. This accounts for unacceptable computational and resource demands in simulation and especially control of these models. To reduce these demands and to be able to compute solutions and controls in acceptable, i.e. applicable, time frames model order reduction is applied. Classically modal truncation is used for this task. The reduced order models (ROMs) generated are normally relatively large and often need manual modification by addition of certain technically motivated modes. That means they are at least partially heuristic and cannot be generated fully automatic. Engineers are therefore searching for alternate reduction methods.Here we will concentrate on the application of balancing based model order reduction techniques. A central topic is to provide a reduced order model for the construction and parameterization of a practicable controller for the application. Our main focus will be on presenting a way to efficiently compute the ROM exploiting the sparsity and second order structure of the FEM semi-discretization, rather than presenting a new reduction technique.

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“…We shall assume throughout this paper that our system is completely controllable and observable. 2 A convenient way to express the balancing transformation is due to [24,25]: Noting that Σ 2 contains the positive eigenvalues of the product W c W o , we decompose the two Gramians according to…”

Section: Balancing Transformationsmentioning

confidence: 99%

“…We thank Volker Mehrmann, Tatjana Stykel, and Timo Reis for valuable discussions and comments. We greatly benefited from the remarks of the anonymous referees, which helped to improve the first version of this article, and we are grateful to them for pointing out [8,24,25].…”

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confidence: 99%

Balanced Truncation of Linear Second-Order Systems: A Hamiltonian Approach

Hartmann¹,

Vulcanov²,

Schütte³

2010

Multiscale Model. Simul.

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We present a formal procedure for structure-preserving model reduction of linear second-order and Hamiltonian control problems that appear in a variety of physical contexts, e.g., vibromechanical systems or electrical circuit design. Typical balanced truncation methods that project onto the subspace of the largest Hankel singular values fail to preserve the problem's physical structure and may suffer from lack of stability. In this paper, we adopt the framework of generalized Hamiltonian systems that covers the class of relevant problems and that allows for a generalization of balanced truncation to second-order problems. It turns out that the Hamiltonian structure, stability, and passivity are preserved if the truncation is done by imposing a holonomic constraint on the system rather than standard Galerkin projection. Abstract. We present a formal procedure for structure-preserving model reduction of linear second-order and Hamiltonian control problems that appear in a variety of physical contexts, e.g., vibromechanical systems or electrical circuit design. Typical balanced truncation methods that project onto the subspace of the largest Hankel singular values fail to preserve the problem's physical structure and may suffer from lack of stability. In this paper, we adopt the framework of generalized Hamiltonian systems that covers the class of relevant problems and that allows for a generalization of balanced truncation to second-order problems. It turns out that the Hamiltonian structure, stability, and passivity are preserved if the truncation is done by imposing a holonomic constraint on the system rather than standard Galerkin projection.Key words. structure-preserving model reduction, generalized Hamiltonian systems, balanced truncation, strong confinement, invariant manifolds, Hankel norm approximation AMS subject classifications. 70Q05, 70J50, 93B11, 93C05, 93C70. DOI. 10.1137/0807327171. Introduction. Model reduction is a major issue for control, optimization, and simulation of large-scale systems. In particular, for linear time-invariant systems, balanced truncation is a well-established tool for deriving reduced (i.e., low-dimensional) models that have an input-output behavior similar to the original model [1, 2]; see also [3] and the references therein. The general idea of balanced truncation is to restrict the system onto the subspace of easily controllable and observable states which can be determined by the computing Hankel singular values associated with the system. Moreover the method is known to preserve certain properties of the original system such as stability or passivity and gives an error bound that is easily computable. One drawback of balanced truncation is that there is no straightforward generalization to second-order systems; see, e.g., [4] or the recent articles [5,6,7,8] for a discussion of various possible strategies. Second-order equations occur in modeling and control of many physical systems, e.g., electrical circuits, structural mechanics, or vibroacoustic models (see...

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Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms

Cited by 492 publications

References 22 publications

Modeling of transitional channel flow using balanced proper orthogonal decomposition

Modeling of transitional channel flow using balanced proper orthogonal decomposition

Efficient balancing-based MOR for large-scale second-order systems

Balanced Truncation of Linear Second-Order Systems: A Hamiltonian Approach

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