Structure-preserving model reduction for mechanical systems

Lall, Sanjay; Krysl, Petr; Marsden, Jerrold E.

doi:10.1016/s0167-2789(03)00227-6

Cited by 143 publications

(116 citation statements)

References 39 publications

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“…Krysl et al [6] and Lall et al [7] reported model reduction techniques in which they limited the space of possible deformations. Wojtan and Turk [3] employed an FEM representation with embedded high-resolution surface mesh.…”

Section: Related Workmentioning

confidence: 99%

Resolution Scaling for Mass Spring Model Simulations

Kot

Nagahashi

Gracki

2014

IEICE Trans. Inf. & Syst.

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SUMMARYThe volumetric representations of deformable objects suffer from high memory and computational costs. In this work we analyze an approach of constructing low-resolution mass spring models (MSMs) on the basis of a high-resolution reference MSM. Preserving the physical properties of the modeled objects is emphasized such that their motion is consistent and independent of the spring network resolution. We varied the node merging algorithm and analyzed how various aspects of the simplification process affected the properties of the model and how these properties translated into visual behavior in a simulation. key words: level of detail, mass spring model, soft body deformation

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Section: Related Workmentioning

confidence: 99%

Resolution Scaling for Mass Spring Model Simulations

Kot

Nagahashi

Gracki

2014

IEICE Trans. Inf. & Syst.

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“…In mechanics, a natural way to restrict a system to a subspace is by means of constraints [26], and, from a physical viewpoint, it makes sense to study the limit of vanishing small HSVs, i.e., to gradually eliminate the least observable and controllable states, thereby forcing the system to the limiting controllable and observable subspace.…”

Section: Strong Confinement Limitmentioning

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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“…All of these properties can be viewed as fundamental to a model of a mechanical system. Methods for model reduction, which take account of this underlying geometry and preserve it, are developed in References [7,40]. The importance of this is evidenced in Figure 2, where each mode shape appears twice; this is a consequence of the underlying correspondence between con"guration variables and their generalized momenta.…”

Section: Empirical Balancingmentioning

confidence: 99%

A subspace approach to balanced truncation for model reduction of nonlinear control systems

Lall

Marsden

Glavaski

2002

Intl J Robust & Nonlinear

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465

421

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SUMMARYIn this paper, we introduce a new method of model reduction for nonlinear control systems. Our approach is to construct an approximately balanced realization. The method requires only standard matrix computations, and we show that when it is applied to linear systems it results in the usual balanced truncation. For nonlinear systems, the method makes use of data from either simulation or experiment to identify the dynamics relevant to the input}output map of the system. An important feature of this approach is that the resulting reduced-order model is nonlinear, and has inputs and outputs suitable for control. We perform an example reduction for a nonlinear mechanical system.

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Structure-preserving model reduction for mechanical systems

Cited by 143 publications

References 39 publications

Resolution Scaling for Mass Spring Model Simulations

Resolution Scaling for Mass Spring Model Simulations

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

A subspace approach to balanced truncation for model reduction of nonlinear control systems

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