Balancing and model reduction for second-order form linear systems

Meyer, David G.; Srinivasan, Sriram

doi:10.1109/9.544000

Cited by 134 publications

(131 citation statements)

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“…Rational interpolation methods have been extended to bilinear [24,36,37,39,58,92,98,192] and quadratic-in-state systems [37,38,97,114]. Even though these methods have proven effective for a wide range of problem settings, they are most widely used in circuit theory, such as [23,44,90,184,195], e.g., to analyze and predict signal propagation and interference in electric circuits; in structural mechanics, such as [53,106,174,198,211], to study, e.g., vibration suppression in large structures or behavior of micro-electromechanical systems; and in (optimal) control and controller reduction, such as [11,21,126,185,215,231], e.g., in LQR/LQG control design.…”

Section: Applicability Of the Basis Computation Methodsmentioning

confidence: 99%

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

2015

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Abstract. 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 which the equations governing the system behavior depend on a set of parameters. Examples include parameterized partial differential equations and large-scale systems of parameterized ordinary differential equations. The goal of parametric model reduction is to generate low-cost but accurate models that characterize system response for different values of the parameters. This paper surveys state-of-the-art methods in projection-based parametric model reduction, describing the different approaches within each class of methods for handling parametric variation and providing a comparative discussion that lends insights to potential advantages and disadvantages in applying each of the methods. We highlight the important role played by parametric model reduction in design, control, optimization, and uncertainty quantification-settings that require repeated model evaluations over different parameter values.

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Section: Applicability Of the Basis Computation Methodsmentioning

confidence: 99%

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

2015

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“…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, e.g., [9,10]), and the main objective of extending balancing methods to such systems is to derive reduced models that remain physically interpretable, i.e., that inherit the interaction structure of the original model.…”

Section: Doi 101137/080732717mentioning

confidence: 99%

“…But this means that ζ must be restricted to the invariant subspace S, since, otherwise, the initial output y(t = 0) diverges as → 0, which might produce unphysical behavior of the system. 4 If we nevertheless drop the assumption on the initial conditions, the solution of the cell problem (3.10) turns out to be…”

Section: Strong Confinement Limitmentioning

confidence: 99%

“…This particular case is certainly rather restrictive; however, the numerical examples discussed in section 6 seem to be of this form, as the dominant Hankel singular values appear in pairs corresponding to positions and their conjugate momenta. The idea of balancing positions and momenta separately is exploited in [4,6,7].…”

Section: ξ(T) = (ĵ −D)∇ĥ(ξ(t)) +Bu(t) Y(t) =ĉ∇ĥ(ξ(T))mentioning

confidence: 99%

“…Although both methods preserve stability and passivity, it turns turns out that they do a bad job in terms of the balanced truncation error bound. In either case (including confinement), going back to the original second-order form is not possible in general, as the reduced Hamiltonian is not a separable sum of potential and kinetic energy; a special situation in which the second-order structure may be preserved occurs when the balancing transformation decays into purely position and momentum (velocity) parts, and, in fact, alternative approaches such as [4,6,7] proceed by treating position and momentum (velocity) parts separately, but they also fail to preserve the stability of the second-order system, as has been pointed out in the recent work [7].…”

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confidence: 99%

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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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Linear Dynamical Systems, Approximation

Antoulas

1999

Wiley Encyclopedia of Electrical and Electronics Engineering

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Balancing and model reduction for second-order form linear systems

Cited by 134 publications

References 8 publications

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

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

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

Linear Dynamical Systems, Approximation

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