All optimal Hankel-norm approximations of linear multivariable systems and their<i>L</i>,<sup>∞</sup>-error bounds†

This paper focuses on methods of constructing of reduced-order models of mechanical systems which preserve the Lagrangian structure of the original system. These methods may be used in combination with standard spatial decomposition methods, such as the Karhunen-Loève expansion, balancing, and wavelet decompositions. The model reduction procedure is implemented for three-dimensional finite-element models of elasticity, and we show that using the standard Newmark implicit integrator, significant savings are obtained in the computational costs of simulation. In particular simulation of the reduced model scales linearly in the number of degrees of freedom, and parallelizes well.

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“…A mathematically precise version of this statement is possible, using the results of [6,9] on the errors obtained using balanced truncation.…”

Section: Previous Workmentioning

confidence: 99%

Structure-preserving model reduction for mechanical systems

Lall¹,

Krysl²,

Marsden³

2003

Physica D: Nonlinear Phenomena

142

112

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“…where the invertibility of (J 22 −D 22 ) is due to Corollary 4.5 below, and we have omitted integration constants C(ζ 1 ). The Fredholm alternative for (3.8) then yields…”

Section: Strong Confinement Limitmentioning

confidence: 99%

“…with the shorthand Q = C∇ 2 H. Moore [21] has shown that, if W c , W o 0 (positive definiteness = complete controllability/observability), there exists a coordinate transformation x → T x such that the two Gramians become equal and diagonal: 1 1 T is a so-called contragredient transformation; see [22] for details.…”

Section: Balancing Transformationsmentioning

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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“…When dealing with high-order models, it is reasonable to look for an approximate stable model ẋ m (t) = A m x m (t) + B m u(t) y m (t) = C m x m (t), (1.2) in which A m ∈ R m×m , B m , C T m ∈ R m×s and x m (t), y m (t) ∈ R m , with m n. Hence, the reduction problem consists in approximating the triplet {A, B, C} by another one {Â,B,Ĉ} of small size. Several approaches in this area have been used as Padé approximation [15,33,34], balanced truncation [29,37], optimal Hankel norm [16,17] and Krylov subspace methods [3,6,11,12,21,22]. These approaches require the solution of coupled Lyapunov matrix equations [1,13,25,27] having the form A P + P A T + B B T = 0 A T Q + Q A + C T C = 0, (1.3) where P, Q are the controllability and the observability Grammians of the system (1.1).…”

Section: Introductionmentioning

confidence: 99%