Boundary Integrability of Multiple Stokes--Dirac Structures

Nishida, Gou; Maschke, Bernhard; Ikeura, Ryojun

doi:10.1137/110856058

Cited by 6 publications

(13 citation statements)

References 8 publications

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“…The definition of boundary port variables plays a crucial role in showing that a PH system is a well-posed boundary control systems [6]. The definition of distributed power variables as in-and outputs is discussed in [7].…”

Section: Introductionmentioning

confidence: 99%

Weak form of Stokes–Dirac structures and geometric discretization of port-Hamiltonian systems

Kotyczka

Maschke²,

Lefèvre³

2018

Journal of Computational Physics

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We present the mixed Galerkin discretization of distributed parameter port-Hamiltonian systems. On the prototypical example of hyperbolic systems of two conservation laws in arbitrary spatial dimension, we derive the main contributions: (i) A weak formulation of the underlying geometric (Stokes-Dirac) structure with a segmented boundary according to the causality of the boundary ports. (ii) The geometric approximation of the Stokes-Dirac structure by a finite-dimensional Dirac structure is realized using a mixed Galerkin approach and power-preserving linear maps, which define minimal discrete power variables. (iii) With a consistent approximation of the Hamiltonian, we obtain finite-dimensional port-Hamiltonian state space models. By the degrees of freedom in the powerpreserving maps, the resulting family of structure-preserving schemes allows for trade-offs between centered approximations and upwinding. We illustrate the method on the example of Whitney finite elements on a 2D simplicial triangulation and compare the eigenvalue approximation in 1D with a related approach.Keywords: Systems of conservation laws with boundary energy flows, port-Hamiltonian systems, mixed Galerkin methods, geometric spatial discretization, structure-preserving discretization. * Accepted version of P. Kotyczka et al., Weak form of Stokes-Dirac structures and geometric discretization of port-Hamiltonian systems, J. Comput. Phys. 361 (2018) 442-476, https://doi.• The power-preserving maps for the discrete power variables offer design degrees of freedom to parametrize the resulting finite-dimensional PH state space models. They can be used to realize upwinding.• Mapping the flow variables instead of the efforts avoids a structural artificial feedthrough, which is not desirable for the approximation of hyperbolic systems.We consider as the prototypical example of distributed parameter PH systems, an open system of two hyperbolic conservation laws in canonical form, as presented in [1]. We use the language of differential forms, see e. g. [23], which highlights the geometric nature of each variable and allows for a unifying representation independent from the dimension of the spatial domain.An important reason for expressing the spatial discretization of PH systems based on the weak form is to make the link with modern geometric discretization methods. Bossavit's work in computational electromagnetism [24], [25] and Tonti's cell method [26] keep track of the geometric nature of the system variables which allows for a direct interpretation of the discrete variables in terms of integral system quantities. This integral point of view is also adopted in discrete exterior calculus [11]. Finite element exterior calculus [27] gives a theoretical frame to describe functional spaces of differential forms and their compatible approximations, which includes the construction of higher order approximation bases that generalize the famous Whitney forms [28], see also [29]. We refer also to the recent article [30] which proposes conforming polynomial ap...

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Section: Introductionmentioning

confidence: 99%

Weak form of Stokes–Dirac structures and geometric discretization of port-Hamiltonian systems

Kotyczka

Maschke²,

Lefèvre³

2018

Journal of Computational Physics

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“…Moreover, the harmonic forms can be classified as tangent or orthogonal by Friedrichs decomposition [6]. As a result, the fourth section shows that an essential property of an extended Stokes-Dirac structure for defining distributed port-Hamiltonian systems on manifolds with non-trivial topology can be derived from our previous results [4,11].…”

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confidence: 80%

“…Hence, a kind of boundary energy controls can be realized. The boundary integrability can be formulated by a particular case of Dirac structures called the Stoke-Dirac structure [3,4], where the Dirac structure is a generalized concept of Poisson and Symplectic structures. The port interconnection of distributed port-Hamiltonian systems determines a network of energy flows between systems domains through their boundaries.…”

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confidence: 99%

Topological geometric extension of Stokes-Dirac structures for global energy flows

Nishida

Maschke

2019

2019 18th European Control Conference (ECC)

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This paper clarifies a global structure of Stokes-Dirac structures used for describing interconnected port-Hamiltonian systems defined on manifolds with non-trivial topology under consistent boundary condition. IntroductionPort-Hamilton systems [1] have been developed as an extended Hamiltonian system, and it is one of most essential system representations for controlling complex physical systems. Port-Hamilton systems are defined by particular variable pairs of an input and an output. The pair is called port, and the variables are called effort and flow. The product of an effort and the corresponding flow has the physical dimension of power, i.e., the time derivative of energy. Thus, the sum of the products of all pairs is equivalent to the time derivative of the total energy of a given system, i.e., Hamiltonian. Indeed, ports can be derived from derivatives of a Hamiltonian. By using port-Hamiltonian systems, physical network systems interconnected through the ports can be described. Then, an energy balance equation can be defined on the terminal of the port-interconnections by the port variables. If the energy balance holds in any interaction with environments, this property can be used for stability analysis, because an energy of systems can be considered as a Lyapunov function. Thus, various control methods using energy flows on such a network, e.g., passivity-based controls [1,2], can be used. Port-Hamilton systems have been extended for systems governed by partial differential equations, and it is called a distributed port-Hamilton system [2, 3]. In the system, ports are defined on a boundary of a control system domain, and they are called boundary ports. Then, the balance equation can be augmented as that on the boundary. As a remarkable property of distributed port-Hamilton systems, Stokes the- *

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“…Then, we derive the Stokes variational differential from the fact that higher order derivatives yield variations of boundary port variables through integration by parts and Stokes theorem. Finally, we shows that the boundary energy balance and the Stokes-Dirac structure [7] [20] that is an extended Dirac structure for distributed port-Hamiltonian systems can be defined in the proposed higher order field port Hamiltonian systems with boundary energy flows.…”

Section: Introductionmentioning

confidence: 91%

“…where Φ corresponds to the boundary term in (31). The term (31) is obtained from the calculation in (20).…”

Section: Theorem 1 the Local Expression Of The Implicit Higher Ordermentioning

confidence: 99%

Hamiltonian Representation of Higher Order Partial Differential Equations with Boundary Energy Flows

Nishida¹

2015

JAMP

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This paper presents a system representation that can be applied to the description of the interaction between systems connected through common boundaries. The systems consist of partial differential equations that are first order with respect to time, but spatially higher order. The representation is derived from the instantaneous multisymplectic Hamiltonian formalism; therefore, it possesses the physical consistency with respect to energy. In the interconnection, particular pairs of control inputs and observing outputs, called port variables, defined on the boundaries are used. The port variables are systematically introduced from the representation.

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Boundary Integrability of Multiple Stokes--Dirac Structures

Cited by 6 publications

References 8 publications

Weak form of Stokes–Dirac structures and geometric discretization of port-Hamiltonian systems

Weak form of Stokes–Dirac structures and geometric discretization of port-Hamiltonian systems

Topological geometric extension of Stokes-Dirac structures for global energy flows

Hamiltonian Representation of Higher Order Partial Differential Equations with Boundary Energy Flows

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