Formal Distributed Port-Hamiltonian Representation of Field Equations

Boundary detection of variational symmetry breaking using port-representation of conservation laws

“…In the case when the system stems from physical modeling, the storage function is the energy of the system and the product has then the dimension of power. This property can be described by a Dirac structure [4], [22]; to be precise, it satisfies a Stokes-Dirac structure on jet bundles [1], [6], [8], [15]. This means that (4) has an integrable quantity given by Stokes's theorem; actually, (4) is characterized by the power balance equation…”

“…In control theory, port-representations [10], [1] are originally based on a conservative system, a Hamiltonian system. One of them, a field port-Lagrangian system [6], [8] can describe a conservation law. We studied a boundary observer for detecting the variational symmetry breaking by the field portLagrangian representation [7].…”

On Casimir Functionals for Infinite-Dimensional Port-Hamiltonian Control Systems

“…The relation (12) for this configuration reads as (13) We choose a power conserving interconnection such that (14) is fulfilled, where and denote the collocated port variables of the finite-dimensional controller system (15) which is modeled on a manifold with local coordinates and is equipped with dual in and output bundles and . The interconnection is chosen according to (14) as a feedback interconnection in the form (16) with an appropriate map and its dual where will denote the components of .…”

“…This will have the consequence, that the variational derivative is interpreted differently compared to [5], [6]- [8] which is not based on a bundle structure at all, i.e., in our setting, the Hamiltonian density explicitly depends on derivative coordinates and furthermore our system class is not necessarily based on the use of skew-adjoint differential operators and the use of energy variables. See also [14] in this context, where Stokes-Dirac Structures on variational complexes on jet-bundles are introduced.…”

Port-based modeling of magnetohydrodynamics equations for Tokamaks

“…Therefore, the Stokes-Dirac structure can express boundary energy control problems, because we can observe boundary integrable energies distributed in internal domains from boundaries. Our present interest is to clarify the counter part of the distributed port-Hamiltonian system representation in the Lagrangian side [12], [13], [14], [15]. The representation will provide us a more general control framework based on variational structures.…”