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The model reduction problem in the Loewner framework for port-Hamiltonian and network systems on graphs is studied. In particular, given a set of right-tangential interpolation data, the (subset of) left-tangential interpolation data that allow constructing an interpolant possessing a port-Hamiltonian structure is characterized. In addition, conditions under which an interpolant retains the underlying port-Hamiltonian structure of the system generating the data are given by requiring a particular structure of the generalized observability matrix. Ipso facto a characterization of the reduced order model in terms of Dirac structure with the aim of relating the Dirac structure of the underlying port-Hamiltonian system with the Dirac structure of the constructed interpolant is given. This result, in turn, is used to solve the model reduction problem in the Loewner framework for network systems described by a weighted graph. The problem is first solved, for a given clustering, by giving conditions on the right- and left-tangential interpolation data that yield an interpolant possessing a network structure. Thereafter, for given tangential data obtained by sampling an underlying network system, we give conditions under which we can select a clustering and construct a reduced model preserving the network structure. Finally, the results are illustrated by means of a second order diffusively coupled system and a first order network system. [ABSTRACT FROM AUTHOR]

Abstract: Port-Hamiltonian systems result from port-based network modeling of physical systems and are an important example of passive state-space systems. In this paper, we develop a framework for model reduction of large-scale multi-input/multi-output port-Hamiltonian systems via tangential rational interpolation. The resulting reduced model is a rational (tangential) interpolant that retains the port-Hamiltonian structure; hence it remains passive. We introduce an -inspired algorithm for effective choice of interpolation points and tangent directions and present several numerical examples illustrating its effectiveness. [Copyright &y& Elsevier]

Abstract: This contribution presents two approximation methods for linear infinite-dimensional systems that ensure the preservation of stability and passivity. The first approach allows one to approximate internal source free infinite-dimensional systems such that the resulting approximation is a port-controlled Hamiltonian system with dissipation. The second method deals with the class of systems that are not required to have conjugated outputs but only a dissipative system operator. It yields approximations with a dissipative system matrix for which bounds of their stability margin are provided. Both approaches are based on a state space formulation of the infinite-dimensional system. This makes it possible to use the Petrov–Galerkin approximation whose free parameters are partly used for achieving the structure preservation. Since still free parameters remain, further application specific objectives, such as, e.g., moment matching, can be achieved. Both approaches are applied to the approximation of an Euler–Bernoulli beam. [Copyright &y& Elsevier]

Abstract: Model reduction of port-Hamiltonian systems by means of the Krylov methods is considered, aiming at port-Hamiltonian structure preservation. It is shown how to employ the Arnoldi method for model reduction in a particular coordinate system in order to preserve not only a specific number of the Markov parameters but also the port-Hamiltonian structure for the reduced order model. Furthermore it is shown how the Lanczos method can be applied in a structure preserving manner to a subclass of port-Hamiltonian systems which is characterized by an algebraic condition. In fact, for the same subclass of port-Hamiltonian systems the Arnoldi method and the Lanczos method turn out to be equivalent in the sense of producing reduced order port-Hamiltonian models with the same transfer function. [Copyright &y& Elsevier]