Structure-Preserving Discretization and Model Order Reduction of Boundary-Controlled 1D Port-Hamiltonian Systems

This paper presents a methodology for the discretization and reduction of a class of one-dimensional Partial Differential Equations (PDEs) with inputs and outputs collocated at the spatial boundaries. The class of system that we consider is known as Boundary-Controlled Port-Hamiltonian Systems (BC-PHSs) and covers a wide class of Hyperbolic PDEs with a large type of boundary inputs and outputs. This is for instance the case of waves and beams with Neumann or Dirichlet boundary conditions at both sides and mixed boundary conditions. In addition, we recall the Loewner framework to reduce the discretized model. We show that if the initial PDE is {\it passive}, the discretized model is also. Moreover, if the initial PDE is {\it impedance energy preserving}, the discretized model is also. The {\it passive} structure is also preserved in the reduced-order if the selected frequency data has positive real part. We use the one-dimensional wave equation and the Timoshenko beam as examples to show the versatility of the proposed approach.