Structure Preserving Spatial Discretization of a 1-D Piezoelectric Timoshenko Beam

In this paper, they authors show how to spatially discretize a distributed model of a piezoelectric beam representing the dynamics of an inflatable space reflector in port-Hamiltonian form. This model can then be used to design a controller for the shape of the inflatable structure. In this paper, they choose lumped pH modeling since these models offer a clear structure for control design. To be able to design a finite dimensional controller for the infinite dimensional system, they need a finite dimensional approximation of the infinite dimensional system which inherits all the structural properties of the infinite dimensional system, e.g., passivity. To achieve this goal first divide the one-dimensional Timoshenko beam with piezoelectric actuation into several finite elements. These finite elements are then interconnected in a physical motivated way. The interconnected system is then a finite dimensional approximation of the beam dynamics in the pH framework. To show the validity of the finite dimensional system, they will present simulation results.