Boundary Control of Distributed Port-Hamiltonian Systems via Generalised Canonical Transformations

This paper presents a novel approach for the development of boundary control laws for a class of linear, distributed port-Hamiltonian systems, with one dimensional spatial domain. The idea is to determine a control action able to map the initial system into a target one, characterised not only by a different Hamiltonian function, but also by new internal dissipative and power-preserving interconnection structures. The methodology consists of two main steps, each associated to a generalised canonical transformation. In the first one, a coordinate change (based on a combination of a linear mapping and a backstepping transformation) is employed to modify the internal structure of the system. Then, in the second step, a generalised canonical transformation capable of properly shaping the Hamiltonian function is introduced. The proposed approach is illustrated with the help of an example, the boundary stabilisation of a lossless transmission line.