Stability Analysis of Repetitive Control: the Port-Hamiltonian Approach

This paper deals with two different topics that, at a first sight, could look quite unrelated. The first one is about repetitive control: the scope is to determine a class of linear systems for which such control technique can be successfully applied, i.e. the resulting closed-loop system is stable. In repetitive control schemes, coupled PDEs and ODEs are present, and the idea is to rely on a port-Hamiltonian formulation, and on the properties of passive/dissipative systems to study the behaviour of the closed-loop dynamic. To perform this analysis, novel results dealing with the exponential stabilisation of linear boundary control system with one-dimensional spatial domain in port-Hamiltonian form via finite dimensional linear controllers are presented. This is in fact the second topic discussed in this paper, and the achieved results are applied in order to characterise a class of linear systems for which repetitive control schemes exponentially converge.