On the control by interconnection and exponential stabilisation of infinite dimensional port-Hamiltonian systems

This paper aims at illustrating how the control by interconnection methodology (energy-Casimir method) can be employed in the development of exponentially stabilising boundary control laws for a class of linear, distributed port-Hamiltonian systems with one dimensional spatial domain. The energy-Casimir method is the starting point to determine a state-feedback law able to shape the closed-loop Hamiltonian and achieve simple stability. Then, it is shown how to design a further control loop that guarantees exponential convergence. Thanks to this result, it is possible to overcome a limitation of standard damping injection strategies that, if combined with energy shaping control laws based on energy-balancing, are able to assure, in general, only asymptotic convergence. The methodology is illustrated with the help of a simple example, the boundary stabilisation of a lossless transmission line.