Boundary Energy-Shaping Control of the Shallow Water Equation

The aim of this paper is to apply new results on the boundary stabilisation via energy-shaping of distributed port-Hamiltonian systems to a nonlinear PDE, i.e. a slightly simplified formulation of the shallow water equation. Usually, stabilisation of non-zero equilibria via energy-balancing has been achieved by looking at, or generating, a set of structural invariants (Casimir functions), in closed-loop. This approach is not successful in case of the shallow water equation because at the equilibrium the regulator is supposed to supply an infinite amount of energy (dissipation obstacle). In this paper, it is shown how to construct a controller that behaves as a state-modulated boundary source and that asymptotically stabilises the desired equilibrium. The proposed approach relies on a parametrisation of the dynamics provided by the image representation of the Dirac structure associated to the distributed port-Hamiltonian system. In this way, the effects of the boundary inputs on the state evolution are explicitly shown, and as a consequence the boundary control action that maps the open-loop system into a target one characterised by the desired stability properties, i.e. by a “new” Hamiltonian with an isolated minimum at the equilibrium, is determined.