On the application of forwarding to generalized Hamiltonian systems

Generalized Hamiltonian systems (GHS) are a class of nonlinear systems of significant interest in control theory, due to the passivity property and the particular structure of the state equations. Control techniques that obtain, by state feedback, a closed-loop generalized Hamiltonian dynamics, are suitable for easy stabilization by passive output feedback, yielding dissipative systems. In Hamiltonian systems with dissipation, the conservative and dissipative parts are clearly identifiable, and the Lyapunov function is chosen as the closed-loop energy function or Hamiltonian. A deeper insight suggest that this convenient Hamiltonian form may also underlie other classical techniques. It has already been observed that some control methodologies preserve the Hamiltonian structure, as long as the open-loop dynamics is also Hamiltonian. This is the case of cascade controllers like backstepping. Whereas the relation between backstepping control and Hamiltonian structures has been recently reported, this works unveils the preservation of the Hamiltonian structure in a class of forwarding-controlled systems. The forwarding procedure is assumed to start from a Hamiltonian subsystem and after each step of the forwarding iteration, a new Hamiltonian structure is obtained, i.e., the GHS structure is preserved. Moreover, the conservative or dissipative nature of the original system is also preserved. The result can be extended to any number of iteration steps, and hence to arbitrary order systems.