Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation

In this paper, we propose a constructive procedure to modify the Hamiltonia n function of forced Hamiltonian systems with dissipation in order to gener ate Lyapunov functions for nonzero equilibria, A key step in the procedure, which is motivated from energy-balance considerations standard in network modeling of physical systems, is to embed the system into a larger Hamilton ian system for which a series of Casimir functions can be easily constructe d. Interestingly enough, for linear systems the resulting Lyapunov function is the incremental energy; thus our derivations provide a physical explana tion to it. An easily verifiable necessary and sufficient condition for the applicability of the technique in the general nonlinear case is given. Som e examples that illustrate the method are given.