Approximate dissipative Hamiltonian realization and construction of local Lyapunov functions

The key in applying energy-based control approach is to be able to express the system under consideration as a dissipative Hamiltonian system, i.e., to obtain Dissipative Hamiltonian Realization (DHR) for the system. In general, the precise DHR form is hard to obtain for nonlinear dynamic systems. When a precise DHR does not exist for a dynamic system or such a precise realization is difficulty to obtain, it is necessary to consider its approximate realization. This paper investigates approximate DHR and construction of local Lyapunov functions for time-invariant nonlinear systems. It is shown that every nonlinear affine system has an approximate DHR if linearization of the system is controllable. Based on the diagonal normal form of nonlinear dynamic systems, a new algorithm is established for the approximate DHR. Finally, we present the concept of kth degree approximate Lyapunov function, and provide a method to construct such a Lyapunov function. Example studies show that the methodology presented in this paper is very effective.