Further results on decoupling with stability for Hamiltonian systems

Abstract

The problem of input-output decoupling with stability by, possibly dynamic, state-feedback is addressed for Hamiltonian systems. As well known, to decide if the problem is solvable, and which class of state-feedback has to be used, the stability properties of the P ^⊥, P* and Δ _mix dynamics are to be investigated. For this reason, on the way to the main result, it is shown that, for general Hamiltonian systems, such dynamics are not necessarily Hamiltonian. On the other hand, it is shown that, for linear simple Hamiltonian systems, both the P ^⊥ and the P* dynamics are Hamiltonian (whereas, as well known, the Δ _mix dynamics are empty). Moreover, for a class of nonlinear simple Hamiltonian systems, a simple to check necessary and sufficient condition for the solvability of the problem via dynamic state-feedback is proposed. Several examples, clarifying the role of different classes of state-feedback control laws (either static or dynamic) in the solution of the problem, are proposed