System-theoretic properties of port-controlled Hamiltonian systems

In our previous paper [1] it has been shown how by using a generalized bond graph formalism the dynamics of non-resistive physical systems (belonging to different domains, i.e., electrical, mechanical, hydraulical, etc.) can be given an intrinsic Hamiltonian formulation of dimension equal to the order of the physical system. Here "Hamiltonian" has to be understood in the generalized sense of defining Hamiltonian equations of motion with respect to a general Poisson bracket (not necessarily of maximal rank). The Poisson bracket is fully determined by the network structure of the physical system (called "junction structure" in bond graph terminology), while the Hamiltonian equals the total internally stored energy. A striking example is the direct Hamiltonian formulation of (nonlinear) LC-circuits [2]. Subsequently in [3] the interaction of non-resistive physical systems with their environment has been formalized by including external ports in the network model, naturally leading to two conjugated sets of external variables: the inputs u represented as generalized flow sources, and the outputs y which are the conjugated efforts. This leads to an interesting class of physical control systems, called port-controlled Hamiltonian systems in [3], formally defined as follows. The state space M (the space of energy variables) is a Poisson manifold, i.e. is endowed with a Poisson bracket. Recall [1], [2], [3] that a Poisson bracket on M is a bilinear map from C∞(M) x C∞(M) into C∞(M) (C∞(M) being the smooth real functions on M), denoted as