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This paper presents a structure-preserving spatial discretization method for distributed parameter port-Hamiltonian systems. The class of considered systems are hyperbolic systems of two conservation laws in arbitrary spatial dimension and geometries. For these systems, a partitioned finite element method (PFEM) is derived, based on the integration by parts of one of the two conservation laws written in weak form. The non-linear one-dimensional shallow-water equation (SWE) is first considered as a motivation example. Then, the method is investigated on the example of the non-linear two-dimensional SWE. Complete derivation of the reduced finite-dimensional port-Hamiltonian system (pHs) is provided and numerical experiments are performed. Extensions to curvilinear (polar) coordinate systems, space-varying coefficients and higher-order pHs (Euler–Bernoulli beam equation) are provided.

In this brief, we develop a mathematical model of a flexible spacecraft system composed of a hub and two symmetrical beams using the port-Hamiltonian framework. This class of system has favorable properties, such as passivity for controller synthesis and stability analysis, where the global Hamiltonian plays the role of a Lyapunov function candidate. The spacecraft model is viewed as a power-conserving interconnection between an infinite (beam) and finite (hub) dimensional system. We show that the interconnection result has a port-Hamiltonian structure and is passive. The introduction of a nonlinear feedback term, which takes into account the beam's flexibility, is developed using the control by an interconnection approach. The closedloop stability is proven; then, through explicitly solving the partial differential equations of the system, asymptotic stability is obtained. Finally, the experimental results are carried out to assess the validity of the proposed design methodology.

No abstract