Discrete-Time Models for Implicit Port-Hamiltonian Systems

Implicit representations of finite-dimensional port-Hamiltonian systems are studied from the perspective of their use in numerical simulation and control design. Implicit representations arise when a system is modeled in Cartesian coordinates and when the system constraints are applied in the form of additional algebraic equations (the system model is in a differential-algebraic equation form). Such representations lend themselves better to sample-data approximations. Given an implicit representation of a port-Hamiltonian system it is shown how to construct a sampled-data model that preserves the port-Hamiltonian (PH) structure under sample and hold. 1. Introduction. The class of Hamiltonian systems has a prominent role in many disciplines. It was recently extended in [6] to include open systems, i.e. systems that interact with the environment via a set of inputs and outputs (called ports), giving rise to port-Hamiltonian (PH) systems. Such extended models immediately reveal the passive properties of the underlying systems, making them particularly well suited for designing passivity-based control (PBC) laws. Two types of model representations of Hamiltonian systems are in widespread use: the explicit representation stated in the form of an ordinary differential equation (ODE) on an abstract manifold [28, 29, 25, 26] and the implicit representation stated in the form of a differential-algebraic equation (DAE) usually evolving in a Euclidean space [4]. While explicit representations are usually preferred in the context of analytical mechanics; see [1, 8, 19], the implicit DAEs models lend themselves better for numerical computations as they lead to simpler expressions for the Hamiltonian functions. The formal relations between the two representations and their equivalence can be established if the system's configuration space is regarded as an embedded submanifold of the Euclidean space; see [4] for a full development. Of principal interest here will be the construction of sampled-data (discrete-time) models of PH systems that best approximate their continuous-time counterparts. Sampled-data models are important in digital control implementations and permit for simpler design of PBC laws directly in discrete time. In this context, the notion of " best approximation " deserves clarification. For linear systems, exact sampled-data models can be constructed by requiring that the solutions of the sampled-data and continuous-time systems coincide at the discretization points. Point-wise model matching is usually impossible in the case of nonlinear systems short of explicit derivation of analytical expressions for their solutions. For general dynamic systems, it is thus the choice of an integration method for generating an approximate solution (like …