Weak Form of Stokes-Dirac Structures and Geometric Discretization of Port-Hamiltonian Systems

We present the mixed Galerkin discretization of distributed parameter port-Hamiltonian systems. On the prototypical example of hyperbolic systems of two conservation laws in arbitrary spatial dimension, we derive the main contributions: (i) A weak formulation of the underlying geometric (Stokes-Dirac) structure with a segmented boundary according to the causality of the boundary ports. (ii) The geometric approximation of the Stokes-Dirac structure by a finite-dimensional Dirac structure is realized using a mixed Galerkin approach and power-preserving linear maps, which define minimal discrete power variables. (iii) With a consistent approximation of the Hamiltonian, we obtain finite-dimensional port-Hamiltonian state space models. By the degrees of freedom in the power-preserving maps, the resulting family of structure-preserving schemes allows for trade-offs between centered approximations and upwinding. We illustrate the method on the example of Whitney finite elements on a 2D simplicial triangulation and compare the eigenvalue approximation in 1D with a related approach.Comment: Copyright 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0

Geometry of Thermodynamic Processes

Since the 1970s contact geometry has been recognized as an appropriate framework for the geometric formulation of the state properties of thermodynamic systems, without, however, addressing the formulation of non-equilibrium thermodynamic processes. In Balian & Valentin (2001) it was shown how the symplectization of contact manifolds provides a new vantage point; enabling, among others, to switch between the energy and entropy representations of a thermodynamic system. In the present paper this is continued towards the global geometric definition of a degenerate Riemannian metric on the homogeneous Lagrangian submanifold describing the state properties, which is overarching the locally defined metrics of Weinhold and Ruppeiner. Next, a geometric formulation is given of non-equilibrium thermodynamic processes, in terms of Hamiltonian dynamics defined by Hamiltonian functions that are homogeneous of degree one in the co-extensive variables and zero on the homogeneous Lagrangian submanifold. The correspondence between objects in contact geometry and their homogeneous counterparts in symplectic geometry, as already largely present in the literature, appears to be elegant and effective. This culminates in the definition of port-thermodynamic systems, and the formulation of interconnection ports. The resulting geometric framework is illustrated on a number of simple examples, already indicating its potential for analysis and control.Comment: 23 page

Exponential stability of a class of boundary control systems

We study a class of partial differential equations (with variable coefficients) on a one dimensional spatial domain with control and observation at the boundary. For this class of systems we provide simple tools to check exponential stability. This class is general enough to include models of flexible structures, traveling waves, heat exchangers, and bioreactors among others. The result is based on the use of a generating function (the energy for physical systems) and an inequality condition at the boundary. Furthermore, based on the port Hamiltonian approach, we give a constructive method to reduce this inequality to a simple matrix inequality

Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation

In this paper, we propose a constructive procedure to modify the Hamiltonian function of forced Hamiltonian systems with dissipation in order to generate Lyapunov functions for nonzero equilibria. A key step in the procedure, which is motivated from energy-balance considerations standard in network modeling of physical systems, is to embed the system into a larger Hamiltonian system for which a series of Casimir functions can be easily constructed. Interestingly enough, for linear systems the resulting Lyapunov function is the incremental energy; thus our derivations provide a physical explanation to it. An easily verifiable necessary and sufficient condition for the applicability of the technique in the general nonlinear case is given. Some examples that illustrate the method are give

Putting energy back in control

A control system design technique using the principle of energy balancing was analyzed. Passivity-based control (PBC) techniques were used to analyze complex systems by decomposing them into simpler sub systems, which upon interconnection and total energy addition were helpful in determining the overall system behavior. An attempt to identify physical obstacles that hampered the use of PBC in applications other than mechanical systems was carried out. The technique was applicable to systems which were stabilized with passive controllers

A port-Hamiltonian formulation of physical swithching systems with varying constraints

International audienceThis paper extends a generic method to design a port-Hamiltonian formulation modeling all geometric interconnection structures of a physical switching system with varying constraints. A non-minimal kernel representation of this family of structures (named Dirac structures) is presented. It is derived from the parameterized incidence matrices which are a mathematical representation of the primal and dual dynamic network graphs associated with the system. This representation has the advantage of making it possible to model complex physical switching systems with varying constraints and to fall within the framework of passivitybased control

Linear Boundary Port-Hamiltonian Systems with Implicitly Defined Energy

In this paper we extend the previously introduced class of boundary port-Hamiltonian systems to boundary control systems where the variational derivative of the Hamiltonian functional is replaced by a pair of reciprocal differential operators. In physical systems modelling, these differential operators naturally represent the constitutive relations associated with the implicitly defined energy of the system and obey Maxwell's reciprocity conditions. On top of the boundary variables associated with the Stokes-Dirac structure, this leads to additional boundary port variables and to the new notion of a Stokes-Lagrange subspace. This extended class of boundary port-Hamiltonian systems is illustrated by a number of examples in the modelling of elastic rods with local and non-local elasticity relations. Finally it shown how a Hamiltonian functional on an extended state space can be associated with the Stokes-Lagrange subspace, and how this leads to an energy balance equation involving the boundary variables of the Stokes-Dirac structure as well as of the Stokes-Lagrange subspace.Comment: 23 page

Generalized Port-Hamiltonian DAE Systems

Motivated by recent work in this area we expand on a generalization of port-Hamiltonian systems that is obtained by replacing the Hamiltonian function representing energy storage by a general Lagrangian subspace. This leads to a new class of algebraic constraints in physical systems modeling, and to an interesting class of DAE systems. It is shown how constant Dirac structures and Lagrangian subspaces allow for similar representations, and how this leads to descriptions of the DAE systems entailing generalized Lagrange multipliers.Comment: 19 page