25 research outputs found
Discrete-time port-Hamiltonian systems: A definition based on symplectic integration
We introduce a new definition of discrete-time port-Hamiltonian systems
(PHS), which results from structure-preserving discretization of explicit PHS
in time. We discretize the underlying continuous-time Dirac structure with the
collocation method and add discrete-time dynamics by the use of symplectic
numerical integration schemes. The conservation of a discrete-time energy
balance - expressed in terms of the discrete-time Dirac structure - extends the
notion of symplecticity of geometric integration schemes to open systems. We
discuss the energy approximation errors in the context of the presented
definition and show that their order is consistent with the order of the
numerical integration scheme. Implicit Gauss-Legendre methods and Lobatto
IIIA/IIIB pairs for partitioned systems are examples for integration schemes
that are covered by our definition. The statements on the numerical energy
errors are illustrated by elementary numerical experiments.Comment: 12 pages. Preprint submitted to Systems & Control Letter
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
Discrete-time port-Hamiltonian systems: A definition based on symplectic integration
International audienceWe introduce a new definition of discrete-time port-Hamiltonian systems (PHS), which results from structure-preserving discretization of explicit PHS in time. We discretize the underlying continuous-time Dirac structure with the collocation method and add discrete-time dynamics by the use of symplectic numerical integration schemes. The conservation of a discrete-time energy balance-expressed in terms of the discrete-time Dirac structure-extends the notion of symplecticity of geometric integration schemes to open systems. We discuss the energy approximation errors in the context of the presented definition and show that their order is consistent with the order of the numerical integration scheme. Implicit Gauss-Legendre methods and Lobatto IIIA/IIIB pairs for partitioned systems are examples for integration schemes that are covered by our definition. The statements on the numerical energy errors are illustrated by elementary numerical experiments
Discrete nonlinear elastodynamics in a port-Hamiltonian framework
We provide a fully nonlinear port-Hamiltonian formulation for discrete
elastodynamical systems as well as a structure-preserving time discretization.
The governing equations are obtained in a variational manner and represent
index-1 differential algebraic equations. Performing an index reduction one
obtains the port-Hamiltonian state space model, which features the nonlinear
strains as an independent state next to position and velocity. Moreover,
hyperelastic material behavior is captured in terms of a nonlinear stored
energy function. The model exhibits passivity and losslessness and has an
underlying symmetry yielding the conservation of angular momentum. We perform
temporal discretization using the midpoint discrete gradient, such that the
beneficial properties are inherited by the developed time stepping scheme in a
discrete sense. The numerical results obtained in a representative example are
demonstrated to validate the findings.Comment: 9 pages, 7 figures, Submitted to Proceedings in Applied Mathematics
and Mechanics 202
Port-Hamiltonian formulation and structure-preserving discretization of hyperelastic strings
Port-Hamiltonian (PH) systems provide a framework for modeling, analysis and
control of complex dynamical systems, where the complexity might result from
multi-physical couplings, non-trivial domains and diverse nonlinearities. A
major benefit of the PH representation is the explicit formulation of power
interfaces, so-called ports, which allow for a power-preserving interconnection
of subsystems to compose flexible multibody systems in a modular way. In this
work, we present a PH representation of geometrically exact strings with
nonlinear material behaviour. Furthermore, using structure-preserving
discretization techniques a corresponding finite-dimensional PH state space
model is developed. Applying mixed finite elements, the semi-discrete model
retains the PH structure and the ports (pairs of velocities and forces) on the
discrete level. Moreover, discrete derivatives are used in order to obtain an
energy-consistent time-stepping method. The numerical properties of the newly
devised model are investigated in a representative example. The developed PH
state space model can be used for structure-preserving simulation and model
order reduction as well as feedforward and feedback control design.Comment: Submitted as a proceeding to the ECCOMAS Thematic Conference on
Multibody Dynamics 202
Partitioned finite element method for structured discretization with mixed boundary conditions
The propagation of acoustic waves in a 2D geometrical domain under mixed boundary control is here described by means of the port-Hamiltonian (pH) formalism. A finite element based method is employed to obtain a consistently discretized model. To construct a model with mixed boundary control, two different methodologies are detailed: one employs Lagrange multipliers, the other relies on a virtual domain decomposition to interconnect models with different causalities. The two approaches are assessed numerically, by comparing the Hamiltonian and the state variables norm for progressively refined meshes
Adaptive Tracking Control with Uncertainty-aware and State-dependent Feedback Action Blending for Robot Manipulators
Adaptive control can significantly improve tracking performance of robot manipulators subject to modeling errors in dynamics. In this letter, we propose a new framework combining the composite adaptive controller using a natural adaptation law and an extension of the adaptive variance algorithm (AVA) for controller blending. The proposed approach not only automatically adjusts the feedback action to reduce the risk of violating actuator constraints but also anticipates substantial modeling errors by means of an uncertainty measure, thus preventing severe performance deterioration. A formal stability analysis of the closed-loop system is conducted. The control scheme is experimentally validated and directly compared with baseline methods on a torque-controlled KUKA LWR IV+