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

Discrete port-controlled Hamiltonian dynamics and average passivation

The paper discusses the modeling and control of port-controlled Hamiltonian dynamics in a pure discrete-time domain. The main result stands in a novel differential-difference representation of discrete port-controlled Hamiltonian systems using the discrete gradient. In these terms, a passive output map is exhibited as well as a passivity based damping controller underlying the natural involvement of discrete-time average passivity

Explicit Simplicial Discretization of Distributed-Parameter Port-Hamiltonian Systems

Simplicial Dirac structures as finite analogues of the canonical Stokes-Dirac structure, capturing the topological laws of the system, are defined on simplicial manifolds in terms of primal and dual cochains related by the coboundary operators. These finite-dimensional Dirac structures offer a framework for the formulation of standard input-output finite-dimensional port-Hamiltonian systems that emulate the behavior of distributed-parameter port-Hamiltonian systems. This paper elaborates on the matrix representations of simplicial Dirac structures and the resulting port-Hamiltonian systems on simplicial manifolds. Employing these representations, we consider the existence of structural invariants and demonstrate how they pertain to the energy shaping of port-Hamiltonian systems on simplicial manifolds

Discrete port-Hamiltonian systems: mixed interconnections

Either from a control theoretic viewpoint or from an analysis viewpoint it is necessary to convert smooth systems to discrete systems, which can then be implemented on computers for numerical simulations. Discrete models can be obtained either by discretizing a smooth model, or by directly modeling at the discrete level itself. The goal of this paper is to apply a previously developed discrete modeling technique to study the interconnection of continuous systems with discrete ones in such a way that passivity is preserved. Such a theory has potential applications, in the field of haptics, telemanipulation etc. It is shown that our discrete modeling theory can be used to formalize previously developed techniques for obtaining passive interconnections of continuous and discrete systems

Aspects of bond graph modelling in control

Abstract available: p. i

Automated Generation of Explicit Port-Hamiltonian Models from Multi-Bond Graphs

Port-Hamiltonian system theory is a well-known framework for the control of complex physical systems. The majority of port-Hamiltonian control design methods base on an \emph{explicit} input-state-output port-Hamiltonian model for the system under consideration. However in the literature, little effort has been made towards a systematic, automatable derivation of such explicit models. In this paper, we present a constructive, formally rigorous method for an explicit port-Hamiltonian formulation of multi-bond graphs. Two conditions, one necessary and one sufficient, for the existence of an explicit port-Hamiltonian formulation of a multi-bond graph are given. We summarise our approach in a fully automated algorithm of which we provide an exemplary implementation along with this publication. The theoretical and practical results are illustrated through an academic example

Implicit IDA-PBC Design and Implementation for a Portal Crane System

Interconnection and damping assignment passivity-based control (IDA-PBC) is a wellknown technique which regulates the behavior of nonlinear systems, assigning a target port-Hamiltonian (pH) structure to the closed-loop. In underactuated mechanical systems (UMSs) its application requires the satisfaction of matching conditions, which in many cases demands to solve partial differential equations (PDEs). Only recently, the IDA-PBC has been extended to UMSs modeled implicitly, where the system dynamics in pH representation are described by a set of differential-algebraic equations (DAEs). In some system classes this implicit approach allows to circumvent the PDE problem and also to design an output-feedback law. The present thesis deals with the design and implementation of the total energy shaping implicit IDA-PBC on a portal crane system located at the laboratory of the Control Engineering Group at TU-Ilmenau. The implicit controller is additionally compared with a simplified (explicit) IDA-PBC [1]. This algorithm shapes the total energy and avoids the PDE problem. However, this thesis reveales a significant implementation flaw in the algorithm, which then could be solved.Interconnection and damping assignment passivity-based control (IDA-PBC) ist eine wohlbekannte Methode zur Regelung von nichtlinearen Systemen, die im geschlossenen Regelkreis eine gewünschte Port-Hamiltonian-Struktur (pH) haben. Die Anwendung auf unteraktuierte mechanische Systeme (UMS) erfordert die Erfüllung von sogenannten Matching Conditions, die meistens die Lösung partieller Differentialgleichungen (PDE) benötigt. Erst kürzlich wurde die IDA-PBC auf implizit modellierte UMS erweitert, bei denen die Systemdynamik in pH-Darstellungen durch Differentialalgebraische Gleichungen (DAE) beschrieben wird. Dieser implizite Ansatz ermöglicht bei einigen Systemklassen, das PDE-Problem zu umgehen und auch eine Ausgangsrückführung zu entwerfen. Die vorliegende Masterarbeit beschäftigt sich mit dem Entwurf und der Implementierung des impliziten IDA-PBC zur Gesamtenergievorgabe auf einem Portalkransystem im Labor des Fachgebiets Regelungstechnik der TU-Ilmenau. Der implizite Regler wird mit einem vereinfachten (expliziten) IDA-PBC verglichen [1]. Dieser Algorithmus gibt ebenso die Gesamtenergie vor und vermeidet das PDE-Problem. In der Masterarbeit wird in diesem Algorithmus ein wesentlicher Implementierungsfehler offengelegt und behoben.Tesi