Karl Rieger

Brief paper: On the observability of discrete-time dynamic systems - A geometric approach

A coordinate-free description for dynamic systems described by explicit (nonlinear) difference equations in one independent variable via a differential geometric framework is presented. Based on this covariant approach suitable geometric objects for discrete-time dynamics are introduced. Especially, the observability along a trajectory is discussed and transformations to normal forms are derived. In addition, the obtained (local) observability criteria can be checked by computer algebra algorithms. Some examples illustrate the proposed approach.

Implicit discrete-time systems and accessibility

An intrinsic description for dynamic systems, whose evolution along discrete time is governed by (nonlinear) implicit difference equations in one independent variable and zero-order (algebraic) equations, is presented by means of differential geometrical methods, where systems are associated with appropriate geometric objects reflecting their dynamics. Dynamic systems given in implicit form have the peculiarity that they may contain so-called hidden restrictions. A normal form is presented which is characterized by the circumstances that there are no further restrictions. In addition, it is illustrated that such a normal form allows for an equivalent system representation in explicit form. Based on the geometric picture of (implicit) discrete-time systems the qualitative property of accessibility along a fixed trajectory is discussed. By applying symmetry groups of discrete-time systems and studying invariants of these groups a formal approach is provided that allows us to gather local accessibility criteria successively, which can be tested by computer algebra. Several examples illustrate the results.

On the exact observability of distributed parameter systems

This contribution is devoted to the observability analysis of distributed parameter systems derived on the basis of the calculus of variations. A system theoretical analysis is motivated by the utilization of a differential geometric framework, which allows a covariant description of infinite dimensional systems. In particular, the (exact) observability along a trajectory of dynamic systems is discussed in general and it is shown that by means of the obtained formal approach (local) observability criteria can be provided.

Local decomposition and accessibility of PDE systems

The local decomposition of (nonlinear) ODE systems, which is obtained in the presence of a codistribution invariant under the system vector field and an associated local partition of the underlying manifold, is well-studied in the literature, and its relevance w.r.t. the local accessibility problem is indisputable. In this contribution we focus on the local decomposition of (nonlinear) PDE systems. In particular, it is shown that in the presence of a codistribution invariant under the so-called generalized system vector field a triangular decomposition, including the decomposition of the boundary conditions under certain conditions, can be obtained. In addition, we highlight the geometric picture behind our approach and that these results can be applied to the accessibility problem, where conditions for the local decomposition of a (non-accessible) system into subsystems are provided. A nonlinear example illustrates the results.

On the Accessibility of Distributed Parameter Systems

This contribution is devoted to the accessibility analysis of distributed parameter systems. A formal system theoretical approach is proposed by means of differential geometry, which allows an intrinsic representation for the class of infinite dimensional systems. Beginning with the introduction of a convenient representation form, in particular, the accessibility along a trajectory is discussed generally. In addition, the derivation of (local) (non-)accessibility criteria via utilizing transformation groups is shown. In order to illustrate the developed theory the proposed method is applied to an example.

Regelung verteilt-parametrischer Hamiltonscher Systeme auf Basis struktureller Invarianten

Zusammenfassung Dieser Beitrag behandelt die Modellierung und Regelung von räumlich eindimensionalen, verteilt-parametrischen Tor-basierten Hamiltonschen Systemen. Motiviert durch die physikalische Interpretation der Tor-basierten Hamiltonschen Systembeschreibung im konzentriert-parametrischen Fall wird eine Erweiterung dieser Systemklasse auf den verteilt-parametrischen Fall vorgeschlagen, welche auf dem klassischen evolutionären Zugang basiert. Weiters wird die aus dem konzentriert-parametrischen Fall bekannte Methode “Regelung auf Basis struktureller Invarianten” auf die vorgestellte verteilt-parametrische Hamiltonsche Darstellung übertragen. Die Effektivität dieser Methode wird anhand der energiebasierten Regelung des Timoshenko Balkens mit Randeingriff gezeigt. Abstract This contribution deals with the modelling and control of distributed-parameter Port-Hamiltonian systems with one-dimensional spatial domains. Motivated by the physical interpretation offered by the Port-Hamiltonian system class in the lumped-parameter case we propose an extension of this framework to the infinite dimensional scenario based on the classical evolutionary approach. Furthermore, we adapt the “control via structural invariants” method, well-known in the lumped-parameter case, with respect to the presented distributed-parameter Port-Hamiltonian description. The efficiency of this method is demonstrated for the energy based boundary control of the Timoshenko beam.

Flatness-based Temperature Control of Metal Sheets.

Abstract In order to enhance the performance of a class of lab-scale annealing testbeds for the steel industries, this paper addresses the temperature tracking task of sheet metal specimen by invoking nonlinear model-based control theory. Based on an accurate mathematical model of the testbed, which uses Ohmic heating via a phase-controlled power converter, a flatness-based control approach is presented. By virtue of the parametrization of the control law in terms of physical parameters, this approach is found to significantly simplify the operation of the annealing testbed for the wide range of samples to be used, by simultaneously offering very high tracking performance for the entire domain of operation.

Accessibility Conditions for PDE Systems and the Adjoint System

Abstract A group-theoretical approach is used to tackle the problem of (local) accessibility along a trajectory of systems described by partial differential equations (PDEs). In particular, a class of first-order PDE systems with boundary conditions and boundary control is applied to illustrate the key ideas of our approach. The methods mainly rely on a coordinate-free formulation of PDE systems, which also incorporates boundary conditions. The accessibility is discussed in general and an approach based on (pointwise, continuous) transformation groups and their invariants is motivated. Using an infinitesimal criterion for invariance we study special group invariants to derive (local) conditions on accessibility. It is highlighted that the basic questions lead to the investigation of a particular adjoint system. A nonlinear example demonstrates the methods and results.

Energy based control of a hydromechanical system

This contribution deals with an energy based controller design for an under-actuated mechanical system with a hydraulic piston actuator. In particular, the example consists of a single acting piston actuator and a rigid mass between two springs. For the resulting system a static feedback control is designed based on energy consideration. Since the control algorithm requires the velocities, which are not measurable in this application, it is extended by a reduced observer. In addition, the stability of the desired equilibrium of the closed loop system is proven and the results are illustrated by simulation.

Beobachtbarkeitsanalyse von verteilt-parametrischen Systemen – Ein differentialgeometrischer Ansatz (Observability Analysis of Distributed Parameter Systems – A Differential Geometrical Approach)

Zur Untersuchung der Systemeigenschaft Beobachtbarkeit von verteilt-parametrischen Systemen wird zunächst eine geeignete Beschreibung für diese Klasse von Systemen im Rahmen der Differentialgeometrie eingeführt. Weiters wird dann insbesondere die Beobachtbarkeit von dynamischen Systemen entlang einer Trajektorie diskutiert, eine geometrische Veranschaulichung für diese Systemeigenschaft gegeben und eine systemtheoretische Analyse basierend auf Transformationsgruppen vorgestellt. Mit Hilfe eines infinitesimalen Invarianzkriteriums können (lokale) Bedingungen für die (Nicht-)Beobachtbarkeit gewonnen werden. Die Anwendung der vorgeschlagenen Methoden auf die Klasse der konzentriert-parametrischen Systeme zeigt, dass diese Kriterien in die bekannten übergehen. Die allgemeine Theorie wird durch ein Beispiel illustriert. In order to investigate the system property observability of distributed parameter systems, first a suitable description for this class of systems via a differential geometrical framework is introduced. In particular, the observability of dynamic systems along a trajectory is discussed, a geometrical illustration of this system property is provided and a system-theoretical analysis based on transformation groups is presented. Then, an infinitesimal invariance principle allows to derive (local) (non)observability conditions. In addition, lumped parameter systems are considered, where the criteria coincide with well known criteria. Finally, the general theory is illustrated by means of an example.