Kurt Zehetleitner

Active Control of Smart Structures Using Port Controlled Hamiltonian Systems

Smart structures based on piezoelectric composites have turned out to be excellent actuators and sensors for active and passive damping in vibration control. In the case of small displacements a linear approach suffices [8], if hysteresis or depolarization of the active material are negligible [7]. This contribution presents a unifying way for the mathematical modeling of smart structures based on Port Controlled Hamiltonian Systems, see [3].

Formale Methoden für implizite dynamische Systeme (A Formal Approach for Implicit Dynamic Systems)

Abstract Die Modellierung komplexer dynamischer Systeme führt oft auf Sätze von algebraischen Gleichungen und gewöhnlichen Differentialgleichungen. Dieser Beitrag präsentiert nun Methoden für implizite Systeme basierend auf der formalen Theorie von gewöhnlichen Differentialgleichungen. Es wird gezeigt, dass gewisse implizite Systeme, man nennt sie formal integrable, Eigenschaften aufweisen, die viele Untersuchungen ohne Überführung in eine explizite Form gestatten. Dabei wird ein dynamisches System als eine Untermannigfaltigkeit mit einer speziellen geometrischen Struktur aufgefasst. Die Form der Gleichungen entspricht dann lediglich einer speziellen Parametrierung dieser Untermannigfaltigkeit. Da man unter gewissen Regularitätsannahmen jedes implizite System in ein formal integrables überführen kann, stellt diese Klasse ein natürliches Bindeglied zwischen den expliziten und den impliziten Systemen dar. Auf Basis dieser Betrachtung werden zuerst Ergebnisse für den allgemeinen Fall präsentiert, dann werden Systeme, die linear in den Ableitungen sind, ausführlicher untersucht. Zum Abschluss wird die Theorie auf den linearen und zeitinvarianten Fall angewandt.

Computer algebra methods for implicit dynamic systems and applications

This contribution deals with the application of computer algebra methods to the analysis of systems of implicit ordinary differential equations. These systems are identified with submanifolds in a suitable jet space. We propose an implementation of the accessibility and observability analysis for implicit ordinary differential equations based on Lie group methods. Since this approach requires that the system of equations is formally integrable, we present an algorithm which converts the given system into this form. In addition, it is shown how the efficiency of the stated algorithms can be improved for polynomial systems with the application of Groebner bases. The presented algorithms are applied to the model of the PVTOL aircraft.

Control of Hydraulic Devices, an Internal Model Approach

Summary. Hydraulic devices are used, where high forces must be generated by small devices. A disadvantage of these devices is the nonlinear behavior, especially of big devices, which are used e.g. in steel industries. Since one cannot measure the whole state in these applications, an output feedback controller is presented, which is based on a reduced observer. In addition disturbance forces, which can be described by exosystems, are taken into account. Their effect is eliminated in the steady state by the internal model approach. The presented design is applied to the hydraulic gap control of mill stands in rolling mills, such that unknown slowly varying disturbance forces and forces caused by eccentricities of the rolls are rejected.

COMPUTER-ALGEBRA ALGORITHMS FOR THE GEOMETRIC CONTROLLER DESIGN OF DAES

Abstract This contribution discusses an approach to extend geometric controller design techniques known for systems of explicit differential equations to computer algebra algorithms for systems of implicit ordinary differential equations also known as DAEs (differential algebraic equations), a system class which arises in some modeling approaches for, e.g., electrical or mechanical systems. Shortly, the methodology used to derive published algorithms on, e.g., observability of implicit systems is reviewed. This geometric approach is based on the formally integrable form of the implicit system. Using this idea it is shown in this contribution how the static feedback linearization of dynamical systems can be adapted to be carried out directly at the formally integrable implicit system without having to deal with the explicit form. The approach is shown with the help of a mechanical example - the Car&Beam system.

Basic Differential Geometry for Mechanics and Control

Differential geometry is an old mathematical discipline which contributed and contributes a lot to mathematical physics. Also its use in mechanics, electrodynamics or thermodynamics has a long history. Nevertheless, the introduction of the geometric methods to nonlinear, model based control occurred only about 25 years ago. The driving force for this development was the same as for physics, that is, geometric methods allow us to deal with dynamic systems in a coordinate free manner. Therefore, this contribution presents an overview on this discipline which starts with smooth manifolds, bundles, vector fields and finishes with jet bundles and jet coordinates.

Implicit control systems: Properties, algorithmic tests and their computer algebra implementation

Abstract This contribution discusses symbolic algorithms for tests on properties of implicit control systems (DAEs), like observability, accessibility or identifiability. The algorithms related to observability and accessibility have already been published. The focus lies on those operations performed in a computer algebra system which are common for all these tests. I particular operations based on Groebner bases are applied to carry out the necessary elimination in the case of polynomial nonlinearities. Furthermore some features of the implemented package DA Egeom in the computer algebra system MAPLE are discussed and finally the package is applied to the mathematical model of the VTOL aircraft. It is available via the institutes homepage.

Symbolic methods for the equivalence problem for systems of implicit ordinary differential equations

This contribution deals with the equivalence problem for systems of implicit ordinary differential equations. Equivalence means that every solution of the original set of equations is a solution of a given normal form and vice versa. Since we describe this system as a submanifold in a suitable jet-space, we present some basics from differential and algebraic geometry and give a short introduction to jet-theory and its application to systems of differential equations. The main results of this contribution are two solutions for the equivalence problem, where time derivatives of the input are admitted or not. Apart from the theoretical results we give a sketch for computer algebra based algorithms necessary to solve these problems efficiently.