Markus Schöberl

CONSTRUCTION OF FLAT OUTPUTS BY REDUCTION AND ELIMINATION

Abstract The construction of flat outputs for nonlinear lumped parameter systems is still an open problem in the general case. An algorithm, based on successive reduction of the number of variables and elimination of variables is presented. It is shown that the reduction requires a dynamic extension in general. The geometric interpretation of this fact is that certain vector fields become projectable on the manifold defined by the extended system. Finally some applications of this approach are presented.

Applications of energy based control methods for the inverted pendulum on a cart

This contribution deals with the application of energy based control methods for the inverted pendulum on a cart model. We will present a swing up controller as well as a nonlinear balancing controller with the focus on the implementation on a laboratory model. Therefore we recapitulate well-known control concepts from the literature which will be adapted such that they work on a concrete experiment with all the undesirable effects like friction and quantisation.

On Casimir Functionals for Infinite-Dimensional Port-Hamiltonian Control Systems

We consider infinite-dimensional port-Hamiltonian systems with respect to control issues. In contrast to the well-established representation relying on Stokes-Dirac structures that are based on skew-adjoint differential operators and the use of energy variables, we employ a different port-Hamiltonian framework. Based on this system representation conditions for Casimir functionals will be derived where in this context the variational derivative plays an extraordinary role. Furthermore the coupling of finite- and infinite-dimensional systems will be analyzed in the spirit of the control by interconnection problem.

Jet bundle formulation of infinite-dimensional port-Hamiltonian systems using differential operators

We consider infinite-dimensional port-Hamiltonian systems described on jet bundles. Based on a power balance relation we introduce the port-Hamiltonian system representation using differential operators regarding the structural mapping, the dissipation mapping and the input mapping. In contrast to the well-known representation on the basis of the underlying Stokes-Dirac structure our approach is not necessarily based on using energy-variables which leads to a different port-Hamiltonian representation of the analyzed partial differential equations. The presented constructions will be specialized to mechanical systems to which class also the presented examples belong.

First-order Hamiltonian field theory and mechanics

This article deals with the geometric analysis of the evolutionary and the polysymplectic approach in first-order Hamiltonian field theory. Based on a variational formulation in the Lagrangian picture, two possible counterparts in a Hamiltonian formulation are discussed. The main difference between these two approaches, which are important for the application, is besides a different bundle construction, the different Legendre transform as well as the analysis of the conserved quantities. Furthermore, the role of the boundary conditions in the Lagrangian and in the Hamiltonian pictures will be addressed. These theoretical investigations will be completed by the analysis of several examples, including the wave equation, a beam equation and a special subclass of continuum mechanics in the presented framework.

On Parametrizations for a Special Class of Nonlinear Systems

Abstract We analyze a constructive procedure to find a system parametrization for a special class of nonlinear control systems. The key feature of the algorithm will be the derivation of coordinate transformations that guarantee that several variables turn out to be not differentiated and can therefore be eliminated. In this context the class of affine derivative systems (AD-systems) will play a prominent role. Elimination of variables in general leads to implicit systems which might not be in affine derivative form. Therefore it is of interest to analyze if elimination is also possible such that the AD-structure is preserved. The suggested ideas will be demonstrated on several examples of nonlinear control systems with three states and two inputs.

Covariant formulation of the governing equations of continuum mechanics in an Eulerian description

We present the balance relations for a continuum in the Eulerian formulation in a pure covariant fashion. Based on the analysis of nonrelativistic particle mechanics, we adapt the covariant description to the case of a continuum. The use of the covariant Nijenhuis differential as well as the splitting of the vertical configuration bundle are the key objects that allow a coordinate-free representation. We state the balance equations such that they are valid, also when time variant transformations are applied, which leads to a nontrivial space-time connection and a metric which explicitly depends on the time.

On an implicit triangular decomposition of nonlinear control systems that are 1-flat-A constructive approach

We study the problem to provide a triangular form based on implicit differential equations for non-linear multi-input systems with respect to the flatness property. Furthermore, we suggest a constructive method for the transformation of a given system into that special triangular shape, if possible. The well known Brunovsky form, which is applicable with regard to the exact linearization problem, can be seen as special case of this implicit triangular form. A key tool in our investigation will be the construction of Cauchy characteristic vector fields that additionally annihilate certain codistributions. In adapted coordinates this construction allows to single out variables whose time-evolution can be derived without any integration.

A jet space approach to check Pfaffian systems for flatness

This contribution uses the representation of control systems, given by a set of ODEs, as Pfaffian systems. It is shown, that flatness for this class of systems is equivalent to the fact, that the unique decomposition of the pull back of the Pfaffian system to a certain jet space into a horizontal and vertical part, has a trivial horizontal part only. We present an algorithm to construct the required jet space, as well as the decomposition. It uses the fact, that a flat Pfaffian system must admit a certain affine input representation after a possible extension by contact forms. Since this representation is not unique, a branching point may appear and all branches have to be checked. This applies also to the decomposition of the system in AI representation into a horizontal and a vertical part. It is shown, how the number of branches can be reduced significantly. The algorithm requires the solution of non linear algebraic equations and the determination of flows generated by distributions. It is worth mentioning, that the presented approach degenerates to the well known one for input to state linearizable system by static feedback. Finally, the application of the algorithm to three examples, all are not input to state linearizable by static feedback, is shown.

On the Port-Hamiltonian Representation of Systems Described by Partial Differential Equations

Abstract We consider infinite dimensional port-Hamiltonian systems. Based on a power balance relation we introduce the port-Hamiltonian system representation where we pay attention to two different scenarios, namely the non-differential operator case and the differential operator case regarding the structural mapping, the dissipation mapping and the in/output mapping. In contrast to the well-known representation on the basis of the underlying Stokes-Dirac structure our approach is not necessarily based on using energy-variables which leads to a different port-Hamiltonian representation of the analyzed partial differential equations.