On the Method of Interconnection and Damping Assignment Passivity-Based Control for the Stabilization of Mechanical Systems

Interconnection and damping assignment passivity-based control (IDA-PBC) is an excellent method to stabilize mechanical systems in the Hamiltonian formalism. In this paper, several improvements are made on the IDA-PBC method. The skew-symmetric interconnection submatrix in the conventional form of IDA-PBC is shown to have some redundancy for systems with the number of degrees of freedom greater than two, containing unnecessary components that do not contribute to the dynamics. To completely remove this redundancy, the use of quadratic gyroscopic forces is proposed in place of the skew-symmetric interconnection submatrix. Reduction of the number of matching partial differential equations in IDA-PBC and simplification of the structure of the matching partial differential equations are achieved by eliminating the gyroscopic force from the matching partial differential equations. In addition, easily verifiable criteria are provided for Lyapunov/exponential stabilizability by IDA-PBC for all linear controlled Hamiltonian systems with arbitrary degrees of underactuation and for all nonlinear controlled Hamiltonian systems with one degree of underactuation. A general design procedure for IDA-PBC is given and illustrated with examples. The duality of the new IDA-PBC method to the method of controlled Lagrangians is discussed. This paper renders the IDA-PBC method as powerful as the controlled Lagrangian method

Input-output linearization and decoupling of mechanical control systems

In this work, we present a problem of simultaneous input-output feedback linearization and decoupling (non-interacting) for mechanical control systems with outputs. We show that the natural requirement of preserving mechanical structure of the system and of transformations imposes supplementary conditions when compared to the classical solution of the same problem for general control systems. These conditions can be expressed using objects on the configuration space only. We illustrate our results with several examples of mechanical control systems

A crystallographic approach to symmetry-breaking in fluid layers

Symmetry-breaking bifurcations, where a flow state with a certain symmetry undergoes a transition to state with a different symmetry, are ubiquitous in fluid mechanics. Much can be understood about the nature of these transitions from symmetry alone, using the theory of groups and their representations. Here we show how the extensive databases on groups in crystallography can be exploited to yield insights into fluid-dynamical problems. In particular, we demonstrate the application of the crystallographic layer groups to problems in fluid layers, using thermal convection as an example. Crystallographic notation provides a concise and unambiguous description of the symmetries involved, and we advocate its broader use by the fluid dynamics community.Comment: 27 pages, 9 figures, 3 supplementary table

Orbit Space Reduction for Symmetric Dynamical Systems with an Application to Laser Dynamics

This work considers the effect of symmetries on analysing bifurcations in dynamical systems. We consider an example of a laser with strong optical feedback which is modelled using coupled non-linear differential equations. A stationary point can be found in space, which can then be continued in parameter space using software such as AUTO. This software will then detect and continue bifurcations which indicate change in dynamics as parameters are varied. Due to symmetries in the equations, using AUTO may require the system of equations to be reduced in order to study periodic orbits of the original system as (relative) equilibria of the reduced system. Reasons for this are explored as well as considering how the equations can be changed or reduced to remove the symmetry. Invariant and Equivariant theory provide the tools for reducing the system of equations to the orbit space, allowing further analysis of the lasers dynamics.Great Western Research in collaboration with Bookham Technology

Resonances in a spring-pendulum: algorithms for equivariant singularity theory

A spring-pendulum in resonance is a time-independent Hamiltonian model system for formal reduction to one degree of freedom, where some symmetry (reversibility) is maintained. The reduction is handled by equivariant singularity theory with a distinguished parameter, yielding an integrable approximation of the Poincaré map. This makes a concise description of certain bifurcations possible. The computation of reparametrizations from normal form to the actual system is performed by Gröbner basis techniques.

Dynamics and Control of Bluff-Body-Wake-Structure Interaction

RÉSUMÉ L’interaction fluide-structure est observée dans la majorité des applications industrielles. L’interaction peut conduire à la génération de forces indésirables affectant les structures et causant de la fatigue ou des dommages. Une approche pour comprendre la dynamique fondamentale et la stabilité des sillages des cylindres consiste à réaliser des simulations utilisant les oscillations forcées. Deux aspects d’un tel écoulement sont considérés. Le premier concerne la dépendance des symétries de configuration de détachement tourbillonnaire avec l’amplitude des oscillations forcées, de la fréquence d’excitation et de la direction des oscillations. Les oscillations forcées, dépendantes de l’amplitude et de la fréquence des oscillations, entrainent la formation de différentes configurations de sillage présentant des symétries spécifiques. Les simulations numériques et les analyses de stabilité sont utilisées pour déterminer les modes instables du sillage et leurs bifurcations. L’oscillation forcée dans la direction d’écoulement d’un cylindre circulaire est étudié pour prédire les modes du sillage sur une plage déterminée d’amplitude et de fréquence. Les calculs numériques en deux dimensions des équations de Navier-Stokes sont réalisés en utilisant la méthode des conditions aux frontières périodiques forcées avec une plage de fréquence de forçage / [1 2] e s f f ; couvrant les régimes d’excitations harmoniques et super-harmoniques avec des ratios d’amplitude dans la plage A/ D[00.5]. Les modes d’accrochage symétriques et asymétriques sont observés pour trois différents ratios d’amplitude et de fréquence d’oscillation. Le mode asymétrique 2S lorsque / 1 e s f f pour A/D=[0.35-0.5], le mode P+S à / 1.5 e s f f pour A/D=[0.175,0.5], et le mode S à / 2 e s f f pour A/D=[0.175,0.5] sont confirmés. À cause de la non-linéarité des équations de Navier-Stokes et de la complexité de la dimension infinie de la dynamique des fluides, les modes primaires sont calculés à l’aide de l’outil de décomposition orthogonale aux valeurs propres (POD). Une procédure de Galerkin est utilisée pour projeter les équations de Navier-Stokes dans un espace de dimension réduite incluant les deux premiers modes POD. Cette méthode réduit la taille du problème d’un espace de dimension infinie à un nombre fini de modes (degrés de liberté) représentant la dynamique du sillage. Les modes dominants du détachement tourbillonnaire sont invariants en fonction des groupes de symétrie. La théorie de la bifurcation équivariante est utilisée pour développer le modèle d’ordre inférieur en utilisant les propriétés de symétrie des modes principaux. Ainsi, la bifurcation équivariante et les théories des formes normales sont combinées avec des calculs numériques des équations de Navier-Stokes pour décrire précisément les propriétés spacio-temporelles des modes de sillage d’écoulement et de leurs bifurcations.----------ABSTRACT Fluid-structure interaction is encountered in most of the engineering and industrial flow applications. The interaction can lead to generation of undesirable forces acting on structures and causing fatigue or damage. One approach to understanding the fundamental wake flow dynamics and stability is to conduct simulations using forced oscillation of a cylinder. Two aspects of such a flow are considered. The first is the dependence of vortex shedding pattern symmetries on the forced oscillation amplitude, forcing frequency ratios and the direction of the oscillation relative to flow direction. The forced oscillation, depending on the amplitude and frequency of the oscillation, causes formation of various patterns which have specific symmetries. Numerical simulations and stability analysis are employed to determine the unstable wake flow modes and their bifurcations. The forced inline oscillation of a circular cylinder is studied to predict the wake modes over a prescribed range of amplitudes and frequencies. The two-dimensional numerical computations of the Navier-Stokes equations are performed using forced periodic boundary condition method in the range of forcing-to-shedding frequency ratios / [1 2] e s f f ; covering the harmonic and superharmonic excitation regimes with amplitude ratio in the range A/ D[00.5]. Symmetric and asymmetric lock-on modes are observed for three different oscillation amplitudes and frequency ratios. The asymmetric 2S mode when / 1 e s f f for A/D=[0.35-0.5], P+S mode at / 1.5 e s f f for A/D=[0.175,0.5] and S mode at / 2 e s f f for A/D=[0.175,0.5] are confirmed. Due to the nonlinearity of the Navier-Stokes équations and the complexity of the infinite dimensional flow dynamics, the primary modes are calculated by the proper orthogonal decomposition (POD) tool. A Galerkin procedure is the used to project the Navier-Stokes equations onto a low-dimensional space spanned by the first two POD modes. This method reduces the problem size from an infinite-dimensional space to a finite number of modes (degree-offreedom)representing the wake dynamics. The vortex shedding dominant modes are invariant Under their symmetry groups. The equivariant bifurcation theory is employed to develop the low order model using the symmetry properties of the primary modes. Thus, equivariant bifurcation and normal form theories are combined with numerical computations of the Navier-Stokes equations to precisely describe the spatio-temporal properties of the wake flow modes and their bifurcations. A linear analysis of the analytical low order model near the bifurcation point is performed to predict the bifurcation sequences observed in simulations

Regelungstheorie

The workshop “Regelungstheorie” (control theory) covered a broad variety of topics that were either concerned with fundamental mathematical aspects of control or with its strong impact in various fields of engineering

Roadmap on Machine learning in electronic structure

AbstractIn recent years, we have been witnessing a paradigm shift in computational materials science. In fact, traditional methods, mostly developed in the second half of the XXth century, are being complemented, extended, and sometimes even completely replaced by faster, simpler, and often more accurate approaches. The new approaches, that we collectively label by machine learning, have their origins in the fields of informatics and artificial intelligence, but are making rapid inroads in all other branches of science. With this in mind, this Roadmap article, consisting of multiple contributions from experts across the field, discusses the use of machine learning in materials science, and share perspectives on current and future challenges in problems as diverse as the prediction of materials properties, the construction of force-fields, the development of exchange correlation functionals for density-functional theory, the solution of the many-body problem, and more. In spite of the already numerous and exciting success stories, we are just at the beginning of a long path that will reshape materials science for the many challenges of the XXIth century