On the Limit Cycles for a Class of Continuous Piecewise Linear Differential Systems with Three Zones

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Optimal Piecewise Linear Function Approximation for GPU-based Applications

Many computer vision and human-computer interaction applications developed in recent years need evaluating complex and continuous mathematical functions as an essential step toward proper operation. However, rigorous evaluation of this kind of functions often implies a very high computational cost, unacceptable in real-time applications. To alleviate this problem, functions are commonly approximated by simpler piecewise-polynomial representations. Following this idea, we propose a novel, efficient, and practical technique to evaluate complex and continuous functions using a nearly optimal design of two types of piecewise linear approximations in the case of a large budget of evaluation subintervals. To this end, we develop a thorough error analysis that yields asymptotically tight bounds to accurately quantify the approximation performance of both representations. It provides an improvement upon previous error estimates and allows the user to control the trade-off between the approximation error and the number of evaluation subintervals. To guarantee real-time operation, the method is suitable for, but not limited to, an efficient implementation in modern Graphics Processing Units (GPUs), where it outperforms previous alternative approaches by exploiting the fixed-function interpolation routines present in their texture units. The proposed technique is a perfect match for any application requiring the evaluation of continuous functions, we have measured in detail its quality and efficiency on several functions, and, in particular, the Gaussian function because it is extensively used in many areas of computer vision and cybernetics, and it is expensive to evaluate.

Path-following analysis of the dynamical response of a piecewise-linear capsule system

Acknowledgements The first author has been supported by a Georg Forster Research Fellowship granted by the Alexander von Humboldt Foundation, Germany

Evaluation of the BEASY program using linear and piecewise linear approaches for the boundary conditions

The boundary element method (BEM) was used to study galvanic corrosion using linear and logarithmic boundary conditions. The linear boundary condition was implemented by using the linear approach and the piecewise linear approach. The logarithmic boundary condition was implemented by the piecewise linear approach. The calculated potential and current density distribution were compared with the prior analytical results. For the linear boundary condition, the BEASY program using the linear approach and the piecewise linear approach gave accurate predictions of the potential and the galvanic current density distributions for varied electrolyte conditions, various film thicknesses, various electrolyte conductivities and various area ratio of anode/cathode. The 50-point piecewise linear method could be used with both linear and logarithmic polarization curves.

On tubular neighborhoods of piecewise linear and topological manifolds.

Analogues of the smooth tubular neighborhood theorem are developed for the topological and piecewise linear categories.

A subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations

We study preconditioning techniques for discontinuous Galerkin discretizations of isotropic linear elasticity problems in primal (displacement) formulation. We propose subspace correction methods based on a splitting of the vector valued piecewise linear discontinuous finite element space, that are optimal with respect to the mesh size and the Lamé parameters. The pure displacement, the mixed and the traction free problems are discussed in detail. We present a convergence analysis of the proposed preconditioners and include numerical examples that validate the theory and assess the performance of the preconditioners.

Techniques en appui des formats de modulation avancés pour les futurs réseaux optiques

Les systèmes de communication optique avec des formats de modulation avancés sont actuellement l’un des sujets de recherche les plus importants dans le domaine de communication optique. Cette recherche est stimulée par les exigences pour des débits de transmission de donnée plus élevés. Dans cette thèse, on examinera les techniques efficaces pour la modulation avancée avec une détection cohérente, et multiplexage par répartition en fréquence orthogonale (OFDM) et multiples tonalités discrètes (DMT) pour la détection directe et la détection cohérente afin d’améliorer la performance de réseaux optiques. Dans la première partie, nous examinons la rétropropagation avec filtre numérique (DFBP) comme une simple technique d’atténuation de nonlinéarité d’amplificateur optique semiconducteur (SOA) dans le système de détection cohérente. Pour la première fois, nous démontrons expérimentalement l’efficacité de DFBP pour compenser les nonlinéarités générées par SOA dans un système de détection cohérente porteur unique 16-QAM. Nous comparons la performance de DFBP avec la méthode de Runge-Kutta quatrième ordre. Nous examinons la sensibilité de performance de DFBP par rapport à ses paramètres. Par la suite, nous proposons une nouvelle méthode d’estimation de paramètre pour DFBP. Finalement, nous démontrons la transmission de signaux de 16-QAM aux taux de 22 Gbaud sur 80km de fibre optique avec la technique d’estimation de paramètre proposée pour DFBP. Dans la deuxième partie, nous nous concentrons sur les techniques afin d’améliorer la performance des systèmes OFDM optiques en examinent OFDM optiques cohérente (CO-OFDM) ainsi que OFDM optiques détection directe (DDO-OFDM). Premièrement, nous proposons une combinaison de coupure et prédistorsion pour compenser les distorsions nonlinéaires d’émetteur de CO-OFDM. Nous utilisons une interpolation linéaire par morceaux (PLI) pour charactériser la nonlinéarité d’émetteur. Dans l’émetteur nous utilisons l’inverse de l’estimation de PLI pour compenser les nonlinéarités induites à l’émetteur de CO-OFDM. Deuxièmement, nous concevons des constellations irrégulières optimisées pour les systèmes DDO-OFDM courte distance en considérant deux modèles de bruit de canal. Nous démontrons expérimentalement 100Gb/s+ OFDM/DMT avec la détection directe en utilisant les constellations QAM optimisées. Dans la troisième partie, nous proposons une architecture réseaux optiques passifs (PON) avec DDO-OFDM pour la liaison descendante et CO-OFDM pour la liaison montante. Nous examinons deux scénarios pour l’allocations de fréquence et le format de modulation des signaux. Nous identifions la détérioration limitante principale du PON bidirectionnelle et offrons des solutions pour minimiser ses effets.

Fuzzy pole placement based on piecewise Lyapunov functions

This paper presents a controller design method for fuzzy dynamic systems based on piecewise Lyapunov functions with constraints on the closed-loop pole location. The main idea is to use switched controllers to locate the poles of the system to obtain a satisfactory transient response. It is shown that the global fuzzy system satisfies the requirements for the design and that the control law can be obtained by solving a set of linear matrix inequalities, which can be efficiently solved with commercially available softwares. An example is given to illustrate the application of the proposed method. Copyright (C) 2009 John Wiley & Sons, Ltd.

A Nonstationary Model of Newborn EEG

The detection of seizure in the newborn is a critical aspect of neurological research. Current automatic detection techniques are difficult to assess due to the problems associated with acquiring and labelling newborn electroencephalogram (EEG) data. A realistic model for newborn EEG would allow confident development, assessment and comparison of these detection techniques. This paper presents a model for newborn EEG that accounts for its self-similar and non-stationary nature. The model consists of background and seizure sub-models. The newborn EEG background model is based on the short-time power spectrum with a time-varying power law. The relationship between the fractal dimension and the power law of a power spectrum is utilized for accurate estimation of the short-time power law exponent. The newborn EEG seizure model is based on a well-known time-frequency signal model. This model addresses all significant time-frequency characteristics of newborn EEG seizure which include; multiple components or harmonics, piecewise linear instantaneous frequency laws and harmonic amplitude modulation. Estimates of the parameters of both models are shown to be random and are modelled using the data from a total of 500 background epochs and 204 seizure epochs. The newborn EEG background and seizure models are validated against real newborn EEG data using the correlation coefficient. The results show that the output of the proposed models has a higher correlation with real newborn EEG than currently accepted models (a 10% and 38% improvement for background and seizure models, respectively).

Synchronization and Basins of Synchronized in Two-Dimensional Piecewise Maps Via Coupling Three Pieces of One-Dimensional Maps

This paper is devoted to the synchronization of a dynamical system defined by two different coupling versions of two identical piecewise linear bimodal maps. We consider both local and global studies, using different tools as natural transversal Lyapunov exponent, Lyapunov functions, eigenvalues and eigenvectors and numerical simulations. We obtain theoretical results for the existence of synchronization on coupling parameter range. We characterize the synchronization manifold as an attractor and measure the synchronization speed. In one coupling version, we give a necessary and sufficient condition for the synchronization. We study the basins of synchronization and show that, depending upon the type of coupling, they can have very different shapes and are not necessarily constituted by the whole phase space; in some cases, they can be riddled.

Modeling of hybrid systems by piecewise decomposition

Piecewise linear models systems arise as mathematical models of systems in many practical applications, often from linearization for nonlinear systems. There are two main approaches of dealing with these systems according to their continuous or discrete-time aspects. We propose an approach which is based on the state transformation, more particularly the partition of the phase portrait in different regions where each subregion is modeled as a two-dimensional linear time invariant system. Then the Takagi-Sugeno model, which is a combination of local model is calculated. The simulation results show that the Alpha partition is well-suited for dealing with such a system