Leonhard Euler: quatre lliçons escollides

Aprofitant la commemoració durant el curs 2006-2007 del tricentenari del naixement d'Euler, es presenten en aquest article quatre temes dels seus treballs: els logaritmes, la fórmula d'Euler, el problema de Basilea i el disseny d'un cabrestant. La tria dels temes s'ha fet amb el criteri que il.lustri de manera assequible part del treball d'Euler i que inclogui un tema proper a les aplicacions.On the occasion of the 300th anniversary of Leonhard Euler’s birth this paper presents four topics from his work: the logarithms, Euler’s formula, the Basel problem and masting of ships. The choice of the topics aims to give a plain illustration of part of Euler’s work and to include a topic on applied mathematics

Leonhard Euler: quatre lIiçons escollides

L'FME dedica el curs acadèmic 2006-2007 a la figura del matemàtic suís Leonhard Euler, una de les ments més importants de la història, comparable a Gauss o Arquímedes. La lliçó inaugural va anar a càrrec d'Enric Fossas, catedràtic i director de l'Institut d'Organització i Control de Sistemes Industrials de la UPCPostprint (published version

Study of Chaos in the buck Converter

A dc-dc buck converter controlled by naturallysampled, constant-frequency pulsewidth modulation in continuous conduction mode gives rise to a great variety of behavior, depending on the values of the parameters of the circuit. We analyze the one-periodic and two-periodic orbits which cross the voltage ramp once per cycle, and we study their stability by computing the characteristic multipliers associated with each one. Subharmonics, bifurcations, and the presence of a strange attractor are also studied. A plot of the number of crossings in the ramp is drawn. This becomes a helpful tool for investigating the evolution of the trajectories when they are close to the attractor. When analytic computations are impossible, we resort to numerical algorithms to simulate the orbits.Peer Reviewe

Leonhard Euler: quatre lliçons escollides

Aprofitant la commemoració durant el curs 2006-2007 del tricentenari del naixement d’Euler, es presenten quatre temes d’entre els seus treballs: els logaritmes, la fórmula d’Euler, el problema de Basilea i el disseny d’un cabrestant. La tria s’ha fet procurant que aquests temes siguin assequibles a estudiants de primer curs i que n’hi hagués algun de matemàtica aplicada.Peer Reviewe

On preserving passivity in sampled-data linear systems

Passivity is a well know phenomenon in several engineering areas. Due to its interesting properties, it is used in several areas of control engineering. Generally, this property is lost under direct discretization. In this work a new methodology which allows to preserve continuous-time passivity is presented. This methodology is based on choosing a proper output, which preserves the passivity structure, while keeping the continuous-time energy function. Analytic formulation and numerical examples, both open and close loop, are provided in the paper

Using optimal control to optimize the extraction rate of a durable non-renewable resource with a monopolistic primary supplier

The problem dealt with in this paper is that of optimizing the path of the extraction rate (and, consequently, the price) for the monopolistic owner of the primary sources of a totally or partially durable non-renewable resource (such as precious metals or gemstones) in a continuous-time frame, assuming that there is an upper bound on the extraction rate and with an interest rate equal to zero. The durability of the resource implies that, unlike the case of non-durable resources, at any time there is a stock of already-used amounts of the resource that are still potentially reusable, in addition to the resource available in the ground for extraction. The problem is addressed using the Maximum Principle of Pontryagin in the framework of optimal control theory, which allows identifying the patterns that the optimal policies can adopt. In this framework, the Hamiltonian is linear in the control input, which implies a bang-bang control policy governed by a switching surface. There is an underlying geometry to the problem that determines the solutions. It is characterized by the switching surface, its time derivative, the intersection point (if any) and the bang-bang trajectories through this point.Peer ReviewedPostprint (published version

A black-box model for neurons

We explore the identification of neuronal voltage traces by artificial neural networks based on wavelets (Wavenet). More precisely, we apply a modification in the representation of dynamical systems by Wavenet which decreases the number of used functions; this approach combines localized and global scope functions (unlike Wavenet, which uses localized functions only). As a proof-of-concept, we focus on the identification of voltage traces obtained by simulation of a paradigmatic neuron model, the Morris-Lecar model. We show that, after training our artificial network with biologically plausible input currents, the network is able to identify the neuron's behaviour with high accuracy, thus obtaining a black box that can be then used for predictive goals. Interestingly, the interval of input currents used for training, ranging from stimuli for which the neuron is quiescent to stimuli that elicit spikes, shows the ability of our network to identify abrupt changes in the bifurcation diagram, from almost linear input-output relationships to highly nonlinear ones. These findings open new avenues to investigate the identification of other neuron models and to provide heuristic models for real neurons by stimulating them in closed-loop experiments, that is, using the dynamic-clamp, a well-known electrophysiology technique.Peer ReviewedPostprint (author's final draft

Galerkin method and approximate tracking in a non-minimum phase bilinear system

The tracking control of non-minimum phase systems is nowadays an open and challenging field, because a general theory is still not available. This article proposes an indirect control strategy in which a key role is played by the inverse problem that arises and their approximate solutions. These are obtained with the Galerkin method, a standard functional analysis tool. A detailed study of the effect on the output caused by the use of an approximate input is performed. Error bounds are also provided. The technique is motivated through its implementation in basic, DC-to-DC nonlinear power converters that are intended to be used as DC-to-AC voltage sources.Peer Reviewe

Técnica ZAD aplicada a un convertidor "buck": análisis de estabilidad y de transición al caos

En este trabajo se presentan algunos resultados obtenidos al analizar el convertidor buck controlado mediante un modulador de anchura de pulsos (PWM) centrado. El ciclo de trabajo se calcula obligando a que una salida, definida como una ecuación diferencial de primer orden en el error de tensión, tenga promedio cero en cada ciclo. Los resultados obtenidos han mostrado que la técnica garantiza frecuencia de conmutación fija y una variedad de fenómenos no lineales como bifurcaciones y caos, en la medida en que el parámetro asociado a la dinámica de primer orden, varía. Estos fenómenos han sido estudiados en el presente trabajo de forma analítica y numérica. Finalmente a nivel de simulación se ha hecho uso de la técnica Time Delay Auto Synchronization para controlar el sistema.Peer Reviewe

Robust controller for a full-bridge rectifier using the IDA approach and GSSA modeling

An interconnection-and-damping assignment passivity-based control (IDA-PBC) for a full-bridge rectifier is presented in this paper. The closed-loop system performance fulfils unity power factor in the ac mains and output dc voltage regulation. The controller design takes advantage of the generalized state space averaging (GSSA) modeling technique to convert the quoted nonstandard problem (in actual variables) into a standard regulation one (in GSSA variables). In this approach, the output current is the measured signal instead of the line current; therefore, the number of sensors does not increase in comparison with traditional approaches. The whole system is robust with respect to load variations.Peer Reviewe