Periodic solutions with nonconstant sign in Abel equations of the second kind

The study of periodic solutions with constant sign in the Abel equation of the second kind can be made through the equation of the first kind. This is because the situation is equivalent under the transformation $x\mapsto x^{-1}$, and there are many results available in the literature for the first kind equation. However, the equivalence breaks down when one seeks for solutions with nonconstant sign. This note is devoted to periodic solutions with nonconstant sign in Abel equations of the second kind. Specifically, we obtain sufficient conditions to ensure the existence of a periodic solution that shares the zeros of the leading coefficient of the Abel equation. Uniqueness and stability features of such solutions are also studied.Comment: 10 page

Novel slow-fast behaviour in an oscillator driven by a frequency-switching force

When an oscillator switches abruptly between different frequencies, there is some ambiguity in deciding how the system should be modelled at the switch. Here we describe two seemingly natural models of a switch in a simple periodically-forced harmonic oscillator, which disagree starkly in their predictions of its long time behaviour. Attempting to resolve the disagreement by `regularizing' the switch not only preserves the disagreement, but shows it increases with time. One of the models corresponds to a conventional `Filippov' description of a nonsmooth system, while the second exhibits a structure that irreversibly ages, developing a number of novel multi-scale behaviours that we believe have not been reported before. These include slow-fast staircases, novel mixed-mode oscillations, and a synchronized canard explosion. These features are proven to exist using asymptotic analysis, but as they involve a slow-fast time-scale separation that increases with time, they lie beyond the reach of numerical methods

Ageing of an oscillator due to frequency switching

If an oscillator is driven by a force that switches between two frequencies, the dynamics it exhibits depends on the precise manner of switching. Here we take a one-dimensional oscillator and consider scenarios in which switching occurs: (i) between two driving forces which have different frequencies, or (ii) as a single forcing whose frequency switches between two values. The difference is subtle, but entirely changes the long term behaviour, and concerns whether the switch can be expressed linearly or nonlinearly in terms of a discontinuous quantity (such as a sign or Heaviside step function that represents the switch between frequencies). In scenario (i) the oscillator has a stable periodic orbit, and the system can be described as a Filippov system. In scenario (ii) the oscillator exhibits hidden dynamics, which lies outside the theory of Filippov's systems, and causes the system to be increasingly (as time passes) dominated by sliding along the frequency-switching threshold, and in particular if periodic orbits do exist, they too exhibit sliding. We show that the behaviour persists, at least asymptotically, if the systems are regularized (i.e. if the switch is modelled as a smooth transition in the manner of (i) or (ii))

Control adaptativo por modelo de referencia para un inversor basado en LCL con síntesis de controlador mínima

[Resumen] Un Control Adaptativo por Modelo de Referencia (MRAC) con Síntesis de Controlador Mínima (MCS), es desarrollado con el objetivo de controlar un Inversor monofásico basado en un Filtro LCL conectado a la Red Eléctrica y adaptar el inductor de la red, que es un parámetro desconocido y que genera incertidumbre, manteniendo las prestaciones y la robustez del sistema. Para ello primero se revisara el algoritmo de MCS y en función de este, se muestran todos los pasos a seguir y el desarrollo de los componentes necesarios para el funcionamiento de MCS en tiempo continuo. Se desarrolla un modelo de simulación promediado y se presentan los resultados obtenidos.Este trabajo ha sido realizado gracias al apoyo de la Secretaría Nacional de Educación Superior, Ciencia y Tecnología e Innovación del Ecuador (SENESCYT), a la Universidad de Cuenca, y a los proyectos DPI2013-41224-P y DPI2015-69286-C3-2-R (MINECO/FEDER) y 2014 SGR 267 de la agencia AGAUR de la Generalitat de CatalunyaGeneralitat de Catalunya; 2014 SGR 267https://doi.org/10.17979/spudc.978849749808

Asymptotic Tracking with DC-to-DC Bilinear Power Converters

Avui en dia la conversió DC-AC té una important aplicació pràctica en el camp dels sistemes de potència ininterrompuda (SPI). Els convertidors commutats bàsics (el buck, lineal, i el boost i el buck-boost, no lineals) presenten una estructura molt simple, i al llarg dels últims quinze anys s'ha estudiat la possibilitat d'usar-los en esquemes de conversió DC-AC. L'objectiu de la tesi és aconseguir que els convertidors DC-DC de potència bàsics puguin seguir referències alternes mitjançant el voltatge de sortida. També es desenvolupen esquemes robustos per tal d'eliminar l'efecte de possibles pertorbacions en la tasca de seguiment. Els modes de lliscament s'usen com a tècnica de control, i es presenten resultats de simulació.La tesi s'organitza en capítols. El primer i el segon contenen una introducció i una revisió de la literatura existent. Els continguts i distribució de la resta de capítols segueix a continuació. El capítol 3 tracta el seguiment exacte i asimptòtic d'una referència variable en el temps per part del voltatge de sortida d'un convertidor reductor, controlat indirectament via el corrent d'entrada. A partir de l'estudi del problema del seguiment en sistemes lineals amb guanys fixos -mitjançant la teoria de mòduls- s'obtenen restriccions sobre els possibles senyals a seguir. A més, es proporciona una estratègia de control lliscant per aconseguir el seguiment, consistent en un procediment per modificar una superfície de lliscament inicialment bona en tasques de regulació i una llei de control. Una adequada elecció de variables d'estat permet que les possibles pertorbacions de la resistència de càrrega satisfacin la condició de superposició. En el capítol 4 s'usa un procediment basat en inversió per aconseguir el seguiment exacte de referències periòdiques amb la resistència de càrrega dels convertidors no lineals boost i buck-boost. També s'obtenen condicions suficients per a possibles senyals a seguir. Es presenta també un marc general per a un tractament via inversió del problema de seguiment exacte en una certa classe de sistemes bilineals de segon ordre: aquells en els quals el problema d'inversió dóna lloc a una EDO del tipus Abel. El capítol 5 estudia l'ús del mètode de Galerkin -una generalització del mètode del Balanç Harmònic- en la solució aproximada del problema invers aparegut al capítol anterior, així com l'efecte que té la seva utilització en el control del sistema. Es demostra l'existència d'una successió de solucions aproximades de l'EDO que representa l'esmentat problema invers. També es prova que aquesta successió convergeix uniformement cap a la solució periòdica de l'EDO, i s'obté una cota d'error. La sortida del sistema presenta un comportament periòdic i asimptòticament estable quan es fa anar la successió d'aproximacions de Galerkin en el control del sistema. A l'hora, la successió de sortides periòdiques presenta convergència uniforme cap a la funció desitjada sota una hipòtesi raonable. També s'obtenen en aquest cas cotes d'error. En el capítol 6 s'aconsegueix seguiment asimptòtic aproximat per a convertidors no lineals bàsics que presenten pertorbacions de càrrega. Això es fa mitjançant un control adaptatiu que estima el paràmetre pertorbat i una aproximació de Galerkin de primer ordre que incorpora l'actualització on-line a una superfície de lliscament apropiada. El capítol 7 proposa exercir un control directe del voltatge de sortida en convertidors boost i buck-boost bidireccionals, tot aprofitant la robustesa davant pertorbacions externes que ofereix aquest tipus de control. Es segueixen referències periòdiques mentre el voltatge de sortida es regula independentment a un nivell prefixat.Nowadays, DC-to-AC conversion has an important practical application in the field of uninterruptible power systems (UPS). Basic DC-to-DC switch mode power converters (the buck, which is linear, and the boost and buck-boost, which are nonlinears) possess a very simple structure, and during the last fifteen years the possibility of using them in DC-to-AC conversion schemes has been studied. The aim of this thesis is to achieve that the output voltage of the DC-to-DC buck, boost and buck-boost power converters can track periodic references. Robust schemes to eliminate disturbance effects in the tracking task are also developed. Sliding modes are used as the control technique, and the obtained results are validated by numeric simulation.The thesis is organized in chapters. The first and the second one contain an introduction and a review of the existing literature. The contents and contributions of the other chapters follow below. Chapter 3 deals with the exact and asymptotic tracking of a time varying reference by the load voltage of a step-down converter, indirectly controlled through the input current. Departing from the study of the tracking problem in linear systems with fixed gains with the aid of module theory, conditions over possible reference signals have been obtained. Moreover, a sliding mode strategy to achieve the control target, consisting in a procedure to modify a switching surface initially good for regulation tasks and a control law, is provided. An approppriate choice of state variables allows possible load perturbations to satisfy the matching condition. In chapter 4, an inversion-based indirect control is used to reach exact tracking of periodic references with the load resistance of nonminimum phase, nonlinear boost and buck-boost converters. Sufficient conditions for candidate references are also obtained. A general frame for an inversion-based treatment of the perfect tracking problem in a certain class of nonminimum phase, second order bilinear systems is proposed: those in which the inversion problem gives raise to an ODE of the Abel type. Chapter 5 studies the use of the Galerkin method -a generalization of the Harmonic Balance method- in the approximate solution of the inverse problem stated in the former chapter, as well as the effect of its use on the control of the system. The existence of a sequence of approximate solutions for the ODE that represents the quoted inverse problem is proved. This sequence is also proved to converge uniformly to the periodic solution of the ODE, and an error bound has been derived. The system output exhibits a periodic and asymptotically stable behavior when the indirect control using the sequence of Galerkin approximations is performed. In turn, the sequence of periodic outputs is shown to exhibit uniform convergence to the original target function under a reasonable hypothesis. Error bounds have also been obtained. In chapter 6, approximate asymptotic tracking is achieved for load perturbed, basic, nonlinear power converters. This is done by means of an adaptive control that estimates the perturbation parameter and a first order Galerkin approximation that incorporates the on-line updating into an appropriate sliding surface. Chapter 7 propounds to exert a direct control of the output voltage in bidirectional boost and buck-boost converters, thus taking advantage of the insensitiveness to external disturbances offered by this type of control. Periodic references are followed, while the unstable inductor current is independently regulated at a prescribed level.Postprint (published version

Linear Robust Output Regulation in a Class of Switched Power Converters

This article addresses the robust output regulation problem for a class of nonlinear switched power converters after its linearization by means of a change of the control vector variable. The methodology employs a dynamic state feedback control law and considers parametric uncertainty due to unknown values of resistive loads. Restrictions arising from the fact that the control gains exhibit fixed values are taken into account. The proposed technique is exemplified with the output voltage regulation of a Noninverting Buck-Boost converter and tested through realistic numerical simulations

Linear Robust Output Regulation in a Class of Switched Power Converters

This article addresses the robust output regulation problem for a class of nonlinear switched power converters after its linearization by means of a change of the control vector variable. The methodology employs a dynamic state feedback control law and considers parametric uncertainty due to unknown values of resistive loads. Restrictions arising from the fact that the control gains exhibit fixed values are taken into account. The proposed technique is exemplified with the output voltage regulation of a Noninverting Buck-Boost converter and tested through realistic numerical simulations

Digital Repetitive Control under Varying Frequency Conditions

XVIII, 159 p. 106 illus.online resource

Traffic Management of Multi-AGV Systems by Improved Dynamic Resource Reservation

Automated guided vehicles (AGVs) are widely used for material handling in warehouses and automated production lines due to their high efficiency and low failure rate with respect to human operated load carriers. However, AGVs usually interact with each other because of the restricted capacity of the layout, and conflicts arise. Although many traffic scheduling algorithms have been proposed to address the AGV fleet control problem, most of them are inefficient for collision and deadlock avoidance in dynamic environments. This paper proposes an improved dynamic resource reservation (IDRR) based method which renders time-efficient task completion and deadlock-free movements of multiple AGVs in a manufacturing system. Unlike traditional approaches, most of which adopt a dynamic single agent reservation of the shared resource points and/or force path deviations, IDRR exploits dynamic multiple reservations of shared resource points. This is combined with a conflict detection and resolution method that accommodates the AGV motions when they meet at a resource point. Extensive, realistic simulation results demonstrate the feasibility and efficiency of the proposed collision and deadlock prevention method in productivity, travelled distance, and time completion of the assigned tasks. The proposal can be implemented on both central and local controllers